Subgroups — C₆₇×#32954 #30712205

Order 17152.

1332 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#475×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1865	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1853	C₂	1	2	yes	maximal
6	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1787	C₂	1	2	yes	maximal
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
13	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
15	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₆₇	—	2	1	no	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#475×C₆₇	—	2	8	no	—
22	C₂×#106×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₂²	1	4	yes	—
25	C₂×#103×C₆₇	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	#391×C₆₇	—	2	1	no	—
28	#389×C₆₇	—	2	1	no	—
29	#389×C₆₇	C₂²	1	2	yes	—
30	#384×C₆₇	C₂²	1	1	yes	characteristic
31	#380×C₆₇	—	2	1	no	—
32	#380×C₆₇	C₂²	1	2	yes	—
33	#340×C₆₇	C₂²	1	4	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₇	—	2	4	no	—
36	C₂²×Q₈×C₆₇	—	2	4	no	—
37	C₂²×Q₈×C₆₇	—	2	1	no	—
38	C₂²×Q₈×C₆₇	—	2	1	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	Q₈	1	4	yes	complemented
41	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	—	2	1	no	—
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
48	C₄⋊Q₈×C₆₇	—	2	4	no	—
49	C₂²⋊Q₈×C₆₇	—	4	4	no	—
50	C₄×Q₈×C₆₇	—	2	8	no	—
51	C₄×Q₈×C₆₇	—	2	8	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	2	4	no	—
54	C₄×Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	D₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	8	no	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	4	no	—
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#106×C₆₇	—	2	8	no	—
78	C₄⋊C₈×C₆₇	—	2	4	no	—
79	#103×C₆₇	—	2	8	no	—
80	#103×C₆₇	—	2	8	no	—
81	C₂²⋊C₈×C₆₇	—	2	4	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Q₈×C₆₇	—	2	4	no	—
86	C₂×Q₈×C₆₇	—	2	16	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	2	16	no	—
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	4	2	no	—
94	C₂×Q₈×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	—	4	2	no	—
96	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
97	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
98	C₂×Q₈×C₆₇	#41	1	4	yes	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	4	1	no	—
114	C₂²×C₄×C₆₇	—	4	2	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
123	C₂²×C₄×C₆₇	#41	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	2	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	—	4	4	no	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²⋊C₄×C₆₇	—	4	4	no	—
140	C₄²×C₆₇	—	2	8	no	—
141	C₄²×C₆₇	—	2	8	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	C₂×D₄	1	4	yes	—
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	1	no	—
148	C₂³×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	1	no	—
150	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
151	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
152	C₂³×C₆₇	#137	1	2	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	16	no	—
155	Q₈×C₆₇	—	2	4	no	—
156	Q₈×C₆₇	—	2	4	no	—
157	Q₈×C₆₇	—	2	4	no	—
158	Q₈×C₆₇	—	8	4	no	—
159	Q₈×C₆₇	C₂²⋊Q₈	1	8	yes	complemented
160	C₂×C₄×C₆₇	—	2	16	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	8	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	4	4	no	—
178	C₂×C₄×C₆₇	—	4	8	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	#120	1	2	yes	—
188	C₂×C₄×C₆₇	#120	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
190	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
191	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
192	C₂×C₄×C₆₇	#137	1	2	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₈×C₆₇	—	4	4	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	#475	1	4	yes	complemented
205	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
206	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₆₇	#525	1	1	yes	characteristic
208	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
209	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
210	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
211	C₂²×C₆₇	#575	1	1	yes	characteristic
212	C₄×C₆₇	—	2	8	no	—
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	4	1	no	—
217	C₄×C₆₇	—	4	4	no	—
218	C₄×C₆₇	—	8	2	no	—
219	C₄×C₆₇	—	8	2	no	—
220	C₄×C₆₇	#448	1	4	yes	—
221	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
222	C₄×C₆₇	#525	1	2	yes	—
223	#32954	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₆₇	—	2	4	no	—
225	C₂×C₆₇	#2848	1	1	yes	characteristic
226	C₂×C₆₇	C₂×#448	1	1	yes	characteristic
227	C₂×C₆₇	#3007	1	1	yes	characteristic
228	C₂×C₆₇	C₂×#475	1	1	yes	characteristic
229	C₂×C₆₇	#3078	1	1	yes	characteristic
230	C₂×C₆₇	#3103	1	1	yes	characteristic
231	C₂×C₆₇	#3113	1	1	yes	characteristic
232	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
233	C₂×#475	C₂×C₆₇	1	2	yes	complemented
234	#2857	C₂×C₆₇	1	1	yes	characteristic
235	#1865	C₂×C₆₇	1	4	yes	complemented
236	#1853	C₂×C₆₇	1	2	yes	—
237	#1803	C₂×C₆₇	1	2	yes	complemented
238	#1797	C₂×C₆₇	1	1	yes	characteristic
239	#1787	C₂×C₆₇	1	2	yes	—
240	C₆₇	#32954	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×C₂²⋊Q₈	—	2	1	no	—
244	C₂×C₄×Q₈	—	2	8	no	—
245	C₂×C₄×Q₈	—	2	1	no	—
