Finite Group Databasebeta

Subgroups — #28390101

Order 14464.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20299001	C₂	1	1	yes	maximal, characteristic
2	#20298894	C₂	1	1	yes	maximal, characteristic
3	#20298890	C₂	1	1	yes	maximal, characteristic
4	#20298725	C₂	1	1	yes	maximal, characteristic
5	#20298348	C₂	1	1	yes	maximal, characteristic
6	#20298346	C₂	1	1	yes	maximal, characteristic
7	#333×C₁₁₃	C₂	1	1	yes	maximal, characteristic
8	#16813622	C₂²	1	1	yes	characteristic
9	C₄×Dic₂₂₆	C₂²	1	1	yes	characteristic
10	#16813556	—	2	1	no	—
11	#16813555	—	2	1	no	—
12	#16813555	—	2	1	no	—
13	#16813555	—	2	1	no	—
14	C₄⋊#11744751	C₂²	1	1	yes	characteristic
15	#16813551	—	2	1	no	—
16	#16813547	—	2	1	no	—
17	#16813545	—	2	1	no	—
18	#16813532	—	2	1	no	—
19	C₄×#11744751	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₂₆	—	2	1	no	—
24	C₂×Dic₂₂₆	D₄	1	1	yes	characteristic
25	C₂×#11744751	—	2	1	no	—
26	C₂×#11744751	D₄	1	1	yes	characteristic
27	C₂×#11744751	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₁₃	—	2	1	no	—
29	C₄⋊Dic₁₁₃	—	2	1	no	—
30	C₄⋊Dic₁₁₃	D₄	1	1	yes	characteristic
31	#14268626	—	2	1	no	—
32	C₄×Dic₁₁₃	—	2	1	no	—
33	C₂×Q₈×C₁₁₃	—	2	1	no	—
34	C₂×C₈×C₁₁₃	—	2	1	no	—
35	C₂×C₈×C₁₁₃	—	2	1	no	—
36	C₄⋊C₄×C₁₁₃	—	2	1	no	—
37	C₄⋊C₄×C₁₁₃	—	2	1	no	—
38	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
39	Q₈×C₁₁₃	—	8	1	no	—
40	C₂×C₄×C₁₁₃	—	4	1	no	—
41	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₁₃	#41	1	1	yes	characteristic
44	C₂×Dic₁₁₃	—	2	1	no	—
45	C₂×Dic₁₁₃	—	2	1	no	—
46	C₂×Dic₁₁₃	—	4	1	no	—
47	Dic₂₂₆	—	2	1	no	—
48	Dic₂₂₆	—	2	1	no	—
49	Dic₂₂₆	—	8	1	no	—
50	C₈×C₁₁₃	—	4	1	no	—
51	C₈×C₁₁₃	—	4	1	no	—
52	#11744751	—	2	1	no	—
53	#11744751	—	4	1	no	—
54	#11744751	Q₁₆	1	2	yes	—
55	C₂²×C₁₁₃	#121	1	1	yes	characteristic
56	C₄×C₁₁₃	—	2	1	no	—
57	C₄×C₁₁₃	—	8	1	no	—
58	C₄×C₁₁₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₁₁₃	#137	1	1	yes	characteristic
60	C₄×C₁₁₃	#137	1	1	yes	characteristic
61	C₄×C₁₁₃	#137	1	1	yes	characteristic
62	Dic₁₁₃	—	4	1	no	—
63	Dic₁₁₃	—	4	1	no	—
64	Dic₁₁₃	—	8	1	no	—
65	C₂×C₁₁₃	#462	1	1	yes	characteristic
66	C₂×C₁₁₃	#470	1	1	yes	characteristic
67	C₂×C₁₁₃	#472	1	1	yes	characteristic
68	#1696	—	113	1	no	maximal, Sylow
69	C₁₁₃	#1696	1	1	yes	characteristic, complemented, Sylow
70	#500	—	113	1	no	—
71	#477	—	113	1	no	—
72	#477	—	113	1	no	—
73	#446	—	113	1	no	—
74	#334	—	113	1	no	—
75	#333	—	113	1	no	—
76	#333	D₁₁₃	1	1	yes	characteristic
77	C₄⋊Q₈	—	113	1	no	—
78	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
79	C₄×Q₈	—	113	1	no	—
80	#107	—	226	1	no	—
81	#107	—	226	1	no	—
82	#107	—	226	1	no	—
83	#107	—	226	1	no	—
84	C₄⋊C₈	—	113	1	no	—
85	C₄⋊C₈	—	226	1	no	—
86	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
88	#103	—	226	1	no	—
89	#103	—	226	1	no	—
90	#97	—	226	1	no	—
91	C₄×C₈	—	113	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	113	1	no	—
94	C₂×Q₈	—	226	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	113	1	no	—
98	C₂×C₈	—	113	1	no	—
99	C₂×C₈	—	226	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	113	1	no	—
103	C₄⋊C₄	—	226	1	no	—
104	C₄⋊C₄	—	226	1	no	—
105	C₄⋊C₄	—	226	1	no	—
106	C₄²	—	226	1	no	—
107	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	904	1	no	—
110	Q₈	—	226	1	no	—
111	Q₈	—	226	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	452	1	no	—
114	C₂×C₄	—	226	1	no	—
115	C₂×C₄	—	226	1	no	—
116	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
117	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
118	C₂×C₄	#14268647	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	113	2	no	—
122	C₈	—	452	1	no	—
123	C₈	—	226	1	no	—
124	C₂²	#16813595	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	904	1	no	—
128	C₄	—	452	1	no	—
129	C₄	—	452	1	no	—
130	C₄	Q₁₆×D₁₁₃	1	1	yes	characteristic
131	C₄	#16813651	1	1	yes	characteristic
132	C₄	#16813651	1	1	yes	characteristic
133	C₄	#16813654	1	1	yes	characteristic
134	C₂	#20298807	1	1	yes	characteristic
135	C₂	#20298855	1	1	yes	characteristic
136	C₂	#20298867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16813595
commutator, cosocle	C₄²×C₁₁₃	C₂³
Frattini	C₄²	C₂²×D₁₁₃
Fitting	#333×C₁₁₃	C₂
socle, prime radical	C₂²×C₁₁₃	#121
hypercenter	#333	D₁₁₃
hypocenter	C₁₁₃	#1696
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
69. C₁₁₃	#1696	semidirect

Maximal subgroups

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

subgroup	index
#20299001	2
#20298894	2
#20298890	2
#20298725	2
#20298348	2
#20298346	2
#333×C₁₁₃	2
#1696	113