Subgroups — C₂×#13067410 #14723129

Order 2816.

910 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622447	C₂	1	2	yes	maximal, complemented
3	#13067410	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622121	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈₈	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₁	—	2	2	no	—
9	C₂×C₄×D₄₄	C₂²	1	2	yes	—
10	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66885	C₄	1	2	yes	—
13	C₂×(C₂×D₁₁)⋊C₈	—	2	2	no	—
14	C₂×#66882	C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
17	#10622447	—	2	4	no	—
18	#10622447	C₂²	1	8	yes	complemented
19	C₄×D₈₈	C₄	1	4	yes	—
20	C₄×D₈₈	C₂²	1	4	yes	—
21	C₁₁×#335	C₂²	1	4	yes	complemented
22	#10622121	C₂²	1	8	yes	complemented
23	C₈×C₁₁⋊C₈	C₂²	1	4	yes	complemented
24	C₂²×D₄₄	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₁	—	2	2	no	—
26	C₂²×C₈×C₁₁	—	2	2	no	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
30	C₂×(C₂×D₁₁)⋊C₄	—	2	2	no	—
31	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
32	C₂²×C₁₁⋊C₈	—	2	1	no	—
33	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
35	C₂×D₈₈	C₈	1	8	yes	complemented
36	C₂×D₈₈	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₁₁	—	2	8	no	—
38	C₂×C₈×D₁₁	—	4	8	no	—
39	C₄×D₄₄	—	2	2	no	—
40	C₄×D₄₄	—	2	2	no	—
41	C₄×D₄₄	C₂×C₄	1	4	yes	—
42	C₄×D₄₄	C₂³	1	4	yes	—
43	C₁₁×C₄⋊C₈	—	2	2	no	—
44	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
45	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₁	C₂³	1	2	yes	—
47	#66885	C₂×C₄	1	8	yes	—
48	(C₂×D₁₁)⋊C₈	—	2	8	no	—
49	#66882	C₂×C₄	1	4	yes	—
50	C₄×C₁₁⋊C₈	—	2	1	no	—
51	C₄×C₁₁⋊C₈	D₄	1	2	yes	—
52	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
53	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
54	C₂×C₈×D₈	—	11	1	no	maximal, Sylow
55	C₂³×D₁₁	—	2	2	no	—
56	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
59	C₂²×Dic₁₁	—	2	2	no	—
60	C₂×D₄₄	—	2	2	no	—
61	C₂×D₄₄	—	2	4	no	—
62	C₂×D₄₄	C₂×C₈	1	8	yes	complemented
63	C₂×D₄₄	C₂²×C₄	1	4	yes	—
64	C₂×C₄×D₁₁	—	2	4	no	—
65	C₂×C₄×D₁₁	—	2	4	no	—
66	C₂×C₄×D₁₁	—	4	4	no	—
67	C₂×C₄×D₁₁	—	4	4	no	—
68	C₂×C₈×C₁₁	—	2	2	no	—
69	C₂×C₈×C₁₁	—	2	4	no	—
70	C₂×C₈×C₁₁	—	4	4	no	—
71	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	complemented
72	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₁	#41	1	1	yes	characteristic
77	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
78	(C₂×D₁₁)⋊C₄	—	2	4	no	—
79	(C₂×D₁₁)⋊C₄	—	2	4	no	—
80	C₄⋊Dic₁₁	—	2	2	no	—
81	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
82	C₂×C₁₁⋊C₈	—	2	4	no	—
83	C₂×C₁₁⋊C₈	—	2	2	no	—
84	C₂×C₁₁⋊C₈	—	2	2	no	—
85	C₂×C₁₁⋊C₈	D₈	1	4	yes	complemented
86	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
87	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
88	D₈₈	C₂×C₈	1	16	yes	complemented
89	C₈×D₁₁	—	4	16	no	—
90	C₂×C₄×D₈	—	11	1	no	—
91	C₂×C₈×D₄	—	11	2	no	—
92	C₈×D₈	—	11	8	no	—
93	C₂×#335	D₁₁	1	1	yes	characteristic, complemented
94	C₂×D₄⋊C₈	—	11	2	no	—
95	C₂×C₈²	—	11	1	no	—
96	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
97	C₂²×D₁₁	—	2	4	no	—
98	C₂²×D₁₁	—	2	8	no	—
99	C₂²×D₁₁	—	4	4	no	—
100	C₂²×D₁₁	—	4	4	no	—
