Subgroups — C₂×#22562286 #31010212

Order 17152.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863834	C₂	1	2	yes	maximal, complemented
2	C₂×#16863773	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863514	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562286	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863210	C₂	1	2	yes	maximal, complemented
6	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524065	C₂²	1	1	yes	characteristic
8	C₂²×D₅₃₆	—	2	1	no	—
9	C₂²×D₅₃₆	C₂²	1	2	yes	—
10	C₂×#14524049	—	2	1	no	—
11	C₂×#14524040	—	2	1	no	—
12	#16863834	—	2	4	no	—
13	#16863834	C₂²	1	8	yes	complemented
14	C₂×#14523988	—	2	2	no	—
15	C₂×D₁₀₇₂	—	2	2	no	—
16	#16863773	—	2	2	no	—
17	#16863773	C₂²	1	4	yes	complemented
18	#16863514	—	2	2	no	—
19	#16863514	C₂²	1	4	yes	complemented
20	C₂×#14523979	C₂²	1	1	yes	characteristic
21	C₂×#14523971	—	2	1	no	—
22	#16863210	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#364×C₆₇	C₂²	1	4	yes	complemented
26	C₂×#11761628	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	2	no	—
28	C₂²×D₂₆₈	—	4	2	no	—
29	#14524065	—	2	1	no	—
30	#14524065	—	2	1	no	—
31	#14524065	D₄	1	2	yes	complemented
32	#14524065	C₂³	1	2	yes	—
33	C₂×Dic₂₆₈	D₄	1	1	yes	characteristic, complemented
34	C₂×#11761617	—	2	1	no	—
35	C₂×D₅₃₆	—	2	2	no	—
36	C₂×D₅₃₆	—	2	2	no	—
37	C₂×D₅₃₆	—	2	2	no	—
38	C₂×D₅₃₆	—	2	4	no	—
39	C₂×D₅₃₆	—	2	2	no	—
40	C₂×D₅₃₆	—	2	2	no	—
41	C₂×D₅₃₆	—	4	1	no	—
42	C₂×D₅₃₆	—	4	1	no	—
43	C₂×D₅₃₆	—	4	2	no	—
44	C₂×D₅₃₆	D₄	1	1	yes	characteristic, complemented
45	C₂×D₅₃₆	C₂³	1	4	yes	—
46	#14524049	—	2	4	no	—
47	#14524049	—	4	2	no	—
48	C₂×#11761612	—	2	1	no	—
49	C₂×#11761611	—	4	1	no	—
50	C₂×C₈×D₆₇	—	4	1	no	—
51	#14524040	—	2	4	no	—
52	#14524040	—	4	2	no	—
53	#14523988	—	2	8	no	—
54	D₁₀₇₂	—	2	8	no	—
55	#14523979	—	2	1	no	—
56	#14523979	C₂³	1	2	yes	—
57	#14523971	—	2	4	no	—
58	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
61	M₃₂×C₆₇	—	2	2	no	—
62	M₃₂×C₆₇	C₂³	1	4	yes	—
63	C₂×C₁₆×C₆₇	—	2	2	no	—
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂³×D₆₇	—	4	2	no	—
67	#11761628	—	2	2	no	—
68	#11761628	—	2	2	no	—
69	#11761628	—	4	1	no	—
70	#11761628	—	4	1	no	—
71	C₂×Dic₁₃₄	—	2	1	no	—
72	C₂×D₂₆₈	—	2	1	no	—
73	C₂×D₂₆₈	—	2	4	no	—
74	C₂×D₂₆₈	—	4	2	no	—
75	C₂×D₂₆₈	—	4	2	no	—
76	C₂×D₂₆₈	—	4	2	no	—
77	C₂×D₂₆₈	—	4	2	no	—
78	C₂×D₂₆₈	—	4	8	no	—
79	C₂×D₂₆₈	—	4	2	no	—
80	C₂×D₂₆₈	—	8	8	no	—
81	C₂×C₂²⋊D₆₇	—	4	1	no	—
82	C₂×C₄×D₆₇	—	4	1	no	—
83	Dic₂₆₈	—	2	1	no	—
84	Dic₂₆₈	C₂×D₄	1	2	yes	complemented
85	#11761617	—	2	2	no	—
86	#11761617	—	2	2	no	—
87	D₅₃₆	—	2	1	no	—
88	D₅₃₆	—	2	4	no	—
89	D₅₃₆	—	2	4	no	—
