Finite Group Databasebeta

Subgroups — #37550109

Order 23168.

146 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 58 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	#26042338	C₂	1	1	yes	maximal, characteristic, complemented
2	#26042337	C₂	1	1	yes	maximal, characteristic, complemented
3	#26042202	C₂	1	1	yes	maximal, characteristic, complemented
4	#26042178	C₂	1	1	yes	maximal, characteristic, complemented
5	#26042177	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15668452	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041393	C₂	1	1	yes	maximal, characteristic
8	#19153891	—	2	1	no	—
9	#19153890	—	2	1	no	—
10	#19153890	C₂²	1	1	yes	characteristic
11	#19153890	C₂²	1	1	yes	characteristic
12	D₁₈₁⋊C₁₆	—	2	1	no	—
13	C₂×#15668452	—	2	1	no	—
14	C₂×#15668452	C₂²	1	1	yes	characteristic
15	C₂×#15668452	C₂²	1	1	yes	characteristic
16	#19153836	C₄	1	1	yes	characteristic
17	#19153824	C₂²	1	1	yes	characteristic
18	#19153823	C₂²	1	1	yes	characteristic
19	#19153751	C₄	1	1	yes	characteristic
20	#19153744	C₄	1	1	yes	characteristic
21	C₄×#13089678	C₂²	1	1	yes	characteristic
22	#104×C₁₈₁	C₄	1	1	yes	characteristic
23	#15668452	—	2	1	no	—
24	#15668452	—	2	1	no	—
25	#15668452	—	2	1	no	—
26	#15668452	D₄	1	1	yes	characteristic
27	#15668452	D₄	1	1	yes	characteristic
28	#15668449	C₂×C₄	1	1	yes	characteristic
29	#15668439	C₈	1	1	yes	characteristic
30	#15668437	C₈	1	1	yes	characteristic
31	#15668436	C₈	1	1	yes	characteristic
32	#15668434	C₈	1	1	yes	characteristic
33	#15668433	—	2	1	no	—
34	#15668433	C₂×C₄	1	1	yes	characteristic
35	#15668432	—	2	1	no	—
36	C₂×#13089678	—	2	1	no	—
37	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
38	C₂×#13089678	C₂³	1	1	yes	characteristic
39	C₈×D₁₈₁	—	2	1	no	—
40	#41×C₁₈₁	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
43	Dic₃₆₂	C₂×C₈	1	1	yes	characteristic
44	D₇₂₄	C₂×C₈	1	1	yes	characteristic
45	C₂²⋊D₁₈₁	—	2	1	no	—
46	C₄×D₁₈₁	—	2	1	no	—
47	#13089678	—	2	1	no	—
48	#13089678	C₂×C₈	1	1	yes	characteristic
49	#13089678	C₂×C₈	1	1	yes	characteristic
50	#13089678	C₂×D₄	1	1	yes	characteristic
51	#13089678	#41	1	1	yes	characteristic
52	Q₈×C₁₈₁	C₂×C₈	1	1	yes	characteristic
53	D₄×C₁₈₁	—	2	1	no	—
54	D₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₈₁	—	2	1	no	—
56	C₂×C₄×C₁₈₁	—	2	1	no	—
57	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₁₈₁	—	2	1	no	—
59	C₂²×C₁₈₁	—	2	1	no	—
60	C₂²×C₁₈₁	#131	1	1	yes	characteristic
61	C₂×D₁₈₁	—	2	1	no	—
62	C₄×C₁₈₁	—	2	2	no	—
63	C₄×C₁₈₁	—	2	1	no	—
64	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
66	Dic₁₈₁	—	2	1	no	—
67	C₂×C₁₈₁	—	2	1	no	—
68	C₂×C₁₈₁	—	4	1	no	—
69	C₂×C₁₈₁	C₈×D₄	1	1	yes	characteristic
70	D₁₈₁	—	4	1	no	—
71	C₁₈₁	#2170	1	1	yes	characteristic, complemented, Sylow
72	#2170	—	181	1	no	maximal, Sylow
73	#504	—	181	1	no	—
74	#504	—	181	1	no	—
75	#443	—	181	1	no	—
76	#364	—	181	1	no	—
77	#350	—	181	1	no	—
78	#350	—	181	1	no	—
79	C₄×C₁₆	—	181	1	no	—
80	#135	—	181	1	no	—
81	#131	—	181	1	no	—
82	#131	—	181	1	no	—
83	M₃₂	—	362	1	no	—
84	M₃₂	—	362	1	no	—
85	M₃₂	—	181	1	no	—
86	M₃₂	—	181	1	no	—
87	C₂×C₁₆	—	362	1	no	—
88	C₂×C₁₆	—	362	1	no	—
89	C₂×C₁₆	—	181	1	no	—
90	C₂×C₁₆	—	181	1	no	—
91	#108	—	181	1	no	—
92	#104	—	181	1	no	—
93	#104	#10624382	1	1	yes	characteristic
94	C₄×C₈	—	181	1	no	—
95	#41	—	181	1	no	—
96	#41	C₂×#10624382	1	1	yes	characteristic
97	Q₁₆	—	181	1	no	—
98	SD₁₆	—	181	1	no	—
99	SD₁₆	—	181	1	no	—
100	D₈	—	181	1	no	—
101	M₁₆	—	362	1	no	—
102	M₁₆	—	362	1	no	—
103	M₁₆	—	181	1	no	—
104	M₁₆	C₂×#10624382	1	1	yes	characteristic
105	C₂×C₈	—	362	1	no	—
106	C₂×C₈	—	362	1	no	—
107	C₂×C₈	—	181	1	no	—
108	C₂×C₈	—	181	1	no	—
109	C₄²	C₂×#10624382	1	1	yes	characteristic
110	C₁₆	—	362	1	no	—
111	C₁₆	—	362	1	no	—
112	C₁₆	—	362	1	no	—
113	C₁₆	—	181	1	no	—
114	C₁₆	—	181	1	no	—
115	Q₈	—	181	1	no	—
116	Q₈	D₁₈₁⋊C₈	1	1	yes	characteristic
117	D₄	—	2	1	no	—
118	D₄	—	362	1	no	—
119	D₄	—	181	1	no	—
120	D₄	D₁₈₁⋊C₈	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	362	1	no	—
124	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	362	1	no	—
127	C₈	—	181	1	no	—
128	C₈	—	181	1	no	—
129	C₈	—	181	1	no	—
130	C₈	—	181	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	362	1	no	—
133	C₂²	#19153903	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	362	1	no	—
137	C₄	D₄×#10624382	1	1	yes	characteristic
138	C₄	C₂×D₁₈₁⋊C₈	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	1	no	—
141	C₂	—	724	1	no	—
142	C₂	#26042306	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#10624382
commutator	C₄×C₁₈₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10624382
Fitting	#104×C₁₈₁	C₄
socle	C₂×C₁₈₁	C₈×D₄
prime radical	#19153836	C₄
cosocle	C₂×#13089678	C₂³
hypercenter	#104	#10624382
hypocenter	C₁₈₁	#2170
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #26042338	C₂	semidirect
2. #26042337	C₂	semidirect
3. #26042202	C₂	semidirect
4. #26042178	C₂	semidirect
5. #26042177	C₂	semidirect
6. C₄×#15668452	C₂	semidirect
71. C₁₈₁	#2170	semidirect

Maximal subgroups

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

subgroup	index
#26042338	2
#26042337	2
#26042202	2
#26042178	2
#26042177	2
C₄×#15668452	2
#26041393	2
#2170	181