Subgroups — D₄×#102×C₆₇ #30701664

Order 17152.

2014 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#102×C₆₇	C₂	1	2	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1786	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₆₇	—	2	1	no	—
13	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
15	D₄²×C₆₇	—	2	4	no	—
16	D₄²×C₆₇	C₄	1	8	yes	complemented
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	—	2	1	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#120×C₆₇	—	2	2	no	—
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
25	C₈×D₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×#102×C₆₇	—	2	8	no	—
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
35	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#389×C₆₇	—	2	1	no	—
38	#385×C₆₇	—	2	2	no	—
39	#384×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#380×C₆₇	C₂²	1	2	yes	complemented
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	#340×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#340×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#340×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	4	no	—
47	C₂²×D₄×C₆₇	—	2	4	no	—
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	—	2	8	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	4	2	no	—
55	C₂²×D₄×C₆₇	—	4	2	no	—
56	C₂²×D₄×C₆₇	—	4	4	no	—
57	C₂²×D₄×C₆₇	—	4	4	no	—
58	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
59	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	—	2	1	no	—
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	—	2	4	no	—
66	C₂²×C₈×C₆₇	—	2	4	no	—
67	C₂²×C₈×C₆₇	—	4	2	no	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	#127×C₆₇	—	2	2	no	—
72	#127×C₆₇	C₂×C₄	1	4	yes	complemented
73	#121×C₆₇	—	2	4	no	—
74	#121×C₆₇	—	2	8	no	—
75	#121×C₆₇	C₂×C₄	1	8	yes	complemented
76	#120×C₆₇	—	2	16	no	—
77	C₄×D₄×C₆₇	—	2	2	no	—
78	C₄×D₄×C₆₇	—	4	4	no	—
79	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
80	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
81	C₄×D₄×C₆₇	D₄	1	4	yes	—
82	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	2	no	—
99	#102×C₆₇	—	2	4	no	—
100	#102×C₆₇	—	2	2	no	—
101	#102×C₆₇	—	2	4	no	—
102	#102×C₆₇	—	2	4	no	—
103	#102×C₆₇	—	4	2	no	—
104	#102×C₆₇	—	4	2	no	—
105	#102×C₆₇	D₄	1	4	yes	direct product factor
106	C₂²⋊C₈×C₆₇	—	2	4	no	—
107	C₄×C₈×C₆₇	—	2	2	no	—
108	#95×C₆₇	—	2	1	no	—
109	#95×C₆₇	—	2	2	no	—
110	C₂⁴×C₆₇	—	2	1	no	—
111	C₂⁴×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	—	2	8	no	—
113	C₂⁴×C₆₇	—	4	4	no	—
114	C₂⁴×C₆₇	—	4	4	no	—
115	C₂⁴×C₆₇	—	4	4	no	—
116	C₂⁴×C₆₇	—	4	4	no	—
117	C₂⁴×C₆₇	—	8	2	no	—
118	C₂⁴×C₆₇	—	8	2	no	—
119	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₆₇	—	2	16	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	8	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	1	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	4	8	no	—
134	C₂×D₄×C₆₇	—	4	8	no	—
135	C₂×D₄×C₆₇	—	4	8	no	—
136	C₂×D₄×C₆₇	—	4	4	no	—
137	C₂×D₄×C₆₇	—	4	4	no	—
138	C₂×D₄×C₆₇	—	4	4	no	—
139	C₂×D₄×C₆₇	—	4	4	no	—
140	C₂×D₄×C₆₇	—	4	32	no	—
141	C₂×D₄×C₆₇	—	4	2	no	—
142	C₂×D₄×C₆₇	—	4	2	no	—
143	C₂×D₄×C₆₇	—	8	1	no	—
144	C₂×D₄×C₆₇	—	8	1	no	—
145	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
146	C₂×D₄×C₆₇	D₈	1	4	yes	—
147	C₂×D₄×C₆₇	SD₁₆	1	4	yes	—
148	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
149	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂×D₄×C₆₇	#41	1	2	yes	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	4	4	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	4	no	—
165	C₂²×C₄×C₆₇	—	8	2	no	—
166	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂×C₈×C₆₇	—	2	1	no	—
173	C₂×C₈×C₆₇	—	2	1	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	1	no	—
178	C₂×C₈×C₆₇	—	2	1	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	4	8	no	—
181	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	complemented
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	4	4	no	—
