Subgroups — #13077712

Order 1408.

322 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10623109	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622453	C₂	1	2	yes	maximal
5	#10622448	C₂	1	1	yes	maximal, characteristic
6	#10622444	C₂	1	2	yes	maximal, complemented
7	#10622443	C₂	1	1	yes	maximal, characteristic, complemented
8	#10622322	C₂	1	2	yes	maximal
9	#10622314	C₂	1	2	yes	maximal, complemented
10	#10622298	C₂	1	1	yes	maximal, characteristic
11	#10622292	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
13	#66999	C₄	1	1	yes	characteristic
14	#66997	C₄	1	1	yes	characteristic
15	#66950	C₄	1	1	yes	characteristic
16	#66949	C₄	1	2	yes	—
17	#66942	C₄	1	2	yes	—
18	#66941	C₄	1	1	yes	characteristic
19	#66925	C₂²	1	2	yes	—
20	C₄²×D₁₁	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₂	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₂	C₂²	1	2	yes	—
23	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₁₁×#97	C₂²	1	2	yes	—
26	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
27	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
28	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₁)⋊C₈	C₂²	1	2	yes	—
30	#66879	C₂²	1	2	yes	—
31	#66879	C₂²	1	2	yes	—
32	Dic₁₁⋊C₈	C₂²	1	2	yes	—
33	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
34	Dic₁₁⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₁₁⋊C₈	C₂²	1	2	yes	—
36	C₈×Dic₁₁	C₂²	1	2	yes	—
37	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
39	C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₂	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₂₂	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₁₁	C₂³	1	2	yes	—
48	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₄²×C₁₁	C₂³	1	1	yes	characteristic
53	C₄²×C₁₁	C₂³	1	2	yes	—
54	(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
58	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₁	C₂³	1	2	yes	—
66	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
70	#2985	—	11	1	no	maximal, Sylow
71	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₁₁	—	2	2	no	—
73	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
83	C₄×D₁₁	—	2	2	no	—
84	C₄×D₁₁	—	2	1	no	—
85	C₄×D₁₁	—	2	1	no	—
86	C₄×D₁₁	—	4	1	no	—
87	Dic₂₂	—	2	2	no	—
88	Dic₂₂	—	4	2	no	—
89	C₈×C₁₁	—	2	1	no	—
90	C₈×C₁₁	—	2	2	no	—
91	C₈×C₁₁	—	2	1	no	—
92	C₁₁⋊C₈	—	2	1	no	—
93	C₁₁⋊C₈	—	2	2	no	—
94	C₁₁⋊C₈	—	2	1	no	—
95	#533	—	11	1	no	—
96	#446	—	11	2	no	—
97	#446	—	11	2	no	—
98	#446	—	11	1	no	—
99	#446	D₁₁	1	1	yes	characteristic, complemented
100	C₈×Q₈	—	11	1	no	—
101	C₈×Q₈	—	11	1	no	—
102	#433	—	11	2	no	—
103	#433	—	11	2	no	—
104	#432	—	11	1	no	—
105	#431	—	11	1	no	—
106	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₁₁	—	2	1	no	—
108	C₂×D₁₁	—	2	1	no	—
109	C₂×D₁₁	—	2	1	no	—
110	C₄×C₁₁	—	2	2	no	—
111	C₄×C₁₁	—	2	2	no	—
112	C₄×C₁₁	—	2	1	no	—
113	C₄×C₁₁	#131	1	2	yes	—
114	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
116	Dic₁₁	—	2	1	no	—
117	Dic₁₁	—	2	2	no	—
118	Dic₁₁	—	2	1	no	—
119	Dic₁₁	—	2	1	no	—
120	Dic₁₁	—	2	2	no	—
121	C₄⋊Q₈	—	11	1	no	—
122	C₄⋊Q₈	—	11	1	no	—
123	#125	—	11	2	no	—
124	C₂²⋊Q₈	—	11	1	no	—
