Subgroups — C₂×C₆₇×#1691 #30698385

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1691	C₂	1	8	yes	maximal, direct product factor
6	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₆₇	—	2	1	no	—
10	C₂²×M₁₆×C₆₇	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#498×C₆₇	—	2	2	no	—
15	#498×C₆₇	C₂²	1	4	yes	complemented
16	#476×C₆₇	—	2	2	no	—
17	#476×C₆₇	C₂²	1	4	yes	—
18	#461×C₆₇	—	2	2	no	—
19	#461×C₆₇	C₂²	1	4	yes	—
20	#436×C₆₇	—	2	2	no	—
21	#436×C₆₇	C₂²	1	4	yes	—
22	C₂×#106×C₆₇	—	2	1	no	—
23	C₂×#106×C₆₇	—	2	1	no	—
24	C₂×#106×C₆₇	—	2	1	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	—	2	1	no	—
29	C₂×#102×C₆₇	—	2	1	no	—
30	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#335×C₆₇	C₂²	1	4	yes	complemented
33	#333×C₆₇	C₂²	1	4	yes	complemented
34	#332×C₆₇	C₂²	1	4	yes	complemented
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	—	2	1	no	—
39	C₂×SD₁₆×C₆₇	—	2	4	no	—
40	C₂×SD₁₆×C₆₇	—	4	4	no	—
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	—	4	4	no	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	C₄⋊Q₈×C₆₇	—	2	1	no	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	C₄⋊Q₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄×D₄×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	C₄×D₄×C₆₇	D₄	1	2	yes	—
59	C₄×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	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	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#106×C₆₇	—	2	4	no	—
68	#106×C₆₇	—	2	4	no	—
69	#106×C₆₇	—	2	4	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	—	2	1	no	—
72	C₄⋊C₈×C₆₇	—	2	1	no	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	4	no	—
78	C₂²⋊C₈×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	D₄	1	2	yes	—
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂⁴×C₆₇	—	2	1	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	—	4	1	no	—
86	C₂×Q₈×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	8	2	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	4	2	no	—
97	C₂²×C₄×C₆₇	—	4	2	no	—
98	C₂²×C₄×C₆₇	—	4	1	no	—
99	C₂²×C₄×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	SD₁₆×C₆₇	—	4	8	no	—
104	M₁₆×C₆₇	—	4	8	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	4	2	no	—
110	C₂×C₈×C₆₇	—	4	2	no	—
111	C₂×C₈×C₆₇	—	4	2	no	—
112	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
113	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₆₇	#41	1	1	yes	characteristic
130	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	#121	1	1	yes	characteristic
136	Q₈×C₆₇	—	4	2	no	—
137	Q₈×C₆₇	—	4	2	no	—
138	Q₈×C₆₇	—	8	2	no	—
139	Q₈×C₆₇	—	8	2	no	—
140	Q₈×C₆₇	—	8	2	no	—
141	D₄×C₆₇	—	2	2	no	—
142	D₄×C₆₇	—	2	2	no	—
143	D₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	8	2	no	—
153	C₂×C₄×C₆₇	—	8	2	no	—
154	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
158	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₄×C₆₇	#137	1	2	yes	—
160	C₂×C₄×C₆₇	#137	1	2	yes	—
161	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
164	C₈×C₆₇	—	2	2	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₈×C₆₇	—	4	2	no	—
167	C₈×C₆₇	—	4	2	no	—
168	C₈×C₆₇	C₂×SD₁₆	1	4	yes	complemented
169	C₂²×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	8	4	no	—
173	C₂²×C₆₇	#461	1	2	yes	—
174	C₂²×C₆₇	#468	1	2	yes	—
175	C₂²×C₆₇	#471	1	2	yes	—
176	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	8	2	no	—
181	C₄×C₆₇	—	8	2	no	—
182	C₄×C₆₇	#461	1	2	yes	—
183	C₄×C₆₇	#468	1	2	yes	—
184	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
185	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
187	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
188	C₂×#1691	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₆₇	—	8	2	no	—
190	C₂×C₆₇	#1691	1	4	yes	direct product factor
191	C₂×C₆₇	C₂×#461	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#468	1	1	yes	characteristic
193	C₂×C₆₇	C₂×#471	1	1	yes	characteristic
194	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
195	C₂×#476	C₂×C₆₇	1	1	yes	characteristic
196	C₂×#461	C₂×C₆₇	1	1	yes	characteristic
197	C₂×#436	C₂×C₆₇	1	1	yes	characteristic
198	#1691	C₂×C₆₇	1	8	yes	direct product factor
199	C₂×#335	C₂×C₆₇	1	1	yes	characteristic, complemented
200	C₂×#333	C₂×C₆₇	1	1	yes	characteristic, complemented
201	C₂×#332	C₂×C₆₇	1	1	yes	characteristic, complemented
202	C₆₇	C₂×#1691	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²×SD₁₆	—	2	1	no	—
204	C₂²×M₁₆	—	2	1	no	—
205	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
206	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
