Subgroups — C₂×C₆₇×#1702 #30698404

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1702	C₂	1	8	yes	maximal, direct product factor
6	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#331×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₇	—	2	1	no	—
10	C₂²×SD₁₆×C₆₇	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#486×C₆₇	—	2	2	no	—
15	#486×C₆₇	C₂²	1	4	yes	—
16	#462×C₆₇	—	2	2	no	—
17	#462×C₆₇	C₂²	1	4	yes	complemented
18	#460×C₆₇	—	2	2	no	—
19	#460×C₆₇	C₂²	1	4	yes	—
20	#446×C₆₇	—	2	2	no	—
21	#446×C₆₇	C₂²	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₆₇	—	2	1	no	—
27	C₂×#102×C₆₇	—	2	1	no	—
28	C₂×#102×C₆₇	—	2	1	no	—
29	C₂×#102×C₆₇	—	2	1	no	—
30	C₂×#97×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#334×C₆₇	C₂²	1	4	yes	complemented
33	#332×C₆₇	C₂²	1	4	yes	complemented
34	#331×C₆₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂×Q₁₆×C₆₇	—	2	4	no	—
41	C₂×Q₁₆×C₆₇	—	4	4	no	—
42	C₂×SD₁₆×C₆₇	—	2	4	no	—
43	C₂×SD₁₆×C₆₇	—	4	4	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
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	C₂³	1	2	yes	—
53	#127×C₆₇	—	2	1	no	—
54	#127×C₆₇	—	2	1	no	—
55	#127×C₆₇	—	2	1	no	—
56	#127×C₆₇	C₂³	1	2	yes	—
57	C₄×Q₈×C₆₇	—	2	1	no	—
58	C₄×Q₈×C₆₇	—	2	1	no	—
59	C₄×Q₈×C₆₇	D₄	1	2	yes	—
60	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
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×C₄²×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₆₇	—	2	4	no	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	#103×C₆₇	—	2	4	no	—
75	#102×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	4	no	—
78	#97×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	D₄	1	2	yes	—
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂⁴×C₆₇	—	4	1	no	—
82	C₂×Q₈×C₆₇	—	2	1	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	4	1	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	1	no	—
95	C₂×D₄×C₆₇	—	4	1	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	—	4	2	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	8	2	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
107	Q₁₆×C₆₇	—	4	8	no	—
108	SD₁₆×C₆₇	—	4	8	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	4	2	no	—
115	C₂×C₈×C₆₇	—	4	2	no	—
116	C₂×C₈×C₆₇	—	4	2	no	—
117	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	4	1	no	—
125	C₄⋊C₄×C₆₇	—	4	1	no	—
126	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₆₇	#41	1	1	yes	characteristic
132	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	—	8	2	no	—
135	C₂³×C₆₇	—	8	2	no	—
136	C₂³×C₆₇	—	8	2	no	—
137	C₂³×C₆₇	#121	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	2	no	—
139	Q₈×C₆₇	—	2	2	no	—
140	Q₈×C₆₇	—	4	2	no	—
141	Q₈×C₆₇	—	4	2	no	—
142	Q₈×C₆₇	—	4	2	no	—
143	Q₈×C₆₇	—	8	2	no	—
144	D₄×C₆₇	—	4	2	no	—
145	D₄×C₆₇	—	4	2	no	—
146	D₄×C₆₇	—	4	2	no	—
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	8	2	no	—
149	D₄×C₆₇	—	8	2	no	—
150	D₄×C₆₇	—	8	2	no	—
151	D₄×C₆₇	—	8	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	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	8	2	no	—
159	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
161	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
162	C₂×C₄×C₆₇	#135	1	2	yes	—
163	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
165	C₂×C₄×C₆₇	#137	1	2	yes	—
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₈×C₆₇	—	2	2	no	—
170	C₈×C₆₇	—	2	2	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₈×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	—	8	2	no	—
175	C₂²×C₆₇	—	8	2	no	—
176	C₂²×C₆₇	—	8	2	no	—
177	C₂²×C₆₇	—	16	4	no	—
178	C₂²×C₆₇	#464	1	2	yes	—
179	C₂²×C₆₇	#469	1	2	yes	—
180	C₂²×C₆₇	#471	1	2	yes	—
181	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	8	2	no	—
187	C₄×C₆₇	#464	1	2	yes	—
188	C₄×C₆₇	#469	1	2	yes	—
189	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
190	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
193	C₂×#1702	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₆₇	—	16	2	no	—
195	C₂×C₆₇	#1702	1	4	yes	direct product factor
196	C₂×C₆₇	C₂×#464	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#471	1	1	yes	characteristic
199	C₂×#486	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
201	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
202	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#1702	C₂×C₆₇	1	8	yes	direct product factor
204	C₂×#334	C₂×C₆₇	1	1	yes	characteristic, complemented
205	C₂×#332	C₂×C₆₇	1	1	yes	characteristic, complemented
206	C₂×#331	C₂×C₆₇	1	1	yes	characteristic
207	C₆₇	C₂×#1702	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×Q₁₆	—	2	1	no	—
209	C₂²×SD₁₆	—	2	1	no	—
210	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
212	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
213	#486	—	2	2	no	—
