Finite Group Databasebeta

Subgroups — #28407770

Order 14464.

152 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂×#16813719	C₂	1	1	yes	maximal, characteristic
2	#20299717	C₂	1	1	yes	maximal, characteristic
3	#20299703	C₂	1	2	yes	maximal, complemented
4	#20299702	C₂	1	2	yes	maximal, complemented
5	#20299564	C₂	1	1	yes	maximal, characteristic
6	#16813719	—	2	1	no	—
7	#16813719	C₂²	1	1	yes	characteristic
8	#16813719	C₂²	1	2	yes	—
9	#16813719	C₂²	1	1	yes	characteristic
10	#16813718	—	2	1	no	—
11	#16813717	—	2	1	no	—
12	C₂×#14268659	—	2	1	no	—
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	—	2	2	no	—
19	#14268659	—	2	1	no	—
20	#14268659	—	2	1	no	—
21	#14268655	C₂×C₄	1	1	yes	characteristic
22	#14268655	D₄	1	1	yes	characteristic
23	#14268655	Q₈	1	1	yes	characteristic
24	#14268655	C₂³	1	1	yes	characteristic
25	C₂×#11744753	D₄	1	1	yes	characteristic
26	C₂×#11744753	Q₈	1	1	yes	characteristic
27	C₄⋊#90673	—	2	1	no	—
28	C₂²⋊#90673	—	2	1	no	—
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	M₁₆	1	1	yes	characteristic
36	C₂×#90673	—	2	2	no	—
37	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₁₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
41	C₄×D₁₁₃	M₁₆	1	1	yes	characteristic
42	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
43	#11744753	C₄⋊C₄	1	1	yes	characteristic
44	#11744753	C₄⋊C₄	1	1	yes	characteristic
45	#11744753	C₂×D₄	1	1	yes	characteristic
46	#11744753	C₂×Q₈	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	—	4	1	no	—
51	C₂×D₁₁₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×D₁₁₃	C₂×M₁₆	1	1	yes	characteristic
53	#90673	—	4	2	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×C₄⋊C₄	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	—	2	1	no	—
62	#2153	—	113	1	no	maximal, Sylow
63	C₁₁₃	#2153	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₃₂	—	113	1	no	—
65	C₂×M₃₂	—	113	1	no	—
66	#405	—	113	1	no	—
67	#364	—	113	2	no	—
68	#364	—	113	2	no	—
69	C₂×M₁₆	—	113	1	no	—
70	C₂²×C₈	—	113	1	no	—
71	#117	—	113	1	no	—
72	M₃₂	—	113	1	no	—
73	M₃₂	—	113	2	no	—
74	M₃₂	—	113	1	no	—
75	M₃₂	—	226	1	no	—
76	M₃₂	—	226	1	no	—
77	C₂×C₁₆	—	226	1	no	—
78	C₂×C₁₆	—	226	1	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	—	226	1	no	—
93	C₂²⋊C₄	—	226	1	no	—
94	C₄²	—	113	2	no	—
95	C₁₆	—	226	2	no	—
96	C₁₆	—	226	1	no	—
97	C₁₆	—	226	1	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	—	226	2	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	—	452	1	no	—
115	C₂²	C₄⋊#11744762	1	1	yes	characteristic
116	C₄	—	113	1	no	—
117	C₄	—	113	1	no	—
118	C₄	—	452	2	no	—
119	C₄	C₄⋊#11744762	1	1	yes	characteristic
120	C₄	C₂²×#11744762	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	226	1	no	—
123	C₂	—	226	1	no	—
124	C₂	C₂×C₄⋊#11744762	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#11744762
commutator	C₄×C₁₁₃	C₂²×C₈
Frattini	C₄	C₂²×#11744762
Fitting	M₁₆×C₁₁₃	C₈
socle	C₂×C₁₁₃	C₂×C₄⋊C₈
prime radical	C₂²×D₁₁₃	M₁₆
cosocle	#14268655	C₂³
hypercenter	M₁₆	#11744762
hypocenter	C₁₁₃	#2153
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #20299703	C₂	semidirect
4. #20299702	C₂	semidirect
63. C₁₁₃	#2153	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813719	2
#20299717	2
#20299703	2
#20299702	2
#20299564	2
#2153	113