Group #36519992 — subgroups

Subgroups — #36519992

Order 22784.

1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 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	#26018523	C₂	1	2	yes	maximal
2	#26018522	C₂	1	12	yes	maximal
3	C₈₉×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₇₈	C₂²	1	4	yes	—
5	C₂×#15667919	C₂²	1	12	yes	—
6	C₂×#15667919	C₂²	1	12	yes	—
7	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₂×#95×C₈₉	C₂²	1	6	yes	—
9	C₂²×Dic₁₇₈	—	4	12	no	—
10	C₂²×Dic₁₇₈	—	4	12	no	—
11	C₂²×Dic₁₇₈	Q₈	1	16	yes	complemented
12	C₂³×Dic₈₉	C₂³	1	8	yes	—
13	#15667919	—	4	12	no	—
14	#15667919	—	4	12	no	—
15	C₂²×Q₈×C₈₉	—	4	6	no	—
16	C₂²×Q₈×C₈₉	Q₈	1	4	yes	complemented
17	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
18	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
19	#95×C₈₉	—	4	6	no	—
20	C₂×Dic₁₇₈	—	4	48	no	—
21	C₂×Dic₁₇₈	—	4	48	no	—
22	C₂×Dic₁₇₈	—	4	16	no	—
23	C₂²×Dic₈₉	—	2	24	no	—
24	C₂²×Dic₈₉	—	2	24	no	—
25	C₂²×Dic₈₉	C₂×Q₈	1	16	yes	—
26	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
27	C₂×Q₈×C₈₉	—	4	12	no	—
28	C₂×Q₈×C₈₉	—	4	16	no	—
29	C₂²×C₄×C₈₉	—	2	12	no	—
30	C₂²×C₄×C₈₉	—	2	12	no	—
31	C₂²×C₄×C₈₉	—	2	6	no	—
32	C₂²×C₄×C₈₉	—	2	12	no	—
33	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
34	C₂²×C₄×C₈₉	C₂×Q₈	1	6	yes	—
35	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
36	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
37	C₂³×C₈₉	H(5,𝔽₂)	1	6	yes	—
38	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
39	Q₈×C₈₉	—	4	16	no	—
40	C₂×C₄×C₈₉	—	2	6	no	—
41	C₂×C₄×C₈₉	—	2	12	no	—
42	C₂×C₄×C₈₉	—	2	24	no	—
43	C₂×C₄×C₈₉	—	4	12	no	—
44	C₂×C₄×C₈₉	—	4	12	no	—
45	C₂×C₄×C₈₉	—	4	4	no	—
46	C₂×Dic₈₉	—	2	48	no	—
47	C₂×Dic₈₉	—	4	24	no	—
48	C₂×Dic₈₉	—	4	8	no	—
49	Dic₁₇₈	—	4	64	no	—
50	C₂²×C₈₉	#543	1	12	yes	—
51	C₂²×C₈₉	#543	1	3	yes	—
52	C₂²×C₈₉	Q₈²	1	6	yes	—
53	C₂²×C₈₉	Q₈²	1	4	yes	—
54	C₂²×C₈₉	H(3,𝔽₄)	1	6	yes	—
55	C₂²×C₈₉	H(3,𝔽₄)	1	4	yes	—
56	C₄×C₈₉	—	4	8	no	—
57	C₄×C₈₉	—	4	6	no	—
58	Dic₈₉	—	4	16	no	—
59	#17309	—	89	1	no	maximal, Sylow
60	C₂×C₈₉	#2811	1	4	yes	—
61	C₂×C₈₉	#2811	1	6	yes	—
62	C₂×C₈₉	#2840	1	4	yes	—
63	C₂×C₈₉	#2840	1	1	yes	characteristic
64	#1446	—	89	12	no	—
65	#1446	—	89	2	no	—
66	#1446	D₈₉	1	1	yes	characteristic
67	C₈₉	#17309	1	1	yes	characteristic, complemented, Sylow
68	C₂³×Q₈	—	89	4	no	—
69	C₂³×Q₈	C₂×D₈₉	1	1	yes	characteristic
70	C₂×#95	—	89	12	no	—
71	C₂×#95	—	89	12	no	—
72	C₂×#95	C₂×D₈₉	1	6	yes	—
73	C₂²×Q₈	—	4	6	no	—
74	C₂²×Q₈	—	89	16	no	—
75	C₂²×Q₈	—	356	12	no	—
76	C₂²×Q₈	—	356	12	no	—
77	C₂²×Q₈	Dic₁₇₈	1	4	yes	complemented
78	C₂³×C₄	—	89	8	no	—
79	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
80	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
81	#95	—	4	6	no	—
82	#95	—	356	12	no	—
83	#95	—	356	12	no	—
84	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
85	C₂×Q₈	—	4	12	no	—
86	C₂×Q₈	—	4	16	no	—
87	C₂×Q₈	—	356	48	no	—
88	C₂×Q₈	—	356	48	no	—
89	C₂×Q₈	—	356	16	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	2	12	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	12	no	—
94	C₂²×C₄	—	89	16	no	—
95	C₂²×C₄	—	178	24	no	—
96	C₂²×C₄	—	178	24	no	—
97	C₂²×C₄	Q₈×D₈₉	1	8	yes	—
98	C₂²×C₄	C₂×Dic₁₇₈	1	6	yes	—
99	C₂³	C₂×Q₈×D₈₉	1	4	yes	—
100	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
101	C₂³	#15668057	1	6	yes	—
102	C₂³	#15668058	1	4	yes	—
103	Q₈	—	4	16	no	—
104	Q₈	—	356	64	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	4	12	no	—
109	C₂×C₄	—	4	12	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	356	24	no	—
112	C₂×C₄	—	356	8	no	—
113	C₂×C₄	—	178	48	no	—
114	C₂²	#19152223	1	12	yes	—
115	C₂²	#19152224	1	3	yes	—
116	C₂²	#19152328	1	6	yes	—
117	C₂²	Q₈×Dic₁₇₈	1	4	yes	—
118	C₂²	#19152340	1	6	yes	—
119	C₂²	#19152341	1	4	yes	—
120	C₄	—	4	8	no	—
121	C₄	—	4	6	no	—
122	C₄	—	356	16	no	—
123	C₂	#26028244	1	4	yes	—
124	C₂	#26028245	1	6	yes	—
125	C₂	#26028504	1	4	yes	—
126	C₂	#26028505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26018523