90	D₅₃₆	—	4	2	no	—
91	D₅₃₆	—	4	2	no	—
92	D₅₃₆	—	4	4	no	—
93	D₅₃₆	—	4	4	no	—
94	D₅₃₆	C₂×D₄	1	2	yes	complemented
95	#11761612	—	2	2	no	—
96	#11761612	—	4	1	no	—
97	#11761611	—	4	4	no	—
98	C₂×C₆₇⋊C₈	—	4	1	no	—
99	C₈×D₆₇	—	4	4	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
103	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	D₈	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
112	C₄²×C₆₇	—	2	1	no	—
113	C₄²×C₆₇	C₂×D₄	1	2	yes	—
114	C₁₆×C₆₇	—	2	4	no	—
115	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂²×D₆₇	—	4	1	no	—
117	C₂²×D₆₇	—	4	4	no	—
118	C₂²×D₆₇	—	8	8	no	—
119	C₂²×D₆₇	—	8	2	no	—
120	C₂²×D₆₇	—	8	2	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₆₇	—	4	2	no	—
130	C₂²⋊D₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	4	1	no	—
132	D₂₆₈	—	2	2	no	—
133	D₂₆₈	—	4	4	no	—
134	D₂₆₈	—	4	4	no	—
135	D₂₆₈	—	4	1	no	—
136	D₂₆₈	—	8	16	no	—
137	D₂₆₈	—	8	2	no	—
138	D₂₆₈	—	8	2	no	—
139	C₄×D₆₇	—	4	2	no	—
140	C₄×D₆₇	—	4	2	no	—
141	Dic₁₃₄	—	2	2	no	—
142	Dic₁₃₄	—	4	1	no	—
143	C₈×C₆₇	#121	1	1	yes	characteristic
144	C₈×C₆₇	#121	1	1	yes	characteristic
145	C₈×C₆₇	C₂×D₈	1	2	yes	—
146	C₈×C₆₇	C₂×D₈	1	2	yes	—
147	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₆₇⋊C₈	—	4	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	#459	1	1	yes	characteristic
152	C₂²×C₆₇	#459	1	1	yes	characteristic
153	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×D₆₇	—	4	2	no	—
155	C₂×D₆₇	—	8	2	no	—
156	C₂×D₆₇	—	8	8	no	—
157	C₂×D₆₇	—	8	4	no	—
158	C₂×D₆₇	—	16	4	no	—
159	C₂×D₆₇	—	16	4	no	—
160	C₄×C₆₇	—	2	4	no	—
161	C₄×C₆₇	#459	1	1	yes	characteristic
162	C₄×C₆₇	#459	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
164	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
165	Dic₆₇	—	4	2	no	—
166	C₂×#2213	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	—	2	1	no	—
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	#2213	1	2	yes	complemented
170	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
171	D₆₇	—	8	2	no	—
172	D₆₇	—	16	4	no	—
173	C₂×#509	—	67	2	no	—
174	C₂×#493	—	67	1	no	—
175	C₂×#443	—	67	1	no	—
176	#2213	—	67	8	no	—
177	C₂×#364	D₆₇	1	1	yes	characteristic, complemented
178	C₂×#361	—	67	2	no	—
179	C₆₇	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	67	1	no	—
181	C₂²×D₈	—	67	2	no	—
182	C₂²×D₈	—	134	1	no	—
183	C₂×#131	—	134	1	no	—
184	C₂×#127	—	134	1	no	—
185	#509	—	67	8	no	—
186	#509	—	134	4	no	—
187	C₂×SD₃₂	—	134	2	no	—
188	C₂×D₁₆	—	134	2	no	—
189	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
190	#493	—	67	4	no	—
191	#493	—	134	2	no	—
192	#443	—	67	4	no	—
193	#443	—	134	2	no	—
194	C₂×#108	—	67	1	no	—
