Group #36476210 — subgroups

Subgroups — #36476210

Order 22784.

1774 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 1774 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₈₉	C₂	1	6	yes	maximal
2	#26018517	C₂	1	8	yes	maximal
3	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₈₉	C₂²	1	12	yes	—
5	C₂²×C₄×Dic₈₉	C₂²	1	4	yes	—
6	C₄×C₄⋊Dic₈₉	—	2	24	no	—
7	C₂×#15667919	C₂²	1	12	yes	—
8	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
9	#376×C₈₉	C₄	1	8	yes	complemented
10	C₂×#95×C₈₉	C₂²	1	4	yes	—
11	C₂³×Dic₈₉	C₂³	1	8	yes	—
12	C₂×C₄⋊Dic₈₉	—	2	48	no	—
13	C₂×C₄⋊Dic₈₉	—	2	24	no	—
14	C₂×C₄×Dic₈₉	—	2	24	no	—
15	#15667919	—	2	24	no	—
16	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
17	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₈₉	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₈₉	D₄	1	12	yes	—
20	C₂×C₄²×C₈₉	Q₈	1	12	yes	—
21	#95×C₈₉	C₂×C₄	1	16	yes	—
22	C₂²×Dic₈₉	—	2	24	no	—
23	C₂²×Dic₈₉	—	2	24	no	—
24	C₂²×Dic₈₉	#41	1	16	yes	—
25	C₄⋊Dic₈₉	—	2	96	no	—
26	C₄×Dic₈₉	—	4	16	no	—
27	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	—	2	24	no	—
29	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
30	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
31	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
32	C₂²×C₄×C₈₉	C₂×D₄	1	6	yes	—
33	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
34	C₂²×C₄×C₈₉	C₂×Q₈	1	6	yes	—
35	C₄²×C₈₉	C₄⋊C₄	1	48	yes	complemented
36	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
37	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
38	C₂³×C₈₉	C₂²×Q₈	1	3	yes	—
39	C₂³×C₈₉	C₂×#41	1	8	yes	—
40	C₂×C₄×C₈₉	—	2	24	no	—
41	C₂×C₄×C₈₉	—	2	24	no	—
42	C₂×C₄×C₈₉	—	2	24	no	—
43	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	24	yes	—
44	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	12	yes	—
45	C₂×C₄×C₈₉	#127	1	4	yes	—
46	C₂×C₄×C₈₉	C₄⋊Q₈	1	12	yes	—
47	C₂×Dic₈₉	—	2	48	no	—
48	C₂×Dic₈₉	—	4	24	no	—
49	C₂×Dic₈₉	—	4	8	no	—
50	C₂²×C₈₉	C₂²×C₄⋊C₄	1	3	yes	—
51	C₂²×C₈₉	C₄×#41	1	4	yes	—
52	C₂²×C₈₉	#525	1	12	yes	—
53	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
54	C₂²×C₈₉	C₂×C₄⋊Q₈	1	3	yes	—
55	C₂²×C₈₉	#533	1	12	yes	—
56	C₄×C₈₉	—	2	24	no	—
57	C₄×C₈₉	#384	1	8	yes	—
58	Dic₈₉	—	4	16	no	—
59	#13856	—	89	1	no	maximal, Sylow
60	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
61	C₂×C₈₉	#2296	1	6	yes	—
62	C₂×C₈₉	#2397	1	2	yes	—
63	C₂×C₈₉	#2399	1	6	yes	—
64	C₂×C₄×C₄⋊C₄	—	89	6	no	—
65	C₂×#376	D₈₉	1	1	yes	characteristic
66	#1444	—	89	8	no	—
67	C₈₉	#13856	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₄⋊C₄	—	89	12	no	—
69	C₂²×C₄²	—	89	4	no	—
70	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
71	C₄×C₄⋊C₄	—	178	24	no	—
72	#376	Dic₈₉	1	8	yes	complemented
73	C₂×#95	—	89	12	no	—
74	C₂×#95	C₂×D₈₉	1	4	yes	—
75	C₂³×C₄	—	89	8	no	—
76	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
77	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
78	C₂×C₄⋊C₄	—	178	48	no	—
79	C₂×C₄⋊C₄	—	178	24	no	—
80	C₂×C₄²	—	178	24	no	—
81	C₂×C₄²	Dic₁₇₈	1	12	yes	—
82	C₂×C₄²	D₃₅₆	1	12	yes	—
83	C₂×C₄²	C₂×Dic₈₉	1	12	yes	—
84	#95	—	178	24	no	—
85	#95	C₂×Dic₈₉	1	16	yes	—
86	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
87	C₂²×C₄	—	2	24	no	—
88	C₂²×C₄	—	89	16	no	—
89	C₂²×C₄	—	178	24	no	—
90	C₂²×C₄	—	178	24	no	—
91	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
92	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
93	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
94	C₂²×C₄	C₂×D₃₅₆	1	6	yes	—
95	C₂²×C₄	C₂×Dic₁₇₈	1	12	yes	—
96	C₂²×C₄	C₂×Dic₁₇₈	1	6	yes	—
97	C₄⋊C₄	—	178	96	no	—
98	C₄²	—	356	16	no	—
99	C₄²	C₄⋊Dic₈₉	1	48	yes	complemented
100	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
101	C₂³	C₂²×D₃₅₆	1	3	yes	—
102	C₂³	C₂²×Dic₁₇₈	1	3	yes	—
103	C₂³	#41×D₈₉	1	8	yes	—
104	C₂×C₄	—	2	24	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	24	no	—
107	C₂×C₄	—	356	8	no	—
108	C₂×C₄	—	356	24	no	—
109	C₂×C₄	—	178	48	no	—
110	C₂×C₄	C₂×C₄⋊Dic₈₉	1	24	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₈₉	1	12	yes	—
112	C₂×C₄	#15668009	1	4	yes	—
113	C₂×C₄	#15668012	1	12	yes	—
114	C₂²	C₂²×C₄⋊Dic₈₉	1	3	yes	—
115	C₂²	#41×Dic₈₉	1	4	yes	—
116	C₂²	#19152110	1	12	yes	—
117	C₂²	C₂×#15668009	1	1	yes	characteristic
118	C₂²	C₂×#15668012	1	3	yes	—
119	C₂²	#19152151	1	12	yes	—
120	C₄	—	2	24	no	—
121	C₄	—	356	16	no	—
122	C₄	#19151468	1	8	yes	—
123	C₂	C₂×#19151468	1	1	yes	characteristic
124	C₂	#26022736	1	6	yes	—
125	C₂	#26023533	1	2	yes	—
126	C₂	#26023545	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄⋊Dic₈₉

