Finite Group Databasebeta

Subgroups — #28407699

Order 14464.

180 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20299711	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11744762	C₂	1	1	yes	maximal, characteristic
3	#20299688	C₂	1	2	yes	maximal, complemented
4	#20299687	C₂	1	2	yes	maximal, complemented
5	M₁₆×#90673	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11744762	—	2	1	no	—
7	C₄⋊#11744762	—	2	1	no	—
8	C₄⋊#11744762	C₂²	1	1	yes	characteristic
9	C₄⋊#11744762	C₂²	1	2	yes	—
10	C₄⋊#11744762	C₂²	1	1	yes	characteristic
11	C₂²⋊#11744762	—	2	1	no	—
12	C₄×#11744762	—	2	1	no	—
13	D₁₁₃⋊M₁₆	C₄	1	1	yes	characteristic
14	C₂×C₄×#90673	C₂²	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	C₂×#11744762	—	2	1	no	—
19	C₂×#11744762	—	2	1	no	—
20	C₂×#11744762	—	2	2	no	—
21	C₂×#11744762	—	4	2	no	—
22	C₂²×#90673	D₄	1	1	yes	characteristic
23	C₂²×#90673	Q₈	1	1	yes	characteristic
24	#14268657	—	2	1	no	—
25	#14268656	—	2	1	no	—
26	#14268655	C₂×C₄	1	2	yes	—
27	C₄×#90673	C₂×C₄	1	1	yes	characteristic
28	C₄×#90673	D₄	1	1	yes	characteristic
29	C₄×#90673	Q₈	1	1	yes	characteristic
30	C₄×#90673	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
32	#14268631	C₈	1	1	yes	characteristic, complemented
33	#14268630	C₈	1	2	yes	complemented
34	C₈×D₁₁₃	C₂×C₄	1	2	yes	—
35	M₁₆×C₁₁₃	C₈	1	1	yes	characteristic, complemented
36	C₂²×D₁₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#90673	—	2	1	no	—
38	C₂×#90673	—	2	1	no	—
39	C₂×#90673	—	4	1	no	—
40	C₂×#90673	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#90673	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#90673	C₂×D₄	1	1	yes	characteristic
43	C₂×#90673	C₂×Q₈	1	1	yes	characteristic
44	#11744762	—	4	1	no	—
45	#11744762	—	4	2	no	—
46	#11744762	—	4	1	no	—
47	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₁₁₃	M₁₆	1	1	yes	characteristic
49	C₄×D₁₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
51	C₄×D₁₁₃	M₁₆	1	1	yes	characteristic
52	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
53	#11744753	—	2	2	no	—
54	C₈×C₁₁₃	C₂×C₈	1	2	yes	complemented
55	#11744751	C₂×C₈	1	2	yes	complemented
56	C₂²×C₁₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×D₁₁₃	—	2	1	no	—
58	C₂×D₁₁₃	—	2	1	no	—
59	C₂×D₁₁₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×D₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	#90673	—	2	1	no	—
62	#90673	—	2	1	no	—
63	#90673	—	2	1	no	—
64	#90673	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#90673	#137	1	1	yes	characteristic
66	C₄×C₁₁₃	C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
68	Dic₁₁₃	C₄⋊C₈	1	1	yes	characteristic
69	Dic₁₁₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₁₁₃	—	2	1	no	—
