Subgroups — C₆₇×#22428 #30701679

Order 17152.

1910 conjugacy classes of subgroups.

The subgroup lattice has 652 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×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2354	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×#1895	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1875	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	C₂²×M₁₆×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	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
24	#436×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	#418×C₆₇	—	2	8	no	—
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
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×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#390×C₆₇	—	2	1	no	—
37	#388×C₆₇	—	2	2	no	—
38	#384×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#380×C₆₇	C₂²	1	2	yes	complemented
40	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
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	—
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	C₂³	1	2	yes	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	—	4	2	no	—
67	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	#127×C₆₇	—	2	2	no	—
71	#127×C₆₇	C₂×C₄	1	4	yes	—
72	#121×C₆₇	—	2	4	no	—
73	#121×C₆₇	—	2	8	no	—
74	#121×C₆₇	C₂×C₄	1	8	yes	—
75	#120×C₆₇	—	2	16	no	—
76	C₄×D₄×C₆₇	—	2	2	no	—
77	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
78	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
79	C₄×D₄×C₆₇	D₄	1	4	yes	—
80	#117×C₆₇	—	2	4	no	—
81	#117×C₆₇	—	4	2	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
94	C₂×C₄²×C₆₇	—	2	2	no	—
95	C₂×C₄²×C₆₇	—	2	1	no	—
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₆₇	—	2	2	no	—
98	#102×C₆₇	—	2	4	no	—
99	#102×C₆₇	—	2	2	no	—
100	#102×C₆₇	—	2	4	no	—
101	#102×C₆₇	—	2	4	no	—
102	#102×C₆₇	—	4	2	no	—
103	#102×C₆₇	—	4	2	no	—
104	#102×C₆₇	D₄	1	4	yes	complemented
105	C₂²⋊C₈×C₆₇	—	2	4	no	—
106	C₄×C₈×C₆₇	—	2	2	no	—
107	#95×C₆₇	—	2	2	no	—
108	#95×C₆₇	—	2	1	no	—
109	C₂⁴×C₆₇	—	2	1	no	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	8	no	—
112	C₂⁴×C₆₇	—	4	4	no	—
113	C₂⁴×C₆₇	—	4	4	no	—
114	C₂⁴×C₆₇	—	4	4	no	—
115	C₂⁴×C₆₇	—	4	4	no	—
116	C₂⁴×C₆₇	—	8	2	no	—
117	C₂⁴×C₆₇	—	8	2	no	—
118	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₆₇	—	2	16	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	1	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	1	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	2	no	—
134	C₂×D₄×C₆₇	—	4	8	no	—
135	C₂×D₄×C₆₇	—	4	8	no	—
136	C₂×D₄×C₆₇	—	4	8	no	—
137	C₂×D₄×C₆₇	—	4	4	no	—
138	C₂×D₄×C₆₇	—	4	4	no	—
139	C₂×D₄×C₆₇	—	4	32	no	—
140	C₂×D₄×C₆₇	—	4	4	no	—
141	C₂×D₄×C₆₇	—	4	4	no	—
142	C₂×D₄×C₆₇	—	4	2	no	—
143	C₂×D₄×C₆₇	—	4	2	no	—
144	C₂×D₄×C₆₇	—	8	1	no	—
145	C₂×D₄×C₆₇	—	8	1	no	—
146	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
147	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
148	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₆₇	#41	1	2	yes	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
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	1	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	4	2	no	—
161	C₂²×C₄×C₆₇	—	4	2	no	—
162	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₆₇	—	4	8	no	—
169	C₂×C₈×C₆₇	—	2	1	no	—
170	C₂×C₈×C₆₇	—	2	1	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	2	1	no	—
174	C₂×C₈×C₆₇	—	2	1	no	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	complemented
178	C₄⋊C₄×C₆₇	—	2	1	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	4	2	no	—
185	C₄⋊C₄×C₆₇	—	8	1	no	—
186	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
187	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
188	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	complemented
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	—	4	4	no	—
192	C₂²⋊C₄×C₆₇	—	8	2	no	—
193	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
194	C₄²×C₆₇	—	4	4	no	—
195	C₄²×C₆₇	—	4	2	no	—
196	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
197	C₄²×C₆₇	C₂×D₄	1	2	yes	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	2	no	—
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	4	no	—