C₂

yes

maximal

#26018522

C₂

yes

maximal

C₈₉×#1446

C₂

yes

maximal, characteristic

C₂³×Dic₁₇₈

C₂²

yes

—

C₂×#15667919

C₂²

yes

—

C₂×#15667919

C₂²

yes

—

C₂³×Q₈×C₈₉

C₂²

yes

characteristic

C₂×#95×C₈₉

C₂²

yes

—

C₂²×Dic₁₇₈

—

C₂²×Dic₁₇₈

—

C₂²×Dic₁₇₈

Q₈

yes

complemented

C₂³×Dic₈₉

C₂³

yes

—

#15667919

—

#15667919

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

Q₈

yes

complemented

C₂³×C₄×C₈₉

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

#95×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

C₂×Q₈

yes

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂²×Q₈

yes

—

C₂³×C₈₉

C₂²×Q₈

yes

characteristic

C₂³×C₈₉

H(5,𝔽₂)

yes

—

C₂³×C₈₉

H(5,𝔽₂)

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

#543

yes

—

C₂²×C₈₉

#543

yes

—

C₂²×C₈₉

Q₈²

yes

—

C₂²×C₈₉

Q₈²

yes

—

C₂²×C₈₉

H(3,𝔽₄)

yes

—

C₂²×C₈₉

H(3,𝔽₄)

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

#17309

—

maximal, Sylow

C₂×C₈₉

#2811

yes

—

C₂×C₈₉

#2811

yes

—

C₂×C₈₉

#2840

yes

—

C₂×C₈₉

yes

—

—

D₈₉

yes

characteristic

C₈₉

#17309

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

C₂³×Q₈

C₂×D₈₉

yes

characteristic

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₈₉

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

356

—

C₂²×Q₈

—

356

—

C₂²×Q₈

Dic₁₇₈

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

—

#95

—

#95

—

356

—

#95

—

356

—

C₂⁴

C₂³×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

Q₈×D₈₉

yes

—

C₂²×C₄

C₂×Dic₁₇₈

yes

—

C₂³

C₂×Q₈×D₈₉

yes

—

100

C₂³

C₂²×Dic₁₇₈

yes

characteristic

101

C₂³

#15668057

yes

—

102

C₂³

#15668058

yes

—

103

Q₈

—

104

Q₈

—

356

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

356

—

113

C₂×C₄

—

178

—

114

C₂²

#19152223

yes

—

115

C₂²

#19152224

yes

—

116

C₂²

#19152328

yes

—

117

C₂²

Q₈×Dic₁₇₈

yes

—

118

C₂²

#19152340

yes

—

119

C₂²

#19152341

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

356

—

123

C₂

#26028244

yes

—

124

C₂

#26028245

yes

—

125

C₂

#26028504

yes

—

126

C₂

#26028505

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂⁴×C₈₉	C₂⁴
Fitting	C₈₉×#1446	C₂
hypercenter	#1446	D₈₉
hypocenter	C₈₉	#17309
LCS	G ▷ C₂⁴×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂⁴×C₈₉

C₂⁴

Fitting

C₈₉×#1446

C₂

D₈₉

C₈₉

G ▷ C₂⁴×C₈₉ ▷ C₈₉

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
11. C₂²×Dic₁₇₈	Q₈	semidirect
16. C₂²×Q₈×C₈₉	Q₈	semidirect
67. C₈₉	#17309	semidirect
77. C₂²×Q₈	Dic₁₇₈	semidirect

H ≅ G/N

type

11. C₂²×Dic₁₇₈

Q₈

semidirect

16. C₂²×Q₈×C₈₉

Q₈

semidirect

67. C₈₉

#17309

semidirect

77. C₂²×Q₈

Dic₁₇₈

semidirect

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

subgroup	index
#26018523	2
#26018522	2
C₈₉×#1446	2
#17309	89

subgroup

index

#26018523

#26018522

C₈₉×#1446

#17309

Subgroups — #36519992

Conjugacy classes of subgroups

Complements

Maximal subgroups