71	C₂×C₁₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	D₁₁₃	—	2	1	no	—
73	D₁₁₃	#428	1	1	yes	characteristic
74	D₁₁₃	#430	1	1	yes	characteristic
75	#1565	—	113	1	no	maximal, Sylow
76	C₁₁₃	#1565	1	1	yes	characteristic, complemented, Sylow
77	#431	—	113	1	no	—
78	C₂×C₄⋊C₈	—	113	1	no	—
79	C₄×M₁₆	—	113	1	no	—
80	#335	—	113	2	no	—
81	#334	—	113	2	no	—
82	C₂×M₁₆	—	113	1	no	—
83	C₂×M₁₆	—	113	1	no	—
84	C₂²×C₈	—	226	1	no	—
85	C₂×C₄²	—	113	1	no	—
86	C₄⋊C₈	—	113	1	no	—
87	C₄⋊C₈	—	113	2	no	—
88	C₄⋊C₈	—	113	1	no	—
89	C₄⋊C₈	—	226	1	no	—
90	C₂²⋊C₈	—	226	1	no	—
91	#97	—	113	2	no	—
92	C₄×C₈	—	113	2	no	—
93	C₄×C₈	—	226	1	no	—
94	C₂²×C₄	—	113	1	no	—
95	C₂²×C₄	—	113	1	no	—
96	C₂²×C₄	—	113	1	no	—
97	M₁₆	—	113	1	no	—
98	M₁₆	—	113	2	no	—
99	M₁₆	—	226	1	no	—
100	M₁₆	—	226	1	no	—
101	M₁₆	#11744762	1	1	yes	characteristic, complemented
102	C₂×C₈	—	113	2	no	—
103	C₂×C₈	—	113	2	no	—
104	C₂×C₈	—	452	2	no	—
105	C₂×C₈	—	226	1	no	—
106	C₂×C₈	—	226	1	no	—
107	C₂×C₈	—	226	2	no	—
108	C₄²	—	113	1	no	—
109	C₄²	—	113	1	no	—
110	C₄²	—	113	1	no	—
111	C₄²	—	113	1	no	—
112	C₂³	—	113	1	no	—
113	C₂×C₄	—	113	1	no	—
114	C₂×C₄	—	113	1	no	—
115	C₂×C₄	—	113	1	no	—
116	C₂×C₄	—	113	1	no	—
117	C₂×C₄	—	113	1	no	—
118	C₂×C₄	—	113	1	no	—
119	C₂×C₄	—	113	1	no	—
120	C₂×C₄	—	113	1	no	—
121	C₂×C₄	—	113	1	no	—
122	C₂×C₄	—	452	1	no	—
123	C₂×C₄	—	226	1	no	—
124	C₂×C₄	—	226	1	no	—
125	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
126	C₈	—	113	2	no	—
127	C₈	—	452	2	no	—
128	C₈	—	452	1	no	—
129	C₈	—	452	1	no	—
130	C₈	—	226	2	no	—
131	C₈	C₂×#11744762	1	2	yes	complemented
132	C₂²	—	113	1	no	—
133	C₂²	—	113	1	no	—
134	C₂²	—	226	1	no	—
135	C₂²	—	226	1	no	—
136	C₂²	C₄⋊#11744762	1	1	yes	characteristic
137	C₄	—	113	1	no	—
138	C₄	—	113	1	no	—
139	C₄	—	113	1	no	—
140	C₄	—	113	1	no	—
141	C₄	—	226	1	no	—
142	C₄	—	226	1	no	—
143	C₄	—	226	1	no	—
144	C₄	C₄⋊#11744762	1	1	yes	characteristic
145	C₄	C₂²×#11744762	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	113	1	no	—
148	C₂	—	113	1	no	—
149	C₂	—	226	1	no	—
150	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	C₄⋊C₄
cosocle	C₄×#90673	C₂³
hypercenter	M₁₆	#11744762
hypocenter	C₁₁₃	#1565
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. #20299688	C₂	semidirect
4. #20299687	C₂	semidirect
32. #14268631	C₈	semidirect
33. #14268630	C₈	semidirect
35. M₁₆×C₁₁₃	C₈	semidirect
54. C₈×C₁₁₃	C₂×C₈	semidirect
55. #11744751	C₂×C₈	semidirect
76. C₁₁₃	#1565	semidirect
101. M₁₆	#11744762	semidirect
131. C₈	C₂×#11744762	semidirect

Maximal subgroups

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

subgroup	index
#20299711	2
C₂×C₄⋊#11744762	2
#20299688	2
#20299687	2
M₁₆×#90673	2
#1565	113