Subgroups — C₆₇×#22485 #30701736

Order 17152.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1823	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#340×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	—	2	1	no	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	#547×C₆₇	—	2	4	no	—
15	#547×C₆₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#123×C₆₇	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	—	2	1	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₈×D₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#389×C₆₇	—	2	1	no	—
33	#385×C₆₇	—	2	2	no	—
34	#384×C₆₇	C₂²	1	1	yes	characteristic
35	#380×C₆₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	1	yes	characteristic
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#41×C₆₇	—	2	8	no	—
45	C₂×#41×C₆₇	—	4	4	no	—
46	C₂×#41×C₆₇	—	4	4	no	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	2	no	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	—	2	4	no	—
56	C₂²×C₈×C₆₇	—	2	4	no	—
57	C₂²×C₈×C₆₇	—	4	2	no	—
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
63	#123×C₆₇	—	2	16	no	—
64	C₂²⋊Q₈×C₆₇	—	2	8	no	—
65	#121×C₆₇	—	2	4	no	—
66	#121×C₆₇	C₂×C₄	1	8	yes	complemented
67	C₄×D₄×C₆₇	—	2	2	no	—
68	C₄×D₄×C₆₇	—	4	4	no	—
69	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
70	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
71	C₄×D₄×C₆₇	D₄	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
81	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂³	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×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	—	2	2	no	—
94	#102×C₆₇	—	2	2	no	—
95	#102×C₆₇	—	2	2	no	—
96	#102×C₆₇	—	4	2	no	—
97	C₂²⋊C₈×C₆₇	—	2	4	no	—
98	C₄×C₈×C₆₇	—	2	2	no	—
99	#95×C₆₇	—	2	1	no	—
100	#95×C₆₇	—	2	2	no	—
101	C₂⁴×C₆₇	—	2	1	no	—
102	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
103	#41×C₆₇	—	4	32	no	—
104	C₂×Q₈×C₆₇	—	2	4	no	—
105	C₂×Q₈×C₆₇	—	4	2	no	—
106	C₂×Q₈×C₆₇	—	4	2	no	—
107	C₂×D₄×C₆₇	—	2	8	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	4	4	no	—
114	C₂×D₄×C₆₇	—	4	4	no	—
115	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
116	C₂×D₄×C₆₇	D₈	1	4	yes	—
117	C₂×D₄×C₆₇	SD₁₆	1	4	yes	—
118	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	8	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	4	4	no	—
131	C₂²×C₄×C₆₇	—	4	4	no	—
132	C₂²×C₄×C₆₇	—	4	4	no	—
133	C₂²×C₄×C₆₇	—	4	2	no	—
134	C₂²×C₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	—	4	4	no	—
136	C₂²×C₄×C₆₇	—	8	2	no	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂×C₈×C₆₇	—	2	1	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	1	no	—
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	4	8	no	—
152	C₂×C₈×C₆₇	#41	1	2	yes	complemented
153	C₄⋊C₄×C₆₇	—	2	8	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	1	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	4	4	no	—
163	C₄⋊C₄×C₆₇	—	4	2	no	—
164	C₄⋊C₄×C₆₇	—	8	1	no	—
165	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₆₇	#41	1	2	yes	complemented
169	C₂²⋊C₄×C₆₇	—	2	4	no	—
170	C₂²⋊C₄×C₆₇	—	2	8	no	—
171	C₂²⋊C₄×C₆₇	—	4	4	no	—
172	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
173	C₄²×C₆₇	—	4	2	no	—
174	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
175	C₄²×C₆₇	C₂×D₄	1	2	yes	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	—	4	1	no	—
183	C₂³×C₆₇	—	4	1	no	—
184	C₂³×C₆₇	—	4	1	no	—
185	C₂³×C₆₇	—	4	1	no	—
186	C₂³×C₆₇	—	4	1	no	—
187	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
188	C₂³×C₆₇	C₂×D₈	1	4	yes	—
189	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
190	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₆₇	—	4	8	no	—
192	D₄×C₆₇	—	2	2	no	—
193	D₄×C₆₇	—	2	2	no	—
194	D₄×C₆₇	—	4	16	no	—
195	D₄×C₆₇	—	4	1	no	—
196	D₄×C₆₇	—	4	1	no	—
197	D₄×C₆₇	#102	1	16	yes	complemented
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	16	no	—
216	C₂×C₄×C₆₇	—	4	8	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	—	4	1	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	8	8	no	—
224	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
226	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
228	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
229	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
235	C₈×C₆₇	—	2	2	no	—
236	C₈×C₆₇	—	4	2	no	—
237	C₈×C₆₇	—	4	1	no	—
238	C₂²×C₆₇	—	2	4	no	—
239	C₂²×C₆₇	—	2	4	no	—
240	C₂²×C₆₇	—	2	4	no	—
241	C₂²×C₆₇	—	2	4	no	—
242	C₂²×C₆₇	—	2	4	no	—
243	C₂²×C₆₇	—	2	4	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	—	4	2	no	—
247	C₂²×C₆₇	—	8	1	no	—
248	C₂²×C₆₇	—	8	1	no	—
249	C₂²×C₆₇	—	8	1	no	—
250	C₂²×C₆₇	—	8	1	no	—
251	C₂²×C₆₇	C₂×#102	1	8	yes	—
252	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₆₇	#546	1	1	yes	characteristic
