Finite Group Databasebeta

Subgroups — #28406979

Order 14464.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#20299645	C₂	1	1	yes	maximal, characteristic
2	#20299638	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299523	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299517	C₂	1	1	yes	maximal, characteristic, complemented
5	#20299514	C₂	1	1	yes	maximal, characteristic
6	#20299488	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299037	C₂	1	1	yes	maximal, characteristic
8	#16813709	—	2	1	no	—
9	Q₈×#90673	—	2	1	no	—
10	#16813697	C₂²	1	1	yes	characteristic
11	#16813692	C₂²	1	1	yes	characteristic
12	#16813690	C₂²	1	1	yes	characteristic
13	#16813686	—	2	1	no	—
14	#16813685	—	2	1	no	—
15	#16813676	C₂²	1	1	yes	characteristic
16	#16813641	C₂²	1	1	yes	characteristic
17	Q₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
18	Dic₉₀₄	C₄	1	1	yes	characteristic, complemented
19	#16813568	C₄	1	1	yes	characteristic, complemented
20	#16813566	C₄	1	1	yes	characteristic, complemented
21	#16813561	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₁₃	C₄	1	1	yes	characteristic, complemented
23	#14268657	—	2	1	no	—
24	#14268656	—	4	1	no	—
25	#14268655	D₄	1	1	yes	characteristic
26	C₂×#11744753	—	4	1	no	—
27	C₄⋊#90673	—	2	1	no	—
28	C₄⋊#90673	—	4	1	no	—
29	C₄×#90673	—	4	1	no	—
30	C₄×#90673	D₄	1	1	yes	characteristic
31	#14268650	—	2	1	no	—
32	#14268646	—	2	1	no	—
33	Q₈×D₁₁₃	—	2	1	no	—
34	Dic₄₅₂	C₂×C₄	1	1	yes	characteristic
35	Dic₄₅₂	C₂×C₄	1	1	yes	characteristic
36	#14268637	—	2	1	no	—
37	#14268635	—	2	1	no	—
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	C₂×#90673	—	4	1	no	—
45	Q₈×C₁₁₃	—	2	1	no	—
46	D₄×C₁₁₃	—	2	1	no	—
47	C₂²⋊D₁₁₃	—	4	1	no	—
48	C₂×Dic₁₁₃	—	4	1	no	—
49	C₄×D₁₁₃	—	4	1	no	—
50	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
51	Dic₂₂₆	—	2	1	no	—
52	Dic₂₂₆	—	2	1	no	—
53	Dic₂₂₆	—	4	1	no	—
54	#11744753	—	2	1	no	—
55	#11744753	—	4	1	no	—
56	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
57	#11744751	#41	1	1	yes	characteristic
58	C₂²×C₁₁₃	—	4	1	no	—
59	C₂×D₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#90673	—	4	1	no	—
61	#90673	—	8	1	no	—
62	C₄×C₁₁₃	—	4	1	no	—
63	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
64	Dic₁₁₃	—	4	1	no	—
65	Dic₁₁₃	—	4	1	no	—
66	Dic₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₁₁₃	—	8	1	no	—
68	C₂×C₁₁₃	#442	1	1	yes	characteristic
69	D₁₁₃	—	2	1	no	—
70	#2180	—	113	1	no	maximal, Sylow
71	C₁₁₃	#2180	1	1	yes	characteristic, complemented, Sylow
72	#510	—	113	1	no	—
73	#444	—	113	1	no	—
74	#441	—	113	1	no	—
75	#365	—	113	1	no	—
76	#362	—	113	1	no	—
77	#360	—	113	1	no	—
78	#347	—	113	1	no	—
79	#135	—	113	1	no	—
80	C₂×Q₁₆	—	113	1	no	—
81	#131	—	226	1	no	—
82	C₄×Q₈	—	226	1	no	—
83	Q₃₂	—	113	1	no	—
84	Q₃₂	—	113	1	no	—
85	SD₃₂	—	113	1	no	—
86	SD₃₂	#90673	1	1	yes	characteristic, complemented
87	M₃₂	—	113	1	no	—
88	M₃₂	—	113	1	no	—
89	#108	—	113	1	no	—
90	#106	—	113	1	no	—
91	#104	—	226	1	no	—
92	#103	—	226	1	no	—
93	#97	—	113	1	no	—
94	#41	—	226	1	no	—
95	C₂×Q₈	—	226	1	no	—
96	Q₁₆	—	113	1	no	—
97	Q₁₆	—	113	1	no	—
98	Q₁₆	—	226	1	no	—
99	Q₁₆	C₂×#90673	1	1	yes	characteristic
100	SD₁₆	—	226	1	no	—
101	D₈	C₂×#90673	1	1	yes	characteristic
102	M₁₆	—	452	1	no	—
103	M₁₆	—	226	1	no	—
104	C₂×C₈	—	113	1	no	—
105	C₂×C₈	—	113	1	no	—
106	C₂×C₈	—	452	1	no	—
107	C₄⋊C₄	—	452	1	no	—
108	C₄⋊C₄	—	226	1	no	—
109	C₄²	—	113	1	no	—
110	C₄²	—	452	1	no	—
111	C₁₆	—	113	1	no	—
112	C₁₆	—	226	1	no	—
113	C₁₆	C₂×#90673	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	452	1	no	—
116	Q₈	—	226	1	no	—
117	Q₈	—	226	1	no	—
118	D₄	—	2	1	no	—
119	D₄	—	452	1	no	—
120	C₂×C₄	—	113	1	no	—
121	C₂×C₄	—	452	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	—	452	1	no	—
127	C₈	—	226	1	no	—
128	C₈	C₂²×#90673	1	1	yes	characteristic
129	C₂²	—	4	1	no	—
130	C₂²	—	113	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	113	1	no	—
133	C₄	—	904	1	no	—
134	C₄	—	452	1	no	—
135	C₄	—	452	1	no	—
136	C₄	—	452	1	no	—
137	C₄	D₄×#90673	1	1	yes	characteristic
138	C₂	—	8	1	no	—
139	C₂	—	226	1	no	—
140	C₂	#20299639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20299639
commutator	C₈×C₁₁₃	C₂²×C₄
Frattini	C₈	C₂²×#90673
Fitting	SD₃₂×C₁₁₃	C₄
socle	C₂×C₁₁₃	#442
prime radical	#16813641	C₂²
cosocle	C₈×D₁₁₃	C₂³
hypercenter	SD₃₂	#90673
hypocenter	C₁₁₃	#2180
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
2. #20299638	C₂	semidirect
3. #20299523	C₂	semidirect
4. #20299517	C₂	semidirect
6. #20299488	C₂	semidirect
18. Dic₉₀₄	C₄	semidirect
19. #16813568	C₄	semidirect
20. #16813566	C₄	semidirect
22. SD₃₂×C₁₁₃	C₄	semidirect
71. C₁₁₃	#2180	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
#20299645	2
#20299638	2
#20299523	2
#20299517	2
#20299514	2
#20299488	2
#20299037	2
#2180	113