Group #26030667 — subgroups

Subgroups — C₂×#19151977 #26030667

Order 11392.

552 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×#13083564	C₂	1	4	yes	maximal
2	C₂×#15668012	C₂	1	1	yes	maximal, characteristic
3	#19151977	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×#13083564	—	2	16	no	—
8	C₂×#13083564	C₂²	1	16	yes	—
9	C₂²×#10623494	C₂²	1	2	yes	—
10	#15668012	—	2	2	no	—
11	#15668012	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₈₉	—	2	2	no	—
13	C₄×#10623494	C₂²	1	4	yes	—
14	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₈₉	—	2	2	no	—
16	C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₈	—	2	8	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	C₂³	1	4	yes	—
22	C₂²×Dic₈₉	—	2	2	no	—
23	#13083564	—	2	32	no	—
24	C₂×#10623494	D₄	1	8	yes	—
25	C₂×#10623494	C₂³	1	4	yes	—
26	C₄⋊Dic₈₉	—	2	8	no	—
27	C₂×Q₈×C₈₉	—	2	2	no	—
28	C₂×Q₈×C₈₉	—	2	8	no	—
29	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₈₉	—	2	2	no	—
31	C₂²×C₄×C₈₉	D₄	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	—	2	8	no	—
34	C₄²×C₈₉	D₄	1	2	yes	—
35	C₄²×C₈₉	C₂³	1	2	yes	—
36	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₈₉	—	2	8	no	—
38	Q₈×C₈₉	—	4	4	no	—
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	—	4	4	no	—
41	C₂×C₄×C₈₉	Q₁₆	1	8	yes	complemented
42	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	—	4	4	no	—
48	Dic₁₇₈	—	2	8	no	—
49	Dic₁₇₈	—	4	4	no	—
50	#10623494	C₂×D₄	1	8	yes	—
51	C₂²×C₈₉	#127	1	2	yes	—
52	C₂²×C₈₉	C₂×Q₁₆	1	4	yes	—
53	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₈₉	—	4	4	no	—
55	C₄×C₈₉	#127	1	2	yes	—
56	C₄×C₈₉	C₂×Q₁₆	1	8	yes	complemented
57	C₄×C₈₉	C₂²×D₄	1	2	yes	—
58	Dic₈₉	—	4	4	no	—
59	C₂×C₈₉	#494	1	4	yes	complemented
60	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
61	C₂×C₈₉	C₂²×Q₁₆	1	2	yes	—
62	C₂×#494	—	89	1	no	maximal, Sylow
63	C₈₉	C₂×#494	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	89	4	no	—
65	C₂×C₄⋊Q₈	—	89	1	no	—
66	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
67	#494	—	89	8	no	—
68	C₂×C₄×C₈	—	89	1	no	—
69	C₂²×Q₈	—	89	2	no	—
70	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
71	C₂×Q₁₆	—	89	16	no	—
72	C₂×Q₁₆	—	178	16	no	—
73	C₂²×C₈	—	89	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	89	4	no	—
76	C₄⋊Q₈	—	178	2	no	—
77	C₄⋊Q₈	C₂×D₈₉	1	4	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	178	2	no	—
80	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
81	C₄×C₈	—	89	4	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	8	no	—
84	C₂×Q₈	—	89	4	no	—
85	C₂×Q₈	—	178	8	no	—
86	C₂×Q₈	—	178	2	no	—
87	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	178	2	no	—
90	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	Q₁₆	—	178	32	no	—
93	C₂×C₈	—	89	8	no	—
94	C₂×C₈	—	89	4	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	178	8	no	—
97	C₄²	C₂²⋊D₈₉	1	2	yes	—
98	C₄²	C₂²×D₈₉	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	Q₈	—	4	4	no	—
102	Q₈	—	356	4	no	—
103	Q₈	—	178	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	356	4	no	—
107	C₂×C₄	—	178	4	no	—
108	C₂×C₄	#13083564	1	8	yes	complemented
109	C₂×C₄	D₄×D₈₉	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
112	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
113	C₈	—	89	8	no	—
114	C₂²	#15668008	1	2	yes	—
115	C₂²	C₂×#13083564	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	356	4	no	—
119	C₄	#15668008	1	2	yes	—
120	C₄	C₂×#13083564	1	8	yes	complemented
121	C₄	C₂×D₄×D₈₉	1	2	yes	—
122	C₂	#19151977	1	4	yes	direct product factor
123	C₂	C₂×#15668008	1	1	yes	characteristic
124	C₂	C₂²×#13083564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083564

C₂

yes

maximal

C₂×#15668012

C₂

yes

maximal, characteristic

#19151977

C₂

yes

maximal, direct product factor

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×#13083564

—

C₂×#13083564

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15668012

—

#15668012

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

C₄×#10623494

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

#13083564

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

#127

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#127

yes

—

C₄×C₈₉

C₂×Q₁₆

yes

complemented

C₄×C₈₉

C₂²×D₄

yes

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

C₂²×Q₁₆

yes

—

C₂×#494

—

maximal, Sylow

C₈₉

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₈₉

yes

characteristic

#494

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²×D₈₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

C₂²⋊D₈₉

yes

—

C₄²

C₂²×D₈₉

yes

—

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

356

—

103

Q₈

—

178

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

178

—

108

C₂×C₄

#13083564

yes

complemented

109

C₂×C₄

D₄×D₈₉

yes

—

110

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

112

C₂×C₄

C₂³×D₈₉

yes

characteristic

113

C₈

—

114

C₂²

#15668008

yes

—

115

C₂²

C₂×#13083564

yes

—

116

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

117

C₄

—

118

C₄

—

356

—

119

C₄

#15668008

yes

—

120

C₄

C₂×#13083564

yes

complemented

121

C₄

C₂×D₄×D₈₉

yes

—

122

C₂

#19151977

yes

direct product factor

123

C₂

C₂×#15668008

yes

characteristic

124

C₂

C₂²×#13083564

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	C₂×#494
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₈₉

hypocenter

C₈₉

C₂×#494

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #19151977	C₂	direct (N × H)
41. C₂×C₄×C₈₉	Q₁₆	semidirect
56. C₄×C₈₉	C₂×Q₁₆	semidirect
59. C₂×C₈₉	#494	semidirect
63. C₈₉	C₂×#494	semidirect
108. C₂×C₄	#13083564	semidirect
120. C₄	C₂×#13083564	semidirect
122. C₂	#19151977	direct (N × H)

H ≅ G/N

type

3. #19151977

C₂

direct (N × H)

41. C₂×C₄×C₈₉

Q₁₆

semidirect

56. C₄×C₈₉

C₂×Q₁₆

semidirect

59. C₂×C₈₉

#494

semidirect

63. C₈₉

C₂×#494

semidirect

108. C₂×C₄

#13083564

semidirect

120. C₄

C₂×#13083564

semidirect

122. C₂

#19151977

direct (N × H)

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

subgroup	index
C₂²×#13083564	2
C₂×#15668012	2
#19151977	2
C₂×C₄×#10623494	2
C₂×C₄⋊Q₈×C₈₉	2
C₂×#494	89

subgroup

index

C₂×C₄⋊Q₈×C₈₉

Subgroups — C₂×#19151977 #26030667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151977 #26030667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151977 #26030667