Finite Group Databasebeta

Subgroups — #28407779

Order 14464.

138 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20299701	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#14268659	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14268659	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊#14268659	C₂	1	1	yes	maximal, characteristic
5	#20299692	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14268659	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299562	C₂	1	1	yes	maximal, characteristic
8	C₂×#14268659	C₂²	1	1	yes	characteristic
9	C₂×#14268659	C₂²	1	1	yes	characteristic
10	C₂×#14268659	C₂²	1	1	yes	characteristic
11	C₂×#14268659	C₂²	1	1	yes	characteristic
12	C₂×#14268656	C₄	1	1	yes	characteristic
13	C₂²×#11744753	C₂²	1	1	yes	characteristic
14	#16813677	C₄	1	1	yes	characteristic
15	#16813677	C₄	1	1	yes	characteristic
16	C₄×#11744753	C₂²	1	1	yes	characteristic
17	C₄×#11744753	C₂²	1	1	yes	characteristic
18	#16813578	C₄	1	1	yes	characteristic
19	#14268659	—	2	1	no	—
20	#14268659	—	2	1	no	—
21	#14268659	—	2	1	no	—
22	#14268659	—	2	1	no	—
23	#14268656	—	2	2	no	—
24	C₂×#11744753	—	2	1	no	—
25	C₂×#11744753	—	2	1	no	—
26	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
27	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
28	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
29	C₂×#11744753	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
31	#14268626	C₈	1	1	yes	characteristic
32	#14268626	C₈	1	1	yes	characteristic
33	C₂²⋊Dic₁₁₃	C₈	1	1	yes	characteristic
34	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²⋊C₄×C₁₁₃	C₈	1	1	yes	characteristic
37	C₂³×C₁₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₁₁₃	—	2	1	no	—
41	C₂×Dic₁₁₃	—	2	1	no	—
42	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
46	#11744753	—	2	1	no	—
47	#11744753	—	2	1	no	—
48	#11744753	M₁₆	1	2	yes	—
49	#11744753	#41	1	1	yes	characteristic
50	#11744753	#41	1	1	yes	characteristic
51	C₂²×C₁₁₃	—	2	1	no	—
52	C₂²×C₁₁₃	—	2	1	no	—
53	C₂²×C₁₁₃	—	2	1	no	—
54	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₁₁₃	—	2	1	no	—
56	C₄×C₁₁₃	—	2	1	no	—
57	Dic₁₁₃	—	2	1	no	—
58	Dic₁₁₃	—	2	1	no	—
59	Dic₁₁₃	—	2	1	no	—
60	Dic₁₁₃	#117	1	1	yes	characteristic
61	Dic₁₁₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₁₁₃	—	4	1	no	—
63	C₂×C₁₁₃	#431	1	1	yes	characteristic
64	C₂×C₁₁₃	#504	1	1	yes	characteristic
65	C₂×C₁₁₃	#504	1	1	yes	characteristic
66	#2164	—	113	1	no	maximal, Sylow
67	C₁₁₃	#2164	1	1	yes	characteristic, complemented, Sylow
68	#405	—	113	1	no	—
69	C₄⋊C₁₆	—	113	1	no	—
70	C₄⋊C₁₆	—	113	1	no	—
71	C₂²⋊C₁₆	—	113	1	no	—
72	C₂²⋊C₁₆	—	113	1	no	—
73	#346	—	113	1	no	—
74	C₄×C₁₆	—	113	1	no	—
75	C₂×M₁₆	—	113	1	no	—
76	C₂²×C₈	—	113	1	no	—
77	#117	—	113	1	no	—
78	C₂×C₁₆	—	113	1	no	—
79	C₂×C₁₆	—	113	1	no	—
80	C₂×C₁₆	—	113	1	no	—
81	C₂×C₁₆	—	113	1	no	—
82	#97	—	113	1	no	—
83	#97	—	113	1	no	—
84	C₄×C₈	—	113	1	no	—
85	C₄×C₈	—	113	1	no	—
86	C₂²×C₄	—	113	1	no	—
87	M₁₆	—	226	2	no	—
88	C₂×C₈	—	113	1	no	—
89	C₂×C₈	—	113	1	no	—
90	C₂×C₈	—	113	1	no	—
91	C₂×C₈	—	113	1	no	—
92	C₂×C₈	—	226	1	no	—
93	C₂×C₈	—	226	1	no	—
94	C₄⋊C₄	—	113	1	no	—
95	C₄⋊C₄	—	113	1	no	—
96	C₂²⋊C₄	—	113	1	no	—
97	C₂²⋊C₄	#11744762	1	1	yes	characteristic
98	C₄²	—	113	1	no	—
99	C₄²	—	113	1	no	—
100	C₁₆	—	226	1	no	—
101	C₁₆	—	226	1	no	—
102	C₁₆	—	226	1	no	—
103	C₁₆	—	226	1	no	—
104	C₂³	C₂×#11744762	1	1	yes	characteristic
105	C₂×C₄	—	113	1	no	—
106	C₂×C₄	—	113	1	no	—
107	C₂×C₄	—	113	1	no	—
108	C₂×C₄	—	113	1	no	—
109	C₂×C₄	—	226	1	no	—
110	C₂×C₄	—	226	1	no	—
111	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
112	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
113	C₈	—	113	1	no	—
114	C₈	—	113	1	no	—
115	C₈	—	113	2	no	—
116	C₈	—	226	1	no	—
117	C₈	—	226	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×#11744762	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	113	1	no	—
125	C₄	—	113	1	no	—
126	C₄	—	226	1	no	—
127	C₄	—	226	1	no	—
128	C₄	—	226	1	no	—
129	C₂	—	4	1	no	—
130	C₂	#20299711	1	1	yes	characteristic
131	C₂	#20299719	1	1	yes	characteristic
132	C₂	#20299719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11744762
commutator, socle	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₂²⋊C₄×C₁₁₃	C₈
prime radical	C₂³×C₁₁₃	C₂×C₈
cosocle	C₂×#11744753	C₂³
hypercenter	C₂²⋊C₄	#11744762
hypocenter	C₁₁₃	#2164
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20299701	C₂	semidirect
2. C₄⋊#14268659	C₂	semidirect
5. #20299692	C₂	semidirect
6. C₄×#14268659	C₂	semidirect
67. C₁₁₃	#2164	semidirect

Maximal subgroups

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

subgroup	index
#20299701	2
C₄⋊#14268659	2
C₂²⋊#14268659	2
#20299692	2
C₄×#14268659	2
#20299562	2
#2164	113