Subgroups — #13079255

Order 1408.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10623065	C₂	1	2	yes	maximal, complemented
2	C₁₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622747	C₂	1	2	yes	maximal, complemented
4	#10622729	C₂	1	1	yes	maximal, characteristic, complemented
5	#10622728	C₂	1	1	yes	maximal, characteristic
6	#10622327	C₂	1	2	yes	maximal, complemented
7	#10622323	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₄₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#10622309	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₈₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#10622295	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₁₁×#117	C₂²	1	1	yes	characteristic
15	#66983	C₂²	1	2	yes	—
16	#66977	C₂²	1	2	yes	—
17	C₂×Dic₄₄	C₂²	1	1	yes	characteristic
18	C₂×D₈₈	C₂²	1	1	yes	characteristic
19	C₂×#4314	C₂²	1	2	yes	—
20	#66951	—	2	2	no	—
21	#66949	—	2	2	no	—
22	#66938	—	2	2	no	—
23	#66936	—	2	2	no	—
24	#66928	C₂²	1	2	yes	—
25	C₄×D₄₄	C₂²	1	2	yes	—
26	#66922	C₂²	1	2	yes	—
27	C₄×Dic₂₂	C₂²	1	2	yes	—
28	C₁₁×#97	C₂²	1	1	yes	characteristic
29	C₁₁×#97	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
32	#66885	C₂²	1	2	yes	—
33	#66885	C₂²	1	2	yes	—
34	#66882	C₂²	1	1	yes	characteristic
35	#66882	C₂²	1	1	yes	characteristic
36	#66882	C₂²	1	2	yes	—
37	#66880	C₂²	1	2	yes	—
38	#66880	C₂²	1	2	yes	—
39	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₁	—	2	2	no	—
41	C₂×D₄₄	C₂³	1	2	yes	—
42	C₂×C₄×D₁₁	—	2	2	no	—
43	C₂×Dic₂₂	—	2	2	no	—
44	C₂×Dic₂₂	C₂³	1	2	yes	—
45	C₁₁×M₁₆	—	2	2	no	—
46	C₂×C₈×C₁₁	—	2	2	no	—
47	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₁₁×C₂²⋊C₄	D₄	1	1	yes	characteristic
54	C₁₁×C₂²⋊C₄	D₄	1	1	yes	characteristic
55	C₄²×C₁₁	C₂³	1	1	yes	characteristic
56	C₄²×C₁₁	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₁	—	2	2	no	—
58	(C₂×D₁₁)⋊C₄	—	2	2	no	—
59	(C₂×D₁₁)⋊C₄	—	2	2	no	—
60	(C₂×D₁₁)⋊C₄	—	2	2	no	—
61	(C₂×D₁₁)⋊C₄	—	2	2	no	—
62	C₄⋊Dic₁₁	—	2	2	no	—
63	C₄⋊Dic₁₁	C₂³	1	2	yes	—
64	C₄⋊Dic₁₁	C₂³	1	2	yes	—
65	Dic₁₁⋊C₄	—	2	2	no	—
66	Dic₁₁⋊C₄	—	2	2	no	—
67	C₄×Dic₁₁	—	2	2	no	—
68	Dic₄₄	—	2	2	no	—
69	D₈₈	—	2	2	no	—
70	#4314	—	2	4	no	—
71	#3175	—	11	1	no	maximal, Sylow
72	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×D₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	1	no	—
75	C₂×C₄×C₁₁	—	2	1	no	—
76	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₁₁	—	8	2	no	—
83	C₂×Dic₁₁	—	2	2	no	—
84	C₂×Dic₁₁	—	2	2	no	—
85	C₂×Dic₁₁	—	2	2	no	—
86	D₄₄	—	2	2	no	—
87	D₄₄	—	2	2	no	—
88	C₄×D₁₁	—	8	2	no	—
89	Dic₂₂	—	2	2	no	—
90	Dic₂₂	—	2	2	no	—
91	Dic₂₂	—	8	2	no	—
92	C₈×C₁₁	—	2	1	no	—
93	C₈×C₁₁	—	2	1	no	—
94	C₈×C₁₁	#41	1	2	yes	—
95	C₈×C₁₁	#41	1	2	yes	—
96	#542	—	11	2	no	—
97	#500	—	11	1	no	—
98	#495	—	11	1	no	—
99	#489	—	11	2	no	—
100	#468	—	11	2	no	—
101	#467	—	11	1	no	—
102	#466	—	11	1	no	—
103	#440	—	11	2	no	—
104	C₄×Q₁₆	—	11	1	no	—
105	C₄×D₈	—	11	1	no	—
106	#405	D₁₁	1	1	yes	characteristic, complemented
107	C₂²×C₁₁	—	2	1	no	—
108	C₂²×C₁₁	—	2	1	no	—
109	C₂²×C₁₁	—	2	1	no	—
110	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₁₁	—	4	2	no	—
112	C₂×D₁₁	—	4	2	no	—
113	C₂×D₁₁	—	4	2	no	—
114	C₄×C₁₁	—	2	1	no	—
115	C₄×C₁₁	—	2	1	no	—
116	C₄×C₁₁	—	2	1	no	—
117	C₄×C₁₁	—	2	1	no	—
118	C₄×C₁₁	—	2	1	no	—
119	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
120	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
121	Dic₁₁	—	4	2	no	—