189	C₄⋊C₄×C₆₇	—	4	2	no	—
190	C₄⋊C₄×C₆₇	—	8	1	no	—
191	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
192	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
193	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	complemented
194	C₂²⋊C₄×C₆₇	—	2	4	no	—
195	C₂²⋊C₄×C₆₇	—	2	8	no	—
196	C₂²⋊C₄×C₆₇	—	4	4	no	—
197	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
198	C₄²×C₆₇	—	4	2	no	—
199	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
200	C₄²×C₆₇	C₂×D₄	1	2	yes	—
201	C₂³×C₆₇	—	2	2	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	2	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	2	2	no	—
207	C₂³×C₆₇	—	2	2	no	—
208	C₂³×C₆₇	—	2	4	no	—
209	C₂³×C₆₇	—	4	16	no	—
210	C₂³×C₆₇	—	4	1	no	—
211	C₂³×C₆₇	—	4	1	no	—
212	C₂³×C₆₇	—	4	1	no	—
213	C₂³×C₆₇	—	4	1	no	—
214	C₂³×C₆₇	—	4	2	no	—
215	C₂³×C₆₇	—	4	2	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	—	4	2	no	—
218	C₂³×C₆₇	—	4	2	no	—
219	C₂³×C₆₇	—	4	2	no	—
220	C₂³×C₆₇	—	4	16	no	—
221	C₂³×C₆₇	—	8	8	no	—
222	C₂³×C₆₇	—	8	8	no	—
223	C₂³×C₆₇	—	8	8	no	—
224	C₂³×C₆₇	—	8	8	no	—
225	C₂³×C₆₇	—	8	8	no	—
226	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
227	C₂³×C₆₇	C₂×D₈	1	4	yes	—
228	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
229	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	2	2	no	—
232	D₄×C₆₇	—	2	4	no	—
233	D₄×C₆₇	—	2	2	no	—
234	D₄×C₆₇	—	2	2	no	—
235	D₄×C₆₇	—	4	16	no	—
236	D₄×C₆₇	—	4	16	no	—
237	D₄×C₆₇	—	4	2	no	—
238	D₄×C₆₇	—	4	2	no	—
239	D₄×C₆₇	—	4	8	no	—
240	D₄×C₆₇	—	8	4	no	—
241	D₄×C₆₇	#102	1	16	yes	direct product factor
242	D₄×C₆₇	C₄×D₄	1	4	yes	complemented
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	1	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	8	no	—
258	C₂×C₄×C₆₇	—	4	2	no	—
259	C₂×C₄×C₆₇	—	4	1	no	—
260	C₂×C₄×C₆₇	—	4	1	no	—
261	C₂×C₄×C₆₇	—	4	2	no	—
262	C₂×C₄×C₆₇	—	4	2	no	—
263	C₂×C₄×C₆₇	—	4	1	no	—
264	C₂×C₄×C₆₇	—	8	8	no	—
265	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
267	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
269	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
270	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
275	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
276	C₈×C₆₇	—	2	2	no	—
277	C₈×C₆₇	—	4	2	no	—
278	C₈×C₆₇	—	4	1	no	—
279	C₂²×C₆₇	—	2	4	no	—
280	C₂²×C₆₇	—	2	4	no	—
281	C₂²×C₆₇	—	2	4	no	—
282	C₂²×C₆₇	—	2	4	no	—
283	C₂²×C₆₇	—	2	4	no	—
284	C₂²×C₆₇	—	2	4	no	—
285	C₂²×C₆₇	—	2	4	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	2	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	4	2	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	—	4	2	no	—
292	C₂²×C₆₇	—	4	4	no	—
293	C₂²×C₆₇	—	4	4	no	—
294	C₂²×C₆₇	—	4	4	no	—
295	C₂²×C₆₇	—	8	32	no	—
296	C₂²×C₆₇	—	8	2	no	—
297	C₂²×C₆₇	—	8	2	no	—
298	C₂²×C₆₇	—	8	2	no	—
299	C₂²×C₆₇	—	8	2	no	—
300	C₂²×C₆₇	C₂×#102	1	8	yes	complemented
301	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
302	C₂²×C₆₇	D₄²	1	1	yes	characteristic
303	C₂²×C₆₇	#546	1	1	yes	characteristic
304	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
305	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
306	C₂²×C₆₇	#576	1	1	yes	characteristic
307	C₂²×C₆₇	#577	1	1	yes	characteristic
308	C₄×C₆₇	—	2	4	no	—
309	C₄×C₆₇	—	2	2	no	—
310	C₄×C₆₇	—	4	2	no	—
311	C₄×C₆₇	—	4	2	no	—
312	C₄×C₆₇	—	8	2	no	—
313	C₄×C₆₇	—	8	1	no	—
314	C₄×C₆₇	C₂×#102	1	4	yes	complemented
315	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
316	C₄×C₆₇	#518	1	1	yes	characteristic
317	C₄×C₆₇	D₄²	1	1	yes	characteristic
318	C₄×C₆₇	#546	1	1	yes	characteristic
319	D₄×#102	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
320	C₂×C₆₇	—	2	8	no	—
321	C₂×C₆₇	—	4	4	no	—
322	C₂×C₆₇	—	8	4	no	—
323	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
324	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
325	C₂×C₆₇	#2897	1	1	yes	characteristic
326	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
