Group #2507 — subgroups

Subgroups — #2507

Order 128.

198 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 112 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	#398	C₂	1	2	yes	maximal
2	#398	C₂	1	2	yes	maximal
3	#389	C₂	1	2	yes	maximal
4	#389	C₂	1	1	yes	maximal, characteristic
5	#389	C₂	1	1	yes	maximal, characteristic
6	#389	C₂	1	1	yes	maximal, characteristic
7	#389	C₂	1	1	yes	maximal, characteristic
8	#387	C₂	1	2	yes	maximal
9	#384	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₄²	C₂²	1	1	yes	characteristic
26	C₂×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₄²	C₂²	1	2	yes	—
28	#95	C₂²	1	1	yes	characteristic
29	#95	C₂²	1	1	yes	characteristic
30	#95	C₂²	1	2	yes	—
31	#95	C₂²	1	2	yes	—
32	#95	C₂²	1	2	yes	—
33	#95	C₂²	1	2	yes	—
34	#95	C₂²	1	2	yes	—
35	C₂²×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₄	C₂³	1	2	yes	—
37	C₂²×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₄	C₂³	1	2	yes	—
39	C₂²×C₄	C₂³	1	2	yes	—
40	C₂²×C₄	C₂³	1	1	yes	characteristic
41	C₂²×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₄	C₂³	1	2	yes	—
44	C₂²×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₄	C₂³	1	1	yes	characteristic
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	4	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄⋊C₄	—	4	2	no	—
53	C₄⋊C₄	—	4	2	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₄⋊C₄	Q₈	1	4	yes	—
57	C₄⋊C₄	Q₈	1	4	yes	—
58	C₄²	—	2	2	no	—
59	C₄²	—	2	2	no	—
60	C₄²	—	2	2	no	—
61	C₄²	—	4	2	no	—
62	C₂³	C₂⁴	1	1	yes	characteristic
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	C₂×Q₈	1	2	yes	—
89	C₂×C₄	C₂×Q₈	1	4	yes	—
90	C₂×C₄	C₂×Q₈	1	2	yes	—
91	C₂×C₄	C₂×Q₈	1	2	yes	—
92	C₂×C₄	C₂×Q₈	1	2	yes	—
93	C₂×C₄	#41	1	2	yes	—
94	C₂×C₄	#41	1	2	yes	—
95	C₂×C₄	#41	1	2	yes	—
96	C₂²	C₂²×Q₈	1	1	yes	characteristic
97	C₂²	C₂²×Q₈	1	1	yes	characteristic
98	C₂²	C₂×#41	1	1	yes	characteristic
99	C₂²	C₂×#41	1	1	yes	characteristic
100	C₂²	C₂×#41	1	1	yes	characteristic
101	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	2	4	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	2	no	—
110	C₄	—	4	2	no	—
111	C₄	—	4	1	no	—
112	C₄	—	4	2	no	—
113	C₄	#125	1	4	yes	—
114	C₂	C₂×#125	1	1	yes	characteristic
115	C₂	#533	1	1	yes	characteristic
116	C₂	#540	1	1	yes	characteristic
117	C₂	#553	1	1	yes	characteristic
118	C₂	#554	1	1	yes	characteristic
119	C₂	#554	1	1	yes	characteristic
120	C₂	Q₈²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#398

C₂

yes

maximal

#398

C₂

yes

maximal

#389

C₂

yes

maximal

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#387

C₂

yes

maximal

#384

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

—

#95

C₂²

yes

characteristic

#95

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₈

yes

—

C₄⋊C₄

Q₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂⁴

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂²

C₂²×Q₈

yes

characteristic

C₂²

C₂²×Q₈

yes

characteristic

C₂²

C₂×#41

yes

characteristic

C₂²

C₂×#41

yes

characteristic

100

C₂²

C₂×#41

yes

characteristic

101

C₂²

H(5,𝔽₂)

yes

characteristic

102

C₂²

H(5,𝔽₂)

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#125

yes

—

114

C₂

C₂×#125

yes

characteristic

115

C₂

#533

yes

characteristic

116

C₂

#540

yes

characteristic

117

C₂

#553

yes

characteristic

118

C₂

#554

yes

characteristic

119

C₂

#554

yes

characteristic

120

C₂

Q₈²

yes

characteristic

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

subgroup

G/H

center, commutator, Frattini, socle, prime radical, cosocle

C₂³

C₂⁴

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

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

subgroup	index
#398	2
#389	2
#387	2
#384	2
C₄×C₄⋊C₄	2

subgroup

index

C₄×C₄⋊C₄

Subgroups — #2507

Conjugacy classes of subgroups

Maximal subgroups