Finite Group Databasebeta

Subgroups — #28407862

Order 14464.

160 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₄⋊#14268661	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14268661	C₂	1	2	yes	maximal
3	C₄×#14268661	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11744762	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268661	C₂²	1	4	yes	—
6	C₂²×#11744762	C₄	1	1	yes	characteristic
7	C₂²×#11744762	C₂²	1	1	yes	characteristic
8	C₄×#11744762	C₄	1	1	yes	characteristic
9	C₄×#11744762	C₄	1	1	yes	characteristic
10	C₄×#11744762	C₂²	1	1	yes	characteristic
11	C₄×#11744762	C₂²	1	1	yes	characteristic
12	C₂×C₄×#90673	C₄	1	1	yes	characteristic
13	#14268661	—	2	4	no	—
14	C₂×#11744762	—	2	2	no	—
15	C₂×#11744762	—	2	2	no	—
16	C₂×#11744762	C₂×C₄	1	1	yes	characteristic
17	C₂×#11744762	C₂×C₄	1	1	yes	characteristic
18	C₂×#11744762	C₂×C₄	1	1	yes	characteristic
19	C₂×#11744762	C₂³	1	1	yes	characteristic
20	C₂²×#90673	C₈	1	1	yes	characteristic
21	C₂²×#90673	C₂×C₄	1	1	yes	characteristic
22	C₄×#90673	C₈	1	1	yes	characteristic
23	C₄×#90673	C₈	1	1	yes	characteristic
24	C₄×#90673	C₂×C₄	1	1	yes	characteristic
25	C₄×#90673	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₁₃	C₈	1	1	yes	characteristic
27	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂×#90673	—	2	2	no	—
29	C₂×#90673	—	2	1	no	—
30	C₂×#90673	—	2	1	no	—
31	C₂×#90673	C₂×C₈	1	1	yes	characteristic
32	C₂×#90673	C₂×C₈	1	1	yes	characteristic
33	C₂×#90673	C₂×C₈	1	1	yes	characteristic
34	C₂×#90673	C₂²×C₄	1	1	yes	characteristic
35	#11744762	—	2	1	no	—
36	#11744762	—	2	1	no	—
37	#11744762	M₁₆	1	2	yes	—
38	#11744762	#41	1	2	yes	—
39	C₂×C₄×C₁₁₃	C₁₆	1	1	yes	characteristic, complemented
40	C₂×Dic₁₁₃	C₁₆	1	1	yes	characteristic, complemented
41	C₄×D₁₁₃	C₁₆	1	1	yes	characteristic, complemented
42	C₄×D₁₁₃	C₁₆	1	1	yes	characteristic, complemented
43	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
44	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
46	C₂×D₁₁₃	—	2	1	no	—
47	C₂×D₁₁₃	—	2	1	no	—
48	C₂×D₁₁₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×D₁₁₃	C₂²×C₈	1	1	yes	characteristic
50	#90673	—	2	1	no	—
51	#90673	—	2	1	no	—
52	#90673	M₃₂	1	2	yes	—
53	#90673	#117	1	1	yes	characteristic
54	#90673	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
56	C₄×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
57	Dic₁₁₃	C₂×C₁₆	1	1	yes	characteristic
58	Dic₁₁₃	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₁₁₃	—	2	1	no	—
60	C₂×C₁₁₃	C₂²×C₁₆	1	1	yes	characteristic
61	D₁₁₃	—	2	1	no	—
62	D₁₁₃	#431	1	1	yes	characteristic
63	D₁₁₃	C₂×M₃₂	1	1	yes	characteristic
64	#2162	—	113	1	no	maximal, Sylow
65	C₁₁₃	#2162	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄×C₈	—	113	1	no	—
67	C₄⋊C₁₆	—	113	2	no	—
68	C₂²⋊C₁₆	—	113	2	no	—
69	C₄×C₁₆	—	113	2	no	—
70	C₂²×C₈	—	113	1	no	—
71	C₂²×C₈	—	113	1	no	—
72	C₂×C₄²	—	113	1	no	—
73	C₂×C₁₆	—	113	4	no	—
74	C₄×C₈	—	113	1	no	—
75	C₄×C₈	—	113	1	no	—
76	C₄×C₈	—	113	1	no	—
77	C₄×C₈	—	113	1	no	—
78	C₂²×C₄	—	113	1	no	—
79	C₂²×C₄	—	113	1	no	—
80	C₂²×C₄	—	113	1	no	—
81	C₂×C₈	—	113	1	no	—
82	C₂×C₈	—	113	1	no	—
83	C₂×C₈	—	113	1	no	—
84	C₂×C₈	—	113	1	no	—
85	C₂×C₈	—	226	2	no	—
86	C₂×C₈	—	226	2	no	—
87	C₄²	—	113	1	no	—
88	C₄²	—	113	1	no	—
89	C₄²	—	113	1	no	—
90	C₄²	—	113	1	no	—
91	C₁₆	—	226	4	no	—
92	C₂³	—	113	1	no	—
93	C₂×C₄	—	113	1	no	—
94	C₂×C₄	—	113	1	no	—
95	C₂×C₄	—	113	1	no	—
96	C₂×C₄	—	113	1	no	—
97	C₂×C₄	—	113	1	no	—
98	C₂×C₄	—	113	1	no	—
99	C₂×C₄	—	113	1	no	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	113	1	no	—
102	C₂×C₄	—	226	1	no	—
103	C₂×C₄	—	226	1	no	—
104	C₂×C₄	—	226	2	no	—
105	C₂×C₄	#14268661	1	1	yes	characteristic, complemented
106	C₈	—	113	2	no	—
107	C₈	—	113	2	no	—
108	C₈	—	226	1	no	—
109	C₈	—	226	1	no	—
110	C₂²	—	113	1	no	—
111	C₂²	—	113	1	no	—
112	C₂²	—	226	1	no	—
113	C₂²	—	226	1	no	—
114	C₂²	C₂×#14268661	1	1	yes	characteristic
115	C₄	—	113	1	no	—
116	C₄	—	113	1	no	—
117	C₄	—	113	2	no	—
118	C₄	—	113	1	no	—
119	C₄	—	113	1	no	—
120	C₄	—	226	1	no	—
121	C₄	—	226	1	no	—
122	C₄	C₂×#14268661	1	1	yes	characteristic
123	C₄	C₂×#14268661	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	113	1	no	—
126	C₂	—	113	1	no	—
127	C₂	—	226	1	no	—
128	C₂	C₂²×#14268661	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14268661
commutator, socle	C₂×C₁₁₃	C₂²×C₁₆
Frattini	C₂	C₂²×#14268661
Fitting	C₂×C₄×C₁₁₃	C₁₆
prime radical	C₂²×D₁₁₃	C₂×C₈
cosocle	C₂×#11744762	C₂³
hypercenter	C₂×C₄	#14268661
hypocenter	C₁₁₃	#2162
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#14268661	C₂	semidirect
3. C₄×#14268661	C₂	semidirect
39. C₂×C₄×C₁₁₃	C₁₆	semidirect
40. C₂×Dic₁₁₃	C₁₆	semidirect
41. C₄×D₁₁₃	C₁₆	semidirect
42. C₄×D₁₁₃	C₁₆	semidirect
65. C₁₁₃	#2162	semidirect
105. C₂×C₄	#14268661	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14268661	2
C₂²⋊#14268661	2
C₄×#14268661	2
C₂×C₄×#11744762	2
#2162	113