195	C₂×#104	—	134	1	no	—
196	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
197	#364	C₂×D₆₇	1	4	yes	complemented
198	#361	—	67	8	no	—
199	C₂×#41	—	134	1	no	—
200	C₂²×D₄	—	268	2	no	—
201	C₂²×D₄	—	134	2	no	—
202	#135	—	67	2	no	—
203	#135	—	67	2	no	—
204	#135	—	134	1	no	—
205	#135	—	134	1	no	—
206	C₂×Q₁₆	—	67	1	no	—
207	C₂×SD₁₆	—	134	1	no	—
208	C₂×D₈	—	268	1	no	—
209	C₂×D₈	—	268	1	no	—
210	C₂×D₈	—	268	2	no	—
211	C₂×D₈	—	67	1	no	—
212	C₂×D₈	—	67	4	no	—
213	C₂×D₈	—	134	2	no	—
214	C₂×D₈	—	134	2	no	—
215	C₂×D₈	—	134	2	no	—
216	C₂×D₈	—	134	4	no	—
217	C₂×D₈	—	134	2	no	—
218	C₂×D₈	—	134	2	no	—
219	#131	—	268	2	no	—
220	#131	—	134	4	no	—
221	C₂×M₁₆	—	268	1	no	—
222	C₂×M₁₆	—	134	1	no	—
223	C₂²×C₈	—	268	1	no	—
224	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
226	#127	—	268	2	no	—
227	#127	—	134	4	no	—
228	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
229	SD₃₂	—	134	8	no	—
230	D₁₆	—	134	8	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	C₂²×D₆₇	1	4	yes	—
233	C₂×C₁₆	—	2	2	no	—
234	#108	—	67	2	no	—
235	#108	—	134	1	no	—
236	#104	—	134	4	no	—
237	C₄×C₈	D₂₆₈	1	2	yes	—
238	C₄×C₈	C₂²×D₆₇	1	2	yes	—
239	C₂⁴	—	268	2	no	—
240	#41	—	268	1	no	—
241	#41	—	268	1	no	—
242	#41	—	134	2	no	—
243	#41	—	134	2	no	—
244	C₂×Q₈	—	134	1	no	—
245	C₂×D₄	—	268	8	no	—
246	C₂×D₄	—	268	1	no	—
247	C₂×D₄	—	268	2	no	—
248	C₂×D₄	—	268	2	no	—
249	C₂×D₄	—	268	2	no	—
250	C₂×D₄	—	268	2	no	—
251	C₂×D₄	—	268	2	no	—
252	C₂×D₄	—	536	8	no	—
253	C₂×D₄	—	134	1	no	—
254	C₂×D₄	—	134	4	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	268	1	no	—
257	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
258	Q₁₆	—	67	2	no	—
259	Q₁₆	—	134	1	no	—
260	SD₁₆	—	134	2	no	—
261	SD₁₆	—	134	2	no	—
262	D₈	—	268	2	no	—
263	D₈	—	268	2	no	—
264	D₈	—	268	4	no	—
265	D₈	—	268	4	no	—
266	D₈	—	67	2	no	—
267	D₈	—	134	1	no	—
268	D₈	—	134	4	no	—
269	D₈	—	134	4	no	—
270	M₁₆	—	268	1	no	—
271	M₁₆	—	268	4	no	—
272	M₁₆	—	134	2	no	—
273	C₂×C₈	—	268	1	no	—
274	C₂×C₈	—	268	4	no	—
275	C₂×C₈	D₅₃₆	1	1	yes	characteristic
276	C₂×C₈	D₅₃₆	1	1	yes	characteristic
277	C₂×C₈	D₅₃₆	1	2	yes	—
278	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
279	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
280	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
281	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
282	C₂×C₈	#11761629	1	1	yes	characteristic
283	C₂×C₈	#11761629	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×D₂₆₈	1	2	yes	—
287	C₁₆	—	2	4	no	—
288	C₂³	—	268	1	no	—
289	C₂³	—	268	4	no	—
290	C₂³	—	536	8	no	—
291	C₂³	—	536	2	no	—
292	C₂³	—	536	2	no	—