208	C₂³×C₆₇	—	4	16	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	—	8	8	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	C₂×C₂²⋊C₄	1	4	yes	—
226	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	D₄×C₆₇	—	2	4	no	—
228	D₄×C₆₇	—	2	2	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	2	no	—
231	D₄×C₆₇	—	2	2	no	—
232	D₄×C₆₇	—	2	8	no	—
233	D₄×C₆₇	—	4	16	no	—
234	D₄×C₆₇	—	4	16	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	4	2	no	—
237	D₄×C₆₇	—	4	8	no	—
238	D₄×C₆₇	—	8	4	no	—
239	D₄×C₆₇	C₄×D₄	1	4	yes	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	8	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	2	no	—
259	C₂×C₄×C₆₇	—	4	1	no	—
260	C₂×C₄×C₆₇	—	4	2	no	—
261	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
262	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
264	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
265	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
266	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
271	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
272	C₈×C₆₇	—	2	2	no	—
273	C₈×C₆₇	—	4	2	no	—
274	C₈×C₆₇	—	4	1	no	—
275	C₂²×C₆₇	—	2	4	no	—
276	C₂²×C₆₇	—	2	4	no	—
277	C₂²×C₆₇	—	2	4	no	—
278	C₂²×C₆₇	—	2	4	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	2	no	—
281	C₂²×C₆₇	—	4	2	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	2	no	—
284	C₂²×C₆₇	—	4	2	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	2	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	4	4	no	—
290	C₂²×C₆₇	—	4	4	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	8	32	no	—
293	C₂²×C₆₇	—	8	2	no	—
294	C₂²×C₆₇	—	8	2	no	—
295	C₂²×C₆₇	—	8	2	no	—
296	C₂²×C₆₇	—	8	2	no	—
297	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
298	C₂²×C₆₇	D₄²	1	1	yes	characteristic
299	C₂²×C₆₇	#546	1	1	yes	characteristic
300	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₂²×C₆₇	#576	1	1	yes	characteristic
303	C₂²×C₆₇	#577	1	1	yes	characteristic
304	C₄×C₆₇	—	2	4	no	—
305	C₄×C₆₇	—	2	2	no	—
306	C₄×C₆₇	—	4	2	no	—
307	C₄×C₆₇	—	4	2	no	—
308	C₄×C₆₇	—	8	2	no	—
309	C₄×C₆₇	—	8	1	no	—
310	C₄×C₆₇	#418	1	4	yes	complemented
311	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
312	C₄×C₆₇	#518	1	1	yes	characteristic
313	C₄×C₆₇	D₄²	1	1	yes	characteristic
314	C₄×C₆₇	#546	1	1	yes	characteristic
315	#22428	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂×C₆₇	—	4	4	no	—
317	C₂×C₆₇	—	4	4	no	—
318	C₂×C₆₇	—	8	4	no	—
319	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
320	C₂×C₆₇	C₂×#418	1	1	yes	characteristic
321	C₂×C₆₇	#2897	1	1	yes	characteristic
322	C₂×C₆₇	#3272	1	1	yes	characteristic
323	C₂×C₆₇	#3274	1	1	yes	characteristic
324	C₂×C₆₇	#3324	1	1	yes	characteristic
325	C₂×C₆₇	#3325	1	1	yes	characteristic
326	C₂×D₄²	C₂×C₆₇	1	1	yes	characteristic
327	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
328	C₂×#418	C₂×C₆₇	1	2	yes	complemented
329	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
331	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
332	#1895	C₂×C₆₇	1	2	yes	complemented
333	#1875	C₂×C₆₇	1	2	yes	complemented
334	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
335	#1786	C₂×C₆₇	1	2	yes	complemented
336	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
337	C₆₇	#22428	1	1	yes	characteristic, direct product factor, Sylow
338	C₂³×D₄	—	2	1	no	—
339	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
340	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
341	D₄²	—	2	4	no	—
342	D₄²	C₄×C₆₇	1	8	yes	complemented
343	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×#121	—	2	1	no	—
345	C₂×#121	C₂²×C₆₇	1	2	yes	—
346	C₂×#120	—	2	2	no	—
347	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
350	#436	—	2	4	no	—
351	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
352	#418	—	2	8	no	—
353	C₂×#102	C₂²×C₆₇	1	2	yes	—
354	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
356	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
357	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
358	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
359	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
360	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
361	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
362	#390	—	2	1	no	—
363	#388	—	2	2	no	—
364	#384	C₂²×C₆₇	1	1	yes	characteristic, complemented
