Subgroups — #13078465

Order 1408.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10623358	C₂	1	1	yes	maximal, characteristic
2	#10623050	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×#416	C₂	1	1	yes	maximal, characteristic
4	#10622845	C₂	1	1	yes	maximal, characteristic
5	C₂×#66882	C₂	1	1	yes	maximal, characteristic, complemented
6	#10622715	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622608	C₂	1	1	yes	maximal, characteristic, complemented
8	#10622409	C₂	1	2	yes	maximal, complemented
9	#10622407	C₂	1	2	yes	maximal, complemented
10	#10622379	C₂	1	2	yes	maximal, complemented
11	#10622372	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₁₁×#117	C₂²	1	1	yes	characteristic
15	Q₈×Dic₁₁	C₂²	1	2	yes	—
16	D₄×Dic₁₁	—	2	2	no	—
17	D₄×Dic₁₁	C₂²	1	2	yes	—
18	#66979	—	2	1	no	—
19	#66979	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₁	—	2	1	no	—
21	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
23	C₂×#4318	C₂²	1	1	yes	characteristic
24	#66943	C₂²	1	2	yes	—
25	#66941	C₂²	1	2	yes	—
26	#66931	—	2	2	no	—
27	#66923	—	2	1	no	—
28	#66922	—	2	1	no	—
29	C₁₁×#103	C₂²	1	1	yes	characteristic
30	C₁₁×#103	C₂²	1	1	yes	characteristic
31	C₁₁×#102	C₂²	1	1	yes	characteristic
32	C₁₁×#102	C₂²	1	1	yes	characteristic
33	#66899	C₂²	1	2	yes	—
34	#66896	C₂²	1	2	yes	—
35	#66882	C₂²	1	2	yes	—
36	#66882	C₂²	1	2	yes	—
37	Dic₁₁⋊C₈	C₂²	1	2	yes	—
38	Dic₁₁⋊C₈	C₂²	1	2	yes	—
39	#66872	C₂²	1	2	yes	—
40	#66871	C₂²	1	2	yes	—
41	C₁₁×#41	—	2	2	no	—
42	C₁₁×#41	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₁	—	2	1	no	—
45	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	—	2	1	no	—
47	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₁	—	2	2	no	—
49	C₂²×Dic₁₁	—	2	1	no	—
50	C₂²×Dic₁₁	—	2	1	no	—
51	C₂×Dic₂₂	—	2	2	no	—
52	C₂×C₈×C₁₁	—	2	2	no	—
53	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
56	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
57	C₁₁×C₂²⋊C₄	—	2	1	no	—
58	C₄²×C₁₁	—	2	1	no	—
59	C₂²⋊Dic₁₁	—	2	2	no	—
60	C₂²⋊Dic₁₁	D₄	1	2	yes	—
61	C₄⋊Dic₁₁	—	2	2	no	—
62	C₄⋊Dic₁₁	—	2	2	no	—
63	C₄⋊Dic₁₁	—	2	1	no	—
64	C₄⋊Dic₁₁	—	2	1	no	—
65	C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
66	C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
67	C₄⋊Dic₁₁	C₂³	1	2	yes	—
68	C₄⋊Dic₁₁	C₂³	1	2	yes	—
69	Dic₁₁⋊C₄	—	2	2	no	—
70	Dic₁₁⋊C₄	—	2	2	no	—
71	C₄×Dic₁₁	—	2	2	no	—
72	C₄×Dic₁₁	C₂³	1	2	yes	—
73	#4318	—	2	2	no	—
74	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
75	#3083	—	11	1	no	maximal, Sylow
76	C₂³×C₁₁	—	2	1	no	—
77	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₁₁	—	2	1	no	—
79	Q₈×C₁₁	C₂×Q₈	1	2	yes	—
80	D₄×C₁₁	—	2	4	no	—
81	D₄×C₁₁	—	2	1	no	—
82	D₄×C₁₁	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₁	—	2	1	no	—
84	C₂×C₄×C₁₁	—	2	4	no	—
85	C₂×C₄×C₁₁	—	2	1	no	—
86	C₂×C₄×C₁₁	—	2	1	no	—
87	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₁₁	#41	1	2	yes	—
90	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₁₁	—	2	2	no	—
92	C₂×Dic₁₁	—	2	2	no	—
93	C₂×Dic₁₁	—	8	2	no	—
94	C₂×Dic₁₁	—	8	1	no	—
95	C₂×Dic₁₁	—	8	1	no	—
96	C₂×Dic₁₁	C₂×D₄	1	2	yes	—
97	C₂×Dic₁₁	C₂×D₄	1	2	yes	—
98	Dic₂₂	—	8	2	no	—
99	C₈×C₁₁	—	2	1	no	—
100	C₈×C₁₁	—	2	1	no	—
101	C₁₁⋊C₈	—	2	2	no	—
102	#544	—	11	1	no	—
103	#520	—	11	1	no	—
104	#479	—	11	2	no	—
105	#478	—	11	2	no	—
106	#477	—	11	2	no	—
107	#474	—	11	2	no	—
108	#428	—	11	1	no	—
109	C₂×#107	—	11	1	no	—
110	#424	—	11	1	no	—
111	#417	—	11	1	no	—
112	#416	D₁₁	1	1	yes	characteristic
113	C₂²×C₁₁	—	2	1	no	—
114	C₂²×C₁₁	—	2	1	no	—
115	C₂²×C₁₁	—	2	2	no	—
116	C₂²×C₁₁	—	4	1	no	—
117	C₂²×C₁₁	—	4	1	no	—
118	C₂²×C₁₁	C₂²⋊Q₈	1	2	yes	—
119	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₁₁	—	2	2	no	—
121	C₄×C₁₁	—	4	1	no	—
122	C₄×C₁₁	—	4	1	no	—
123	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
126	Dic₁₁	—	2	2	no	—
127	Dic₁₁	—	2	2	no	—
128	Dic₁₁	—	4	2	no	—
