Finite Group Databasebeta

Subgroups — #28392814

Order 14464.

152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20298999	C₂	1	1	yes	maximal, characteristic, complemented
2	#20298973	C₂	1	1	yes	maximal, characteristic
3	#20298400	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298399	C₂	1	1	yes	maximal, characteristic, complemented
5	#20298398	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298373	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₉₀₄	C₂²	1	1	yes	characteristic
9	C₂×#14268632	—	2	1	no	—
10	#16813618	—	2	1	no	—
11	#16813613	—	2	1	no	—
12	C₂×#14268622	C₂²	1	1	yes	characteristic
13	#16813556	C₂²	1	1	yes	characteristic
14	#16813553	—	2	1	no	—
15	C₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
18	#107×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×D₄₅₂	—	2	1	no	—
20	C₂×C₂²⋊D₁₁₃	—	4	1	no	—
21	C₂×C₂²⋊D₁₁₃	—	4	1	no	—
22	D₉₀₄	—	2	1	no	—
23	D₉₀₄	—	2	1	no	—
24	#14268632	—	4	2	no	—
25	C₂×#11744751	D₄	1	1	yes	characteristic
26	C₄⋊Dic₁₁₃	—	2	1	no	—
27	C₄⋊Dic₁₁₃	—	4	1	no	—
28	#14268626	—	4	1	no	—
29	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
30	#14268622	—	2	1	no	—
31	C₂×D₄×C₁₁₃	—	2	1	no	—
32	D₈×C₁₁₃	—	2	1	no	—
33	D₈×C₁₁₃	—	2	1	no	—
34	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₁₃	—	2	1	no	—
36	C₁₆×C₁₁₃	—	2	1	no	—
37	C₂³×C₁₁₃	—	4	1	no	—
38	C₂²×D₁₁₃	—	4	1	no	—
39	D₄×C₁₁₃	—	4	1	no	—
40	D₄×C₁₁₃	—	4	1	no	—
41	C₂×C₄×C₁₁₃	—	4	1	no	—
42	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²⋊D₁₁₃	—	8	2	no	—
44	C₂²⋊D₁₁₃	—	8	2	no	—
45	C₂×Dic₁₁₃	—	4	1	no	—
46	C₂×Dic₁₁₃	D₈	1	1	yes	characteristic
47	C₂×Dic₁₁₃	D₈	1	1	yes	characteristic
48	D₄₅₂	—	4	1	no	—
49	D₄₅₂	—	4	1	no	—
50	C₈×C₁₁₃	#41	1	1	yes	characteristic
51	C₈×C₁₁₃	#41	1	1	yes	characteristic
52	#11744751	SD₁₆	1	2	yes	—
53	C₂²×C₁₁₃	—	8	1	no	—
54	C₂²×C₁₁₃	—	8	1	no	—
55	C₂²×C₁₁₃	—	8	1	no	—
56	C₂²×C₁₁₃	C₂×D₈	1	1	yes	characteristic
57	C₂×D₁₁₃	—	8	1	no	—
58	C₂×D₁₁₃	—	8	1	no	—
59	C₂×D₁₁₃	—	8	1	no	—
60	C₄×C₁₁₃	—	8	1	no	—
61	C₄×C₁₁₃	#124	1	1	yes	characteristic
62	C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₁₁₃	—	2	1	no	—
64	Dic₁₁₃	—	2	1	no	—
65	Dic₁₁₃	—	8	1	no	—
66	C₂×C₁₁₃	—	16	1	no	—
67	C₂×C₁₁₃	#486	1	1	yes	characteristic
68	C₂×C₁₁₃	#509	1	1	yes	characteristic
69	C₂×C₁₁₃	#509	1	1	yes	characteristic
70	D₁₁₃	—	16	1	no	—
71	#2244	—	113	1	no	maximal, Sylow
72	C₁₁₃	#2244	1	1	yes	characteristic, complemented, Sylow
73	#499	—	113	1	no	—
74	#493	—	113	1	no	—
75	#357	—	113	1	no	—
76	#357	—	113	1	no	—
77	#357	—	113	1	no	—
78	#357	D₁₁₃	1	1	yes	characteristic, complemented
79	#346	—	113	1	no	—
80	C₂×D₈	—	113	1	no	—
81	C₂×D₈	—	226	1	no	—
82	C₂×D₈	C₂×D₁₁₃	1	1	yes	characteristic
83	#127	—	226	1	no	—
84	#125	—	226	1	no	—
85	C₂×C₁₆	—	113	1	no	—
86	C₂×C₁₆	C₂×D₁₁₃	1	1	yes	characteristic
87	#107	—	113	1	no	—
88	#107	C₂×D₁₁₃	1	1	yes	characteristic
89	#106	—	226	1	no	—
90	C₄×C₈	—	113	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	452	1	no	—
93	C₂×D₄	—	452	1	no	—
94	C₂×D₄	—	226	1	no	—
95	D₈	—	2	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	452	2	no	—
98	D₈	—	226	1	no	—
99	D₈	—	226	1	no	—
100	C₂×C₈	—	113	1	no	—
101	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	452	1	no	—
104	C₄⋊C₄	—	452	1	no	—
105	C₄⋊C₄	—	226	1	no	—
106	C₄²	—	113	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	226	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	452	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	904	2	no	—
114	D₄	—	904	2	no	—
115	D₄	—	452	1	no	—
116	D₄	—	452	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	113	1	no	—
119	C₂×C₄	—	113	1	no	—
120	C₂×C₄	—	452	1	no	—
121	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
122	C₈	—	113	2	no	—
123	C₈	#14268646	1	1	yes	characteristic
124	C₈	#14268647	1	1	yes	characteristic
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	904	1	no	—
129	C₂²	—	904	1	no	—
130	C₂²	—	904	1	no	—
131	C₂²	D₈×D₁₁₃	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	904	1	no	—
134	C₄	—	226	1	no	—
135	C₄	—	226	1	no	—
136	C₄	#16813608	1	1	yes	characteristic
137	C₄	SD₁₆×D₁₁₃	1	1	yes	characteristic
138	C₂	—	16	1	no	—
139	C₂	—	1808	1	no	—
140	C₂	#20298943	1	1	yes	characteristic
141	C₂	#20299032	1	1	yes	characteristic
142	C₂	#20299035	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₁₃
commutator, cosocle	C₂×C₈×C₁₁₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₁₃
Fitting	#357×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₈
prime radical	#20298973	C₂
hypercenter	#357	D₁₁₃
hypocenter	C₁₁₃	#2244
LCS	G ▷ C₂×C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
1. #20298999	C₂	semidirect
3. #20298400	C₂	semidirect
4. #20298399	C₂	semidirect
5. #20298398	C₂	semidirect
6. #20298373	C₂	semidirect
7. #357×C₁₁₃	C₂	semidirect
72. C₁₁₃	#2244	semidirect
78. #357	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20298999	2
#20298973	2
#20298400	2
#20298399	2
#20298398	2
#20298373	2
#357×C₁₁₃	2
#2244	113