Finite Group Databasebeta

Subgroups — #37550128

Order 23168.

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	#26042325	C₂	1	1	yes	maximal, characteristic
2	#26042319	C₂	1	1	yes	maximal, characteristic, complemented
3	#26042203	C₂	1	1	yes	maximal, characteristic, complemented
4	#26042198	C₂	1	1	yes	maximal, characteristic, complemented
5	#26042196	C₂	1	1	yes	maximal, characteristic
6	#26042169	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041721	C₂	1	1	yes	maximal, characteristic
8	#19153904	—	2	1	no	—
9	Q₈×#10624382	—	2	1	no	—
10	#19153891	C₂²	1	1	yes	characteristic
11	#19153886	C₂²	1	1	yes	characteristic
12	#19153884	C₂²	1	1	yes	characteristic
13	#19153881	—	2	1	no	—
14	#19153879	—	2	1	no	—
15	#19153870	C₂²	1	1	yes	characteristic
16	#19153839	C₂²	1	1	yes	characteristic
17	Q₁₆×D₁₈₁	C₂²	1	1	yes	characteristic
18	#19153762	C₄	1	1	yes	characteristic, complemented
19	#19153761	C₄	1	1	yes	characteristic, complemented
20	#19153759	C₄	1	1	yes	characteristic, complemented
21	#19153755	C₂²	1	1	yes	characteristic
22	Q₃₂×C₁₈₁	C₄	1	1	yes	characteristic, complemented
23	#15668459	—	2	1	no	—
24	#15668459	—	4	1	no	—
25	D₁₈₁⋊C₈	—	4	1	no	—
26	D₁₈₁⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊#10624382	—	2	1	no	—
28	C₄⋊#10624382	—	4	1	no	—
29	C₄×#10624382	—	4	1	no	—
30	C₄×#10624382	D₄	1	1	yes	characteristic
31	#15668452	—	2	1	no	—
32	#15668450	—	2	1	no	—
33	Q₈×D₁₈₁	—	2	1	no	—
34	Dic₇₂₄	C₂×C₄	1	1	yes	characteristic
35	#15668439	—	2	1	no	—
36	#15668436	—	2	1	no	—
37	D₁₄₄₈	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₈₁	C₂³	1	1	yes	characteristic
39	#15668424	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
41	Q₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
43	C₂×#10624382	—	2	1	no	—
44	C₂×#10624382	—	4	1	no	—
45	#13089685	—	2	1	no	—
46	#13089685	—	4	1	no	—
47	Dic₃₆₂	—	2	1	no	—
48	Dic₃₆₂	—	4	1	no	—
49	D₇₂₄	—	2	1	no	—
50	D₇₂₄	—	4	1	no	—
51	C₄×D₁₈₁	—	4	1	no	—
52	C₄×D₁₈₁	—	4	1	no	—
53	C₄×D₁₈₁	C₂×D₄	1	1	yes	characteristic
54	#13089678	#41	1	1	yes	characteristic
55	Q₈×C₁₈₁	—	2	1	no	—
56	Q₈×C₁₈₁	—	2	1	no	—
57	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×D₁₈₁	—	4	1	no	—
59	C₂×D₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#10624382	—	4	1	no	—
61	#10624382	—	8	1	no	—
62	C₄×C₁₈₁	—	4	1	no	—
63	C₄×C₁₈₁	—	4	1	no	—
64	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
65	Dic₁₈₁	—	4	1	no	—
66	Dic₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₁₈₁	#442	1	1	yes	characteristic
