Subgroups — C₆₇×#10748 #30689999

Order 17152.

1190 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#410×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1729	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	#568×C₆₇	—	2	2	no	—
8	#568×C₆₇	—	2	4	no	—
9	#568×C₆₇	C₄	1	4	yes	complemented
10	C₂×#131×C₆₇	—	2	4	no	—
11	C₂²×M₁₆×C₆₇	—	2	2	no	—
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	#410×C₆₇	—	2	2	no	—
17	#410×C₆₇	—	2	2	no	—
18	C₂×#99×C₆₇	C₄	1	4	yes	—
19	#408×C₆₇	—	2	2	no	—
20	#408×C₆₇	C₄	1	4	yes	complemented
21	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
22	#344×C₆₇	—	2	4	no	—
23	#336×C₆₇	—	2	4	no	—
24	C₂×#41×C₆₇	—	2	2	no	—
25	C₂×#41×C₆₇	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×#41×C₆₇	D₄	1	2	yes	—
31	C₂×#41×C₆₇	D₄	1	2	yes	—
32	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	D₄	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	#131×C₆₇	—	2	16	no	—
38	#131×C₆₇	—	2	16	no	—
39	C₂×M₁₆×C₆₇	—	2	4	no	—
40	C₂×M₁₆×C₆₇	—	2	8	no	—
41	C₂×M₁₆×C₆₇	—	2	8	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	1	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
52	C₂²×C₈×C₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₆₇	D₄	1	4	yes	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	—	2	2	no	—
62	#117×C₆₇	—	2	2	no	—
63	#117×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	#99×C₆₇	—	2	8	no	—
68	C₂²⋊C₈×C₆₇	—	2	8	no	—
69	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	#41×C₆₇	—	2	4	no	—
72	#41×C₆₇	—	2	4	no	—
73	#41×C₆₇	—	2	4	no	—
74	#41×C₆₇	—	2	4	no	—
75	#41×C₆₇	—	2	4	no	—
76	#41×C₆₇	—	2	4	no	—
77	#41×C₆₇	—	2	4	no	—
78	#41×C₆₇	—	2	4	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₂×D₄×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	C₄²	1	4	yes	—
88	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
89	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
97	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	2	yes	—
108	C₂²×C₄×C₆₇	#41	1	2	yes	—
109	M₁₆×C₆₇	—	2	16	no	—
110	M₁₆×C₆₇	—	2	4	no	—
111	M₁₆×C₆₇	—	2	4	no	—
112	M₁₆×C₆₇	—	2	4	no	—
113	M₁₆×C₆₇	—	2	4	no	—
114	M₁₆×C₆₇	—	2	4	no	—
115	M₁₆×C₆₇	—	4	2	no	—
116	C₂×C₈×C₆₇	—	2	16	no	—
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	2	4	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	4	no	—
122	C₂×C₈×C₆₇	—	4	1	no	—
123	C₂×C₈×C₆₇	—	4	1	no	—
124	C₂×C₈×C₆₇	—	4	2	no	—
125	C₂×C₈×C₆₇	—	4	1	no	—
126	C₂×C₈×C₆₇	—	4	1	no	—
127	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₂²⋊C₄×C₆₇	—	2	4	no	—
131	C₂²⋊C₄×C₆₇	—	2	4	no	—
132	C₄²×C₆₇	—	4	2	no	—
133	C₄²×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	4	2	no	—
137	C₂³×C₆₇	—	4	4	no	—
138	C₂³×C₆₇	#95	1	4	yes	—
139	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
140	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂³×C₆₇	C₄×D₄	1	2	yes	—
143	C₂³×C₆₇	C₄×D₄	1	2	yes	—
144	Q₈×C₆₇	—	2	4	no	—
145	Q₈×C₆₇	—	2	4	no	—
146	D₄×C₆₇	—	2	8	no	—
147	D₄×C₆₇	—	2	8	no	—
148	D₄×C₆₇	—	2	4	no	—
149	D₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	4	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	4	8	no	—
157	C₂×C₄×C₆₇	—	4	4	no	—
158	C₂×C₄×C₆₇	#95	1	4	yes	—
159	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	#117	1	2	yes	—
165	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₆₇	#120	1	2	yes	—
168	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
170	C₂×C₄×C₆₇	#121	1	2	yes	—
171	C₂×C₄×C₆₇	#121	1	2	yes	—
172	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
173	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