254	C₂²×C₆₇	#547	1	1	yes	characteristic
255	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
256	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
257	C₂²×C₆₇	#577	1	1	yes	characteristic
258	C₂²×C₆₇	#578	1	1	yes	characteristic
259	C₄×C₆₇	—	2	4	no	—
260	C₄×C₆₇	—	2	2	no	—
261	C₄×C₆₇	—	4	2	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	—	4	2	no	—
264	C₄×C₆₇	—	8	1	no	—
265	C₄×C₆₇	—	8	2	no	—
266	C₄×C₆₇	C₂×#102	1	4	yes	—
267	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
268	C₄×C₆₇	#520	1	1	yes	characteristic
269	C₄×C₆₇	#546	1	1	yes	characteristic
270	C₄×C₆₇	#547	1	1	yes	characteristic
271	#22485	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂×C₆₇	—	2	8	no	—
273	C₂×C₆₇	—	8	2	no	—
274	C₂×C₆₇	#2339	1	1	yes	characteristic
275	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
276	C₂×C₆₇	#2898	1	1	yes	characteristic
277	C₂×C₆₇	#3294	1	1	yes	characteristic
278	C₂×C₆₇	#3298	1	1	yes	characteristic
279	C₂×C₆₇	#3334	1	1	yes	characteristic
280	C₂×C₆₇	#3335	1	1	yes	characteristic
281	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
282	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
283	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
284	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
285	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#1893	C₂×C₆₇	1	2	yes	complemented
287	#1874	C₂×C₆₇	1	2	yes	complemented
288	#1823	C₂×C₆₇	1	2	yes	complemented
289	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₂×#340	C₂×C₆₇	1	2	yes	complemented
292	C₆₇	#22485	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²×#41	—	2	1	no	—
294	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
295	#547	—	2	4	no	—
296	#547	C₄×C₆₇	1	8	yes	complemented
297	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×#123	—	2	2	no	—
299	C₂×C₂²⋊Q₈	—	2	1	no	—
300	C₂×#121	C₂²×C₆₇	1	2	yes	—
301	C₂×C₄×D₄	—	2	1	no	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
304	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
306	C₈×D₄	—	2	4	no	—
307	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×#102	C₂²×C₆₇	1	2	yes	—
311	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
312	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
313	#389	—	2	1	no	—
314	#385	—	2	2	no	—
315	#384	C₂²×C₆₇	1	1	yes	characteristic
316	#380	C₂²×C₆₇	1	2	yes	—
317	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
318	#340	C₂²×C₆₇	1	2	yes	—
319	#340	C₂²×C₆₇	1	1	yes	characteristic
320	#340	C₂²×C₆₇	1	2	yes	—
321	#340	C₂²×C₆₇	1	2	yes	—
322	#340	C₂²×C₆₇	1	2	yes	—
323	#340	C₂²×C₆₇	1	2	yes	—
324	#340	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×#41	—	2	8	no	—
326	C₂×#41	—	4	4	no	—
327	C₂×#41	—	4	4	no	—
328	C₂²×Q₈	—	2	1	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
331	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
332	C₂³×C₄	—	2	2	no	—
333	C₂³×C₄	—	2	1	no	—
334	C₂³×C₄	—	2	2	no	—
335	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
336	C₂²×C₈	—	2	4	no	—
337	C₂²×C₈	—	2	4	no	—
338	C₂²×C₈	—	4	2	no	—
339	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
340	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
344	#123	—	2	16	no	—
345	C₂²⋊Q₈	—	2	8	no	—
346	#121	—	2	4	no	—
347	#121	C₂×C₄×C₆₇	1	8	yes	complemented
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	4	4	no	—
350	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
351	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
352	C₄×D₄	D₄×C₆₇	1	4	yes	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	4	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	4	no	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	4	2	no	—
361	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
362	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×C₂²⋊C₄	—	2	1	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
374	C₄⋊C₈	—	2	2	no	—
375	#102	—	2	2	no	—
376	#102	—	2	2	no	—
377	#102	—	4	2	no	—
378	C₂²⋊C₈	—	2	4	no	—
379	C₄×C₈	—	2	2	no	—
380	#95	—	2	1	no	—
381	#95	—	2	2	no	—
382	C₂⁴	—	2	1	no	—
383	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
384	#41	—	4	32	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	4	2	no	—
387	C₂×Q₈	—	4	2	no	—
388	C₂×D₄	—	2	8	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	4	4	no	—
395	C₂×D₄	—	4	4	no	—
396	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
397	C₂×D₄	D₈×C₆₇	1	4	yes	—
398	C₂×D₄	SD₁₆×C₆₇	1	4	yes	—
399	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	8	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	4	4	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	—	4	4	no	—
414	C₂²×C₄	—	4	2	no	—
415	C₂²×C₄	—	4	2	no	—
416	C₂²×C₄	—	4	4	no	—
417	C₂²×C₄	—	8	2	no	—
418	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