122	Dic₁₁	—	4	2	no	—
123	Dic₁₁	—	4	2	no	—
124	C₂×Q₁₆	—	11	1	no	—
125	C₂×SD₁₆	—	11	2	no	—
126	C₂×D₈	—	11	1	no	—
127	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
129	C₄⋊Q₈	—	11	2	no	—
130	#126	—	22	2	no	—
131	#124	—	11	2	no	—
132	#124	—	22	2	no	—
133	#123	—	22	2	no	—
134	C₂²⋊Q₈	—	11	2	no	—
135	C₂²⋊Q₈	—	22	2	no	—
136	#121	—	11	2	no	—
137	C₄×Q₈	—	11	2	no	—
138	C₄×D₄	—	11	2	no	—
139	#117	C₂×D₁₁	1	1	yes	characteristic
140	#107	—	11	1	no	—
141	#107	—	11	1	no	—
142	#107	—	11	2	no	—
143	#103	—	11	2	no	—
144	#103	—	11	2	no	—
145	#102	—	11	2	no	—
146	#102	—	11	2	no	—
147	#97	C₂×D₁₁	1	1	yes	characteristic
148	#97	C₂×D₁₁	1	1	yes	characteristic
149	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
150	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
151	C₂×C₁₁	—	4	1	no	—
152	C₂×C₁₁	#532	1	1	yes	characteristic
153	C₂×C₁₁	#576	1	1	yes	characteristic
154	C₂×C₁₁	#578	1	1	yes	characteristic
155	D₁₁	—	8	2	no	—
156	C₂×Q₈	—	11	2	no	—
157	C₂×Q₈	—	22	2	no	—
158	C₂×D₄	—	11	2	no	—
159	C₂×D₄	—	22	2	no	—
160	C₂²×C₄	—	22	2	no	—
161	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
162	Q₁₆	—	22	2	no	—
163	SD₁₆	—	22	4	no	—
164	D₈	—	22	2	no	—
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
168	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
171	C₄⋊C₄	—	11	2	no	—
172	C₄⋊C₄	—	11	2	no	—
173	C₄⋊C₄	—	22	2	no	—
174	C₄⋊C₄	—	22	2	no	—
175	C₄⋊C₄	—	22	2	no	—
176	C₄⋊C₄	D₄₄	1	1	yes	characteristic
177	C₄⋊C₄	D₄₄	1	1	yes	characteristic
178	C₂²⋊C₄	—	22	2	no	—
179	C₂²⋊C₄	—	22	2	no	—
180	C₂²⋊C₄	—	22	2	no	—
181	C₂²⋊C₄	—	22	2	no	—
182	C₂²⋊C₄	—	22	2	no	—
183	C₂²⋊C₄	D₄₄	1	1	yes	characteristic
184	C₂²⋊C₄	D₄₄	1	1	yes	characteristic
185	C₄²	—	22	2	no	—
186	C₄²	C₂²×D₁₁	1	1	yes	characteristic
187	C₄²	C₂²×D₁₁	1	1	yes	characteristic
188	C₁₁	#3175	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	22	2	no	—
190	C₂³	C₂×D₄₄	1	1	yes	characteristic
191	Q₈	—	22	2	no	—
192	Q₈	—	22	2	no	—
193	Q₈	—	88	2	no	—
194	D₄	—	22	2	no	—
195	D₄	—	22	2	no	—
196	D₄	—	88	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	22	2	no	—
200	C₂×C₄	—	22	2	no	—
201	C₂×C₄	—	22	2	no	—
202	C₂×C₄	—	88	2	no	—
203	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#4339	1	2	yes	—
212	C₈	#4339	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	44	2	no	—
217	C₂²	—	44	2	no	—
218	C₂²	—	44	2	no	—
219	C₂²	C₂²×D₄₄	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	44	2	no	—
226	C₄	—	44	2	no	—
227	C₄	—	44	2	no	—
228	C₄	C₂×#4339	1	1	yes	characteristic
229	C₄	C₂×#4339	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	88	2	no	—
232	C₂	#10623008	1	1	yes	characteristic
233	C₂	#10623283	1	1	yes	characteristic
234	C₂	#10623284	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₁₁×#405	C₂
socle	C₂²×C₁₁	C₂²×D₄
prime radical	#10622728	C₂
hypercenter	#405	D₁₁
hypocenter	C₁₁	#3175
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #10623065	C₂	semidirect
2. C₁₁×#405	C₂	semidirect
3. #10622747	C₂	semidirect
4. #10622729	C₂	semidirect
6. #10622327	C₂	semidirect
7. #10622323	C₂	semidirect
8. C₄×Dic₄₄	C₂	semidirect
9. #10622309	C₂	semidirect
10. C₄×D₈₈	C₂	semidirect
11. #10622295	C₂	semidirect
106. #405	D₁₁	semidirect
188. C₁₁	#3175	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10623065	2
C₁₁×#405	2
#10622747	2
#10622729	2
#10622728	2
#10622327	2
#10622323	2
C₄×Dic₄₄	2
#10622309	2
C₄×D₈₈	2
#10622295	2
#3175	11