68	D₁₈₁	—	2	1	no	—
69	D₁₈₁	—	8	1	no	—
70	C₁₈₁	#2180	1	1	yes	characteristic, complemented, Sylow
71	#2180	—	181	1	no	maximal, Sylow
72	#510	—	181	1	no	—
73	#444	—	181	1	no	—
74	#441	—	181	1	no	—
75	#365	—	181	1	no	—
76	#362	—	181	1	no	—
77	#360	—	181	1	no	—
78	#347	—	181	1	no	—
79	#135	—	181	1	no	—
80	C₂×Q₁₆	—	181	1	no	—
81	#131	—	362	1	no	—
82	C₄×Q₈	—	362	1	no	—
83	Q₃₂	—	181	1	no	—
84	Q₃₂	#10624382	1	1	yes	characteristic, complemented
85	SD₃₂	—	181	1	no	—
86	SD₃₂	—	181	1	no	—
87	M₃₂	—	181	1	no	—
88	M₃₂	—	181	1	no	—
89	#108	—	181	1	no	—
90	#106	—	181	1	no	—
91	#104	—	362	1	no	—
92	#103	—	362	1	no	—
93	#97	—	181	1	no	—
94	#41	—	362	1	no	—
95	C₂×Q₈	—	362	1	no	—
96	Q₁₆	—	362	1	no	—
97	Q₁₆	—	181	1	no	—
98	Q₁₆	C₂×#10624382	1	1	yes	characteristic
99	Q₁₆	C₂×#10624382	1	1	yes	characteristic
100	SD₁₆	—	362	1	no	—
101	D₈	—	181	1	no	—
102	M₁₆	—	362	1	no	—
103	M₁₆	—	724	1	no	—
104	C₂×C₈	—	181	1	no	—
105	C₂×C₈	—	181	1	no	—
106	C₂×C₈	—	724	1	no	—
107	C₄⋊C₄	—	362	1	no	—
108	C₄⋊C₄	—	724	1	no	—
109	C₄²	—	181	1	no	—
110	C₄²	—	724	1	no	—
111	C₁₆	—	362	1	no	—
112	C₁₆	—	181	1	no	—
113	C₁₆	C₂×#10624382	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	362	1	no	—
117	Q₈	—	724	1	no	—
118	D₄	—	362	1	no	—
119	D₄	—	724	1	no	—
120	C₂×C₄	—	362	1	no	—
121	C₂×C₄	—	181	1	no	—
122	C₂×C₄	—	724	1	no	—
123	C₂×C₄	—	724	1	no	—
124	C₂×C₄	—	724	1	no	—
125	C₈	—	362	1	no	—
126	C₈	—	181	1	no	—
127	C₈	—	724	1	no	—
128	C₈	C₂²×#10624382	1	1	yes	characteristic
129	C₂²	—	181	1	no	—
130	C₂²	—	724	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	1448	1	no	—
134	C₄	—	181	1	no	—
135	C₄	—	724	1	no	—
136	C₄	—	724	1	no	—
137	C₄	D₄×#10624382	1	1	yes	characteristic
138	C₂	—	362	1	no	—
139	C₂	—	1448	1	no	—
140	C₂	#26042320	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26042320
commutator	C₈×C₁₈₁	C₂²×C₄
Frattini	C₈	C₂²×#10624382
Fitting	Q₃₂×C₁₈₁	C₄
socle	C₂×C₁₈₁	#442
prime radical	#19153839	C₂²
cosocle	C₈×D₁₈₁	C₂³
hypercenter	Q₃₂	#10624382
hypocenter	C₁₈₁	#2180
LCS	G ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. #26042319	C₂	semidirect
3. #26042203	C₂	semidirect
4. #26042198	C₂	semidirect
6. #26042169	C₂	semidirect
18. #19153762	C₄	semidirect
19. #19153761	C₄	semidirect
20. #19153759	C₄	semidirect
22. Q₃₂×C₁₈₁	C₄	semidirect
70. C₁₈₁	#2180	semidirect
84. Q₃₂	#10624382	semidirect

Maximal subgroups

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

subgroup	index
#26042325	2
#26042319	2
#26042203	2
#26042198	2
#26042196	2
#26042169	2
#26041721	2
#2180	181