208	#498	—	2	2	no	—
209	#498	C₂²×C₆₇	1	4	yes	complemented
210	#476	—	2	2	no	—
211	#476	C₂²×C₆₇	1	4	yes	—
212	#461	—	2	2	no	—
213	#461	C₂²×C₆₇	1	4	yes	—
214	#436	—	2	2	no	—
215	#436	C₂²×C₆₇	1	4	yes	—
216	C₂×#106	—	2	1	no	—
217	C₂×#106	—	2	1	no	—
218	C₂×#106	—	2	1	no	—
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#103	—	2	1	no	—
223	C₂×#102	—	2	1	no	—
224	C₂×C₂²⋊C₈	—	2	1	no	—
225	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
226	#335	C₂²×C₆₇	1	4	yes	complemented
227	#333	C₂²×C₆₇	1	4	yes	complemented
228	#332	C₂²×C₆₇	1	4	yes	complemented
229	C₂²×Q₈	—	2	1	no	—
230	C₂²×Q₈	—	2	1	no	—
231	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
232	C₂³×C₄	—	2	1	no	—
233	C₂×SD₁₆	—	2	4	no	—
234	C₂×SD₁₆	—	4	4	no	—
235	C₂×M₁₆	—	2	4	no	—
236	C₂×M₁₆	—	4	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	2	1	no	—
240	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
249	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
250	C₄×D₄	—	2	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	D₄×C₆₇	1	2	yes	—
253	C₄×D₄	C₂³×C₆₇	1	2	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	—	2	1	no	—
260	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
261	#106	—	2	4	no	—
262	#106	—	2	4	no	—
263	#106	—	2	4	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
268	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
269	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
270	#103	—	2	4	no	—
271	#102	—	2	4	no	—
272	C₂²⋊C₈	—	2	4	no	—
273	C₄×C₈	D₄×C₆₇	1	2	yes	—
274	C₄×C₈	C₂³×C₆₇	1	2	yes	—
275	C₂⁴	—	2	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	—	8	2	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
297	SD₁₆	—	4	8	no	—
298	M₁₆	—	4	8	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	SD₁₆×C₆₇	1	4	yes	complemented
307	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
308	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₄²	#41×C₆₇	1	1	yes	characteristic
324	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
325	C₂³	—	2	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	2	no	—
329	C₂³	#121×C₆₇	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	8	2	no	—
333	Q₈	—	8	2	no	—
334	Q₈	—	8	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	2	2	no	—
337	D₄	—	4	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
349	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
350	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
352	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
353	C₂×C₄	#137×C₆₇	1	2	yes	—
354	C₂×C₄	#137×C₆₇	1	2	yes	—
355	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
358	C₈	—	2	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	4	2	no	—
362	C₈	C₂×SD₁₆×C₆₇	1	4	yes	complemented
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	8	4	no	—
367	C₂²	#461×C₆₇	1	2	yes	—
368	C₂²	#468×C₆₇	1	2	yes	—
369	C₂²	#471×C₆₇	1	2	yes	—
370	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	8	2	no	—
375	C₄	—	8	2	no	—
376	C₄	#461×C₆₇	1	2	yes	—
377	C₄	#468×C₆₇	1	2	yes	—
378	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
379	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
380	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
381	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
382	C₂	—	8	2	no	—
383	C₂	C₆₇×#1691	1	4	yes	direct product factor
384	C₂	C₂×#461×C₆₇	1	1	yes	characteristic
385	C₂	C₂×#468×C₆₇	1	1	yes	characteristic
386	C₂	C₂×#471×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#498×C₆₇	C₂	semidirect
5. C₆₇×#1691	C₂	direct (N × H)
6. C₂×#335×C₆₇	C₂	semidirect
7. C₂×#333×C₆₇	C₂	semidirect
8. C₂×#332×C₆₇	C₂	semidirect
15. #498×C₆₇	C₂²	semidirect
32. #335×C₆₇	C₂²	semidirect
33. #333×C₆₇	C₂²	semidirect
34. #332×C₆₇	C₂²	semidirect
112. C₂×C₈×C₆₇	SD₁₆	semidirect
168. C₈×C₆₇	C₂×SD₁₆	semidirect
188. C₂×#1691	C₆₇	direct (N × H)
190. C₂×C₆₇	#1691	direct (N × H)
194. C₂×#498	C₂×C₆₇	semidirect
198. #1691	C₂×C₆₇	direct (N × H)
199. C₂×#335	C₂×C₆₇	semidirect
200. C₂×#333	C₂×C₆₇	semidirect
201. C₂×#332	C₂×C₆₇	semidirect
202. C₆₇	C₂×#1691	direct (N × H)
209. #498	C₂²×C₆₇	semidirect
226. #335	C₂²×C₆₇	semidirect
227. #333	C₂²×C₆₇	semidirect
228. #332	C₂²×C₆₇	semidirect
306. C₂×C₈	SD₁₆×C₆₇	semidirect
362. C₈	C₂×SD₁₆×C₆₇	semidirect
383. C₂	C₆₇×#1691	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#498×C₆₇	2
C₂×#476×C₆₇	2
C₂×#461×C₆₇	2
C₂×#436×C₆₇	2
C₆₇×#1691	2
C₂×#335×C₆₇	2
C₂×#333×C₆₇	2
C₂×#332×C₆₇	2
C₂×#1691	67

Subgroups — C2×C67×#1691 #30698385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1691 #30698385