Subgroups — C₆₇×#32964 #30712215

Order 17152.

1206 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×#519×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2628	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1854	C₂	1	2	yes	maximal
8	C₆₇×#1805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1805	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1787	C₂	1	2	yes	maximal
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#519×C₆₇	—	2	4	no	—
15	#519×C₆₇	—	2	4	no	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	—	2	1	no	—
23	C₂×#117×C₆₇	—	2	1	no	—
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
26	#477×C₆₇	—	2	4	no	—
27	#475×C₆₇	—	2	4	no	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	C₂×#106×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₂²	1	2	yes	—
33	C₂×#103×C₆₇	C₂²	1	2	yes	—
34	C₂×#103×C₆₇	C₂²	1	2	yes	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
37	#400×C₆₇	—	2	1	no	—
38	#399×C₆₇	—	2	1	no	—
39	#395×C₆₇	—	2	1	no	—
40	#393×C₆₇	—	2	1	no	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#382×C₆₇	C₂²	1	1	yes	characteristic
43	#380×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#340×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₆₇	—	2	2	no	—
48	C₂²×Q₈×C₆₇	—	2	2	no	—
49	C₂²×Q₈×C₆₇	—	2	2	no	—
50	C₂²×Q₈×C₆₇	—	2	2	no	—
51	C₂²×Q₈×C₆₇	—	2	1	no	—
52	C₂²×Q₈×C₆₇	—	2	1	no	—
53	C₂²×Q₈×C₆₇	—	2	2	no	—
54	C₂²×Q₈×C₆₇	Q₈	1	4	yes	complemented
55	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	—	2	2	no	—
63	C₄×Q₈×C₆₇	—	2	8	no	—
64	C₄×Q₈×C₆₇	—	2	8	no	—
65	C₄×Q₈×C₆₇	—	2	2	no	—
66	C₄×Q₈×C₆₇	—	2	2	no	—
67	C₄×Q₈×C₆₇	—	2	2	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	—	2	2	no	—
70	C₄×Q₈×C₆₇	—	2	2	no	—
71	C₄×Q₈×C₆₇	D₄	1	4	yes	—
72	C₄×Q₈×C₆₇	D₄	1	4	yes	—
73	#117×C₆₇	—	2	8	no	—
74	#117×C₆₇	—	2	8	no	—
75	#117×C₆₇	—	2	2	no	—
76	#117×C₆₇	—	2	2	no	—
77	#117×C₆₇	—	2	2	no	—
78	#117×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	—	2	1	no	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	—	2	1	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#107×C₆₇	—	2	4	no	—
103	#106×C₆₇	—	2	4	no	—
104	C₄⋊C₈×C₆₇	—	2	2	no	—
105	C₄⋊C₈×C₆₇	—	2	2	no	—
106	#103×C₆₇	—	2	4	no	—
107	#103×C₆₇	—	2	4	no	—
108	#103×C₆₇	—	2	4	no	—
109	#103×C₆₇	—	2	4	no	—
110	C₂²⋊C₈×C₆₇	—	2	4	no	—
111	#95×C₆₇	—	2	2	no	—
112	#95×C₆₇	—	2	1	no	—
113	#95×C₆₇	—	2	1	no	—
114	#95×C₆₇	—	2	2	no	—
115	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	8	no	—
118	C₂×Q₈×C₆₇	—	2	8	no	—
119	C₂×Q₈×C₆₇	—	2	4	no	—
120	C₂×Q₈×C₆₇	—	2	8	no	—
121	C₂×Q₈×C₆₇	—	2	8	no	—
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	—	2	1	no	—
124	C₂×Q₈×C₆₇	—	2	1	no	—
125	C₂×Q₈×C₆₇	—	2	1	no	—
126	C₂×Q₈×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	4	2	no	—
130	C₂×Q₈×C₆₇	—	8	2	no	—
131	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
132	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
134	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
135	C₂×Q₈×C₆₇	#41	1	2	yes	—
136	C₂×Q₈×C₆₇	#41	1	2	yes	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
156	C₂²×C₄×C₆₇	#41	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	4	no	—
167	C₄⋊C₄×C₆₇	—	2	4	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	4	2	no	—
177	C₄⋊C₄×C₆₇	—	8	2	no	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
180	C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂²⋊C₄×C₆₇	—	2	2	no	—
182	C₂²⋊C₄×C₆₇	—	2	2	no	—
183	C₂²⋊C₄×C₆₇	—	8	2	no	—
184	C₄²×C₆₇	—	2	4	no	—
185	C₄²×C₆₇	—	2	4	no	—
186	C₄²×C₆₇	—	2	4	no	—
187	C₄²×C₆₇	—	2	4	no	—
188	C₄²×C₆₇	—	2	4	no	—
189	C₄²×C₆₇	—	2	1	no	—
190	C₄²×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	C₂×D₄	1	2	yes	—
192	C₄²×C₆₇	C₂×D₄	1	2	yes	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
198	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₆₇	—	2	16	no	—
