Subgroups — C₆₇×#32937 #30712188

Order 17152.

1502 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#470×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2848	C₂	1	2	yes	maximal
5	C₆₇×#1894	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1892	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1809	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
11	C₂²×M₁₆×C₆₇	—	2	8	no	—
12	C₂²×M₁₆×C₆₇	—	2	4	no	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
19	#470×C₆₇	—	2	8	no	—
20	#467×C₆₇	—	2	4	no	—
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
23	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
25	#385×C₆₇	—	2	4	no	—
26	#385×C₆₇	—	2	2	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	#380×C₆₇	C₂²	1	4	yes	—
29	#379×C₆₇	—	2	2	no	—
30	#340×C₆₇	C₂²	1	2	yes	complemented
31	#340×C₆₇	C₂²	1	4	yes	complemented
32	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	—	2	4	no	—
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	—	2	4	no	—
37	C₂³×C₄×C₆₇	—	2	2	no	—
38	C₂³×C₄×C₆₇	D₄	1	2	yes	—
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₆₇	—	2	2	no	—
43	C₂×Q₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	—	2	32	no	—
45	C₂×M₁₆×C₆₇	—	2	4	no	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	—	2	4	no	—
49	C₂×M₁₆×C₆₇	—	2	4	no	—
50	C₂×M₁₆×C₆₇	D₄	1	8	yes	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	—	2	4	no	—
54	C₂²×C₈×C₆₇	D₄	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²⋊Q₈×C₆₇	—	2	4	no	—
59	C₂²⋊Q₈×C₆₇	—	2	8	no	—
60	C₂²⋊Q₈×C₆₇	—	2	4	no	—
61	C₂²⋊Q₈×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	#107×C₆₇	—	2	2	no	—
71	#106×C₆₇	—	2	4	no	—
72	#103×C₆₇	—	2	4	no	—
73	#103×C₆₇	—	2	8	no	—
74	#95×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	4	no	—
76	C₂⁴×C₆₇	—	2	4	no	—
77	C₂⁴×C₆₇	—	2	2	no	—
78	C₂⁴×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	—	4	2	no	—
80	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
81	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	8	no	—
87	C₂²×C₄×C₆₇	—	2	8	no	—
88	C₂²×C₄×C₆₇	—	2	16	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₆₇	#41	1	4	yes	—
108	C₂²×C₄×C₆₇	#41	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₆₇	—	4	4	no	—
111	M₁₆×C₆₇	—	2	16	no	—
112	M₁₆×C₆₇	—	2	16	no	—
113	C₂×C₈×C₆₇	—	2	8	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	8	no	—
118	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
119	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	4	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	4	no	—
127	C₄⋊C₄×C₆₇	—	4	8	no	—
128	C₂²⋊C₄×C₆₇	—	4	8	no	—
129	C₂²⋊C₄×C₆₇	—	4	8	no	—
130	C₂²⋊C₄×C₆₇	—	4	4	no	—
131	C₂²⋊C₄×C₆₇	—	4	4	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	—	2	8	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	4	no	—
138	C₂³×C₆₇	—	2	4	no	—
139	C₂³×C₆₇	—	4	16	no	—
140	C₂³×C₆₇	—	4	4	no	—
141	C₂³×C₆₇	—	4	4	no	—
142	C₂³×C₆₇	#121	1	4	yes	—
143	C₂³×C₆₇	#121	1	2	yes	—
144	C₂³×C₆₇	#137	1	4	yes	—
145	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	2	8	no	—
150	C₂×C₄×C₆₇	—	2	16	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	4	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	4	no	—
165	C₂×C₄×C₆₇	—	4	4	no	—
166	C₂×C₄×C₆₇	—	4	4	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	#121	1	2	yes	—
169	C₂×C₄×C₆₇	#121	1	4	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
174	C₈×C₆₇	—	2	4	no	—
175	C₈×C₆₇	—	2	2	no	—
176	C₈×C₆₇	#120	1	4	yes	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	8	no	—
