Group #26025625 — subgroups

Subgroups — #26025625

Order 11392.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×Dic₁₇₈	C₂	1	2	yes	maximal
2	#19151528	C₂	1	4	yes	maximal
3	#19151503	C₂	1	4	yes	maximal
4	#19151497	C₂	1	4	yes	maximal
5	#384×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₄×Dic₁₇₈	—	2	8	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂×#13083553	C₂²	1	4	yes	—
11	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
12	#15667919	C₂²	1	8	yes	—
13	#15667919	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	4	no	—
18	C₂×Dic₁₇₈	—	2	4	no	—
19	C₂×Dic₁₇₈	Q₈	1	8	yes	complemented
20	C₂²×Dic₈₉	C₂³	1	4	yes	—
21	C₂²×Dic₈₉	C₂³	1	4	yes	—
22	C₄⋊Dic₈₉	—	2	4	no	—
23	C₄⋊Dic₈₉	—	4	4	no	—
24	#13083553	—	2	8	no	—
25	C₄×Dic₈₉	—	2	8	no	—
26	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
29	C₄⋊C₄×C₈₉	—	2	8	no	—
30	C₄⋊C₄×C₈₉	—	4	2	no	—
31	C₄²×C₈₉	D₄	1	4	yes	—
32	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈₉	—	2	8	no	—
36	C₂×C₄×C₈₉	—	2	4	no	—
37	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
40	C₂×C₄×C₈₉	#41	1	4	yes	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	2	4	no	—
43	C₂×Dic₈₉	—	2	4	no	—
44	C₂×Dic₈₉	—	2	4	no	—
45	C₂×Dic₈₉	—	2	4	no	—
46	C₂×Dic₈₉	C₂×Q₈	1	8	yes	—
47	Dic₁₇₈	—	2	16	no	—
48	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
50	C₂²×C₈₉	C₂×#41	1	2	yes	—
51	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
53	C₄×C₈₉	—	2	2	no	—
54	C₄×C₈₉	—	4	4	no	—
55	C₄×C₈₉	C₂²⋊Q₈	1	8	yes	—
56	Dic₈₉	—	2	8	no	—
57	Dic₈₉	—	4	4	no	—
58	C₂×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
59	C₂×C₈₉	#537	1	1	yes	characteristic
60	C₂×C₈₉	#557	1	2	yes	—
61	C₂×C₈₉	Q₈²	1	1	yes	characteristic
62	C₂×C₈₉	#559	1	1	yes	characteristic
63	#2582	—	89	1	no	maximal, Sylow
64	C₈₉	#2582	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄×Q₈	—	89	2	no	—
66	#398	—	89	4	no	—
67	#391	—	89	4	no	—
68	#389	—	89	4	no	—
69	#384	D₈₉	1	1	yes	characteristic
70	C₂²×Q₈	—	89	2	no	—
71	C₄×Q₈	—	178	8	no	—
72	C₂×C₄⋊C₄	—	89	4	no	—
73	C₂×C₄⋊C₄	—	89	2	no	—
74	C₂×C₄⋊C₄	—	89	4	no	—
75	C₂×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
76	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
77	C₂×C₄²	—	89	4	no	—
78	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
79	#95	—	89	8	no	—
80	#95	—	89	4	no	—
81	C₂×Q₈	—	89	8	no	—
82	C₂×Q₈	—	178	4	no	—
83	C₂×Q₈	—	178	4	no	—
84	C₂²×C₄	—	89	4	no	—
85	C₂²×C₄	—	89	4	no	—
86	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
87	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	356	4	no	—
92	C₄⋊C₄	—	178	8	no	—
93	C₄⋊C₄	—	178	4	no	—
94	C₄²	—	178	8	no	—
95	C₄²	D₃₅₆	1	4	yes	—
96	C₂³	C₂³×D₈₉	1	1	yes	characteristic
97	Q₈	—	178	16	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	89	8	no	—
103	C₂×C₄	—	178	4	no	—
104	C₂×C₄	—	178	4	no	—
105	C₂×C₄	—	178	4	no	—
106	C₂×C₄	—	178	4	no	—
107	C₂×C₄	—	178	4	no	—
108	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
109	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
110	C₂×C₄	Q₈×D₈₉	1	4	yes	—
111	C₂×C₄	#13083573	1	4	yes	—
112	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
113	C₂²	C₂×Q₈×D₈₉	1	2	yes	—
114	C₂²	C₂×#13083573	1	2	yes	—
115	C₂²	#15668057	1	1	yes	characteristic
116	C₂²	#15668060	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	4	4	no	—
119	C₄	—	356	4	no	—
120	C₄	—	178	8	no	—
121	C₄	#15667992	1	8	yes	—
122	C₂	C₂×#15667992	1	2	yes	—
123	C₂	#19152179	1	1	yes	characteristic
124	C₂	#19152323	1	2	yes	—
125	C₂	#19152328	1	1	yes	characteristic
126	C₂	#19152334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₇₈

C₂

yes

maximal

#19151528

C₂

yes

maximal

#19151503

C₂

yes

maximal

#19151497

C₂

yes

maximal

#384×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₄×Dic₁₇₈

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

complemented

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×Q₈

yes

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

—

C₂×C₈₉

#537

yes

characteristic

C₂×C₈₉

#557

yes

—

C₂×C₈₉

Q₈²

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₄×Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

D₃₅₆

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

178

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

178

—

104

C₂×C₄

—

178

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

—

178

—

108

C₂×C₄

C₂×D₃₅₆

yes

—

109

C₂×C₄

C₂×D₃₅₆

yes

—

110

C₂×C₄

Q₈×D₈₉

yes

—

111

C₂×C₄

#13083573

yes

—

112

C₂²

C₂²×D₃₅₆

yes

characteristic

113

C₂²

C₂×Q₈×D₈₉

yes

—

114

C₂²

C₂×#13083573

yes

—

115

C₂²

#15668057

yes

characteristic

116

C₂²

#15668060

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

356

—

120

C₄

—

178

—

121

C₄

#15667992

yes

—

122

C₂

C₂×#15667992

yes

—

123

C₂

#19152179

yes

characteristic

124

C₂

#19152323

yes

—

125

C₂

#19152328

yes

characteristic

126

C₂

#19152334

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#384×C₈₉	C₂
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2582
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#384×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
19. C₂×Dic₁₇₈	Q₈	semidirect
64. C₈₉	#2582	semidirect

H ≅ G/N

type

19. C₂×Dic₁₇₈

Q₈

semidirect

64. C₈₉

#2582

semidirect

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

subgroup	index
C₂×C₄×Dic₁₇₈	2
#19151528	2
#19151503	2
#19151497	2
#384×C₈₉	2
#2582	89

subgroup

index

C₂×C₄×Dic₁₇₈

#384×C₈₉

Subgroups — #26025625

Conjugacy classes of subgroups

Complements

Maximal subgroups