200	Q₈×C₆₇	—	2	2	no	—
201	Q₈×C₆₇	—	2	2	no	—
202	Q₈×C₆₇	—	2	2	no	—
203	Q₈×C₆₇	—	2	2	no	—
204	Q₈×C₆₇	—	2	2	no	—
205	Q₈×C₆₇	—	2	2	no	—
206	Q₈×C₆₇	—	8	4	no	—
207	Q₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
208	Q₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
209	C₂×C₄×C₆₇	—	2	16	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	2	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	#120	1	2	yes	—
239	C₂×C₄×C₆₇	#120	1	2	yes	—
240	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
247	C₈×C₆₇	—	4	2	no	—
248	C₈×C₆₇	—	4	2	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	4	1	no	—
253	C₂²×C₆₇	—	4	1	no	—
254	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
255	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂²×C₆₇	#525	1	1	yes	characteristic
257	C₂²×C₆₇	#577	1	2	yes	—
258	C₂²×C₆₇	#578	1	2	yes	—
259	C₄×C₆₇	—	2	4	no	—
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	2	2	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	2	2	no	—
264	C₄×C₆₇	—	4	1	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	4	1	no	—
267	C₄×C₆₇	—	4	2	no	—
268	C₄×C₆₇	—	8	2	no	—
269	C₄×C₆₇	—	8	2	no	—
270	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
271	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
272	C₄×C₆₇	#525	1	1	yes	characteristic
273	C₄×C₆₇	#525	1	1	yes	characteristic
274	#32964	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂×C₆₇	—	4	2	no	—
276	C₂×C₆₇	#2848	1	1	yes	characteristic
277	C₂×C₆₇	#3010	1	1	yes	characteristic
278	C₂×C₆₇	#3011	1	1	yes	characteristic
279	C₂×C₆₇	#3082	1	2	yes	—
280	C₂×C₆₇	#3121	1	2	yes	—
281	C₂×#519	C₂×C₆₇	1	1	yes	characteristic
282	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
283	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
284	#2628	C₂×C₆₇	1	1	yes	characteristic
285	#1865	C₂×C₆₇	1	2	yes	complemented
286	#1863	C₂×C₆₇	1	2	yes	complemented
287	#1854	C₂×C₆₇	1	2	yes	—
288	#1805	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#1805	C₂×C₆₇	1	1	yes	characteristic, complemented
290	#1798	C₂×C₆₇	1	1	yes	characteristic
291	#1787	C₂×C₆₇	1	2	yes	—
292	C₆₇	#32964	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
295	#519	—	2	4	no	—
296	#519	—	2	4	no	—
297	C₂×C₄×Q₈	—	2	1	no	—
298	C₂×C₄×Q₈	—	2	1	no	—
299	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#117	—	2	1	no	—
304	C₂×#117	—	2	1	no	—
305	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
307	#477	—	2	4	no	—
308	#475	—	2	4	no	—
309	C₂×#107	C₂²×C₆₇	1	2	yes	—
310	C₂×#106	C₂²×C₆₇	1	2	yes	—
311	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#103	C₂²×C₆₇	1	2	yes	—
314	C₂×#103	C₂²×C₆₇	1	2	yes	—
315	C₂×#103	C₂²×C₆₇	1	2	yes	—
316	C₂×#103	C₂²×C₆₇	1	2	yes	—
317	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
318	#400	—	2	1	no	—
319	#399	—	2	1	no	—
320	#395	—	2	1	no	—
321	#393	—	2	1	no	—
322	#389	C₂²×C₆₇	1	2	yes	—
323	#382	C₂²×C₆₇	1	1	yes	characteristic
324	#380	C₂²×C₆₇	1	2	yes	—
325	#340	C₂²×C₆₇	1	2	yes	—
326	#340	C₂²×C₆₇	1	2	yes	—
327	#336	C₂²×C₆₇	1	2	yes	—
328	C₂²×Q₈	—	2	2	no	—
329	C₂²×Q₈	—	2	2	no	—
330	C₂²×Q₈	—	2	2	no	—
331	C₂²×Q₈	—	2	2	no	—
332	C₂²×Q₈	—	2	1	no	—
333	C₂²×Q₈	—	2	1	no	—
334	C₂²×Q₈	—	2	2	no	—
335	C₂²×Q₈	Q₈×C₆₇	1	4	yes	complemented
336	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
337	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
338	C₂³×C₄	—	2	1	no	—
339	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
341	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	8	no	—
345	C₄×Q₈	—	2	8	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	D₄×C₆₇	1	4	yes	—
353	C₄×Q₈	D₄×C₆₇	1	4	yes	—
354	#117	—	2	8	no	—
355	#117	—	2	8	no	—
356	#117	—	2	2	no	—
357	#117	—	2	2	no	—
358	#117	—	2	2	no	—
359	#117	—	2	2	no	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
376	C₂×C₄²	—	2	1	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