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₁	#131	1	2	yes	—
107	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₁	#135	1	2	yes	—
109	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₁₁	—	2	4	no	—
113	C₂×Dic₁₁	—	4	4	no	—
114	D₄₄	—	2	4	no	—
115	D₄₄	—	2	8	no	—
116	D₄₄	C₂²×C₈	1	8	yes	complemented
117	C₄×D₁₁	—	4	4	no	—
118	C₄×D₁₁	—	4	4	no	—
119	C₄×D₁₁	—	4	8	no	—
120	C₈×C₁₁	—	2	2	no	—
121	C₈×C₁₁	—	4	4	no	—
122	C₈×C₁₁	C₂²×C₈	1	4	yes	complemented
123	C₁₁⋊C₈	—	2	4	no	—
124	C₁₁⋊C₈	—	2	2	no	—
125	C₁₁⋊C₈	C₂×D₈	1	4	yes	complemented
126	C₂²×D₈	—	11	1	no	—
127	C₂³×C₈	—	22	2	no	—
128	C₂×C₄×D₄	—	11	2	no	—
129	C₄×D₈	—	11	4	no	—
130	C₄×D₈	—	11	4	no	—
131	C₈×D₄	—	11	8	no	—
132	C₈×D₄	—	22	4	no	—
133	C₂×#107	—	11	1	no	—
134	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
135	C₂×#102	—	11	2	no	—
136	C₂×C₂²⋊C₈	—	22	2	no	—
137	C₂×C₄×C₈	—	11	1	no	—
138	C₂×C₄×C₈	—	11	1	no	—
139	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
140	#335	C₂×D₁₁	1	4	yes	complemented
141	D₄⋊C₈	—	11	8	no	—
142	C₈²	—	11	4	no	—
143	C₂²×C₁₁	C₈×D₄	1	2	yes	—
144	C₂²×C₁₁	C₄×D₈	1	2	yes	—
145	C₂²×C₁₁	#443	1	2	yes	—
146	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₁	—	2	8	no	—
148	C₂×D₁₁	—	4	16	no	—
149	C₂×D₁₁	—	4	4	no	—
150	C₂×D₁₁	—	4	4	no	—
151	C₄×C₁₁	—	2	1	no	—
152	C₄×C₁₁	—	2	1	no	—
153	C₄×C₁₁	C₈×D₄	1	2	yes	—
154	C₄×C₁₁	C₄×D₈	1	2	yes	—
155	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
156	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
157	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
159	Dic₁₁	—	4	4	no	—
160	C₂²×D₄	—	11	2	no	—
161	C₂³×C₄	—	22	2	no	—
162	C₂×D₈	—	11	8	no	—
163	C₂×D₈	—	11	4	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	11	1	no	—
166	C₂²×C₈	—	11	1	no	—
167	C₂²×C₈	—	22	8	no	—
168	C₂²×C₈	—	22	1	no	—
169	C₂²×C₈	—	44	8	no	—
170	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
171	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
172	C₄×D₄	—	11	4	no	—
173	C₄×D₄	—	11	4	no	—
174	C₄×D₄	—	22	2	no	—
175	C₄×D₄	—	22	2	no	—
176	C₂×C₄⋊C₄	—	11	2	no	—
177	C₂×C₂²⋊C₄	—	22	2	no	—
178	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
179	#107	—	11	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
182	#102	—	11	8	no	—
183	C₂²⋊C₈	—	22	8	no	—
184	C₄×C₈	—	11	2	no	—
185	C₄×C₈	—	11	2	no	—
186	C₄×C₈	—	11	2	no	—
187	C₄×C₈	—	22	1	no	—
188	C₄×C₈	C₄×D₁₁	1	2	yes	—
189	C₄×C₈	C₂²×D₁₁	1	2	yes	—
190	C₂×C₁₁	C₈×D₈	1	4	yes	complemented
191	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
192	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
193	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
194	D₁₁	—	4	8	no	—
195	C₂⁴	—	22	2	no	—
196	C₂×D₄	—	11	8	no	—
197	C₂×D₄	—	11	4	no	—
198	C₂×D₄	—	22	2	no	—
199	C₂×D₄	—	22	4	no	—
200	C₂²×C₄	—	22	2	no	—
201	C₂²×C₄	—	22	4	no	—
202	C₂²×C₄	—	22	4	no	—
203	C₂²×C₄	—	44	4	no	—
204	C₂²×C₄	—	44	4	no	—
205	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
207	C₂²×C₄	#4343	1	1	yes	characteristic
208	D₈	—	11	16	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	11	4	no	—
213	C₂×C₈	—	11	2	no	—
214	C₂×C₈	—	11	2	no	—
215	C₂×C₈	—	22	4	no	—
216	C₂×C₈	—	22	2	no	—