C₂

yes

maximal

#26018517

C₂

yes

maximal

C₂×#376×C₈₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₈₉

C₂²

yes

—

C₂²×C₄×Dic₈₉

C₂²

yes

—

C₄×C₄⋊Dic₈₉

—

C₂×#15667919

C₂²

yes

—

C₂²×C₄²×C₈₉

C₂²

yes

—

#376×C₈₉

C₄

yes

complemented

C₂×#95×C₈₉

C₂²

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄×Dic₈₉

—

#15667919

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

#95×C₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

#41

yes

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

C₄⋊C₄

yes

complemented

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂³×C₈₉

C₂²×D₄

yes

—

C₂³×C₈₉

C₂²×Q₈

yes

—

C₂³×C₈₉

C₂×#41

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

#127

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

C₂²×C₄⋊C₄

yes

—

C₂²×C₈₉

C₄×#41

yes

—

C₂²×C₈₉

#525

yes

—

C₂²×C₈₉

C₂×#127

yes

characteristic

C₂²×C₈₉

C₂×C₄⋊Q₈

yes

—

C₂²×C₈₉

#533

yes

—

C₄×C₈₉

—

C₄×C₈₉

#384

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2397

yes

—

C₂×C₈₉

#2399

yes

—

C₂×C₄×C₄⋊C₄

—

C₂×#376

D₈₉

yes

characteristic

#1444

—

C₈₉

#13856

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₂²×C₄²

C₂×D₈₉

yes

—

C₄×C₄⋊C₄

—

178

—

#376

Dic₈₉

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₈₉

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

—

178

—

C₂×C₄²

Dic₁₇₈

yes

—

C₂×C₄²

D₃₅₆

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

—

#95

—

178

—

#95

C₂×Dic₈₉

yes

—

C₂⁴

C₂³×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×Dic₈₉

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

—

C₂²×C₄

C₂×Dic₁₇₈

yes

—

C₂²×C₄

C₂×Dic₁₇₈

yes

—

C₄⋊C₄

—

178

—

C₄²

—

356

—

C₄²

C₄⋊Dic₈₉

yes

complemented

100

C₂³

C₂³×Dic₈₉

yes

characteristic

101

C₂³

C₂²×D₃₅₆

yes

—

102

C₂³

C₂²×Dic₁₇₈

yes

—

103

C₂³

#41×D₈₉

yes

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

356

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

178

—

110

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

111

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

112

C₂×C₄

#15668009

yes

—

113

C₂×C₄

#15668012

yes

—

114

C₂²

C₂²×C₄⋊Dic₈₉

yes

—

115

C₂²

#41×Dic₈₉

yes

—

116

C₂²

#19152110

yes

—

117

C₂²

C₂×#15668009

yes

characteristic

118

C₂²

C₂×#15668012

yes

—

119

C₂²

#19152151

yes

—

120

C₄

—

121

C₄

—

356

—

122

C₄

#19151468

yes

—

123

C₂

C₂×#19151468

yes

characteristic

124

C₂

#26022736

yes

—

125

C₂

#26023533

yes

—

126

C₂

#26023545

yes

—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₈₉
commutator	C₂³×C₈₉	C₂³×C₄
Frattini	C₂⁴	C₂³×D₈₉
Fitting	C₂×#376×C₈₉	C₂
socle, prime radical, cosocle	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	#13856
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

subgroup

G/H

center

C₂³×C₄

C₂²×D₈₉

commutator

C₂³×C₈₉

C₂³×C₄

Frattini

C₂⁴

C₂³×D₈₉

Fitting

C₂×#376×C₈₉

C₂

socle, prime radical, cosocle

C₂⁴×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³×C₄ ◁ C₂×#376

N	H ≅ G/N	type
9. #376×C₈₉	C₄	semidirect
35. C₄²×C₈₉	C₄⋊C₄	semidirect
67. C₈₉	#13856	semidirect
72. #376	Dic₈₉	semidirect
99. C₄²	C₄⋊Dic₈₉	semidirect

H ≅ G/N

type

9. #376×C₈₉

C₄

semidirect

35. C₄²×C₈₉

C₄⋊C₄

semidirect

67. C₈₉

#13856

semidirect

72. #376

Dic₈₉

semidirect

99. C₄²

C₄⋊Dic₈₉

semidirect

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

subgroup	index
C₂×C₄×C₄⋊Dic₈₉	2
#26018517	2
C₂×#376×C₈₉	2
#13856	89

subgroup

index

C₂×C₄×C₄⋊Dic₈₉

#26018517

C₂×#376×C₈₉

#13856

Subgroups — #36476210

Conjugacy classes of subgroups

Complements

Maximal subgroups