174	C₂×C₄×C₆₇	#123	1	2	yes	—
175	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
176	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
177	C₈×C₆₇	—	2	8	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	2	4	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	8	2	no	—
189	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
190	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
191	C₂²×C₆₇	#385	1	1	yes	characteristic
192	C₂²×C₆₇	#385	1	1	yes	characteristic
193	C₂²×C₆₇	#386	1	2	yes	—
194	C₄×C₆₇	—	2	4	no	—
195	C₄×C₆₇	—	8	4	no	—
196	C₄×C₆₇	#379	1	1	yes	characteristic
197	C₄×C₆₇	#380	1	2	yes	—
198	C₄×C₆₇	#381	1	1	yes	characteristic
199	C₄×C₆₇	#385	1	2	yes	—
200	C₄×C₆₇	#386	1	1	yes	characteristic
201	C₄×C₆₇	#386	1	1	yes	characteristic
202	#10748	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	2	2	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	2	2	no	—
207	C₂×C₆₇	—	4	4	no	—
208	C₂×C₆₇	C₂²⋊#95	1	1	yes	characteristic
209	C₂×C₆₇	#1868	1	2	yes	—
210	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
211	C₂×#410	C₂×C₆₇	1	1	yes	characteristic
212	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
213	C₂×#344	C₂×C₆₇	1	2	yes	complemented
214	#1729	C₂×C₆₇	1	2	yes	complemented
215	C₆₇	#10748	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
217	#568	—	2	2	no	—
218	#568	—	2	4	no	—
219	#568	C₄×C₆₇	1	4	yes	complemented
220	C₂×#131	—	2	4	no	—
221	C₂²×M₁₆	—	2	2	no	—
222	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
223	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
224	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#117	C₂²×C₆₇	1	2	yes	—
226	#410	—	2	2	no	—
227	#410	—	2	2	no	—
228	C₂×#99	C₄×C₆₇	1	4	yes	—
229	#408	—	2	2	no	—
230	#408	C₄×C₆₇	1	4	yes	complemented
231	C₂×C₂²⋊C₈	—	2	2	no	—
232	#344	—	2	4	no	—
233	#336	—	2	4	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	2	2	no	—
236	C₂×#41	—	2	2	no	—
237	C₂×#41	—	2	2	no	—
238	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
239	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
240	C₂×#41	D₄×C₆₇	1	2	yes	—
241	C₂×#41	D₄×C₆₇	1	2	yes	—
242	C₂²×Q₈	Q₈×C₆₇	1	1	yes	characteristic
243	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
244	C₂²×D₄	Q₈×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	D₄×C₆₇	1	2	yes	—
246	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
247	#131	—	2	16	no	—
248	#131	—	2	16	no	—
249	C₂×M₁₆	—	2	4	no	—
250	C₂×M₁₆	—	2	8	no	—
251	C₂×M₁₆	—	2	8	no	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
260	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
261	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
262	C₂²×C₈	—	2	4	no	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
269	C₂²×C₈	D₄×C₆₇	1	4	yes	—
270	#117	—	2	2	no	—
271	#117	—	2	2	no	—
272	#117	—	2	2	no	—
273	#117	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
276	C₂×C₄²	—	2	2	no	—
277	#99	—	2	8	no	—
278	C₂²⋊C₈	—	2	8	no	—
279	C₂⁴	C₄⋊C₄×C₆₇	1	2	yes	—
280	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
281	#41	—	2	4	no	—
282	#41	—	2	4	no	—
283	#41	—	2	4	no	—
284	#41	—	2	4	no	—
285	#41	—	2	4	no	—
286	#41	—	2	4	no	—
287	#41	—	2	4	no	—
288	#41	—	2	4	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
292	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
293	C₂×D₄	—	2	4	no	—
294	C₂×D₄	—	2	4	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	C₄²×C₆₇	1	4	yes	—
298	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
299	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