293	C₂³	#14524071	1	1	yes	characteristic
294	Q₈	—	268	1	no	—
295	Q₈	—	134	2	no	—
296	D₄	—	268	2	no	—
297	D₄	—	268	2	no	—
298	D₄	—	268	1	no	—
299	D₄	—	268	4	no	—
300	D₄	—	268	4	no	—
301	D₄	—	536	16	no	—
302	D₄	—	536	2	no	—
303	D₄	—	536	2	no	—
304	D₄	—	134	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	268	1	no	—
308	C₂×C₄	—	268	2	no	—
309	C₂×C₄	—	268	2	no	—
310	C₂×C₄	#14524018	1	1	yes	characteristic
311	C₂×C₄	#14524018	1	1	yes	characteristic
312	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
313	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
314	C₂×C₄	#14524071	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
316	C₈	—	268	2	no	—
317	C₈	#14524018	1	1	yes	characteristic
318	C₈	#14524018	1	1	yes	characteristic
319	C₈	C₂×D₅₃₆	1	2	yes	—
320	C₈	C₂×D₅₃₆	1	2	yes	—
321	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
322	C₈	C₂×#11761629	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	268	2	no	—
325	C₂²	—	536	8	no	—
326	C₂²	—	536	2	no	—
327	C₂²	—	536	4	no	—
328	C₂²	—	1072	4	no	—
329	C₂²	—	1072	4	no	—
330	C₂²	#16863592	1	1	yes	characteristic
331	C₂²	#16863592	1	1	yes	characteristic
332	C₂²	C₂×#14524071	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	268	2	no	—
335	C₄	#16863592	1	1	yes	characteristic
336	C₄	#16863592	1	1	yes	characteristic
337	C₄	C₂×#14524018	1	1	yes	characteristic
338	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	536	2	no	—
342	C₂	—	1072	4	no	—
343	C₂	#22562286	1	2	yes	direct product factor
344	C₂	C₂×#16863592	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863592
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#459
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	C₂×#2213
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#16863834	C₂	semidirect
2. C₂×#16863773	C₂	semidirect
3. C₂×#16863514	C₂	semidirect
4. #22562286	C₂	direct (N × H)
5. C₂×#16863210	C₂	semidirect
6. C₂×#364×C₆₇	C₂	semidirect
13. #16863834	C₂²	semidirect
17. #16863773	C₂²	semidirect
19. #16863514	C₂²	semidirect
22. #16863210	C₂²	semidirect
25. #364×C₆₇	C₂²	semidirect
31. #14524065	D₄	semidirect
33. C₂×Dic₂₆₈	D₄	semidirect
44. C₂×D₅₃₆	D₄	semidirect
84. Dic₂₆₈	C₂×D₄	semidirect
94. D₅₃₆	C₂×D₄	semidirect
169. C₂×C₆₇	#2213	semidirect
177. C₂×#364	D₆₇	semidirect
179. C₆₇	C₂×#2213	semidirect
197. #364	C₂×D₆₇	semidirect
343. C₂	#22562286	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×#16863834	2
C₂×#16863773	2
C₂×#16863514	2
#22562286	2
C₂×#16863210	2
C₂×#364×C₆₇	2
C₂×#2213	67

Subgroups — C2×#22562286 #31010212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562286 #31010212