Finite Group Databasebeta

Subgroups — #37546869

Order 23168.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#26042096	C₂	1	1	yes	maximal, characteristic, complemented
2	#26042084	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041722	C₂	1	3	yes	maximal, complemented
4	#26041717	C₂	1	3	yes	maximal, complemented
5	#26041711	C₂	1	3	yes	maximal, complemented
6	C₂×#19153761	C₂	1	3	yes	maximal, complemented
7	#504×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#19153863	—	2	1	no	—
9	#19153862	—	2	1	no	—
10	#19153849	—	2	3	no	—
11	#19153844	—	2	3	no	—
12	#19153838	C₂²	1	3	yes	—
13	#19153838	C₂²	1	3	yes	complemented
14	C₂×Dic₇₂₄	C₂²	1	3	yes	complemented
15	C₂×D₁₄₄₈	C₂²	1	3	yes	—
16	Dic₁₄₄₈	C₂²	1	3	yes	complemented
17	#19153761	C₂²	1	3	yes	complemented
18	#19153761	C₂²	1	6	yes	complemented
19	#19153761	C₂²	1	1	yes	characteristic, complemented
20	D₂₈₉₆	C₂²	1	3	yes	complemented
21	#131×C₁₈₁	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₈₁	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₈₁	C₂²	1	3	yes	complemented
24	#15668450	—	4	1	no	—
25	#15668449	—	2	3	no	—
26	#15668449	—	2	3	no	—
27	#15668448	—	4	3	no	—
28	C₂×Dic₃₆₂	—	2	3	no	—
29	Q₈×D₁₈₁	—	4	1	no	—
30	C₂×D₇₂₄	—	2	3	no	—
31	D₄×D₁₈₁	—	4	3	no	—
32	Dic₇₂₄	—	2	3	no	—
33	Dic₇₂₄	C₂³	1	3	yes	—
34	Dic₇₂₄	C₂³	1	1	yes	characteristic
35	#15668438	—	2	3	no	—
36	#15668438	—	2	3	no	—
37	D₁₄₄₈	—	2	3	no	—
38	D₁₄₄₈	C₂³	1	1	yes	characteristic
39	D₁₄₄₈	C₂³	1	3	yes	—
40	#41×C₁₈₁	D₄	1	1	yes	characteristic
41	M₁₆×C₁₈₁	D₄	1	3	yes	—
42	C₂×C₈×C₁₈₁	C₂³	1	3	yes	—
43	C₁₆×C₁₈₁	C₂³	1	1	yes	characteristic
44	C₁₆×C₁₈₁	C₂³	1	3	yes	—
45	C₂²×D₁₈₁	—	4	3	no	—
46	Dic₃₆₂	—	2	1	no	—
47	Dic₃₆₂	—	2	3	no	—
48	Dic₃₆₂	—	4	3	no	—
49	D₇₂₄	—	2	3	no	—
50	D₇₂₄	—	2	1	no	—
51	D₇₂₄	—	4	3	no	—
52	C₂²⋊D₁₈₁	—	4	3	no	—
53	C₂²⋊D₁₈₁	—	4	3	no	—
54	C₂×Dic₁₈₁	—	4	3	no	—
55	C₄×D₁₈₁	—	4	3	no	—
56	C₄×D₁₈₁	—	4	3	no	—
57	Q₈×C₁₈₁	D₈	1	1	yes	characteristic
58	D₄×C₁₈₁	D₈	1	3	yes	—
59	C₂×C₄×C₁₈₁	C₂×D₄	1	3	yes	—
60	C₈×C₁₈₁	C₂×D₄	1	3	yes	—
61	C₈×C₁₈₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₈₁	C₂×D₈	1	3	yes	—
63	C₂×D₁₈₁	—	4	1	no	—
64	C₂×D₁₈₁	—	4	3	no	—
65	C₂×D₁₈₁	—	16	3	no	—
66	C₄×C₁₈₁	C₂×D₈	1	3	yes	—
67	C₄×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₈₁	—	4	1	no	—
69	Dic₁₈₁	—	4	3	no	—
70	C₂×C₁₈₁	—	2	3	no	—
71	C₂×C₁₈₁	C₂²×D₈	1	1	yes	characteristic
72	D₁₈₁	—	8	1	no	—
73	D₁₈₁	—	8	3	no	—
74	C₁₈₁	#3416	1	1	yes	characteristic, complemented, Sylow
75	#3416	—	181	1	no	maximal, Sylow
76	#578	—	181	1	no	—
77	#576	—	181	1	no	—
78	#510	—	181	3	no	—
79	#509	—	181	3	no	—
80	#508	—	181	3	no	—
81	C₂×SD₃₂	—	181	3	no	—
