Finite Group Databasebeta

Subgroups — #28407737

Order 14464.

184 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20299717	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813717	C₂	1	1	yes	maximal, characteristic
3	#20299693	C₂	1	2	yes	maximal, complemented
4	#20299691	C₂	1	2	yes	maximal, complemented
5	#20299564	C₂	1	1	yes	maximal, characteristic
6	#16813719	C₄	1	2	yes	complemented
7	#16813718	C₄	1	2	yes	complemented
8	#16813717	C₄	1	2	yes	complemented
9	#16813717	C₂²	1	1	yes	characteristic
10	#16813717	C₂²	1	2	yes	—
11	#16813717	C₂²	1	1	yes	characteristic
12	C₂×#14268659	C₄	1	2	yes	complemented
13	C₂×#14268655	C₂²	1	1	yes	characteristic
14	#16813701	C₄	1	1	yes	characteristic
15	#16813676	C₄	1	2	yes	—
16	C₈×#90673	C₂²	1	2	yes	—
17	M₁₆×D₁₁₃	C₄	1	1	yes	characteristic
18	#14268659	C₂×C₄	1	2	yes	—
19	#14268659	C₂×C₄	1	2	yes	—
20	#14268659	C₂×C₄	1	4	yes	—
21	#14268655	C₂×C₄	1	1	yes	characteristic
22	#14268655	C₂×C₄	1	1	yes	characteristic
23	#14268655	C₂×C₄	1	1	yes	characteristic
24	#14268655	C₂³	1	1	yes	characteristic
25	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
26	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
27	C₄⋊#90673	C₈	1	2	yes	—
28	C₂²⋊#90673	C₈	1	2	yes	—
29	C₄×#90673	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
31	#14268631	C₈	1	1	yes	characteristic
32	#14268630	C₈	1	2	yes	—
33	C₈×D₁₁₃	C₂×C₄	1	2	yes	—
34	M₁₆×C₁₁₃	C₈	1	1	yes	characteristic
35	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×#90673	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₁₃	C₄²	1	1	yes	characteristic
39	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
40	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
41	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
42	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
43	#11744753	C₄²	1	1	yes	characteristic
44	#11744753	C₄²	1	1	yes	characteristic
45	#11744753	C₂²×C₄	1	1	yes	characteristic
46	#11744753	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₁₁₃	C₂×C₈	1	2	yes	—
48	#11744751	C₂×C₈	1	2	yes	—
49	C₂²×C₁₁₃	C₄×C₈	1	1	yes	characteristic
50	C₂×D₁₁₃	—	2	2	no	—
51	C₂×D₁₁₃	C₄×C₈	1	1	yes	characteristic
52	C₂×D₁₁₃	C₂²×C₈	1	1	yes	characteristic
53	#90673	—	2	4	no	—
54	C₄×C₁₁₃	C₄×C₈	1	1	yes	characteristic
55	C₄×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
56	Dic₁₁₃	C₄×C₈	1	1	yes	characteristic
57	Dic₁₁₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₁₁₃	—	2	1	no	—
59	C₂×C₁₁₃	C₂×C₄×C₈	1	1	yes	characteristic
60	D₁₁₃	—	2	1	no	—
61	D₁₁₃	#504	1	2	yes	—
62	#2108	—	113	1	no	maximal, Sylow
63	C₁₁₃	#2108	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₃₂	—	113	1	no	—
65	C₂²×C₁₆	—	113	1	no	—
66	#405	—	113	1	no	—
67	#346	—	113	2	no	—
68	C₄×C₁₆	—	113	2	no	—
69	C₂×M₁₆	—	113	1	no	—
70	C₂²×C₈	—	113	1	no	—
71	#117	—	113	1	no	—
72	M₃₂	—	113	2	no	—
73	M₃₂	—	113	2	no	—
74	C₂×C₁₆	—	113	1	no	—
75	C₂×C₁₆	—	113	2	no	—
76	C₂×C₁₆	—	113	1	no	—
77	C₂×C₁₆	—	113	2	no	—
78	C₂×C₁₆	—	113	2	no	—
79	#97	—	113	2	no	—
80	C₄×C₈	—	113	2	no	—
81	C₂²×C₄	—	113	1	no	—
82	M₁₆	—	113	1	no	—
83	M₁₆	—	113	2	no	—
84	M₁₆	#11744762	1	1	yes	characteristic
85	C₂×C₈	—	113	1	no	—
86	C₂×C₈	—	113	1	no	—
87	C₂×C₈	—	113	1	no	—
88	C₂×C₈	—	113	1	no	—
89	C₂×C₈	—	113	1	no	—
90	C₂×C₈	—	113	1	no	—
91	C₂×C₈	—	113	2	no	—
92	C₄⋊C₄	—	113	2	no	—
93	C₂²⋊C₄	—	113	2	no	—
94	C₄²	—	113	2	no	—
95	C₁₆	—	113	2	no	—
96	C₁₆	—	113	2	no	—
97	C₁₆	—	113	4	no	—
98	C₂³	—	113	1	no	—
99	C₂×C₄	—	113	1	no	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	113	1	no	—
102	C₂×C₄	—	113	1	no	—
103	C₂×C₄	—	113	1	no	—
104	C₂×C₄	—	113	4	no	—
105	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
106	C₈	—	113	1	no	—
107	C₈	—	113	1	no	—
108	C₈	—	113	1	no	—
109	C₈	—	113	1	no	—
110	C₈	—	113	2	no	—
111	C₈	C₂×#11744762	1	2	yes	—
112	C₂²	—	113	1	no	—
113	C₂²	—	113	1	no	—
114	C₂²	—	226	2	no	—
115	C₂²	C₄×#11744762	1	1	yes	characteristic
116	C₄	—	113	1	no	—
117	C₄	—	113	1	no	—
118	C₄	—	226	4	no	—
119	C₄	C₄×#11744762	1	1	yes	characteristic
120	C₄	C₂²×#11744762	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	113	2	no	—
123	C₂	—	226	1	no	—
124	C₂	C₂×C₄×#11744762	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11744762
commutator, socle	C₂×C₁₁₃	C₂×C₄×C₈
Fitting	M₁₆×C₁₁₃	C₈
prime radical	C₂²×D₁₁₃	C₂×C₈
cosocle	#14268655	C₂³
hypercenter	M₁₆	#11744762
hypocenter	C₁₁₃	#2108
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #20299693	C₂	semidirect
4. #20299691	C₂	semidirect
6. #16813719	C₄	semidirect
7. #16813718	C₄	semidirect
8. #16813717	C₄	semidirect
12. C₂×#14268659	C₄	semidirect
63. C₁₁₃	#2108	semidirect

Maximal subgroups

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

subgroup	index
#20299717	2
C₂×#16813717	2
#20299693	2
#20299691	2
#20299564	2
#2108	113