365	#380	C₂²×C₆₇	1	2	yes	complemented
366	C₄×C₄⋊C₄	—	2	1	no	—
367	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
368	#340	C₂²×C₆₇	1	2	yes	complemented
369	#340	C₂²×C₆₇	1	1	yes	characteristic, complemented
370	#340	C₂²×C₆₇	1	1	yes	characteristic, complemented
371	C₂⁵	—	2	2	no	—
372	C₂²×D₄	—	2	4	no	—
373	C₂²×D₄	—	2	4	no	—
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	8	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	—	2	1	no	—
378	C₂²×D₄	—	2	4	no	—
379	C₂²×D₄	—	2	1	no	—
380	C₂²×D₄	—	4	2	no	—
381	C₂²×D₄	—	4	2	no	—
382	C₂²×D₄	—	4	4	no	—
383	C₂²×D₄	—	4	4	no	—
384	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
385	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
386	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
387	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
388	C₂³×C₄	—	2	1	no	—
389	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
390	C₂×M₁₆	—	2	4	no	—
391	C₂×M₁₆	—	2	4	no	—
392	C₂×M₁₆	—	4	2	no	—
393	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
394	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
396	#127	—	2	2	no	—
397	#127	C₂×C₄×C₆₇	1	4	yes	—
398	#121	—	2	4	no	—
399	#121	—	2	8	no	—
400	#121	C₂×C₄×C₆₇	1	8	yes	—
401	#120	—	2	16	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
404	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
405	C₄×D₄	D₄×C₆₇	1	4	yes	—
406	#117	—	2	4	no	—
407	#117	—	4	2	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	2	no	—
417	C₂×C₂²⋊C₄	—	2	2	no	—
418	C₂×C₂²⋊C₄	—	2	2	no	—
419	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
420	C₂×C₄²	—	2	2	no	—
421	C₂×C₄²	—	2	1	no	—
422	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
423	C₄⋊C₈	—	2	2	no	—
424	#102	—	2	4	no	—
425	#102	—	2	2	no	—
426	#102	—	2	4	no	—
427	#102	—	2	4	no	—
428	#102	—	4	2	no	—
429	#102	—	4	2	no	—
430	#102	D₄×C₆₇	1	4	yes	complemented
431	C₂²⋊C₈	—	2	4	no	—
432	C₄×C₈	—	2	2	no	—
433	#95	—	2	2	no	—
434	#95	—	2	1	no	—
435	C₂⁴	—	2	1	no	—
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	2	8	no	—
438	C₂⁴	—	4	4	no	—
439	C₂⁴	—	4	4	no	—
440	C₂⁴	—	4	4	no	—
441	C₂⁴	—	4	4	no	—
442	C₂⁴	—	8	2	no	—
443	C₂⁴	—	8	2	no	—
444	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
445	C₂×D₄	—	2	16	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	4	8	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂×D₄	—	4	8	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	4	32	no	—
466	C₂×D₄	—	4	4	no	—
467	C₂×D₄	—	4	4	no	—
468	C₂×D₄	—	4	2	no	—
469	C₂×D₄	—	4	2	no	—
470	C₂×D₄	—	8	1	no	—
471	C₂×D₄	—	8	1	no	—
472	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
473	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
474	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
475	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
476	C₂×D₄	#41×C₆₇	1	2	yes	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
492	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
494	M₁₆	—	4	8	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	complemented
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	8	1	no	—
512	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
513	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
514	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	complemented
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	4	4	no	—
518	C₂²⋊C₄	—	8	2	no	—
519	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
520	C₄²	—	4	4	no	—
521	C₄²	—	4	2	no	—
522	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
523	C₄²	C₂×D₄×C₆₇	1	2	yes	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	4	16	no	—
535	C₂³	—	4	16	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	4	1	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	2	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	2	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	2	no	—
546	C₂³	—	8	8	no	—
547	C₂³	—	8	8	no	—
548	C₂³	—	8	8	no	—
549	C₂³	—	8	8	no	—
550	C₂³	—	8	8	no	—
551	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
552	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
553	D₄	—	2	4	no	—
554	D₄	—	2	2	no	—
555	D₄	—	2	4	no	—
556	D₄	—	2	2	no	—
557	D₄	—	2	2	no	—
558	D₄	—	2	8	no	—
559	D₄	—	4	16	no	—
560	D₄	—	4	16	no	—