217	C₂×C₈	—	22	2	no	—
218	C₂×C₈	—	44	16	no	—
219	C₂×C₈	C₈×D₁₁	1	4	yes	complemented
220	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
221	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
222	C₄⋊C₄	—	11	4	no	—
223	C₄⋊C₄	—	22	2	no	—
224	C₂²⋊C₄	—	22	4	no	—
225	C₂²⋊C₄	—	22	4	no	—
226	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
227	C₄²	D₄×D₁₁	1	1	yes	characteristic
228	C₄²	#4343	1	1	yes	characteristic
229	C₄²	C₂³×D₁₁	1	1	yes	characteristic
230	C₁₁	C₂×C₈×D₈	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	22	4	no	—
232	C₂³	—	22	8	no	—
233	C₂³	—	44	4	no	—
234	C₂³	—	44	4	no	—
235	C₂³	#66946	1	1	yes	characteristic
236	D₄	—	11	8	no	—
237	D₄	—	22	4	no	—
238	D₄	—	22	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	22	4	no	—
241	C₂×C₄	—	44	4	no	—
242	C₂×C₄	—	44	8	no	—
243	C₂×C₄	—	44	4	no	—
244	C₂×C₄	—	44	4	no	—
245	C₂×C₄	#66946	1	1	yes	characteristic
246	C₂×C₄	#66946	1	1	yes	characteristic
247	C₂×C₄	#66946	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
249	C₂×C₄	#66960	1	2	yes	—
250	C₂×C₄	D₈×D₁₁	1	2	yes	—
251	C₂×C₄	#66972	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
254	C₂×C₄	C₂×#4343	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	—	11	4	no	—
258	C₈	—	22	4	no	—
259	C₈	—	22	2	no	—
260	C₈	C₂×C₈×D₁₁	1	4	yes	complemented
261	C₂²	—	22	8	no	—
262	C₂²	—	44	4	no	—
263	C₂²	—	44	4	no	—
264	C₂²	—	44	16	no	—
265	C₂²	#10622447	1	2	yes	—
266	C₂²	#10622485	1	2	yes	—
267	C₂²	#10622508	1	2	yes	—
268	C₂²	C₂×#66946	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	44	4	no	—
272	C₄	#10622447	1	2	yes	—
273	C₄	#10622485	1	2	yes	—
274	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
275	C₄	C₂×#66960	1	1	yes	characteristic
276	C₄	C₂×D₈×D₁₁	1	1	yes	characteristic
277	C₄	C₂×#66972	1	1	yes	characteristic
278	C₂	—	44	8	no	—
279	C₂	#13067410	1	4	yes	direct product factor
280	C₂	C₂×#10622447	1	1	yes	characteristic
281	C₂	C₂×#10622485	1	1	yes	characteristic
282	C₂	C₂×#10622508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle	C₂³×C₁₁	C₄×D₄
prime radical	C₂²×D₈₈	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	C₂×C₈×D₈
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×#10622447	C₂	semidirect
3. #13067410	C₂	direct (N × H)
4. C₂×#10622121	C₂	semidirect
5. C₂×C₈×C₁₁⋊C₈	C₂	semidirect
6. C₂×C₁₁×#335	C₂	semidirect
18. #10622447	C₂²	semidirect
21. C₁₁×#335	C₂²	semidirect
22. #10622121	C₂²	semidirect
23. C₈×C₁₁⋊C₈	C₂²	semidirect
35. C₂×D₈₈	C₈	semidirect
62. C₂×D₄₄	C₂×C₈	semidirect
71. C₂×C₈×C₁₁	C₂×C₈	semidirect
85. C₂×C₁₁⋊C₈	D₈	semidirect
88. D₈₈	C₂×C₈	semidirect
93. C₂×#335	D₁₁	semidirect
116. D₄₄	C₂²×C₈	semidirect
122. C₈×C₁₁	C₂²×C₈	semidirect
125. C₁₁⋊C₈	C₂×D₈	semidirect
140. #335	C₂×D₁₁	semidirect
190. C₂×C₁₁	C₈×D₈	semidirect
219. C₂×C₈	C₈×D₁₁	semidirect
230. C₁₁	C₂×C₈×D₈	semidirect
260. C₈	C₂×C₈×D₁₁	semidirect
279. C₂	#13067410	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×C₄×D₈₈	2
C₂×#10622447	2
#13067410	2
C₂×#10622121	2
C₂×C₈×C₁₁⋊C₈	2
C₂×C₁₁×#335	2
C₂×C₈×D₈	11

Subgroups — C2×#13067410 #14723129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067410 #14723129