Subgroups — C₆₇×#22762 #30702013

Order 17152.

902 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1935	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1920	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1877	C₂	1	2	yes	maximal
7	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1762	C₂	1	2	yes	maximal
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₄×SD₁₆×C₆₇	C₄	1	8	yes	complemented
18	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₆₇	C₂²	1	2	yes	—
21	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#391×C₆₇	C₂²	1	2	yes	—
27	#389×C₆₇	C₂²	1	2	yes	—
28	#387×C₆₇	—	2	2	no	—
29	#385×C₆₇	—	2	2	no	—
30	#384×C₆₇	C₂²	1	1	yes	characteristic
31	#384×C₆₇	C₂²	1	1	yes	characteristic
32	#382×C₆₇	—	2	1	no	—
33	#380×C₆₇	—	2	1	no	—
34	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×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	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂×SD₁₆×C₆₇	—	2	2	no	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	4	2	no	—
57	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#106×C₆₇	C₂×C₄	1	4	yes	complemented
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	4	2	no	—
80	#103×C₆₇	C₂×C₄	1	4	yes	complemented
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	C₂×C₄	1	4	yes	—
83	#97×C₆₇	—	4	2	no	—
84	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	D₄	1	2	yes	—
90	C₂⁴×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	2	1	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₆₇	—	2	1	no	—
98	C₂×D₄×C₆₇	—	2	1	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₆₇	#41	1	2	yes	—
102	C₂×D₄×C₆₇	#41	1	2	yes	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	4	4	no	—
110	C₂²×C₄×C₆₇	—	4	4	no	—
111	C₂²×C₄×C₆₇	—	4	2	no	—
112	C₂²×C₄×C₆₇	—	4	2	no	—
113	C₂²×C₄×C₆₇	—	8	2	no	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	SD₁₆×C₆₇	—	2	8	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	4	4	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	—	4	2	no	—
141	C₄⋊C₄×C₆₇	—	8	1	no	—
142	C₄⋊C₄×C₆₇	—	8	1	no	—
143	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
145	C₂²⋊C₄×C₆₇	—	2	4	no	—
146	C₂²⋊C₄×C₆₇	—	4	4	no	—
147	C₄²×C₆₇	—	2	4	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	8	2	no	—
151	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
152	C₄²×C₆₇	#41	1	2	yes	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	1	no	—
158	C₂³×C₆₇	—	4	1	no	—
159	C₂³×C₆₇	—	4	1	no	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	2	2	no	—
162	Q₈×C₆₇	—	2	2	no	—
163	Q₈×C₆₇	—	4	1	no	—
164	Q₈×C₆₇	—	4	1	no	—
165	D₄×C₆₇	—	2	2	no	—
166	D₄×C₆₇	—	2	2	no	—
167	D₄×C₆₇	—	2	2	no	—
168	D₄×C₆₇	#117	1	4	yes	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	8	8	no	—
194	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₄×C₆₇	#137	1	2	yes	—
198	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
204	C₈×C₆₇	—	2	2	no	—
205	C₈×C₆₇	—	2	2	no	—
206	C₈×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	4	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	4	2	no	—
214	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₆₇	#546	1	1	yes	characteristic
216	C₂²×C₆₇	#548	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
219	C₂²×C₆₇	#577	1	1	yes	characteristic
220	C₂²×C₆₇	#577	1	1	yes	characteristic
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	4	2	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	4	2	no	—
225	C₄×C₆₇	—	4	2	no	—
226	C₄×C₆₇	—	4	2	no	—
227	C₄×C₆₇	—	8	2	no	—
228	C₄×C₆₇	—	8	2	no	—
229	C₄×C₆₇	#440	1	4	yes	—
230	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
231	C₄×C₆₇	#520	1	1	yes	characteristic
232	C₄×C₆₇	#552	1	1	yes	characteristic
233	C₄×C₆₇	#553	1	1	yes	characteristic
234	#22762	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₆₇	—	4	4	no	—
236	C₂×C₆₇	#2352	1	1	yes	characteristic
237	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
238	C₂×C₆₇	#2952	1	1	yes	characteristic
239	C₂×C₆₇	#3324	1	1	yes	characteristic
240	C₂×C₆₇	#3331	1	1	yes	characteristic
241	C₂×C₆₇	#3372	1	1	yes	characteristic
242	C₂×C₆₇	#3373	1	1	yes	characteristic
243	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
244	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
245	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
246	#1935	C₂×C₆₇	1	2	yes	complemented
247	#1920	C₂×C₆₇	1	2	yes	complemented
248	#1877	C₂×C₆₇	1	2	yes	—
249	#1834	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1803	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#1802	C₂×C₆₇	1	1	yes	characteristic
252	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#1762	C₂×C₆₇	1	2	yes	—
254	C₆₇	#22762	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
258	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₂²×C₄⋊C₄	—	2	1	no	—
260	C₄×SD₁₆	C₄×C₆₇	1	8	yes	complemented
261	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×#103	C₂²×C₆₇	1	2	yes	—
264	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×#102	C₂²×C₆₇	1	2	yes	—
266	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
267	C₂×#97	C₂²×C₆₇	1	2	yes	—
268	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
269	#391	C₂²×C₆₇	1	2	yes	—
270	#389	C₂²×C₆₇	1	2	yes	—
271	#387	—	2	2	no	—
