Finite Group Databasebeta

Subgroups — #28406974

Order 14464.

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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	#20299642	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299640	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299527	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299513	C₂	1	1	yes	maximal, characteristic, complemented
5	#20299512	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#90673	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299029	C₂	1	1	yes	maximal, characteristic
8	#16813710	—	2	1	no	—
9	#16813709	—	2	1	no	—
10	#16813695	C₂²	1	1	yes	characteristic
11	#16813693	C₂²	1	1	yes	characteristic
12	#16813693	C₂²	1	1	yes	characteristic
13	#16813687	—	2	1	no	—
14	#16813686	—	2	1	no	—
15	C₈×#90673	C₂²	1	1	yes	characteristic
16	#16813645	C₂²	1	1	yes	characteristic
17	#16813641	C₂²	1	1	yes	characteristic
18	#16813568	C₄	1	1	yes	characteristic, complemented
19	#16813567	C₄	1	1	yes	characteristic, complemented
20	#16813563	C₄	1	1	yes	characteristic, complemented
21	C₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₁₃	C₄	1	1	yes	characteristic, complemented
23	#14268657	—	2	1	no	—
24	#14268657	—	2	1	no	—
25	#14268657	—	4	1	no	—
26	#14268656	—	4	1	no	—
27	#14268655	—	4	1	no	—
28	#14268655	D₄	1	1	yes	characteristic
29	C₂×#11744753	—	4	1	no	—
30	C₄×#90673	D₄	1	1	yes	characteristic
31	#14268650	—	2	1	no	—
32	#14268648	—	2	1	no	—
33	#14268646	—	2	1	no	—
34	Dic₄₅₂	C₂×C₄	1	1	yes	characteristic
35	#14268635	—	2	1	no	—
36	#14268634	—	2	1	no	—
37	D₉₀₄	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
39	#14268622	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic
41	D₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#90673	—	2	1	no	—
44	Q₈×C₁₁₃	—	2	1	no	—
45	D₄×C₁₁₃	—	2	1	no	—
46	C₂²⋊D₁₁₃	—	4	1	no	—
47	C₂×Dic₁₁₃	—	4	1	no	—
48	D₄₅₂	—	2	1	no	—
49	D₄₅₂	—	4	1	no	—
50	C₄×D₁₁₃	—	4	1	no	—
51	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
52	Dic₂₂₆	—	2	1	no	—
53	#11744753	—	4	1	no	—
54	#11744753	—	4	1	no	—
55	#11744753	D₈	1	1	yes	characteristic
56	#11744753	D₈	1	1	yes	characteristic
57	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
58	#11744751	#41	1	1	yes	characteristic
59	C₂²×C₁₁₃	—	4	1	no	—
60	C₂×D₁₁₃	—	4	1	no	—
61	C₂×D₁₁₃	#135	1	1	yes	characteristic
62	#90673	—	2	2	no	—
63	C₄×C₁₁₃	—	4	1	no	—
64	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
65	Dic₁₁₃	—	4	1	no	—
66	Dic₁₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₁₁₃	—	8	1	no	—
68	C₂×C₁₁₃	C₄×D₈	1	1	yes	characteristic
69	D₁₁₃	—	2	1	no	—
70	D₁₁₃	—	8	1	no	—
71	#2178	—	113	1	no	maximal, Sylow
72	C₁₁₃	#2178	1	1	yes	characteristic, complemented, Sylow
73	#508	—	113	1	no	—
74	#443	—	113	1	no	—
75	#443	—	113	1	no	—
76	#368	—	113	1	no	—
77	#359	—	113	1	no	—
78	#359	—	113	1	no	—
79	C₄×C₁₆	—	113	1	no	—
80	#135	—	113	1	no	—
81	#135	—	113	1	no	—
82	#131	—	226	1	no	—
83	#131	—	226	1	no	—
84	Q₃₂	—	113	1	no	—
85	SD₃₂	—	113	1	no	—
86	SD₃₂	#90673	1	1	yes	characteristic, complemented
87	D₁₆	—	113	1	no	—
88	C₂×C₁₆	—	113	1	no	—
89	C₂×C₁₆	—	113	1	no	—
90	#108	—	113	1	no	—
91	#108	—	113	1	no	—
92	#104	—	226	1	no	—
93	#104	—	226	1	no	—
94	C₄×C₈	—	113	1	no	—
95	#41	—	226	1	no	—
96	#41	—	226	1	no	—
97	Q₁₆	—	113	1	no	—
98	Q₁₆	C₂×#90673	1	1	yes	characteristic
99	SD₁₆	—	226	1	no	—
100	SD₁₆	—	226	1	no	—
101	D₈	—	113	1	no	—
102	D₈	C₂×#90673	1	1	yes	characteristic
103	M₁₆	—	452	1	no	—
104	M₁₆	—	452	1	no	—
105	M₁₆	—	226	1	no	—
106	M₁₆	—	226	1	no	—
107	C₂×C₈	—	113	1	no	—
108	C₂×C₈	—	113	1	no	—
109	C₂×C₈	—	452	1	no	—
110	C₂×C₈	—	452	1	no	—
111	C₄²	—	113	1	no	—
112	C₁₆	—	113	1	no	—
113	C₁₆	—	226	1	no	—
114	C₁₆	C₂×#90673	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	226	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	452	1	no	—
119	D₄	—	452	1	no	—
120	D₄	—	226	1	no	—
121	C₂×C₄	—	113	1	no	—
122	C₂×C₄	—	452	1	no	—
123	C₂×C₄	—	452	1	no	—
124	C₂×C₄	—	226	1	no	—
125	C₈	—	113	1	no	—
126	C₈	—	113	1	no	—
127	C₈	—	113	1	no	—
128	C₈	—	452	1	no	—
129	C₈	—	452	1	no	—
130	C₈	C₂²×#90673	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	113	1	no	—
133	C₂²	—	452	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	113	1	no	—
136	C₄	—	452	1	no	—
137	C₄	—	226	2	no	—
138	C₄	D₄×#90673	1	1	yes	characteristic
139	C₂	—	8	1	no	—
140	C₂	—	904	1	no	—
141	C₂	—	226	1	no	—
142	C₂	D₈×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#90673
commutator	C₈×C₁₁₃	C₂²×C₄
Frattini	C₈	C₂²×#90673
Fitting	SD₃₂×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₄×D₈
prime radical	#20299029	C₂
cosocle	C₈×D₁₁₃	C₂³
hypercenter	SD₃₂	#90673
hypocenter	C₁₁₃	#2178
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20299642	C₂	semidirect
2. #20299640	C₂	semidirect
3. #20299527	C₂	semidirect
4. #20299513	C₂	semidirect
5. #20299512	C₂	semidirect
6. C₁₆×#90673	C₂	semidirect
18. #16813568	C₄	semidirect
19. #16813567	C₄	semidirect
20. #16813563	C₄	semidirect
22. SD₃₂×C₁₁₃	C₄	semidirect
72. C₁₁₃	#2178	semidirect
86. SD₃₂	#90673	semidirect

Maximal subgroups

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

subgroup	index
#20299642	2
#20299640	2
#20299527	2
#20299513	2
#20299512	2
C₁₆×#90673	2
#20299029	2
#2178	113