180	C₂²×C₆₇	—	2	4	no	—
181	C₂²×C₆₇	—	2	8	no	—
182	C₂²×C₆₇	—	4	16	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	4	4	no	—
186	C₂²×C₆₇	#470	1	4	yes	complemented
187	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×#121	1	2	yes	—
189	C₂²×C₆₇	C₂×#137	1	2	yes	—
190	C₂²×C₆₇	#575	1	2	yes	—
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	—	2	4	no	—
193	C₄×C₆₇	—	8	2	no	—
194	C₄×C₆₇	—	8	2	no	—
195	C₄×C₆₇	—	8	4	no	—
196	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
197	C₄×C₆₇	#525	1	1	yes	characteristic
198	C₄×C₆₇	#525	1	2	yes	—
199	#32937	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₆₇	—	2	4	no	—
201	C₂×C₆₇	—	4	4	no	—
202	C₂×C₆₇	#2847	1	1	yes	characteristic
203	C₂×C₆₇	C₂×#470	1	1	yes	characteristic
204	C₂×C₆₇	#3054	1	1	yes	characteristic
205	C₂×C₆₇	#3056	1	4	yes	—
206	C₂³×M₁₆	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#470	C₂×C₆₇	1	2	yes	complemented
208	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#2848	C₂×C₆₇	1	2	yes	—
210	#1894	C₂×C₆₇	1	4	yes	complemented
211	#1892	C₂×C₆₇	1	2	yes	complemented
212	#1809	C₂×C₆₇	1	2	yes	complemented
213	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₆₇	#32937	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
217	C₂²×M₁₆	—	2	8	no	—
218	C₂²×M₁₆	—	2	4	no	—
219	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
220	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
221	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
224	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
225	#470	—	2	8	no	—
226	#467	—	2	4	no	—
227	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
228	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
229	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
230	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
231	#385	—	2	4	no	—
232	#385	—	2	2	no	—
233	#380	C₂²×C₆₇	1	2	yes	—
234	#380	C₂²×C₆₇	1	4	yes	—
235	#379	—	2	2	no	—
236	#340	C₂²×C₆₇	1	2	yes	complemented
237	#340	C₂²×C₆₇	1	4	yes	complemented
238	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
239	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
240	C₂³×C₄	—	2	4	no	—
241	C₂³×C₄	—	2	2	no	—
242	C₂³×C₄	—	2	4	no	—
243	C₂³×C₄	—	2	2	no	—
244	C₂³×C₄	D₄×C₆₇	1	2	yes	—
245	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
247	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
248	C₂×Q₁₆	—	2	2	no	—
249	C₂×Q₁₆	—	2	4	no	—
250	C₂×M₁₆	—	2	32	no	—
251	C₂×M₁₆	—	2	4	no	—
252	C₂×M₁₆	—	2	4	no	—
253	C₂×M₁₆	—	2	4	no	—
254	C₂×M₁₆	—	2	4	no	—
255	C₂×M₁₆	—	2	4	no	—
256	C₂×M₁₆	D₄×C₆₇	1	8	yes	—
257	C₂²×C₈	—	2	2	no	—
258	C₂²×C₈	—	2	4	no	—
259	C₂²×C₈	—	2	4	no	—
260	C₂²×C₈	D₄×C₆₇	1	4	yes	—
261	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
262	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
264	C₂²⋊Q₈	—	2	4	no	—
265	C₂²⋊Q₈	—	2	8	no	—
266	C₂²⋊Q₈	—	2	4	no	—
267	C₂²⋊Q₈	—	2	8	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	4	no	—
270	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
271	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	4	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	#107	—	2	2	no	—
277	#106	—	2	4	no	—
278	#103	—	2	4	no	—
279	#103	—	2	8	no	—
280	#95	—	2	2	no	—
281	#95	—	2	4	no	—
282	C₂⁴	—	2	4	no	—
283	C₂⁴	—	2	2	no	—
284	C₂⁴	—	2	4	no	—
285	C₂⁴	—	4	2	no	—
286	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
287	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	4	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	8	no	—
