Finite Group Databasebeta

Subgroups — #20298613

Order 7232.

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

normal subgroups only

This group has 154 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	#41×D₁₁₃	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813579	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813550	C₂	1	1	yes	maximal, characteristic, complemented
5	#16813549	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813548	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14268648	C₄	1	1	yes	characteristic, complemented
9	#14268647	—	2	1	no	—
10	#14268647	C₂²	1	1	yes	characteristic
11	#14268646	—	2	1	no	—
12	#14268646	C₄	1	1	yes	characteristic, complemented
13	Q₈×D₁₁₃	C₄	1	1	yes	characteristic, complemented
14	D₄×D₁₁₃	—	2	1	no	—
15	D₄×D₁₁₃	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×D₁₁₃	—	2	1	no	—
17	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
18	#14268631	C₂²	1	1	yes	characteristic, complemented
19	#14268630	—	2	1	no	—
20	C₈×D₁₁₃	—	2	1	no	—
21	#14268626	—	2	1	no	—
22	#14268624	—	2	1	no	—
23	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic, complemented
24	#41×C₁₁₃	C₂²	1	1	yes	characteristic
25	M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
26	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₁₁₃	—	2	1	no	—
28	C₂²×D₁₁₃	D₄	1	1	yes	characteristic
29	Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
30	D₄×C₁₁₃	—	2	1	no	—
31	D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₁₃	—	2	1	no	—
33	C₂×C₄×C₁₁₃	—	2	1	no	—
34	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₂²⋊D₁₁₃	—	2	1	no	—
36	C₂²⋊D₁₁₃	—	2	1	no	—
37	C₂²⋊D₁₁₃	—	2	1	no	—
38	C₂×Dic₁₁₃	—	2	1	no	—
39	C₂×Dic₁₁₃	—	2	1	no	—
40	C₂×Dic₁₁₃	D₄	1	1	yes	characteristic
41	D₄₅₂	—	2	1	no	—
42	D₄₅₂	C₂×C₄	1	1	yes	characteristic
43	C₄×D₁₁₃	—	2	1	no	—
44	C₄×D₁₁₃	—	2	1	no	—
45	C₄×D₁₁₃	—	2	1	no	—
46	C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×D₁₁₃	D₄	1	1	yes	characteristic
49	C₄×D₁₁₃	D₄	1	1	yes	characteristic
50	Dic₂₂₆	—	2	1	no	—
51	Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
52	C₈×C₁₁₃	—	2	1	no	—
53	#11744751	—	2	1	no	—
54	C₂²×C₁₁₃	—	2	1	no	—
55	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×D₁₁₃	—	2	1	no	—
57	C₂×D₁₁₃	—	4	1	no	—
58	C₂×D₁₁₃	—	4	2	no	—
59	C₂×D₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₁₃	—	2	1	no	—
62	C₄×C₁₁₃	—	4	1	no	—
63	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
65	Dic₁₁₃	—	2	1	no	—
66	Dic₁₁₃	—	4	1	no	—
67	Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₁₁₃	—	2	1	no	—
70	C₂×C₁₁₃	—	4	1	no	—
71	C₂×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	D₁₁₃	—	2	1	no	—
73	D₁₁₃	—	2	1	no	—
74	D₁₁₃	—	4	1	no	—
75	C₁₁₃	#421	1	1	yes	characteristic, complemented, Sylow
76	#421	—	113	1	no	maximal, Sylow
77	C₂×#41	—	113	1	no	—
78	C₂×M₁₆	—	113	1	no	—
79	#117	—	113	1	no	—
80	#104	—	113	1	no	—
81	#104	—	113	1	no	—
82	#104	—	113	1	no	—
83	#104	D₁₁₃	1	1	yes	characteristic, complemented
84	#41	—	113	1	no	—
85	#41	—	113	1	no	—
86	#41	—	113	1	no	—
87	#41	—	226	1	no	—
88	#41	—	226	1	no	—
89	#41	C₂×D₁₁₃	1	1	yes	characteristic
90	C₂×Q₈	—	113	1	no	—
91	C₂×D₄	—	113	1	no	—
92	C₂×D₄	—	226	1	no	—
93	C₂²×C₄	—	113	1	no	—
94	C₂²×C₄	—	226	1	no	—
95	M₁₆	—	113	1	no	—
96	M₁₆	—	226	1	no	—
97	M₁₆	C₂×D₁₁₃	1	1	yes	characteristic, complemented
98	C₂×C₈	—	226	1	no	—
99	C₄⋊C₄	—	226	1	no	—
100	C₂²⋊C₄	—	226	1	no	—
101	C₄²	—	113	1	no	—
102	C₄²	C₂×D₁₁₃	1	1	yes	characteristic, complemented
103	C₂³	—	113	1	no	—
104	C₂³	—	226	1	no	—
105	Q₈	—	113	1	no	—
106	Q₈	—	226	1	no	—
107	Q₈	C₄×D₁₁₃	1	1	yes	characteristic
108	D₄	—	2	1	no	—
109	D₄	—	113	1	no	—
110	D₄	—	226	1	no	—
111	D₄	—	226	1	no	—
112	D₄	—	226	1	no	—
113	D₄	—	226	1	no	—
114	D₄	C₄×D₁₁₃	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	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	—	226	1	no	—
123	C₂×C₄	—	226	1	no	—
124	C₂×C₄	—	226	1	no	—
125	C₂×C₄	—	226	1	no	—
126	C₂×C₄	—	226	1	no	—
127	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	226	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	113	1	no	—
132	C₂²	—	113	1	no	—
133	C₂²	—	452	1	no	—
134	C₂²	—	452	2	no	—
135	C₂²	—	226	1	no	—
136	C₂²	D₄×D₁₁₃	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	113	1	no	—
140	C₄	—	113	1	no	—
141	C₄	—	452	1	no	—
142	C₄	—	226	1	no	—
143	C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
144	C₄	D₄×D₁₁₃	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	452	1	no	—
148	C₂	—	226	1	no	—
149	C₂	—	226	1	no	—
150	C₂	C₂²⋊C₄×D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#104×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂×C₂²⋊C₄
hypercenter	#104	D₁₁₃
hypocenter	C₁₁₃	#421
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₁₃	C₂	semidirect
3. #16813579	C₂	semidirect
4. #16813550	C₂	semidirect
5. #16813549	C₂	semidirect
6. #16813548	C₂	semidirect
7. #104×C₁₁₃	C₂	semidirect
8. #14268648	C₄	semidirect
12. #14268646	C₄	semidirect
13. Q₈×D₁₁₃	C₄	semidirect
15. D₄×D₁₁₃	C₄	semidirect
18. #14268631	C₂²	semidirect
23. C₄×Dic₁₁₃	C₂²	semidirect
25. M₁₆×C₁₁₃	C₂²	semidirect
26. C₄²×C₁₁₃	C₂²	semidirect
75. C₁₁₃	#421	semidirect
83. #104	D₁₁₃	semidirect
97. M₁₆	C₂×D₁₁₃	semidirect
102. C₄²	C₂×D₁₁₃	semidirect