129	Dic₁₁	—	4	2	no	—
130	C₂×#41	C₂×D₁₁	1	1	yes	characteristic
131	C₂×M₁₆	—	11	1	no	—
132	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
133	C₄⋊Q₈	—	11	2	no	—
134	C₄⋊Q₈	—	22	1	no	—
135	#125	—	11	2	no	—
136	#125	—	22	1	no	—
137	C₂²⋊Q₈	—	22	2	no	—
138	C₄×Q₈	—	11	2	no	—
139	C₄×D₄	—	11	2	no	—
140	C₄×D₄	—	22	2	no	—
141	#117	—	11	1	no	—
142	#117	—	22	1	no	—
143	#117	C₂×D₁₁	1	1	yes	characteristic
144	C₂×C₄⋊C₄	—	11	1	no	—
145	C₂×C₄⋊C₄	—	11	1	no	—
146	C₂×C₄⋊C₄	—	22	1	no	—
147	#107	—	11	2	no	—
148	#107	—	11	2	no	—
149	#107	—	11	2	no	—
150	#106	—	11	2	no	—
151	C₄⋊C₈	—	11	2	no	—
152	C₄⋊C₈	—	11	2	no	—
153	#103	—	11	2	no	—
154	#103	C₂×D₁₁	1	1	yes	characteristic
155	#103	C₂×D₁₁	1	1	yes	characteristic
156	#102	—	11	2	no	—
157	#102	C₂×D₁₁	1	1	yes	characteristic
158	#102	C₂×D₁₁	1	1	yes	characteristic
159	C₂×C₁₁	—	2	2	no	—
160	C₂×C₁₁	—	4	2	no	—
161	C₂×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₁	#576	1	1	yes	characteristic
163	C₂×C₁₁	#578	1	1	yes	characteristic
164	#41	—	2	2	no	—
165	#41	Dic₂₂	1	4	yes	complemented
166	C₂×Q₈	—	22	2	no	—
167	C₂×Q₈	C₂²×D₁₁	1	1	yes	characteristic
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂²×D₁₁	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	22	2	no	—
172	C₂²×C₄	—	22	1	no	—
173	C₂²×C₄	—	22	1	no	—
174	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
175	M₁₆	—	22	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	11	2	no	—
178	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
180	C₄⋊C₄	—	11	1	no	—
181	C₄⋊C₄	—	11	1	no	—
182	C₄⋊C₄	—	11	2	no	—
183	C₄⋊C₄	—	11	2	no	—
184	C₄⋊C₄	—	22	2	no	—
185	C₄⋊C₄	—	22	2	no	—
186	C₄⋊C₄	—	22	2	no	—
187	C₄⋊C₄	—	22	2	no	—
188	C₄⋊C₄	—	22	1	no	—
189	C₄⋊C₄	—	22	1	no	—
190	C₄⋊C₄	C₂²×D₁₁	1	1	yes	characteristic
191	C₄⋊C₄	C₂²×D₁₁	1	1	yes	characteristic
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	11	2	no	—
194	C₂²⋊C₄	—	22	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	11	2	no	—
197	C₄²	—	22	2	no	—
198	C₁₁	#3083	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	2	1	no	—
200	C₂³	D₄×D₁₁	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	88	2	no	—
203	Q₈	C₂×Dic₂₂	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₂₂	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	11	2	no	—
212	C₂×C₄	—	11	2	no	—
213	C₂×C₄	—	22	2	no	—
214	C₂×C₄	—	22	2	no	—
215	C₂×C₄	—	88	2	no	—
216	C₂×C₄	—	88	1	no	—
217	C₂×C₄	—	88	1	no	—
218	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
219	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
220	C₂×C₄	#4341	1	2	yes	—
221	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	22	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#66931	1	2	yes	—
231	C₂²	C₂×D₄×D₁₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	22	2	no	—
236	C₄	—	22	2	no	—
237	C₄	—	44	2	no	—
238	C₄	—	44	2	no	—
239	C₄	#66931	1	2	yes	—
240	C₄	C₂²×Dic₂₂	1	1	yes	characteristic
241	C₄	C₂×#4341	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#66931	1	1	yes	characteristic
245	C₂	#10623300	1	1	yes	characteristic
246	C₂	#10623302	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₁
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₁₁×#416	C₂
socle	C₂²×C₁₁	C₂²×D₄
prime radical	C₂×C₁₁×#41	C₂²
hypercenter	#416	D₁₁
hypocenter	C₁₁	#3083
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #10623050	C₂	semidirect
5. C₂×#66882	C₂	semidirect
6. #10622715	C₂	semidirect
7. #10622608	C₂	semidirect
8. #10622409	C₂	semidirect
9. #10622407	C₂	semidirect
10. #10622379	C₂	semidirect
11. #10622372	C₂	semidirect
42. C₁₁×#41	Q₈	semidirect
165. #41	Dic₂₂	semidirect
198. C₁₁	#3083	semidirect

Maximal subgroups

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

subgroup	index
#10623358	2
#10623050	2
C₁₁×#416	2
#10622845	2
C₂×#66882	2
#10622715	2
#10622608	2
#10622409	2
#10622407	2
#10622379	2
#10622372	2
#3083	11