294	C₂²×C₄	—	2	16	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
313	C₂²×C₄	#41×C₆₇	1	4	yes	—
314	C₂²×C₄	#41×C₆₇	1	2	yes	—
315	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
316	Q₁₆	—	4	4	no	—
317	M₁₆	—	2	16	no	—
318	M₁₆	—	2	16	no	—
319	C₂×C₈	—	2	8	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	8	no	—
324	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
325	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
326	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
327	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	4	no	—
333	C₄⋊C₄	—	4	8	no	—
334	C₂²⋊C₄	—	4	8	no	—
335	C₂²⋊C₄	—	4	8	no	—
336	C₂²⋊C₄	—	4	4	no	—
337	C₂²⋊C₄	—	4	4	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	8	no	—
341	C₂³	—	2	2	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	4	no	—
344	C₂³	—	2	4	no	—
345	C₂³	—	4	16	no	—
346	C₂³	—	4	4	no	—
347	C₂³	—	4	4	no	—
348	C₂³	#121×C₆₇	1	4	yes	—
349	C₂³	#121×C₆₇	1	2	yes	—
350	C₂³	#137×C₆₇	1	4	yes	—
351	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
352	Q₈	—	4	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	4	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	16	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	#121×C₆₇	1	2	yes	—
375	C₂×C₄	#121×C₆₇	1	4	yes	—
376	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
378	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
380	C₈	—	2	4	no	—
381	C₈	—	2	2	no	—
382	C₈	#120×C₆₇	1	4	yes	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	8	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	8	no	—
387	C₂²	—	2	4	no	—
388	C₂²	—	4	16	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	4	no	—
392	C₂²	#470×C₆₇	1	4	yes	complemented
393	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
394	C₂²	C₂×#121×C₆₇	1	2	yes	—
395	C₂²	C₂×#137×C₆₇	1	2	yes	—
396	C₂²	#575×C₆₇	1	2	yes	—
397	C₄	—	2	2	no	—
398	C₄	—	2	4	no	—
399	C₄	—	8	2	no	—
400	C₄	—	8	4	no	—
401	C₄	—	8	2	no	—
402	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
403	C₄	#525×C₆₇	1	1	yes	characteristic
404	C₄	#525×C₆₇	1	2	yes	—
405	C₂	—	2	4	no	—
406	C₂	—	4	4	no	—
407	C₂	C₆₇×#2847	1	1	yes	characteristic
408	C₂	C₂×#470×C₆₇	1	1	yes	characteristic
409	C₂	C₆₇×#3054	1	1	yes	characteristic
410	C₂	C₆₇×#3056	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁵×C₆₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#470×C₆₇	C₂	semidirect
3. C₂×#467×C₆₇	C₂	semidirect
5. C₆₇×#1894	C₂	semidirect
6. C₆₇×#1892	C₂	semidirect
7. C₆₇×#1809	C₂	semidirect
8. C₆₇×#1807	C₂	semidirect
10. C₂²×Q₁₆×C₆₇	C₂²	semidirect
21. C₂×#107×C₆₇	C₂²	semidirect
22. C₂×#106×C₆₇	C₂²	semidirect
23. C₂×#103×C₆₇	C₂²	semidirect
24. C₂×#103×C₆₇	C₂²	semidirect
30. #340×C₆₇	C₂²	semidirect
31. #340×C₆₇	C₂²	semidirect
186. C₂²×C₆₇	#470	semidirect
199. #32937	C₆₇	direct (N × H)
207. C₂×#470	C₂×C₆₇	semidirect
208. C₂×#467	C₂×C₆₇	semidirect
210. #1894	C₂×C₆₇	semidirect
211. #1892	C₂×C₆₇	semidirect
212. #1809	C₂×C₆₇	semidirect
213. #1807	C₂×C₆₇	semidirect
214. C₆₇	#32937	direct (N × H)
216. C₂²×Q₁₆	C₂²×C₆₇	semidirect
227. C₂×#107	C₂²×C₆₇	semidirect
228. C₂×#106	C₂²×C₆₇	semidirect
229. C₂×#103	C₂²×C₆₇	semidirect
230. C₂×#103	C₂²×C₆₇	semidirect
236. #340	C₂²×C₆₇	semidirect
237. #340	C₂²×C₆₇	semidirect
392. C₂²	#470×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₇	2
C₂×#470×C₆₇	2
C₂×#467×C₆₇	2
C₆₇×#2848	2
C₆₇×#1894	2
C₆₇×#1892	2
C₆₇×#1809	2
C₆₇×#1807	2
#32937	67

Subgroups — C67×#32937 #30712188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32937 #30712188