Finite Group Databasebeta

Subgroups — #13070066

Order 1408.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#421	C₂	1	1	yes	maximal, characteristic
2	#10622850	C₂	1	1	yes	maximal, characteristic
3	#10622846	C₂	1	1	yes	maximal, characteristic
4	#10622612	C₂	1	1	yes	maximal, characteristic, complemented
5	#10622196	C₂	1	1	yes	maximal, characteristic, complemented
6	#10622194	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622169	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
10	C₁₁×#117	C₂²	1	1	yes	characteristic
11	#66979	C₂²	1	1	yes	characteristic
12	C₂×#4318	C₂²	1	1	yes	characteristic
13	C₂×#4318	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₁₁×#104	—	2	2	no	—
16	#66900	—	2	1	no	—
17	#66899	—	2	1	no	—
18	#66897	—	2	1	no	—
19	#66896	—	2	1	no	—
20	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
21	C₁₁×#41	—	2	1	no	—
22	C₁₁×#41	—	2	1	no	—
23	C₁₁×#41	—	2	1	no	—
24	C₁₁×#41	—	2	1	no	—
25	C₂×Q₈×C₁₁	D₄	1	1	yes	characteristic
26	C₂×D₄×C₁₁	—	2	1	no	—
27	C₂×D₄×C₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₁	—	2	1	no	—
29	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
30	C₁₁×M₁₆	—	2	1	no	—
31	C₁₁×M₁₆	—	2	1	no	—
32	C₂×C₈×C₁₁	—	2	1	no	—
33	C₁₁×C₄⋊C₄	—	2	1	no	—
34	C₁₁×C₂²⋊C₄	—	2	1	no	—
35	C₄²×C₁₁	D₄	1	2	yes	—
36	C₂²⋊Dic₁₁	—	2	1	no	—
37	C₄⋊Dic₁₁	—	2	1	no	—
38	C₄×Dic₁₁	—	2	1	no	—
39	#4318	—	2	1	no	—
40	#4318	—	2	1	no	—
41	#4318	—	2	1	no	—
42	#4318	—	2	1	no	—
43	C₂×C₁₁⋊C₈	—	2	1	no	—
44	C₂×C₁₁⋊C₈	—	2	1	no	—
45	C₂×C₁₁⋊C₈	—	2	1	no	—
46	C₂×C₁₁⋊C₈	—	2	1	no	—
47	C₂×C₁₁⋊C₈	—	2	1	no	—
48	#2080	—	11	1	no	maximal, Sylow
49	C₂³×C₁₁	—	2	1	no	—
50	C₂³×C₁₁	#41	1	1	yes	characteristic
51	Q₈×C₁₁	—	2	1	no	—
52	Q₈×C₁₁	—	2	1	no	—
53	D₄×C₁₁	—	2	1	no	—
54	D₄×C₁₁	—	2	1	no	—
55	D₄×C₁₁	—	4	1	no	—
56	D₄×C₁₁	—	4	1	no	—
57	C₂×C₄×C₁₁	—	2	2	no	—
58	C₂×C₄×C₁₁	—	2	1	no	—
59	C₂×C₄×C₁₁	—	4	1	no	—
60	C₂×C₄×C₁₁	—	4	1	no	—
61	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
64	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
65	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
66	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
67	C₂×Dic₁₁	—	4	1	no	—
68	C₈×C₁₁	—	4	1	no	—
69	C₁₁⋊C₈	—	4	1	no	—
70	C₁₁⋊C₈	—	4	1	no	—
71	C₁₁⋊C₈	—	4	1	no	—
72	#424	—	11	1	no	—
73	#421	D₁₁	1	1	yes	characteristic
74	#416	—	11	1	no	—
75	#413	—	11	1	no	—
76	#344	—	11	1	no	—
77	#341	—	11	1	no	—
78	H(3,ℤ/4)	—	11	1	no	—
79	C₂²×C₁₁	—	4	1	no	—
80	C₂²×C₁₁	—	4	1	no	—
81	C₂²×C₁₁	—	8	1	no	—
82	C₂²×C₁₁	#123	1	1	yes	characteristic
83	C₂²×C₁₁	#123	1	1	yes	characteristic
84	C₂²×C₁₁	#126	1	1	yes	characteristic
85	C₄×C₁₁	—	4	2	no	—
86	C₄×C₁₁	—	4	1	no	—
87	C₄×C₁₁	#121	1	1	yes	characteristic
88	C₄×C₁₁	#123	1	1	yes	characteristic
89	C₄×C₁₁	#124	1	1	yes	characteristic