327	C₂×C₆₇	D₄×SD₁₆	1	1	yes	characteristic
328	C₂×C₆₇	#3294	1	1	yes	characteristic
329	C₂×C₆₇	#3295	1	1	yes	characteristic
330	C₂×D₄²	C₂×C₆₇	1	1	yes	characteristic
331	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₂²×#102	C₂×C₆₇	1	2	yes	complemented
333	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
334	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
335	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
336	#1893	C₂×C₆₇	1	2	yes	complemented
337	#1874	C₂×C₆₇	1	2	yes	complemented
338	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
339	#1786	C₂×C₆₇	1	2	yes	complemented
340	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
341	C₆₇	D₄×#102	1	1	yes	characteristic, direct product factor, Sylow
342	C₂³×D₄	—	2	1	no	—
343	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
344	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
345	D₄²	—	2	4	no	—
346	D₄²	C₄×C₆₇	1	8	yes	complemented
347	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×#121	—	2	1	no	—
349	C₂×#121	C₂²×C₆₇	1	2	yes	—
350	C₂×#120	—	2	2	no	—
351	C₂×C₄×D₄	—	2	1	no	—
352	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
353	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
354	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
355	C₈×D₄	—	2	4	no	—
356	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
357	C₂×#102	—	2	8	no	—
358	C₂×#102	C₂²×C₆₇	1	2	yes	—
359	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
362	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
363	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
364	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
365	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
366	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
367	#389	—	2	1	no	—
368	#385	—	2	2	no	—
369	#384	C₂²×C₆₇	1	1	yes	characteristic, complemented
370	#380	C₂²×C₆₇	1	2	yes	complemented
371	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
372	#340	C₂²×C₆₇	1	2	yes	complemented
373	#340	C₂²×C₆₇	1	1	yes	characteristic, complemented
374	#340	C₂²×C₆₇	1	1	yes	characteristic, complemented
375	C₂⁵	—	2	2	no	—
376	C₂²×D₄	—	2	4	no	—
377	C₂²×D₄	—	2	4	no	—
378	C₂²×D₄	—	2	2	no	—
379	C₂²×D₄	—	2	8	no	—
380	C₂²×D₄	—	2	2	no	—
381	C₂²×D₄	—	2	1	no	—
382	C₂²×D₄	—	2	4	no	—
383	C₂²×D₄	—	2	1	no	—
384	C₂²×D₄	—	4	2	no	—
385	C₂²×D₄	—	4	2	no	—
386	C₂²×D₄	—	4	4	no	—
387	C₂²×D₄	—	4	4	no	—
388	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
389	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
390	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
391	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
392	C₂³×C₄	—	2	1	no	—
393	C₂³×C₄	—	2	2	no	—
394	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
395	C₂²×C₈	—	2	4	no	—
396	C₂²×C₈	—	2	4	no	—
397	C₂²×C₈	—	4	2	no	—
398	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
399	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
400	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
401	#127	—	2	2	no	—
402	#127	C₂×C₄×C₆₇	1	4	yes	complemented
403	#121	—	2	4	no	—
404	#121	—	2	8	no	—
405	#121	C₂×C₄×C₆₇	1	8	yes	complemented
406	#120	—	2	16	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	4	4	no	—
409	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
410	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
411	C₄×D₄	D₄×C₆₇	1	4	yes	—
412	C₂×C₄⋊C₄	—	2	4	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	4	2	no	—
417	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	2	no	—
425	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
428	C₄⋊C₈	—	2	2	no	—
429	#102	—	2	4	no	—
430	#102	—	2	2	no	—
431	#102	—	2	4	no	—
432	#102	—	2	4	no	—
433	#102	—	4	2	no	—
434	#102	—	4	2	no	—
435	#102	D₄×C₆₇	1	4	yes	direct product factor
436	C₂²⋊C₈	—	2	4	no	—
437	C₄×C₈	—	2	2	no	—
438	#95	—	2	1	no	—
439	#95	—	2	2	no	—
440	C₂⁴	—	2	1	no	—
441	C₂⁴	—	2	2	no	—
442	C₂⁴	—	2	8	no	—
443	C₂⁴	—	4	4	no	—
444	C₂⁴	—	4	4	no	—
