Subgroups — C₂×#13070785 #14908501

Order 2816.

558 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622468	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10622809	C₂	1	1	yes	maximal, characteristic
3	#13070785	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622151	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622125	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10622149	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10622204	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#360	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#135	C₂²	1	1	yes	characteristic
10	C₂×#67001	—	2	1	no	—
11	C₂×#66941	—	2	1	no	—
12	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#106	C₂²	1	1	yes	characteristic
14	C₂×#66901	—	2	1	no	—
15	#10622809	—	2	2	no	—
16	#10622809	C₂²	1	4	yes	—
17	C₂×#66883	C₂²	1	1	yes	characteristic
18	C₂×#66881	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66876	C₂²	1	1	yes	characteristic
21	C₂×#66872	—	2	1	no	—
22	#10622468	—	2	2	no	—
23	#10622468	C₂²	1	4	yes	complemented
24	C₁₁×#360	C₂²	1	4	yes	—
25	#10622204	C₂²	1	4	yes	—
26	#10622151	C₂²	1	4	yes	complemented
27	#10622149	C₂²	1	4	yes	complemented
28	#10622125	C₂²	1	4	yes	complemented
29	C₂×C₁₁×#41	—	2	1	no	—
30	C₂²×Dic₂₂	—	4	1	no	—
31	C₁₁×#135	—	2	1	no	—
32	C₁₁×#135	—	2	1	no	—
33	C₁₁×#135	Q₈	1	2	yes	complemented
34	C₁₁×#135	C₂³	1	2	yes	—
35	C₂×C₁₁×Q₁₆	Q₈	1	1	yes	characteristic, complemented
36	C₂×C₁₁×SD₁₆	—	2	1	no	—
37	C₂×C₁₁×D₈	Q₈	1	1	yes	characteristic, complemented
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
40	#67001	—	2	4	no	—
41	#67001	—	4	2	no	—
42	C₂×C₄⋊Dic₁₁	—	2	1	no	—
43	C₂×Dic₁₁⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
45	C₂×#4318	—	2	1	no	—
46	C₂×#4318	—	4	1	no	—
47	C₂²×C₁₁⋊C₈	—	4	1	no	—
48	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
49	#66941	—	2	4	no	—
50	#66941	—	4	2	no	—
51	C₁₁×M₃₂	—	2	1	no	—
52	C₁₁×M₃₂	C₂³	1	2	yes	—
53	C₂×C₁₁×C₁₆	—	2	1	no	—
54	C₁₁×#106	—	2	1	no	—
55	C₁₁×#106	C₂³	1	2	yes	—
56	#66901	—	2	4	no	—
57	#66883	—	2	1	no	—
58	#66883	C₂³	1	2	yes	—
59	#66881	—	2	1	no	—
60	#66881	C₂³	1	2	yes	—
61	C₈×Dic₁₁	D₄	1	2	yes	—
62	C₈×Dic₁₁	C₂³	1	2	yes	—
63	#66876	—	2	1	no	—
64	#66876	C₂³	1	2	yes	—
65	C₂×C₁₁⋊C₁₆	—	2	1	no	—
66	#66872	—	2	4	no	—
67	C₂×#2230	—	11	1	no	maximal, Sylow
68	C₁₁×#41	—	2	2	no	—
69	C₁₁×#41	—	2	2	no	—
70	C₁₁×#41	—	4	1	no	—
71	C₁₁×#41	—	4	1	no	—
72	C₂×Q₈×C₁₁	—	2	1	no	—
73	C₂×D₄×C₁₁	—	2	1	no	—
74	C₂×D₄×C₁₁	—	4	1	no	—
75	C₂²×C₄×C₁₁	—	4	1	no	—
76	C₂²×C₄×C₁₁	—	4	1	no	—
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₁	—	2	1	no	—
79	C₂²×Dic₁₁	—	4	1	no	—
80	C₂×Dic₂₂	—	4	4	no	—
81	C₂×Dic₂₂	—	8	4	no	—
82	C₁₁×Q₁₆	—	2	1	no	—
83	C₁₁×Q₁₆	C₂×Q₈	1	2	yes	complemented
84	C₁₁×SD₁₆	—	2	2	no	—
85	C₁₁×SD₁₆	—	2	2	no	—
86	C₁₁×D₈	—	2	1	no	—
87	C₁₁×D₈	C₂×Q₈	1	2	yes	complemented
88	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
94	C₁₁×C₄⋊C₄	—	2	2	no	—
95	C₁₁×C₄⋊C₄	—	4	1	no	—
96	C₄⋊Dic₁₁	—	2	2	no	—
97	C₄⋊Dic₁₁	—	4	1	no	—
98	Dic₁₁⋊C₄	—	4	4	no	—
99	C₄×Dic₁₁	—	2	1	no	—
100	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
101	#4318	—	2	2	no	—
102	#4318	—	4	4	no	—
103	#4318	—	4	1	no	—
104	C₂×C₁₁⋊C₈	—	4	4	no	—
105	C₂×C₁₁⋊C₈	—	4	1	no	—
106	C₂×C₁₁⋊C₈	SD₁₆	1	1	yes	characteristic
107	C₂×C₁₁⋊C₈	SD₁₆	1	1	yes	characteristic
108	C₂×C₁₁⋊C₈	SD₁₆	1	2	yes	—
109	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
