Subgroups — C₂×#13070787 #14908503

Order 2816.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622293	C₂	1	1	yes	maximal, characteristic, complemented
2	#13070787	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10622132	C₂	1	2	yes	maximal, complemented
4	C₂×#10622101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622204	C₂	1	2	yes	maximal
6	C₂×C₁₁×#443	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₁×#135	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×#131	—	2	1	no	—
9	C₂×#66922	—	2	1	no	—
10	C₁₁×#443	—	2	2	no	—
11	C₁₁×#443	C₂²	1	4	yes	—
12	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#104	—	2	1	no	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×#66881	—	2	1	no	—
16	C₂×#66881	C₂²	1	2	yes	—
17	C₂×#66876	C₂²	1	2	yes	—
18	#10622293	—	2	2	no	—
19	#10622293	C₂²	1	4	yes	complemented
20	#10622204	C₂²	1	8	yes	—
21	#10622132	C₂²	1	8	yes	complemented
22	#10622101	C₂²	1	4	yes	complemented
23	C₂×C₁₁×#41	—	2	1	no	—
24	C₂²×Dic₂₂	—	4	1	no	—
25	C₁₁×#135	—	2	1	no	—
26	C₁₁×#135	—	2	1	no	—
27	C₁₁×#135	Q₈	1	2	yes	complemented
28	C₁₁×#135	C₂³	1	2	yes	—
29	C₂×C₁₁×Q₁₆	Q₈	1	1	yes	characteristic, complemented
30	C₂×C₁₁×SD₁₆	—	2	1	no	—
31	C₂×C₁₁×D₈	Q₈	1	1	yes	characteristic, complemented
32	C₁₁×#131	—	2	4	no	—
33	C₁₁×#131	—	4	2	no	—
34	C₂×C₁₁×M₁₆	—	2	1	no	—
35	C₂×C₁₁×M₁₆	—	4	1	no	—
36	C₂²×C₈×C₁₁	—	4	1	no	—
37	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₁	—	2	2	no	—
41	C₂×C₄⋊Dic₁₁	—	4	1	no	—
42	#66922	—	2	4	no	—
43	#66922	—	4	2	no	—
44	C₁₁×#108	—	2	1	no	—
45	C₁₁×#108	C₂³	1	2	yes	—
46	C₁₁×#104	—	2	4	no	—
47	C₄×C₈×C₁₁	D₄	1	2	yes	—
48	C₄×C₈×C₁₁	C₂³	1	2	yes	—
49	#66881	—	2	2	no	—
50	#66881	—	2	4	no	—
51	#66881	C₂³	1	4	yes	—
52	#66876	—	2	2	no	—
53	#66876	C₂³	1	4	yes	—
54	C₂×C₁₁⋊C₁₆	—	2	2	no	—
55	C₂×#2230	—	11	1	no	maximal, Sylow
56	C₁₁×#41	—	2	2	no	—
57	C₁₁×#41	—	2	2	no	—
58	C₁₁×#41	—	4	1	no	—
59	C₁₁×#41	—	4	1	no	—
60	C₂×Q₈×C₁₁	—	2	1	no	—
61	C₂×D₄×C₁₁	—	2	1	no	—
62	C₂×D₄×C₁₁	—	4	1	no	—
63	C₂²×C₄×C₁₁	—	2	1	no	—
64	C₂²×C₄×C₁₁	—	4	1	no	—
65	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₁₁	—	4	2	no	—
67	C₂×Dic₂₂	—	4	4	no	—
68	C₂×Dic₂₂	—	8	4	no	—
69	C₁₁×Q₁₆	—	2	1	no	—
70	C₁₁×Q₁₆	C₂×Q₈	1	2	yes	complemented
71	C₁₁×SD₁₆	—	2	2	no	—
72	C₁₁×SD₁₆	—	2	2	no	—
73	C₁₁×D₈	—	2	1	no	—
74	C₁₁×D₈	C₂×Q₈	1	2	yes	complemented
75	C₁₁×M₁₆	—	2	2	no	—
76	C₁₁×M₁₆	—	4	4	no	—
77	C₁₁×M₁₆	—	4	1	no	—
78	C₂×C₈×C₁₁	—	4	4	no	—
79	C₂×C₈×C₁₁	—	4	1	no	—
80	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
82	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
83	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
90	C₄²×C₁₁	—	2	1	no	—
91	C₄²×C₁₁	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₁₁	—	2	4	no	—
93	C₄⋊Dic₁₁	—	4	4	no	—
94	C₄⋊Dic₁₁	—	4	2	no	—
95	C₁₁⋊C₁₆	—	2	4	no	—
96	C₂×#498	—	11	1	no	—
97	C₂×#443	D₁₁	1	1	yes	characteristic
98	#2230	—	11	8	no	—
99	C₂×#365	—	11	2	no	—
100	C₂×#364	—	11	1	no	—
101	C₂×#360	—	11	2	no	—