561	D₄	—	4	2	no	—
562	D₄	—	4	2	no	—
563	D₄	—	4	8	no	—
564	D₄	—	8	4	no	—
565	D₄	C₄×D₄×C₆₇	1	4	yes	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
588	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
589	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
590	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
591	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
592	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
594	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
595	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
596	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
597	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
598	C₈	—	2	2	no	—
599	C₈	—	4	2	no	—
600	C₈	—	4	1	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	4	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	4	4	no	—
616	C₂²	—	4	4	no	—
617	C₂²	—	4	4	no	—
618	C₂²	—	8	32	no	—
619	C₂²	—	8	2	no	—
620	C₂²	—	8	2	no	—
621	C₂²	—	8	2	no	—
622	C₂²	—	8	2	no	—
623	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
624	C₂²	D₄²×C₆₇	1	1	yes	characteristic
625	C₂²	#546×C₆₇	1	1	yes	characteristic
626	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
627	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
628	C₂²	#576×C₆₇	1	1	yes	characteristic
629	C₂²	#577×C₆₇	1	1	yes	characteristic
630	C₄	—	2	4	no	—
631	C₄	—	2	2	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	2	no	—
634	C₄	—	8	2	no	—
635	C₄	—	8	1	no	—
636	C₄	#418×C₆₇	1	4	yes	complemented
637	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
638	C₄	#518×C₆₇	1	1	yes	characteristic
639	C₄	D₄²×C₆₇	1	1	yes	characteristic
640	C₄	#546×C₆₇	1	1	yes	characteristic
641	C₂	—	4	4	no	—
642	C₂	—	4	4	no	—
643	C₂	—	8	4	no	—
644	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
645	C₂	C₂×#418×C₆₇	1	1	yes	characteristic
646	C₂	C₆₇×#2897	1	1	yes	characteristic
647	C₂	C₆₇×#3272	1	1	yes	characteristic
648	C₂	C₆₇×#3274	1	1	yes	characteristic
649	C₂	C₆₇×#3324	1	1	yes	characteristic
650	C₂	C₆₇×#3325	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×#436×C₆₇	C₂	semidirect
3. C₂×#418×C₆₇	C₂	semidirect
4. C₆₇×#2354	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1957	C₂	semidirect
7. C₆₇×#1895	C₂	semidirect
8. C₆₇×#1875	C₂	semidirect
9. C₆₇×#1802	C₂	semidirect
10. C₆₇×#1786	C₂	semidirect
11. C₄×#102×C₆₇	C₂	semidirect
14. C₂²×M₁₆×C₆₇	C₂²	semidirect
16. D₄²×C₆₇	C₄	semidirect
23. C₂×#117×C₆₇	C₂²	semidirect
25. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
30. C₂×#102×C₆₇	C₂²	semidirect
31. C₂×#102×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
34. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
35. C₂×C₄×C₈×C₆₇	C₂²	semidirect
38. #384×C₆₇	C₂²	semidirect
39. #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
104. #102×C₆₇	D₄	semidirect
177. C₂×C₈×C₆₇	C₂×D₄	semidirect
188. C₄⋊C₄×C₆₇	C₂×D₄	semidirect
310. C₄×C₆₇	#418	semidirect
315. #22428	C₆₇	direct (N × H)
327. C₂×#436	C₂×C₆₇	semidirect
328. C₂×#418	C₂×C₆₇	semidirect
329. #2354	C₂×C₆₇	semidirect
330. #1982	C₂×C₆₇	semidirect
331. #1957	C₂×C₆₇	semidirect
332. #1895	C₂×C₆₇	semidirect
333. #1875	C₂×C₆₇	semidirect
334. #1802	C₂×C₆₇	semidirect
335. #1786	C₂×C₆₇	semidirect
336. C₄×#102	C₂×C₆₇	semidirect
337. C₆₇	#22428	direct (N × H)
340. C₂²×M₁₆	C₂²×C₆₇	semidirect
342. D₄²	C₄×C₆₇	semidirect
349. C₂×#117	C₂²×C₆₇	semidirect
351. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
356. C₂×#102	C₂²×C₆₇	semidirect
357. C₂×#102	C₂²×C₆₇	semidirect
358. C₂×#102	C₂²×C₆₇	semidirect
359. C₂×#102	C₂²×C₆₇	semidirect
360. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
361. C₂×C₄×C₈	C₂²×C₆₇	semidirect
364. #384	C₂²×C₆₇	semidirect
365. #380	C₂²×C₆₇	semidirect
367. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
368. #340	C₂²×C₆₇	semidirect
369. #340	C₂²×C₆₇	semidirect
370. #340	C₂²×C₆₇	semidirect
430. #102	D₄×C₆₇	semidirect
503. C₂×C₈	C₂×D₄×C₆₇	semidirect
514. C₄⋊C₄	C₂×D₄×C₆₇	semidirect
636. C₄	#418×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×#436×C₆₇	2
C₂×#418×C₆₇	2
C₆₇×#2354	2
C₆₇×#1982	2
C₆₇×#1957	2
C₆₇×#1895	2
C₆₇×#1875	2
C₆₇×#1802	2
C₆₇×#1786	2
C₄×#102×C₆₇	2
#22428	67

Subgroups — C67×#22428 #30701679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22428 #30701679