82	#504	D₁₈₁	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	362	1	no	—
84	H(5,𝔽₂)	—	362	1	no	—
85	#137	—	362	3	no	—
86	AGL(1,ℤ/8)	—	362	3	no	—
87	#135	—	181	3	no	—
88	#135	—	181	3	no	—
89	C₂×Q₁₆	—	181	3	no	—
90	C₂×D₈	—	181	3	no	—
91	#131	C₂×D₁₈₁	1	1	yes	characteristic
92	Q₃₂	—	181	3	no	—
93	SD₃₂	—	181	3	no	—
94	SD₃₂	—	181	6	no	—
95	SD₃₂	—	181	1	no	—
96	D₁₆	—	181	3	no	—
97	M₃₂	C₂×D₁₈₁	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₈₁	1	3	yes	complemented
99	#41	—	362	3	no	—
100	#41	—	362	3	no	—
101	#41	—	724	1	no	—
102	#41	—	724	3	no	—
103	#41	D₇₂₄	1	1	yes	characteristic
104	C₂×Q₈	—	362	3	no	—
105	C₂×Q₈	—	724	1	no	—
106	C₂×D₄	—	362	3	no	—
107	C₂×D₄	—	724	3	no	—
108	Q₁₆	—	362	3	no	—
109	Q₁₆	—	181	3	no	—
110	Q₁₆	—	181	1	no	—
111	SD₁₆	—	362	3	no	—
112	SD₁₆	—	362	3	no	—
113	D₈	—	362	3	no	—
114	D₈	—	181	1	no	—
115	D₈	—	181	3	no	—
116	M₁₆	D₇₂₄	1	3	yes	—
117	C₂×C₈	C₂²×D₁₈₁	1	3	yes	—
118	C₁₆	C₂²×D₁₈₁	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₈₁	1	3	yes	—
120	C₂³	—	724	3	no	—
121	Q₈	—	362	1	no	—
122	Q₈	—	362	3	no	—
123	Q₈	—	724	3	no	—
124	Q₈	D₁₄₄₈	1	1	yes	characteristic
125	D₄	—	362	3	no	—
126	D₄	—	362	1	no	—
127	D₄	—	724	3	no	—
128	D₄	—	724	3	no	—
129	D₄	—	724	3	no	—
130	D₄	D₁₄₄₈	1	3	yes	—
131	C₂×C₄	—	724	3	no	—
132	C₂×C₄	—	724	3	no	—
133	C₂×C₄	—	724	3	no	—
134	C₂×C₄	C₂×D₇₂₄	1	3	yes	—
135	C₈	C₂×D₇₂₄	1	3	yes	—
136	C₈	C₂³×D₁₈₁	1	1	yes	characteristic
137	C₂²	—	2896	3	no	—
138	C₂²	—	724	1	no	—
139	C₂²	—	724	3	no	—
140	C₂²	C₂×D₁₄₄₈	1	3	yes	—
141	C₄	—	724	1	no	—
142	C₄	—	724	3	no	—
143	C₄	C₂×D₁₄₄₈	1	3	yes	—
144	C₄	C₂²×D₇₂₄	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	1448	3	no	—
147	C₂	—	1448	1	no	—
148	C₂	C₂²×D₁₄₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₄₄₈
commutator, cosocle	C₈×C₁₈₁	C₂⁴
Frattini	C₈	C₂³×D₁₈₁
Fitting	#504×C₁₈₁	C₂
socle	C₂×C₁₈₁	C₂²×D₈
hypercenter	#504	D₁₈₁
hypocenter	C₁₈₁	#3416
LCS	G ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #26042096	C₂	semidirect
2. #26042084	C₂	semidirect
3. #26041722	C₂	semidirect
4. #26041717	C₂	semidirect
5. #26041711	C₂	semidirect
6. C₂×#19153761	C₂	semidirect
7. #504×C₁₈₁	C₂	semidirect
13. #19153838	C₂²	semidirect
14. C₂×Dic₇₂₄	C₂²	semidirect
16. Dic₁₄₄₈	C₂²	semidirect
17. #19153761	C₂²	semidirect
18. #19153761	C₂²	semidirect
19. #19153761	C₂²	semidirect
20. D₂₈₉₆	C₂²	semidirect
22. M₃₂×C₁₈₁	C₂²	semidirect
23. C₂×C₁₆×C₁₈₁	C₂²	semidirect
74. C₁₈₁	#3416	semidirect
82. #504	D₁₈₁	semidirect
97. M₃₂	C₂×D₁₈₁	semidirect
98. C₂×C₁₆	C₂×D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26042096	2
#26042084	2
#26041722	2
#26041717	2
#26041711	2
C₂×#19153761	2
#504×C₁₈₁	2
#3416	181