272	#385	—	2	2	no	—
273	#384	C₂²×C₆₇	1	1	yes	characteristic
274	#384	C₂²×C₆₇	1	1	yes	characteristic
275	#382	—	2	1	no	—
276	#380	—	2	1	no	—
277	C₄×C₄⋊C₄	—	2	1	no	—
278	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
279	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
280	#340	C₂²×C₆₇	1	2	yes	—
281	#340	C₂²×C₆₇	1	2	yes	—
282	#340	C₂²×C₆₇	1	2	yes	—
283	#340	C₂²×C₆₇	1	2	yes	—
284	#336	C₂²×C₆₇	1	2	yes	—
285	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
286	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	—	2	1	no	—
288	C₂³×C₄	—	2	2	no	—
289	C₂×SD₁₆	—	2	2	no	—
290	C₂×SD₁₆	—	2	2	no	—
291	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
292	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
293	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₄×Q₈	—	2	2	no	—
296	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
297	C₄×D₄	—	2	4	no	—
298	C₄×D₄	—	2	2	no	—
299	C₄×D₄	—	4	2	no	—
300	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	4	4	no	—
308	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	—	2	1	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	#106	C₂×C₄×C₆₇	1	4	yes	complemented
321	C₄⋊C₈	—	2	2	no	—
322	#103	—	4	2	no	—
323	#103	C₂×C₄×C₆₇	1	4	yes	complemented
324	#102	—	2	4	no	—
325	#102	C₂×C₄×C₆₇	1	4	yes	—
326	#97	—	4	2	no	—
327	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
328	#95	—	2	2	no	—
329	#95	—	2	2	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	#95	D₄×C₆₇	1	2	yes	—
333	C₂⁴	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
344	C₂×D₄	#41×C₆₇	1	2	yes	—
345	C₂×D₄	#41×C₆₇	1	2	yes	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	—	4	4	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	—	4	2	no	—
356	C₂²×C₄	—	8	2	no	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
358	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
359	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
362	SD₁₆	—	2	8	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	4	4	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	8	1	no	—
385	C₄⋊C₄	—	8	1	no	—
386	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
387	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₄²	—	2	4	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	8	2	no	—
394	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
395	C₄²	#41×C₆₇	1	2	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	1	no	—
402	C₂³	—	4	1	no	—
403	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
404	Q₈	—	2	2	no	—
405	Q₈	—	2	2	no	—
406	Q₈	—	4	1	no	—
407	Q₈	—	4	1	no	—
408	D₄	—	2	2	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	#117×C₆₇	1	4	yes	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	8	8	no	—
437	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
438	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
439	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
440	C₂×C₄	#137×C₆₇	1	2	yes	—
441	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
445	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
446	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
447	C₈	—	2	2	no	—
448	C₈	—	2	2	no	—
449	C₈	—	4	2	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
458	C₂²	#546×C₆₇	1	1	yes	characteristic
459	C₂²	#548×C₆₇	1	1	yes	characteristic
460	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
461	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
462	C₂²	#577×C₆₇	1	1	yes	characteristic
463	C₂²	#577×C₆₇	1	1	yes	characteristic
464	C₄	—	2	2	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	2	no	—
471	C₄	—	8	2	no	—
472	C₄	#440×C₆₇	1	4	yes	—
473	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
474	C₄	#520×C₆₇	1	1	yes	characteristic
475	C₄	#552×C₆₇	1	1	yes	characteristic
476	C₄	#553×C₆₇	1	1	yes	characteristic
477	C₂	—	4	4	no	—
478	C₂	C₆₇×#2352	1	1	yes	characteristic
479	C₂	C₂×#440×C₆₇	1	1	yes	characteristic
480	C₂	C₆₇×#2952	1	1	yes	characteristic
481	C₂	C₆₇×#3324	1	1	yes	characteristic
482	C₂	C₆₇×#3331	1	1	yes	characteristic
483	C₂	C₆₇×#3372	1	1	yes	characteristic
484	C₂	C₆₇×#3373	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	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2355	C₂	semidirect
4. C₆₇×#1935	C₂	semidirect
5. C₆₇×#1920	C₂	semidirect
7. C₆₇×#1834	C₂	semidirect
8. C₆₇×#1803	C₂	semidirect
10. C₆₇×#1770	C₂	semidirect
17. C₄×SD₁₆×C₆₇	C₄	semidirect
53. C₄×Q₈×C₆₇	C₂×C₄	semidirect
77. #106×C₆₇	C₂×C₄	semidirect
80. #103×C₆₇	C₂×C₄	semidirect
84. C₄×C₈×C₆₇	C₂×C₄	semidirect
234. #22762	C₆₇	direct (N × H)
244. #2355	C₂×C₆₇	semidirect
246. #1935	C₂×C₆₇	semidirect
247. #1920	C₂×C₆₇	semidirect
249. #1834	C₂×C₆₇	semidirect
250. #1803	C₂×C₆₇	semidirect
252. #1770	C₂×C₆₇	semidirect
254. C₆₇	#22762	direct (N × H)
260. C₄×SD₁₆	C₄×C₆₇	semidirect
296. C₄×Q₈	C₂×C₄×C₆₇	semidirect
320. #106	C₂×C₄×C₆₇	semidirect
323. #103	C₂×C₄×C₆₇	semidirect
327. C₄×C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×#2355	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1935	2
C₆₇×#1920	2
C₆₇×#1877	2
C₆₇×#1834	2
C₆₇×#1803	2
C₆₇×#1802	2
C₆₇×#1770	2
C₆₇×#1762	2
#22762	67

Subgroups — C67×#22762 #30702013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22762 #30702013