102	C₂³×C₁₁	—	4	1	no	—
103	C₂³×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
104	Q₈×C₁₁	—	2	2	no	—
105	Q₈×C₁₁	—	4	1	no	—
106	D₄×C₁₁	—	2	2	no	—
107	D₄×C₁₁	—	4	2	no	—
108	D₄×C₁₁	—	4	2	no	—
109	D₄×C₁₁	—	4	1	no	—
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	—	4	1	no	—
113	C₂×C₄×C₁₁	—	4	2	no	—
114	C₂×C₄×C₁₁	—	4	2	no	—
115	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₁₁	—	4	4	no	—
122	C₂×Dic₁₁	—	8	4	no	—
123	Dic₂₂	—	8	8	no	—
124	C₈×C₁₁	—	4	2	no	—
125	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
127	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
128	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
129	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
130	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
131	C₂×#135	C₂×D₁₁	1	1	yes	characteristic
132	C₂×#131	—	2	1	no	—
133	C₂×C₄⋊Q₈	—	22	1	no	—
134	C₂×M₃₂	—	11	2	no	—
135	#498	—	11	4	no	—
136	#498	—	22	2	no	—
137	#443	—	2	2	no	—
138	#443	C₂×D₁₁	1	4	yes	—
139	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
140	C₂×#106	—	11	2	no	—
141	C₂×#106	—	22	1	no	—
142	C₂×#104	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
144	#365	—	11	8	no	—
145	#364	—	11	4	no	—
146	#360	—	11	8	no	—
147	C₂²×C₁₁	—	2	2	no	—
148	C₂²×C₁₁	—	4	2	no	—
149	C₂²×C₁₁	—	8	2	no	—
150	C₂²×C₁₁	#476	1	1	yes	characteristic
151	C₂²×C₁₁	#476	1	1	yes	characteristic
152	C₂²×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₁₁	—	2	4	no	—
154	C₄×C₁₁	—	4	2	no	—
155	C₄×C₁₁	#476	1	1	yes	characteristic
156	C₄×C₁₁	#476	1	1	yes	characteristic
157	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₁₁	—	8	4	no	—
160	C₂×#41	—	2	1	no	—
161	C₂²×Q₈	—	44	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	Dic₂₂	1	2	yes	complemented
165	#135	C₂²×D₁₁	1	2	yes	—
166	C₂×Q₁₆	Dic₂₂	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	Dic₂₂	1	1	yes	characteristic, complemented
169	#131	—	2	4	no	—
170	#131	—	4	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂²×C₈	—	4	1	no	—
174	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
176	C₄⋊Q₈	—	22	4	no	—
177	C₄⋊Q₈	—	44	2	no	—
178	C₂×C₄⋊C₄	—	22	2	no	—
179	C₂×C₄⋊C₄	—	44	1	no	—
180	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
181	M₃₂	—	11	4	no	—
182	M₃₂	—	22	2	no	—
183	C₂×C₁₆	—	22	2	no	—
184	#108	—	2	1	no	—
185	#108	C₂²×D₁₁	1	2	yes	—
186	#106	—	11	4	no	—
187	#106	—	22	2	no	—
188	#106	—	22	4	no	—
189	#104	—	2	4	no	—
190	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
191	C₄×C₈	C₂²×D₁₁	1	2	yes	—
192	C₂×C₁₁	—	2	1	no	—
193	C₂×C₁₁	—	2	1	no	—
194	C₂×C₁₁	—	8	2	no	—
195	C₂×C₁₁	#2230	1	2	yes	complemented
196	C₂×C₁₁	C₂×#476	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×Q₈	—	44	4	no	—
203	C₂×Q₈	—	88	4	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	44	2	no	—
209	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
210	Q₁₆	—	2	1	no	—
211	Q₁₆	C₂×Dic₂₂	1	2	yes	complemented
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	D₈	—	2	1	no	—