125	C₂²⋊Q₈	—	11	2	no	—
126	C₂²⋊Q₈	—	11	1	no	—
127	C₄×Q₈	—	11	1	no	—
128	C₄×Q₈	—	11	2	no	—
129	C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
130	#117	—	11	2	no	—
131	C₂×C₄²	—	11	1	no	—
132	C₄⋊C₈	—	11	1	no	—
133	C₄⋊C₈	—	11	1	no	—
134	C₄⋊C₈	—	11	2	no	—
135	C₄⋊C₈	—	11	2	no	—
136	C₄⋊C₈	—	11	1	no	—
137	C₄⋊C₈	—	11	2	no	—
138	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
139	C₄⋊C₈	C₂×D₁₁	1	2	yes	—
140	C₂²⋊C₈	—	11	1	no	—
141	C₂²⋊C₈	—	11	1	no	—
142	C₂²⋊C₈	—	11	2	no	—
143	#97	—	11	2	no	—
144	#97	—	11	2	no	—
145	#97	C₂×D₁₁	1	2	yes	—
146	C₄×C₈	—	11	2	no	—
147	C₄×C₈	—	11	1	no	—
148	C₄×C₈	—	11	2	no	—
149	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
150	C₂×C₁₁	#519	1	1	yes	characteristic
151	C₂×C₁₁	C₂×#131	1	1	yes	characteristic
152	C₂×C₁₁	#568	1	1	yes	characteristic
153	D₁₁	—	4	1	no	—
154	C₂×Q₈	—	11	1	no	—
155	C₂×Q₈	—	11	2	no	—
156	C₂×Q₈	C₄×D₁₁	1	1	yes	characteristic
157	C₂²×C₄	—	11	1	no	—
158	C₂²×C₄	—	11	1	no	—
159	C₂²×C₄	—	11	1	no	—
160	C₂×C₈	—	11	1	no	—
161	C₂×C₈	—	11	1	no	—
162	C₂×C₈	—	11	2	no	—
163	C₂×C₈	C₂²×D₁₁	1	2	yes	—
164	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
166	C₄⋊C₄	—	11	2	no	—
167	C₄⋊C₄	—	11	2	no	—
168	C₄⋊C₄	—	11	2	no	—
169	C₄⋊C₄	—	11	2	no	—
170	C₄⋊C₄	—	11	2	no	—
171	C₄⋊C₄	—	11	2	no	—
172	C₄⋊C₄	—	11	1	no	—
173	C₄⋊C₄	C₄×D₁₁	1	2	yes	—
174	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
175	C₂²⋊C₄	—	11	2	no	—
176	C₂²⋊C₄	—	11	2	no	—
177	C₄²	—	11	1	no	—
178	C₄²	—	11	1	no	—
179	C₄²	—	11	2	no	—
180	C₄²	—	11	1	no	—
181	C₄²	C₂²×D₁₁	1	1	yes	characteristic
182	C₄²	C₂²×D₁₁	1	2	yes	—
183	C₁₁	#2985	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	11	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	22	2	no	—
187	Q₈	—	44	2	no	—
188	C₂×C₄	—	11	1	no	—
189	C₂×C₄	—	11	1	no	—
190	C₂×C₄	—	11	1	no	—
191	C₂×C₄	—	11	2	no	—
192	C₂×C₄	—	11	2	no	—
193	C₂×C₄	—	22	2	no	—
194	C₂×C₄	—	22	1	no	—
195	C₂×C₄	—	22	1	no	—
196	C₂×C₄	—	44	1	no	—
197	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	22	1	no	—
206	C₈	—	22	2	no	—
207	C₈	—	22	1	no	—
208	C₂²	—	22	1	no	—
209	C₂²	—	22	1	no	—
210	C₂²	—	22	1	no	—
211	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	22	1	no	—
216	C₄	—	22	2	no	—
217	C₄	—	22	1	no	—
218	C₄	—	22	2	no	—
219	C₄	—	22	1	no	—
220	C₄	#66960	1	2	yes	—
221	C₄	#67040	1	1	yes	characteristic
222	C₄	#67043	1	1	yes	characteristic
223	C₂	—	44	1	no	—
224	C₂	#10623103	1	1	yes	characteristic
225	C₂	C₂×#66960	1	1	yes	characteristic
226	C₂	#10623281	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#446	C₂
prime radical	C₂²×D₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#446	D₁₁
hypocenter	C₁₁	#2985
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₁₁×#446	C₂	semidirect
3. Q₈×C₁₁⋊C₈	C₂	semidirect
6. #10622444	C₂	semidirect
7. #10622443	C₂	semidirect
9. #10622314	C₂	semidirect
11. #10622292	C₂	semidirect
99. #446	D₁₁	semidirect
183. C₁₁	#2985	semidirect

Maximal subgroups

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

subgroup	index
#10623109	2
C₁₁×#446	2
Q₈×C₁₁⋊C₈	2
#10622453	2
#10622448	2
#10622444	2
#10622443	2
#10622322	2
#10622314	2
#10622298	2
#10622292	2
#2985	11