300	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
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	4	no	—
306	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
307	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
308	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
316	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	#41×C₆₇	1	2	yes	—
318	C₂²×C₄	#41×C₆₇	1	2	yes	—
319	M₁₆	—	2	16	no	—
320	M₁₆	—	2	4	no	—
321	M₁₆	—	2	4	no	—
322	M₁₆	—	2	4	no	—
323	M₁₆	—	2	4	no	—
324	M₁₆	—	2	4	no	—
325	M₁₆	—	4	2	no	—
326	C₂×C₈	—	2	16	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	4	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	—	4	1	no	—
334	C₂×C₈	—	4	2	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	C₂²⋊C₄×C₆₇	1	8	yes	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	2	4	no	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₄²	—	4	2	no	—
343	C₄²	—	4	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	4	no	—
348	C₂³	#95×C₆₇	1	4	yes	—
349	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
350	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
351	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
352	C₂³	C₄×D₄×C₆₇	1	2	yes	—
353	C₂³	C₄×D₄×C₆₇	1	2	yes	—
354	Q₈	—	2	4	no	—
355	Q₈	—	2	4	no	—
356	D₄	—	2	8	no	—
357	D₄	—	2	8	no	—
358	D₄	—	2	4	no	—
359	D₄	—	2	4	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	8	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	#95×C₆₇	1	4	yes	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	#117×C₆₇	1	2	yes	—
375	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
376	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
377	C₂×C₄	#120×C₆₇	1	2	yes	—
378	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
379	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
380	C₂×C₄	#121×C₆₇	1	2	yes	—
381	C₂×C₄	#121×C₆₇	1	2	yes	—
382	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
383	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
384	C₂×C₄	#123×C₆₇	1	2	yes	—
385	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
386	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
387	C₈	—	2	8	no	—
388	C₈	—	4	2	no	—
389	C₈	—	4	2	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	8	2	no	—
399	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
400	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
401	C₂²	#385×C₆₇	1	1	yes	characteristic
402	C₂²	#385×C₆₇	1	1	yes	characteristic
403	C₂²	#386×C₆₇	1	2	yes	—
404	C₄	—	2	4	no	—
405	C₄	—	8	4	no	—
406	C₄	#379×C₆₇	1	1	yes	characteristic
407	C₄	#380×C₆₇	1	2	yes	—
408	C₄	#381×C₆₇	1	1	yes	characteristic
409	C₄	#385×C₆₇	1	2	yes	—
410	C₄	#386×C₆₇	1	1	yes	characteristic
411	C₄	#386×C₆₇	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	—	2	2	no	—
416	C₂	—	4	4	no	—
417	C₂	C₆₇×C₂²⋊#95	1	1	yes	characteristic
418	C₂	C₆₇×#1868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	#120
commutator	C₂³	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#385
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#344×C₆₇	C₂	semidirect
5. C₆₇×#1729	C₂	semidirect
9. #568×C₆₇	C₄	semidirect
20. #408×C₆₇	C₄	semidirect
202. #10748	C₆₇	direct (N × H)
213. C₂×#344	C₂×C₆₇	semidirect
214. #1729	C₂×C₆₇	semidirect
215. C₆₇	#10748	direct (N × H)
219. #568	C₄×C₆₇	semidirect
230. #408	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×#568×C₆₇	2
C₂×#410×C₆₇	2
C₂×#408×C₆₇	2
C₂×#344×C₆₇	2
C₆₇×#1729	2
#10748	67

Subgroups — C67×#10748 #30689999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10748 #30689999