214	#486	C₂²×C₆₇	1	4	yes	—
215	#462	—	2	2	no	—
216	#462	C₂²×C₆₇	1	4	yes	complemented
217	#460	—	2	2	no	—
218	#460	C₂²×C₆₇	1	4	yes	—
219	#446	—	2	2	no	—
220	#446	C₂²×C₆₇	1	4	yes	complemented
221	C₂×C₄⋊C₈	—	2	1	no	—
222	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#103	—	2	1	no	—
226	C₂×#102	—	2	1	no	—
227	C₂×#102	—	2	1	no	—
228	C₂×#102	—	2	1	no	—
229	C₂×#97	—	2	1	no	—
230	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
231	#334	C₂²×C₆₇	1	4	yes	complemented
232	#332	C₂²×C₆₇	1	4	yes	complemented
233	#331	C₂²×C₆₇	1	4	yes	—
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
236	C₂²×D₄	—	2	1	no	—
237	C₂²×D₄	—	2	1	no	—
238	C₂²×D₄	—	2	1	no	—
239	C₂×Q₁₆	—	2	4	no	—
240	C₂×Q₁₆	—	4	4	no	—
241	C₂×SD₁₆	—	2	4	no	—
242	C₂×SD₁₆	—	4	4	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
247	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
252	#127	—	2	1	no	—
253	#127	—	2	1	no	—
254	#127	—	2	1	no	—
255	#127	C₂³×C₆₇	1	2	yes	—
256	C₄×Q₈	—	2	1	no	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	D₄×C₆₇	1	2	yes	—
259	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	—	2	1	no	—
265	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
266	C₄⋊C₈	—	2	4	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
271	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
272	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
273	#103	—	2	4	no	—
274	#102	—	2	4	no	—
275	#102	—	2	4	no	—
276	#102	—	2	4	no	—
277	#97	—	2	4	no	—
278	C₄×C₈	D₄×C₆₇	1	2	yes	—
279	C₄×C₈	C₂³×C₆₇	1	2	yes	—
280	C₂⁴	—	4	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×Q₈	—	4	2	no	—
288	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	2	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	8	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
306	Q₁₆	—	4	8	no	—
307	SD₁₆	—	4	8	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	4	2	no	—
314	C₂×C₈	—	4	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
317	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₄²	#41×C₆₇	1	1	yes	characteristic
331	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
332	C₂³	—	4	2	no	—
333	C₂³	—	8	2	no	—
334	C₂³	—	8	2	no	—
335	C₂³	—	8	2	no	—
336	C₂³	#121×C₆₇	1	1	yes	characteristic
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	4	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	8	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	4	2	no	—
345	D₄	—	4	2	no	—
346	D₄	—	4	2	no	—
347	D₄	—	8	2	no	—
348	D₄	—	8	2	no	—
349	D₄	—	8	2	no	—
350	D₄	—	8	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	2	no	—
358	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
359	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
360	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
361	C₂×C₄	#135×C₆₇	1	2	yes	—
362	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
363	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
364	C₂×C₄	#137×C₆₇	1	2	yes	—
365	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	2	no	—
370	C₈	—	4	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	16	4	no	—
377	C₂²	#464×C₆₇	1	2	yes	—
378	C₂²	#469×C₆₇	1	2	yes	—
379	C₂²	#471×C₆₇	1	2	yes	—
380	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	4	2	no	—
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	#464×C₆₇	1	2	yes	—
387	C₄	#469×C₆₇	1	2	yes	—
388	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
389	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
390	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
391	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
392	C₂	—	16	2	no	—
393	C₂	C₆₇×#1702	1	4	yes	direct product factor
394	C₂	C₂×#464×C₆₇	1	1	yes	characteristic
395	C₂	C₂×#469×C₆₇	1	1	yes	characteristic
396	C₂	C₂×#471×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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
2. C₂×#462×C₆₇	C₂	semidirect
4. C₂×#446×C₆₇	C₂	semidirect
5. C₆₇×#1702	C₂	direct (N × H)
6. C₂×#334×C₆₇	C₂	semidirect
7. C₂×#332×C₆₇	C₂	semidirect
17. #462×C₆₇	C₂²	semidirect
21. #446×C₆₇	C₂²	semidirect
32. #334×C₆₇	C₂²	semidirect
33. #332×C₆₇	C₂²	semidirect
193. C₂×#1702	C₆₇	direct (N × H)
195. C₂×C₆₇	#1702	direct (N × H)
200. C₂×#462	C₂×C₆₇	semidirect
202. C₂×#446	C₂×C₆₇	semidirect
203. #1702	C₂×C₆₇	direct (N × H)
204. C₂×#334	C₂×C₆₇	semidirect
205. C₂×#332	C₂×C₆₇	semidirect
207. C₆₇	C₂×#1702	direct (N × H)
216. #462	C₂²×C₆₇	semidirect
220. #446	C₂²×C₆₇	semidirect
231. #334	C₂²×C₆₇	semidirect
232. #332	C₂²×C₆₇	semidirect
393. C₂	C₆₇×#1702	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×#486×C₆₇	2
C₂×#462×C₆₇	2
C₂×#460×C₆₇	2
C₂×#446×C₆₇	2
C₆₇×#1702	2
C₂×#334×C₆₇	2
C₂×#332×C₆₇	2
C₂×#331×C₆₇	2
C₂×#1702	67

Subgroups — C2×C67×#1702 #30698404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1702 #30698404