90	C₄×C₁₁	#126	1	1	yes	characteristic
91	Dic₁₁	—	8	1	no	—
92	C₂×#41	C₂×D₁₁	1	1	yes	characteristic
93	C₂×M₁₆	—	11	1	no	—
94	C₂×M₁₆	—	11	1	no	—
95	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
96	C₂²×C₈	—	11	1	no	—
97	#117	—	11	1	no	—
98	#117	C₂×D₁₁	1	1	yes	characteristic
99	C₄⋊C₈	—	22	2	no	—
100	#104	—	2	2	no	—
101	#103	—	22	1	no	—
102	#102	—	22	1	no	—
103	#101	—	22	1	no	—
104	#100	—	22	1	no	—
105	C₂×C₁₁	—	2	1	no	—
106	C₂×C₁₁	—	2	1	no	—
107	C₂×C₁₁	—	2	1	no	—
108	C₂×C₁₁	—	8	1	no	—
109	C₂×C₁₁	#397	1	1	yes	characteristic
110	#41	—	2	1	no	—
111	#41	—	2	1	no	—
112	#41	—	2	1	no	—
113	#41	—	2	1	no	—
114	C₂×Q₈	D₄₄	1	1	yes	characteristic
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	D₄₄	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
119	M₁₆	—	2	1	no	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	22	1	no	—
122	M₁₆	—	22	1	no	—
123	M₁₆	—	22	1	no	—
124	M₁₆	—	22	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	22	1	no	—
127	C₂×C₈	—	22	1	no	—
128	C₂×C₈	—	22	1	no	—
129	C₂×C₈	—	22	1	no	—
130	C₂×C₈	—	22	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	22	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	22	1	no	—
135	C₄²	—	22	1	no	—
136	C₄²	C₂²⋊D₁₁	1	2	yes	—
137	C₁₁	#2080	1	1	yes	characteristic, complemented, Sylow
138	C₂³	—	2	1	no	—
139	C₂³	#4341	1	1	yes	characteristic
140	Q₈	—	2	1	no	—
141	Q₈	—	2	1	no	—
142	D₄	—	2	1	no	—
143	D₄	—	2	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	44	1	no	—
151	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
152	C₂×C₄	#4339	1	1	yes	characteristic
153	C₂×C₄	#4341	1	1	yes	characteristic
154	C₂×C₄	#4341	1	1	yes	characteristic
155	C₂×C₄	#4341	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	44	1	no	—
159	C₈	—	44	1	no	—
160	C₈	—	44	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	#66932	1	1	yes	characteristic
165	C₂²	#66939	1	1	yes	characteristic
166	C₂²	#66988	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	88	1	no	—
170	C₄	#66932	1	1	yes	characteristic
171	C₄	#66939	1	1	yes	characteristic
172	C₄	#66983	1	1	yes	characteristic
173	C₄	#66989	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#10622566	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#66989
commutator, cosocle	C₂²×C₄×C₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×#421	C₂
socle	C₂×C₁₁	#397
prime radical	C₂×C₁₁×#41	C₂²
hypercenter	#421	D₁₁
hypocenter	C₁₁	#2080
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
4. #10622612	C₂	semidirect
5. #10622196	C₂	semidirect
6. #10622194	C₂	semidirect
7. #10622169	C₂	semidirect
137. C₁₁	#2080	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×#421	2
#10622850	2
#10622846	2
#10622612	2
#10622196	2
#10622194	2
#10622169	2
#2080	11