383	#107	—	2	4	no	—
384	#106	—	2	4	no	—
385	C₄⋊C₈	—	2	2	no	—
386	C₄⋊C₈	—	2	2	no	—
387	#103	—	2	4	no	—
388	#103	—	2	4	no	—
389	#103	—	2	4	no	—
390	#103	—	2	4	no	—
391	C₂²⋊C₈	—	2	4	no	—
392	#95	—	2	2	no	—
393	#95	—	2	1	no	—
394	#95	—	2	1	no	—
395	#95	—	2	2	no	—
396	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	2	8	no	—
399	C₂×Q₈	—	2	8	no	—
400	C₂×Q₈	—	2	4	no	—
401	C₂×Q₈	—	2	8	no	—
402	C₂×Q₈	—	2	8	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	4	2	no	—
411	C₂×Q₈	—	8	2	no	—
412	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
413	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
414	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
415	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
416	C₂×Q₈	#41×C₆₇	1	2	yes	—
417	C₂×Q₈	#41×C₆₇	1	2	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
437	C₂²×C₄	#41×C₆₇	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	4	2	no	—
458	C₄⋊C₄	—	8	2	no	—
459	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₂²⋊C₄	—	8	2	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	4	no	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	4	no	—
469	C₄²	—	2	4	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	C₂×D₄×C₆₇	1	2	yes	—
473	C₄²	C₂×D₄×C₆₇	1	2	yes	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
479	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
480	Q₈	—	2	16	no	—
481	Q₈	—	2	2	no	—
482	Q₈	—	2	2	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	2	2	no	—
485	Q₈	—	2	2	no	—
486	Q₈	—	2	2	no	—
487	Q₈	—	8	4	no	—
488	Q₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
489	Q₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
490	C₂×C₄	—	2	16	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	#120×C₆₇	1	2	yes	—
520	C₂×C₄	#120×C₆₇	1	2	yes	—
521	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
524	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
526	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
527	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
528	C₈	—	4	2	no	—
529	C₈	—	4	2	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	2	no	—
535	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
536	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
537	C₂²	#525×C₆₇	1	1	yes	characteristic
538	C₂²	#577×C₆₇	1	2	yes	—
539	C₂²	#578×C₆₇	1	2	yes	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	4	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	2	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	2	no	—
549	C₄	—	8	2	no	—
550	C₄	—	8	2	no	—
551	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
552	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
553	C₄	#525×C₆₇	1	1	yes	characteristic
554	C₄	#525×C₆₇	1	1	yes	characteristic
555	C₂	—	4	2	no	—
556	C₂	C₆₇×#2848	1	1	yes	characteristic
557	C₂	C₆₇×#3010	1	1	yes	characteristic
558	C₂	C₆₇×#3011	1	1	yes	characteristic
559	C₂	C₆₇×#3082	1	2	yes	—
560	C₂	C₆₇×#3121	1	2	yes	—

	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 ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#477×C₆₇	C₂	semidirect
3. C₂×#475×C₆₇	C₂	semidirect
5. C₆₇×#1865	C₂	semidirect
6. C₆₇×#1863	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
9. C₆₇×#1805	C₂	semidirect
54. C₂²×Q₈×C₆₇	Q₈	semidirect
274. #32964	C₆₇	direct (N × H)
282. C₂×#477	C₂×C₆₇	semidirect
283. C₂×#475	C₂×C₆₇	semidirect
285. #1865	C₂×C₆₇	semidirect
286. #1863	C₂×C₆₇	semidirect
288. #1805	C₂×C₆₇	semidirect
289. #1805	C₂×C₆₇	semidirect
292. C₆₇	#32964	direct (N × H)
335. C₂²×Q₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#519×C₆₇	2
C₂×#477×C₆₇	2
C₂×#475×C₆₇	2
C₆₇×#2628	2
C₆₇×#1865	2
C₆₇×#1863	2
C₆₇×#1854	2
C₆₇×#1805	2
C₆₇×#1798	2
C₆₇×#1787	2
#32964	67

Subgroups — C67×#32964 #30712215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32964 #30712215