419	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	4	8	no	—
433	C₂×C₈	#41×C₆₇	1	2	yes	complemented
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	4	4	no	—
444	C₄⋊C₄	—	4	2	no	—
445	C₄⋊C₄	—	8	1	no	—
446	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
447	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
448	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
449	C₄⋊C₄	#41×C₆₇	1	2	yes	complemented
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	4	4	no	—
453	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
454	C₄²	—	4	2	no	—
455	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
456	C₄²	C₂×D₄×C₆₇	1	2	yes	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	4	1	no	—
463	C₂³	—	4	1	no	—
464	C₂³	—	4	1	no	—
465	C₂³	—	4	1	no	—
466	C₂³	—	4	1	no	—
467	C₂³	—	4	1	no	—
468	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
469	C₂³	C₂×D₈×C₆₇	1	4	yes	—
470	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
471	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
472	Q₈	—	4	8	no	—
473	D₄	—	2	2	no	—
474	D₄	—	2	2	no	—
475	D₄	—	4	16	no	—
476	D₄	—	4	1	no	—
477	D₄	—	4	1	no	—
478	D₄	#102×C₆₇	1	16	yes	complemented
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	16	no	—
497	C₂×C₄	—	4	8	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	8	8	no	—
505	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
506	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
507	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
508	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
509	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
510	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
513	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
514	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
515	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
516	C₈	—	2	2	no	—
517	C₈	—	4	2	no	—
518	C₈	—	4	1	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	8	1	no	—
529	C₂²	—	8	1	no	—
530	C₂²	—	8	1	no	—
531	C₂²	—	8	1	no	—
532	C₂²	C₂×#102×C₆₇	1	8	yes	—
533	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
534	C₂²	#546×C₆₇	1	1	yes	characteristic
535	C₂²	#547×C₆₇	1	1	yes	characteristic
536	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
537	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
538	C₂²	#577×C₆₇	1	1	yes	characteristic
539	C₂²	#578×C₆₇	1	1	yes	characteristic
540	C₄	—	2	4	no	—
541	C₄	—	2	2	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	4	no	—
545	C₄	—	8	2	no	—
546	C₄	—	8	1	no	—
547	C₄	C₂×#102×C₆₇	1	4	yes	—
548	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
549	C₄	#520×C₆₇	1	1	yes	characteristic
550	C₄	#546×C₆₇	1	1	yes	characteristic
551	C₄	#547×C₆₇	1	1	yes	characteristic
552	C₂	—	2	8	no	—
553	C₂	—	8	2	no	—
554	C₂	C₆₇×#2339	1	1	yes	characteristic
555	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic
556	C₂	C₆₇×#2898	1	1	yes	characteristic
557	C₂	C₆₇×#3294	1	1	yes	characteristic
558	C₂	C₆₇×#3298	1	1	yes	characteristic
559	C₂	C₆₇×#3334	1	1	yes	characteristic
560	C₂	C₆₇×#3335	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×#547×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. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1988	C₂	semidirect
6. C₆₇×#1893	C₂	semidirect
7. C₆₇×#1874	C₂	semidirect
8. C₆₇×#1823	C₂	semidirect
9. C₆₇×#1802	C₂	semidirect
10. C₄×#102×C₆₇	C₂	semidirect
11. C₂×#340×C₆₇	C₂	semidirect
15. #547×C₆₇	C₄	semidirect
62. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
66. #121×C₆₇	C₂×C₄	semidirect
70. C₄×D₄×C₆₇	C₂×C₄	semidirect
80. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
152. C₂×C₈×C₆₇	#41	semidirect
168. C₄⋊C₄×C₆₇	#41	semidirect
197. D₄×C₆₇	#102	semidirect
271. #22485	C₆₇	direct (N × H)
282. C₂×C₈×D₄	C₂×C₆₇	semidirect
283. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
284. #1990	C₂×C₆₇	semidirect
285. #1988	C₂×C₆₇	semidirect
286. #1893	C₂×C₆₇	semidirect
287. #1874	C₂×C₆₇	semidirect
288. #1823	C₂×C₆₇	semidirect
289. #1802	C₂×C₆₇	semidirect
290. C₄×#102	C₂×C₆₇	semidirect
291. C₂×#340	C₂×C₆₇	semidirect
292. C₆₇	#22485	direct (N × H)
296. #547	C₄×C₆₇	semidirect
343. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
347. #121	C₂×C₄×C₆₇	semidirect
351. C₄×D₄	C₂×C₄×C₆₇	semidirect
361. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
433. C₂×C₈	#41×C₆₇	semidirect
449. C₄⋊C₄	#41×C₆₇	semidirect
478. D₄	#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×#547×C₆₇	2
C₂×C₈×D₄×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1990	2
C₆₇×#1988	2
C₆₇×#1893	2
C₆₇×#1874	2
C₆₇×#1823	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
C₂×#340×C₆₇	2
#22485	67

Subgroups — C67×#22485 #30701736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22485 #30701736