Group #2499 — subgroups

Subgroups — #2499

Order 128.

242 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 104 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#117	C₂	1	1	yes	maximal, characteristic
3	#397	C₂	1	4	yes	maximal
4	#391	C₂	1	1	yes	maximal, characteristic, complemented
5	#391	C₂	1	1	yes	maximal, characteristic, complemented
6	#387	C₂	1	4	yes	maximal, complemented
7	#384	C₂	1	1	yes	maximal, characteristic, complemented
8	#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₄	C₂²	1	1	yes	characteristic
12	C₂²⋊Q₈	—	2	4	no	—
13	#117	—	2	4	no	—
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²	C₂²	1	2	yes	—
23	C₂×C₄²	C₂²	1	1	yes	characteristic
24	C₂×C₄²	C₂²	1	1	yes	characteristic
25	#95	C₂²	1	2	yes	—
26	#95	C₂²	1	4	yes	—
27	#95	C₂²	1	4	yes	—
28	#95	C₂²	1	4	yes	—
29	C₂⁴	C₂³	1	1	yes	characteristic
30	C₂×Q₈	—	2	2	no	—
31	C₂×Q₈	—	2	2	no	—
32	C₂×Q₈	—	2	2	no	—
33	C₂²×C₄	—	2	2	no	—
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	D₄	1	4	yes	—
36	C₂²×C₄	C₂³	1	2	yes	—
37	C₂²×C₄	C₂³	1	2	yes	—
38	C₂²×C₄	C₂³	1	2	yes	—
39	C₂²×C₄	C₂³	1	2	yes	—
40	C₂²×C₄	C₂³	1	2	yes	—
41	C₂²×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₄	C₂³	1	2	yes	—
43	C₂²×C₄	C₂³	1	1	yes	characteristic
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	2	2	no	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	4	no	—
48	C₄⋊C₄	—	2	8	no	—
49	C₂²⋊C₄	—	2	4	no	—
50	C₂²⋊C₄	—	2	4	no	—
51	C₂²⋊C₄	—	2	4	no	—
52	C₄²	—	2	4	no	—
53	C₄²	—	2	2	no	—
54	C₄²	—	2	2	no	—
55	C₂³	—	2	2	no	—
56	C₂³	—	2	1	no	—
57	C₂³	—	2	1	no	—
58	C₂³	—	2	2	no	—
59	C₂³	C₂×D₄	1	2	yes	—
60	C₂³	C₂⁴	1	1	yes	characteristic
61	Q₈	—	4	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	8	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	C₂×D₄	1	2	yes	—
80	C₂×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₄	#41	1	4	yes	—
82	C₂×C₄	#41	1	2	yes	—
83	C₂×C₄	#41	1	2	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	—	4	2	no	—
89	C₂²	C₂²×D₄	1	1	yes	characteristic
90	C₂²	C₂×#41	1	1	yes	characteristic
91	C₂²	C₂×#41	1	1	yes	characteristic
92	C₂²	C₂×#41	1	2	yes	—
93	C₂²	2₋¹⁺⁴	1	2	yes	—
94	C₄	—	2	2	no	—
95	C₄	—	2	4	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	4	2	no	—
99	C₄	—	4	2	no	—
100	C₄	—	4	2	no	—
101	C₄	#123	1	4	yes	—
102	C₂	—	4	2	no	—
103	C₂	C₂×#123	1	1	yes	characteristic
104	C₂	#532	1	1	yes	characteristic
105	C₂	#536	1	1	yes	characteristic
106	C₂	#552	1	2	yes	—
107	C₂	#556	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂×#117

C₂

yes

maximal, characteristic

#397

C₂

yes

maximal

#391

C₂

yes

maximal, characteristic, complemented

#391

C₂

yes

maximal, characteristic, complemented

#387

C₂

yes

maximal, complemented

#384

C₂

yes

maximal, characteristic, complemented

#382

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊C₄

C₂

yes

maximal, characteristic

C₂²×Q₈

C₂²

yes

characteristic

C₂³×C₄

C₂²

yes

characteristic

C₂²⋊Q₈

—

#117

—

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×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂⁴

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄

yes

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

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×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₄

yes

characteristic

C₂²

C₂×#41

yes

characteristic

C₂²

C₂×#41

yes

characteristic

C₂²

C₂×#41

yes

—

C₂²

2₋¹⁺⁴

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

#123

yes

—

102

C₂

—

103

C₂

C₂×#123

yes

characteristic

104

C₂

#532

yes

characteristic

105

C₂

#536

yes

characteristic

106

C₂

#552

yes

—

107

C₂

#556

yes

—

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

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂³

C₂⁴

prime radical

C₂⁴

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
4. #391	C₂	semidirect
5. #391	C₂	semidirect
6. #387	C₂	semidirect
7. #384	C₂	semidirect
8. #382	C₂	semidirect

H ≅ G/N

type

4. #391

C₂

semidirect

5. #391

C₂

semidirect

6. #387

C₂

semidirect

7. #384

C₂

semidirect

8. #382

C₂

semidirect

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

subgroup	index
C₂×C₂²⋊Q₈	2
C₂×#117	2
#397	2
#391	2
#387	2
#384	2
#382	2
C₄×C₂²⋊C₄	2

subgroup

index

C₂×C₂²⋊Q₈

C₄×C₂²⋊C₄

Subgroups — #2499

Conjugacy classes of subgroups

Complements

Maximal subgroups