215	D₈	C₂×Dic₂₂	1	2	yes	complemented
216	M₁₆	—	2	2	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	4	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	#4324	1	1	yes	characteristic
222	C₂×C₈	#4324	1	1	yes	characteristic
223	C₂×C₈	#4324	1	2	yes	—
224	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
225	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
226	C₂×C₈	#4339	1	1	yes	characteristic
227	C₂×C₈	#4339	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
230	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
231	C₄⋊C₄	—	22	4	no	—
232	C₄⋊C₄	—	44	4	no	—
233	C₄⋊C₄	—	44	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
236	C₁₆	—	22	4	no	—
237	C₁₁	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	4	1	no	—
239	C₂³	#67002	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	—	88	8	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	44	4	no	—
253	C₂×C₄	—	88	4	no	—
254	C₂×C₄	#66977	1	1	yes	characteristic
255	C₂×C₄	#66977	1	1	yes	characteristic
256	C₂×C₄	C₂×#4324	1	1	yes	characteristic
257	C₂×C₄	C₂×#4324	1	1	yes	characteristic
258	C₂×C₄	#67002	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	#66977	1	1	yes	characteristic
262	C₈	#66977	1	1	yes	characteristic
263	C₈	C₂×#4324	1	2	yes	—
264	C₈	C₂×#4324	1	2	yes	—
265	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
266	C₈	C₂×#4339	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	#10622621	1	1	yes	characteristic
271	C₂²	#10622621	1	1	yes	characteristic
272	C₂²	C₂×#67002	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	88	4	no	—
276	C₄	#10622621	1	1	yes	characteristic
277	C₄	#10622621	1	1	yes	characteristic
278	C₄	C₂×#66977	1	1	yes	characteristic
279	C₄	C₂²×#4324	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	#13070787	1	2	yes	direct product factor
284	C₂	C₂×#10622621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622621
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#443	C₂
socle	C₂²×C₁₁	#476
prime radical	C₂×C₁₁×#135	C₂²
hypercenter	C₂×#443	D₁₁
hypocenter	C₁₁	C₂×#2230
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#10622293	C₂	semidirect
2. #13070787	C₂	direct (N × H)
3. C₂×#10622132	C₂	semidirect
4. C₂×#10622101	C₂	semidirect
19. #10622293	C₂²	semidirect
21. #10622132	C₂²	semidirect
22. #10622101	C₂²	semidirect
27. C₁₁×#135	Q₈	semidirect
29. C₂×C₁₁×Q₁₆	Q₈	semidirect
31. C₂×C₁₁×D₈	Q₈	semidirect
70. C₁₁×Q₁₆	C₂×Q₈	semidirect
74. C₁₁×D₈	C₂×Q₈	semidirect
164. #135	Dic₂₂	semidirect
166. C₂×Q₁₆	Dic₂₂	semidirect
168. C₂×D₈	Dic₂₂	semidirect
195. C₂×C₁₁	#2230	semidirect
211. Q₁₆	C₂×Dic₂₂	semidirect
215. D₈	C₂×Dic₂₂	semidirect
237. C₁₁	C₂×#2230	semidirect
283. C₂	#13070787	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×#10622293	2
#13070787	2
C₂×#10622132	2
C₂×#10622101	2
C₂×#10622204	2
C₂×C₁₁×#443	2
C₂×#2230	11

Subgroups — C2×#13070787 #14908503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070787 #14908503