110	C₁₁×C₁₆	—	2	2	no	—
111	C₁₁⋊C₁₆	—	2	2	no	—
112	C₂×#498	—	11	1	no	—
113	C₂×#443	—	11	1	no	—
114	#2230	—	11	8	no	—
115	C₂×#365	—	11	1	no	—
116	C₂×#365	—	11	1	no	—
117	C₂×#364	—	11	1	no	—
118	C₂×#360	—	11	1	no	—
119	C₂×#360	D₁₁	1	1	yes	characteristic
120	C₂³×C₁₁	—	4	1	no	—
121	C₂³×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₁₁	—	2	2	no	—
123	Q₈×C₁₁	—	4	1	no	—
124	D₄×C₁₁	—	2	2	no	—
125	D₄×C₁₁	—	4	2	no	—
126	D₄×C₁₁	—	4	2	no	—
127	D₄×C₁₁	—	4	1	no	—
128	C₂×C₄×C₁₁	—	4	1	no	—
129	C₂×C₄×C₁₁	—	4	2	no	—
130	C₂×C₄×C₁₁	—	4	2	no	—
131	C₂×C₄×C₁₁	—	4	2	no	—
132	C₂×C₄×C₁₁	—	8	2	no	—
133	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₁	—	2	4	no	—
140	C₂×Dic₁₁	—	2	2	no	—
141	C₂×Dic₁₁	—	4	2	no	—
142	C₂×Dic₁₁	—	8	2	no	—
143	Dic₂₂	—	8	8	no	—
144	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
148	C₁₁⋊C₈	—	4	2	no	—
149	C₁₁⋊C₈	C₂×SD₁₆	1	2	yes	—
150	C₁₁⋊C₈	C₂×SD₁₆	1	2	yes	—
151	C₂×#135	C₂×D₁₁	1	1	yes	characteristic
152	C₂×#131	—	22	1	no	—
153	C₂×C₄⋊Q₈	—	22	1	no	—
154	C₂×M₃₂	—	11	1	no	—
155	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
156	#498	—	11	4	no	—
157	#498	—	22	2	no	—
158	#443	—	11	4	no	—
159	#443	—	22	2	no	—
160	C₂×#108	—	11	1	no	—
161	C₂×#106	—	11	1	no	—
162	C₂×#106	—	22	1	no	—
163	C₂×#106	C₂×D₁₁	1	1	yes	characteristic
164	C₂×#104	—	22	1	no	—
165	C₂×C₄×C₈	—	11	1	no	—
166	#365	—	11	4	no	—
167	#365	—	11	4	no	—
168	#364	—	11	4	no	—
169	#360	—	11	4	no	—
170	#360	C₂×D₁₁	1	4	yes	—
171	C₂²×C₁₁	—	2	2	no	—
172	C₂²×C₁₁	—	4	2	no	—
173	C₂²×C₁₁	—	8	2	no	—
174	C₂²×C₁₁	#476	1	1	yes	characteristic
175	C₂²×C₁₁	#476	1	1	yes	characteristic
176	C₂²×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₄×C₁₁	—	4	2	no	—
178	C₄×C₁₁	—	8	2	no	—
179	C₄×C₁₁	#476	1	1	yes	characteristic
180	C₄×C₁₁	#476	1	1	yes	characteristic
181	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
182	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
183	Dic₁₁	—	2	4	no	—
184	Dic₁₁	—	8	2	no	—
185	C₂×#41	—	2	1	no	—
186	C₂²×Q₈	—	44	1	no	—
187	#135	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	Dic₂₂	1	2	yes	complemented
190	#135	C₂²×D₁₁	1	2	yes	—
191	C₂×Q₁₆	Dic₂₂	1	1	yes	characteristic, complemented
192	C₂×SD₁₆	—	2	1	no	—
193	C₂×D₈	Dic₂₂	1	1	yes	characteristic, complemented
194	#131	—	22	4	no	—
195	#131	—	44	2	no	—
196	C₂×M₁₆	—	22	1	no	—
197	C₂×M₁₆	—	44	1	no	—
198	C₂²×C₈	—	11	1	no	—
199	C₂²×C₈	—	44	1	no	—
200	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
201	C₄⋊Q₈	—	22	4	no	—
202	C₄⋊Q₈	—	44	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	22	1	no	—
205	C₂×C₄⋊C₄	—	44	1	no	—
206	C₂×C₄²	—	11	1	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	—	11	2	no	—
209	M₃₂	—	22	1	no	—
210	M₃₂	C₂²×D₁₁	1	2	yes	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	22	1	no	—
213	#108	—	11	2	no	—
214	#108	—	22	1	no	—
215	#106	—	2	1	no	—
216	#106	—	11	2	no	—
217	#106	—	22	4	no	—
218	#106	—	22	1	no	—
219	#106	C₂²×D₁₁	1	2	yes	—
220	#104	—	22	4	no	—
221	C₄×C₈	—	11	2	no	—
222	C₄×C₈	—	11	2	no	—
223	C₂×C₁₁	—	2	1	no	—
224	C₂×C₁₁	—	2	1	no	—
225	C₂×C₁₁	—	8	2	no	—
226	C₂×C₁₁	#2230	1	2	yes	complemented
227	C₂×C₁₁	C₂×#476	1	1	yes	characteristic
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	#41	—	4	1	no	—
231	#41	—	4	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	44	4	no	—
234	C₂×Q₈	—	88	4	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	22	1	no	—