445	C₂⁴	—	4	4	no	—
446	C₂⁴	—	4	4	no	—
447	C₂⁴	—	8	2	no	—
448	C₂⁴	—	8	2	no	—
449	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
450	C₂×D₄	—	2	16	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	4	8	no	—
464	C₂×D₄	—	4	8	no	—
465	C₂×D₄	—	4	8	no	—
466	C₂×D₄	—	4	4	no	—
467	C₂×D₄	—	4	4	no	—
468	C₂×D₄	—	4	4	no	—
469	C₂×D₄	—	4	4	no	—
470	C₂×D₄	—	4	32	no	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	—	4	2	no	—
473	C₂×D₄	—	8	1	no	—
474	C₂×D₄	—	8	1	no	—
475	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
476	C₂×D₄	D₈×C₆₇	1	4	yes	—
477	C₂×D₄	SD₁₆×C₆₇	1	4	yes	—
478	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
479	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
480	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
481	C₂×D₄	#41×C₆₇	1	2	yes	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	4	4	no	—
495	C₂²×C₄	—	8	2	no	—
496	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
497	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
499	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
500	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	2	no	—
510	C₂×C₈	—	4	8	no	—
511	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	complemented
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	4	4	no	—
519	C₄⋊C₄	—	4	2	no	—
520	C₄⋊C₄	—	8	1	no	—
521	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
522	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
523	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	complemented
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	8	no	—
526	C₂²⋊C₄	—	4	4	no	—
527	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
528	C₄²	—	4	2	no	—
529	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
530	C₄²	C₂×D₄×C₆₇	1	2	yes	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	4	16	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	—	4	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	2	no	—
546	C₂³	—	4	2	no	—
547	C₂³	—	4	2	no	—
548	C₂³	—	4	2	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	16	no	—
551	C₂³	—	8	8	no	—
552	C₂³	—	8	8	no	—
553	C₂³	—	8	8	no	—
554	C₂³	—	8	8	no	—
555	C₂³	—	8	8	no	—
556	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
557	C₂³	C₂×D₈×C₆₇	1	4	yes	—
558	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
559	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
560	D₄	—	2	4	no	—
561	D₄	—	2	2	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	2	no	—
564	D₄	—	2	2	no	—
565	D₄	—	4	16	no	—
566	D₄	—	4	16	no	—
567	D₄	—	4	2	no	—
568	D₄	—	4	2	no	—
569	D₄	—	4	8	no	—
570	D₄	—	8	4	no	—
571	D₄	#102×C₆₇	1	16	yes	direct product factor
572	D₄	C₄×D₄×C₆₇	1	4	yes	complemented
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	8	8	no	—
595	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
596	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
597	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
598	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
599	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
600	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
602	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
603	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
604	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
605	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
606	C₈	—	2	2	no	—
607	C₈	—	4	2	no	—
608	C₈	—	4	1	no	—
609	C₂²	—	2	4	no	—
610	C₂²	—	2	4	no	—
611	C₂²	—	2	4	no	—
612	C₂²	—	2	4	no	—
613	C₂²	—	2	4	no	—
614	C₂²	—	2	4	no	—
615	C₂²	—	2	4	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	4	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	4	no	—
625	C₂²	—	8	32	no	—
626	C₂²	—	8	2	no	—
627	C₂²	—	8	2	no	—
628	C₂²	—	8	2	no	—
629	C₂²	—	8	2	no	—
630	C₂²	C₂×#102×C₆₇	1	8	yes	complemented
631	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
632	C₂²	D₄²×C₆₇	1	1	yes	characteristic
633	C₂²	#546×C₆₇	1	1	yes	characteristic
