Finite Group Databasebeta

Subgroups — #28406125

Order 14464.

180 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 92 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	#20299614	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11744753	C₂	1	1	yes	maximal, characteristic
3	#20299461	C₂	1	2	yes	maximal, complemented
4	#20299459	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11744753	—	2	1	no	—
7	C₄⋊#11744753	—	2	1	no	—
8	C₄⋊#11744753	C₂²	1	2	yes	—
9	C₄⋊#11744753	C₂²	1	1	yes	characteristic
10	C₄⋊#11744753	C₂²	1	1	yes	characteristic
11	C₂²⋊#11744753	—	2	1	no	—
12	C₄×#11744753	—	2	1	no	—
13	C₂×#14268631	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
15	#16813532	C₄	1	2	yes	—
16	C₈×Dic₁₁₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₁₁₃	C₄	1	1	yes	characteristic
18	C₂×#11744753	—	2	1	no	—
19	C₂×#11744753	—	2	1	no	—
20	C₂×#11744753	—	2	2	no	—
21	C₂×#11744753	—	4	2	no	—
22	C₂²×Dic₁₁₃	D₄	1	1	yes	characteristic
23	C₂²×Dic₁₁₃	Q₈	1	1	yes	characteristic
24	#14268631	—	2	2	no	—
25	C₂×#11744751	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
28	C₄×Dic₁₁₃	Q₈	1	1	yes	characteristic
29	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
31	M₁₆×C₁₁₃	C₈	1	4	yes	complemented
32	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
33	C₂³×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₁₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₁₃	M₁₆	1	2	yes	—
37	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₁₃	—	2	1	no	—
39	C₂×Dic₁₁₃	—	2	1	no	—
40	C₂×Dic₁₁₃	—	4	1	no	—
41	C₂×Dic₁₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₁₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₁₁₃	C₂×Q₈	1	1	yes	characteristic
45	#11744753	—	4	1	no	—
46	#11744753	—	4	2	no	—
47	#11744753	—	4	1	no	—
48	C₈×C₁₁₃	C₂×C₈	1	4	yes	complemented
49	#11744751	—	2	2	no	—
50	C₂²×C₁₁₃	—	2	1	no	—
51	C₂²×C₁₁₃	—	2	1	no	—
52	C₂²×C₁₁₃	C₄⋊C₈	1	2	yes	—
53	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₁₃	C₄⋊C₈	1	2	yes	—
55	C₄×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
57	Dic₁₁₃	—	2	1	no	—
58	Dic₁₁₃	—	2	1	no	—
59	Dic₁₁₃	—	2	1	no	—
60	Dic₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₁₁₃	#137	1	1	yes	characteristic
62	C₂×C₁₁₃	—	2	2	no	—
63	C₂×C₁₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₁₃	#428	1	1	yes	characteristic
65	C₂×C₁₁₃	#430	1	1	yes	characteristic
66	#1565	—	113	1	no	maximal, Sylow
67	C₁₁₃	#1565	1	1	yes	characteristic, complemented, Sylow
68	#431	—	113	1	no	—
69	C₂×C₄⋊C₈	—	113	1	no	—
70	C₄×M₁₆	—	113	1	no	—
71	#335	—	113	2	no	—
72	#334	—	113	2	no	—
73	C₂×M₁₆	—	113	1	no	—
74	C₂×M₁₆	#90673	1	1	yes	characteristic
75	C₂²×C₈	—	226	1	no	—
76	C₂×C₄²	—	113	1	no	—
77	C₄⋊C₈	—	113	1	no	—
78	C₄⋊C₈	—	113	2	no	—
79	C₄⋊C₈	—	113	1	no	—
80	C₄⋊C₈	—	226	1	no	—
81	C₂²⋊C₈	—	226	1	no	—
82	#97	—	113	2	no	—
83	C₄×C₈	—	113	2	no	—
84	C₄×C₈	—	226	1	no	—
85	C₂²×C₄	—	113	1	no	—
86	C₂²×C₄	—	113	1	no	—
87	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
88	M₁₆	—	226	2	no	—
89	M₁₆	#11744753	1	4	yes	complemented
90	C₂×C₈	—	113	2	no	—
91	C₂×C₈	—	452	2	no	—
92	C₂×C₈	—	226	1	no	—
93	C₂×C₈	—	226	1	no	—
94	C₂×C₈	—	226	2	no	—
95	C₂×C₈	C₂×#90673	1	2	yes	—
96	C₄²	—	113	1	no	—
97	C₄²	—	113	1	no	—
98	C₄²	—	113	1	no	—
99	C₄²	—	113	1	no	—
100	C₂³	C₄⋊#90673	1	1	yes	characteristic
101	C₂×C₄	—	113	1	no	—
102	C₂×C₄	—	113	1	no	—
103	C₂×C₄	—	113	1	no	—
104	C₂×C₄	—	113	1	no	—
105	C₂×C₄	—	452	1	no	—
106	C₂×C₄	—	226	1	no	—
107	C₂×C₄	—	226	1	no	—
108	C₂×C₄	C₄⋊#90673	1	1	yes	characteristic
109	C₂×C₄	C₂×#11744753	1	2	yes	—
110	C₂×C₄	#14268657	1	2	yes	—
111	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
112	C₈	—	452	2	no	—
113	C₈	—	452	1	no	—
114	C₈	—	452	1	no	—
115	C₈	—	226	2	no	—
116	C₈	C₂×#11744753	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊#11744753	1	2	yes	—
120	C₂²	C₂×C₄⋊#90673	1	1	yes	characteristic
121	C₄	—	113	1	no	—
122	C₄	—	113	1	no	—
123	C₄	—	226	1	no	—
124	C₄	—	226	1	no	—
125	C₄	—	226	1	no	—
126	C₄	C₄⋊#11744753	1	2	yes	—
127	C₄	C₂²×#11744753	1	1	yes	characteristic
128	C₄	C₂×#14268657	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊#11744753	1	1	yes	characteristic
131	C₂	#20299609	1	1	yes	characteristic
132	C₂	#20299613	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#90673
commutator	C₄×C₁₁₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#90673
Fitting	C₂×M₁₆×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₁₃	C₂³
hypercenter	C₂×M₁₆	#90673
hypocenter	C₁₁₃	#1565
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20299461	C₂	semidirect
4. #20299459	C₂	semidirect
31. M₁₆×C₁₁₃	C₈	semidirect
48. C₈×C₁₁₃	C₂×C₈	semidirect
67. C₁₁₃	#1565	semidirect
89. M₁₆	#11744753	semidirect
116. C₈	C₂×#11744753	semidirect

Maximal subgroups

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

subgroup	index
#20299614	2
C₂×C₄⋊#11744753	2
#20299461	2
#20299459	2
M₁₆×Dic₁₁₃	2
#1565	113