240	C₂²×C₄	—	44	1	no	—
241	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
242	Q₁₆	—	2	1	no	—
243	Q₁₆	C₂×Dic₂₂	1	2	yes	complemented
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	2	2	no	—
246	D₈	—	2	1	no	—
247	D₈	C₂×Dic₂₂	1	2	yes	complemented
248	M₁₆	—	22	2	no	—
249	M₁₆	—	44	4	no	—
250	M₁₆	—	44	1	no	—
251	C₂×C₈	—	11	1	no	—
252	C₂×C₈	—	11	1	no	—
253	C₂×C₈	—	11	2	no	—
254	C₂×C₈	—	11	2	no	—
255	C₂×C₈	—	44	4	no	—
256	C₂×C₈	—	44	1	no	—
257	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
258	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
259	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
260	C₂×C₈	#4341	1	1	yes	characteristic
261	C₂×C₈	#4341	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	22	2	no	—
266	C₄⋊C₄	—	44	4	no	—
267	C₄⋊C₄	—	44	1	no	—
268	C₄²	—	11	2	no	—
269	C₄²	—	22	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	22	2	no	—
272	C₁₁	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	4	1	no	—
274	C₂³	#66964	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	88	8	no	—
278	D₄	—	2	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	22	4	no	—
288	C₂×C₄	—	22	2	no	—
289	C₂×C₄	—	44	2	no	—
290	C₂×C₄	—	88	2	no	—
291	C₂×C₄	#66931	1	1	yes	characteristic
292	C₂×C₄	#66931	1	1	yes	characteristic
293	C₂×C₄	#66964	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₁₁	1	1	yes	characteristic
295	C₂×C₄	SD₁₆×D₁₁	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
297	C₈	—	11	2	no	—
298	C₈	—	11	2	no	—
299	C₈	—	44	2	no	—
300	C₈	#66931	1	1	yes	characteristic
301	C₈	#66931	1	1	yes	characteristic
302	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
303	C₈	C₂×#4341	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#10622374	1	1	yes	characteristic
308	C₂²	#10622374	1	1	yes	characteristic
309	C₂²	C₂×#66964	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	22	4	no	—
313	C₄	—	88	2	no	—
314	C₄	#10622374	1	1	yes	characteristic
315	C₄	#10622374	1	1	yes	characteristic
316	C₄	C₂×#66931	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₁₁	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#13070785	1	2	yes	direct product factor
322	C₂	C₂×#10622374	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622374
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#360	C₂
socle	C₂²×C₁₁	#476
prime radical	C₂×C₁₁×#135	C₂²
hypercenter	C₂×#360	D₁₁
hypocenter	C₁₁	C₂×#2230
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#10622468	C₂	semidirect
3. #13070785	C₂	direct (N × H)
4. C₂×#10622151	C₂	semidirect
5. C₂×#10622125	C₂	semidirect
6. C₂×#10622149	C₂	semidirect
23. #10622468	C₂²	semidirect
26. #10622151	C₂²	semidirect
27. #10622149	C₂²	semidirect
28. #10622125	C₂²	semidirect
33. C₁₁×#135	Q₈	semidirect
35. C₂×C₁₁×Q₁₆	Q₈	semidirect
37. C₂×C₁₁×D₈	Q₈	semidirect
83. C₁₁×Q₁₆	C₂×Q₈	semidirect
87. C₁₁×D₈	C₂×Q₈	semidirect
189. #135	Dic₂₂	semidirect
191. C₂×Q₁₆	Dic₂₂	semidirect
193. C₂×D₈	Dic₂₂	semidirect
226. C₂×C₁₁	#2230	semidirect
243. Q₁₆	C₂×Dic₂₂	semidirect
247. D₈	C₂×Dic₂₂	semidirect
272. C₁₁	C₂×#2230	semidirect
321. C₂	#13070785	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×#10622468	2
C₂×#10622809	2
#13070785	2
C₂×#10622151	2
C₂×#10622125	2
C₂×#10622149	2
C₂×#10622204	2
C₂×C₁₁×#360	2
C₂×#2230	11

Subgroups — C2×#13070785 #14908501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070785 #14908501