Finite Group Databasebeta

Subgroups — #37549332

Order 23168.

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

normal subgroups only

This group has 152 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	#26042329	C₂	1	1	yes	maximal, characteristic
2	Q₈×#13089685	C₂	1	1	yes	maximal, characteristic
3	#26042142	C₂	1	1	yes	maximal, characteristic
4	#26042129	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#13089685	C₂	1	1	yes	maximal, characteristic
6	#13089685⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#26041371	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#13089685	—	2	1	no	—
9	C₄⋊#13089685	—	2	1	no	—
10	C₄⋊#13089685	C₂²	1	1	yes	characteristic
11	C₄⋊#13089685	C₂²	1	1	yes	characteristic
12	#19153871	—	2	1	no	—
13	C₄×#13089685	—	2	1	no	—
14	C₄×#13089685	C₂²	1	1	yes	characteristic
15	C₄×#13089685	C₂²	1	1	yes	characteristic
16	C₂×#15668439	C₄	1	1	yes	characteristic
17	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
18	#19153781	C₂²	1	1	yes	characteristic
19	#19153749	C₄	1	1	yes	characteristic
20	#19153741	C₄	1	1	yes	characteristic
21	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
22	#103×C₁₈₁	C₄	1	1	yes	characteristic
23	C₂×#13089685	—	2	1	no	—
24	C₂×#13089685	—	2	1	no	—
25	C₂×#13089685	—	2	1	no	—
26	C₂×#13089685	D₄	1	1	yes	characteristic
27	C₂×#13089685	D₄	1	1	yes	characteristic
28	C₂×Dic₃₆₂	C₂×C₄	1	1	yes	characteristic
29	#15668439	—	2	2	no	—
30	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₁₈₁	—	2	1	no	—
32	C₄⋊Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
33	#15668428	—	2	1	no	—
34	C₄×Dic₁₈₁	—	2	1	no	—
35	C₄×Dic₁₈₁	—	2	1	no	—
36	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
40	#13089685	—	2	1	no	—
41	#13089685	—	4	1	no	—
42	#13089685	—	4	1	no	—
43	#13089685	—	4	1	no	—
44	#13089685	Q₁₆	1	2	yes	complemented
45	Dic₃₆₂	—	2	1	no	—
46	Dic₃₆₂	—	2	1	no	—
47	C₂×Dic₁₈₁	—	2	1	no	—
48	C₂×Dic₁₈₁	—	2	1	no	—
49	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₈₁	#41	1	1	yes	characteristic
51	#13089678	—	2	1	no	—
52	Q₈×C₁₈₁	—	2	1	no	—
53	Q₈×C₁₈₁	M₁₆	1	2	yes	—
54	C₂×C₄×C₁₈₁	—	2	1	no	—
55	C₂×C₄×C₁₈₁	—	2	1	no	—
56	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
57	C₈×C₁₈₁	—	2	1	no	—
58	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₈₁	—	2	2	no	—
60	C₄×C₁₈₁	—	4	1	no	—
61	C₄×C₁₈₁	C₂×M₁₆	1	1	yes	characteristic
62	C₄×C₁₈₁	#131	1	1	yes	characteristic
63	Dic₁₈₁	—	2	1	no	—
64	Dic₁₈₁	—	4	1	no	—
65	Dic₁₈₁	—	4	1	no	—
66	Dic₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
67	Dic₁₈₁	#135	1	1	yes	characteristic
68	C₂×C₁₈₁	#435	1	1	yes	characteristic
69	C₂×C₁₈₁	C₄×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₈₁	#444	1	1	yes	characteristic
71	C₁₈₁	#1584	1	1	yes	characteristic, complemented, Sylow
72	#1584	—	181	1	no	maximal, Sylow
73	#446	—	181	1	no	—
74	C₈×Q₈	—	181	1	no	—
75	C₄×Q₁₆	—	181	1	no	—
76	#335	—	181	1	no	—
77	Q₈⋊C₈	—	181	1	no	—
78	Q₈⋊C₈	—	181	1	no	—
79	#322	—	181	1	no	—
80	C₂×Q₁₆	—	181	1	no	—
81	C₄×Q₈	—	181	1	no	—
82	C₄×Q₈	—	181	1	no	—
83	#107	—	181	1	no	—
84	C₄⋊C₈	—	362	1	no	—
85	C₄⋊C₈	—	362	1	no	—
86	C₄⋊C₈	—	181	1	no	—
87	C₄⋊C₈	—	181	1	no	—
88	#103	—	181	1	no	—
89	#103	#10624382	1	1	yes	characteristic
90	#97	—	362	1	no	—
91	C₄×C₈	—	362	1	no	—
92	C₄×C₈	—	181	1	no	—
93	C₄×C₈	—	181	1	no	—
94	C₄×C₈	—	181	1	no	—
95	C₂×Q₈	—	181	1	no	—
96	C₂×Q₈	C₂×#10624382	1	1	yes	characteristic
97	Q₁₆	—	362	2	no	—
98	C₂×C₈	—	362	1	no	—
99	C₂×C₈	—	362	1	no	—
100	C₂×C₈	—	362	1	no	—
101	C₂×C₈	—	181	1	no	—
102	C₂×C₈	—	181	1	no	—
103	C₂×C₈	—	181	1	no	—
104	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
105	C₄⋊C₄	—	362	1	no	—
106	C₄⋊C₄	—	362	1	no	—
107	C₄⋊C₄	—	181	1	no	—
108	C₄⋊C₄	C₂×#10624382	1	1	yes	characteristic
109	C₄²	—	362	1	no	—
110	C₄²	—	362	1	no	—
111	C₄²	—	181	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	362	1	no	—
114	Q₈	—	362	1	no	—
115	Q₈	#15668459	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	362	1	no	—
119	C₂×C₄	—	362	1	no	—
120	C₂×C₄	—	181	1	no	—
121	C₂×C₄	—	181	1	no	—
122	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	362	1	no	—
125	C₈	—	362	1	no	—
126	C₈	—	181	2	no	—
127	C₈	—	724	1	no	—
128	C₈	—	724	1	no	—
129	C₈	—	724	1	no	—
130	C₂²	D₄×#10624382	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	362	1	no	—
134	C₄	—	181	1	no	—
135	C₄	—	181	1	no	—
136	C₄	—	724	1	no	—
137	C₄	—	724	1	no	—
138	C₄	C₂×#15668459	1	1	yes	characteristic
139	C₄	#19153903	1	1	yes	characteristic
140	C₂	#26042308	1	1	yes	characteristic
141	C₂	Q₁₆×#10624382	1	1	yes	characteristic
142	C₂	#26042324	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#10624382
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10624382
Fitting	#103×C₁₈₁	C₄
socle, prime radical	C₂²×C₁₈₁	C₄×D₄
cosocle	C₄×Dic₁₈₁	C₂³
hypercenter	#103	#10624382
hypocenter	C₁₈₁	#1584
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
44. #13089685	Q₁₆	semidirect
71. C₁₈₁	#1584	semidirect

Maximal subgroups

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

subgroup	index
#26042329	2
Q₈×#13089685	2
#26042142	2
#26042129	2
Q₈⋊#13089685	2
#13089685⋊C₈	2
#26041371	2
#1584	181