246	C₂×C₄×Q₈	—	2	1	no	—
247	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
248	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
249	C₂²×C₄⋊C₄	—	2	1	no	—
250	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
251	C₂²×C₄²	—	2	1	no	—
252	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
253	#475	—	2	8	no	—
254	C₂×#106	C₂²×C₆₇	1	4	yes	—
255	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
256	C₂×#103	C₂²×C₆₇	1	4	yes	—
257	C₂×#103	C₂²×C₆₇	1	4	yes	—
258	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
259	#391	—	2	1	no	—
260	#389	—	2	1	no	—
261	#389	C₂²×C₆₇	1	2	yes	—
262	#384	C₂²×C₆₇	1	1	yes	characteristic
263	#380	—	2	1	no	—
264	#380	C₂²×C₆₇	1	2	yes	—
265	#340	C₂²×C₆₇	1	4	yes	—
266	#336	C₂²×C₆₇	1	2	yes	—
267	C₂²×Q₈	—	2	4	no	—
268	C₂²×Q₈	—	2	4	no	—
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×Q₈	—	2	1	no	—
271	C₂²×Q₈	—	2	2	no	—
272	C₂²×Q₈	Q₈×C₆₇	1	4	yes	complemented
273	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
274	C₂³×C₄	—	2	1	no	—
275	C₂³×C₄	—	2	1	no	—
276	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
277	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
280	C₄⋊Q₈	—	2	4	no	—
281	C₂²⋊Q₈	—	4	4	no	—
282	C₄×Q₈	—	2	8	no	—
283	C₄×Q₈	—	2	8	no	—
284	C₄×Q₈	—	2	4	no	—
285	C₄×Q₈	—	2	4	no	—
286	C₄×Q₈	—	2	4	no	—
287	C₄×Q₈	D₄×C₆₇	1	8	yes	—
288	C₂×C₄⋊C₄	—	2	8	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	4	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	—	2	2	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	—	2	4	no	—
305	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
309	#106	—	2	8	no	—
310	C₄⋊C₈	—	2	4	no	—
311	#103	—	2	8	no	—
312	#103	—	2	8	no	—
313	C₂²⋊C₈	—	2	4	no	—
314	#95	—	2	2	no	—
315	#95	—	2	2	no	—
316	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	16	no	—
319	C₂×Q₈	—	2	4	no	—
320	C₂×Q₈	—	2	16	no	—
321	C₂×Q₈	—	2	4	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	—	4	2	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
329	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
330	C₂×Q₈	#41×C₆₇	1	4	yes	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
355	C₂²×C₄	#41×C₆₇	1	2	yes	—
356	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
357	C₂×C₈	—	2	4	no	—
358	C₂×C₈	—	2	4	no	—
359	C₂×C₈	—	2	4	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
371	C₂²⋊C₄	—	4	4	no	—
372	C₄²	—	2	8	no	—
373	C₄²	—	2	8	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	C₂×D₄×C₆₇	1	4	yes	—
378	C₂³	—	2	1	no	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	1	no	—
381	C₂³	—	2	1	no	—
382	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
383	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
384	C₂³	#137×C₆₇	1	2	yes	—
385	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
386	Q₈	—	2	16	no	—
387	Q₈	—	2	4	no	—
388	Q₈	—	2	4	no	—
389	Q₈	—	2	4	no	—
390	Q₈	—	8	4	no	—
391	Q₈	C₂²⋊Q₈×C₆₇	1	8	yes	complemented
392	C₂×C₄	—	2	16	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	4	4	no	—
410	C₂×C₄	—	4	8	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	#120×C₆₇	1	2	yes	—
420	C₂×C₄	#120×C₆₇	1	2	yes	—
421	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
422	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
423	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
424	C₂×C₄	#137×C₆₇	1	2	yes	—
425	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
426	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
428	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
429	C₈	—	4	4	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	#475×C₆₇	1	4	yes	complemented
437	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
438	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
439	C₂²	#525×C₆₇	1	1	yes	characteristic
440	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
441	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
442	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
443	C₂²	#575×C₆₇	1	1	yes	characteristic
444	C₄	—	2	8	no	—
445	C₄	—	2	2	no	—
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	4	no	—
450	C₄	—	8	2	no	—
451	C₄	—	8	2	no	—
452	C₄	#448×C₆₇	1	4	yes	—
453	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
454	C₄	#525×C₆₇	1	2	yes	—
455	C₂	—	2	4	no	—
456	C₂	C₆₇×#2848	1	1	yes	characteristic
457	C₂	C₂×#448×C₆₇	1	1	yes	characteristic
458	C₂	C₆₇×#3007	1	1	yes	characteristic
459	C₂	C₂×#475×C₆₇	1	1	yes	characteristic
460	C₂	C₆₇×#3078	1	1	yes	characteristic
461	C₂	C₆₇×#3103	1	1	yes	characteristic
462	C₂	C₆₇×#3113	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#475×C₆₇	C₂	semidirect
4. C₆₇×#1865	C₂	semidirect
6. C₆₇×#1803	C₂	semidirect
40. C₂²×Q₈×C₆₇	Q₈	semidirect
159. Q₈×C₆₇	C₂²⋊Q₈	semidirect
204. C₂²×C₆₇	#475	semidirect
223. #32954	C₆₇	direct (N × H)
233. C₂×#475	C₂×C₆₇	semidirect
235. #1865	C₂×C₆₇	semidirect
237. #1803	C₂×C₆₇	semidirect
240. C₆₇	#32954	direct (N × H)
272. C₂²×Q₈	Q₈×C₆₇	semidirect
391. Q₈	C₂²⋊Q₈×C₆₇	semidirect
436. C₂²	#475×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#475×C₆₇	2
C₆₇×#2857	2
C₆₇×#1865	2
C₆₇×#1853	2
C₆₇×#1803	2
C₆₇×#1797	2
C₆₇×#1787	2
#32954	67

Subgroups — C67×#32954 #30712205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32954 #30712205