634	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
635	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
636	C₂²	#576×C₆₇	1	1	yes	characteristic
637	C₂²	#577×C₆₇	1	1	yes	characteristic
638	C₄	—	2	4	no	—
639	C₄	—	2	2	no	—
640	C₄	—	4	2	no	—
641	C₄	—	4	2	no	—
642	C₄	—	8	2	no	—
643	C₄	—	8	1	no	—
644	C₄	C₂×#102×C₆₇	1	4	yes	complemented
645	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
646	C₄	#518×C₆₇	1	1	yes	characteristic
647	C₄	D₄²×C₆₇	1	1	yes	characteristic
648	C₄	#546×C₆₇	1	1	yes	characteristic
649	C₂	—	2	8	no	—
650	C₂	—	4	4	no	—
651	C₂	—	8	4	no	—
652	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
653	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic
654	C₂	C₆₇×#2897	1	1	yes	characteristic
655	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
656	C₂	D₄×SD₁₆×C₆₇	1	1	yes	characteristic
657	C₂	C₆₇×#3294	1	1	yes	characteristic
658	C₂	C₆₇×#3295	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×D₄²×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₆₇	C₂	semidirect
3. C₂²×#102×C₆₇	C₂	semidirect
4. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1957	C₂	semidirect
7. C₆₇×#1893	C₂	semidirect
8. C₆₇×#1874	C₂	semidirect
9. C₆₇×#1802	C₂	semidirect
10. C₆₇×#1786	C₂	semidirect
11. C₄×#102×C₆₇	C₂	semidirect
14. C₂³×C₈×C₆₇	C₂²	semidirect
16. D₄²×C₆₇	C₄	semidirect
24. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
26. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
31. C₂×#102×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
34. C₂×#102×C₆₇	C₂²	semidirect
35. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
36. C₂×C₄×C₈×C₆₇	C₂²	semidirect
39. #384×C₆₇	C₂²	semidirect
40. #380×C₆₇	C₂²	semidirect
41. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
42. #340×C₆₇	C₂²	semidirect
43. #340×C₆₇	C₂²	semidirect
44. #340×C₆₇	C₂²	semidirect
58. C₂²×D₄×C₆₇	C₂×C₄	semidirect
72. #127×C₆₇	C₂×C₄	semidirect
75. #121×C₆₇	C₂×C₄	semidirect
80. C₄×D₄×C₆₇	C₂×C₄	semidirect
105. #102×C₆₇	D₄	direct (N × H)
181. C₂×C₈×C₆₇	C₂×D₄	semidirect
193. C₄⋊C₄×C₆₇	C₂×D₄	semidirect
241. D₄×C₆₇	#102	direct (N × H)
242. D₄×C₆₇	C₄×D₄	semidirect
300. C₂²×C₆₇	C₂×#102	semidirect
314. C₄×C₆₇	C₂×#102	semidirect
319. D₄×#102	C₆₇	direct (N × H)
331. C₂×C₈×D₄	C₂×C₆₇	semidirect
332. C₂²×#102	C₂×C₆₇	semidirect
333. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
334. #1982	C₂×C₆₇	semidirect
335. #1957	C₂×C₆₇	semidirect
336. #1893	C₂×C₆₇	semidirect
337. #1874	C₂×C₆₇	semidirect
338. #1802	C₂×C₆₇	semidirect
339. #1786	C₂×C₆₇	semidirect
340. C₄×#102	C₂×C₆₇	semidirect
341. C₆₇	D₄×#102	direct (N × H)
344. C₂³×C₈	C₂²×C₆₇	semidirect
346. D₄²	C₄×C₆₇	semidirect
354. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
356. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
361. C₂×#102	C₂²×C₆₇	semidirect
362. C₂×#102	C₂²×C₆₇	semidirect
363. C₂×#102	C₂²×C₆₇	semidirect
364. C₂×#102	C₂²×C₆₇	semidirect
365. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
366. C₂×C₄×C₈	C₂²×C₆₇	semidirect
369. #384	C₂²×C₆₇	semidirect
370. #380	C₂²×C₆₇	semidirect
371. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
372. #340	C₂²×C₆₇	semidirect
373. #340	C₂²×C₆₇	semidirect
374. #340	C₂²×C₆₇	semidirect
388. C₂²×D₄	C₂×C₄×C₆₇	semidirect
402. #127	C₂×C₄×C₆₇	semidirect
405. #121	C₂×C₄×C₆₇	semidirect
410. C₄×D₄	C₂×C₄×C₆₇	semidirect
435. #102	D₄×C₆₇	direct (N × H)
511. C₂×C₈	C₂×D₄×C₆₇	semidirect
523. C₄⋊C₄	C₂×D₄×C₆₇	semidirect
571. D₄	#102×C₆₇	direct (N × H)
572. D₄	C₄×D₄×C₆₇	semidirect
630. C₂²	C₂×#102×C₆₇	semidirect
644. C₄	C₂×#102×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄²×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂²×#102×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1982	2
C₆₇×#1957	2
C₆₇×#1893	2
C₆₇×#1874	2
C₆₇×#1802	2
C₆₇×#1786	2
C₄×#102×C₆₇	2
D₄×#102	67

Subgroups — D4×#102×C67 #30701664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#102×C₆₇ #30701664