Group #34002 — subgroups

Subgroups — C₄×#519 #34002

Order 256.

1701 conjugacy classes of subgroups; only the 1519 normal subgroups are shown.

normal subgroups only

This group has 1701 conjugacy classes of subgroups — too many to draw. Only its 1519 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×#519	C₂	1	2	yes	maximal
2	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	#2276	C₂	1	8	yes	maximal, complemented
4	#2274	C₂	1	6	yes	maximal
5	C₄²×Q₈	C₂	1	8	yes	maximal, complemented
6	C₄×#117	C₂	1	6	yes	maximal
7	C₂³×Q₈	C₂²	1	1	yes	characteristic
8	#519	C₄	1	32	yes	direct product factor
9	C₂×C₄×Q₈	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈	C₂²	1	4	yes	—
11	C₂×C₄×Q₈	C₂²	1	4	yes	—
12	C₂×C₄×Q₈	C₂²	1	16	yes	—
13	C₂×#117	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₄²	C₂²	1	3	yes	—
16	C₄×C₄⋊C₄	C₂²	1	12	yes	—
17	C₄×C₄⋊C₄	C₂²	1	12	yes	—
18	C₄×C₄⋊C₄	C₂²	1	48	yes	—
19	C₄×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₄×C₂²⋊C₄	C₂²	1	12	yes	—
21	#376	C₂²	1	12	yes	—
22	C₄³	C₂²	1	12	yes	—
23	C₂²×Q₈	C₂×C₄	1	8	yes	—
24	C₂²×Q₈	C₂×C₄	1	16	yes	—
25	C₂²×Q₈	C₂³	1	4	yes	—
26	C₂³×C₄	C₂³	1	1	yes	characteristic
27	C₂³×C₄	C₂³	1	3	yes	—
28	C₂³×C₄	C₂³	1	3	yes	—
29	C₄×Q₈	C₂×C₄	1	32	yes	—
30	C₄×Q₈	C₂×C₄	1	32	yes	—
31	C₄×Q₈	C₂×C₄	1	128	yes	complemented
32	#117	C₂×C₄	1	96	yes	—
33	C₂×C₄⋊C₄	C₂×C₄	1	24	yes	—
34	C₂×C₄⋊C₄	C₂³	1	12	yes	—
35	C₂×C₄⋊C₄	C₂³	1	24	yes	—
36	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
37	C₂×C₄²	C₂×C₄	1	24	yes	—
38	C₂×C₄²	C₂³	1	8	yes	—
39	C₂×C₄²	C₂³	1	24	yes	—
40	C₂×C₄²	C₂³	1	24	yes	—
41	C₂×C₄²	C₂³	1	6	yes	—
42	C₂×C₄²	C₂³	1	6	yes	—
43	#95	C₂³	1	8	yes	—
44	#95	C₂³	1	24	yes	—
45	C₂⁴	C₂⁴	1	1	yes	characteristic
46	C₂×Q₈	C₄²	1	64	yes	complemented
47	C₂×Q₈	C₂²×C₄	1	16	yes	—
48	C₂×Q₈	C₂²×C₄	1	32	yes	—
49	C₂²×C₄	—	2	4	no	—
50	C₂²×C₄	—	4	6	no	—
51	C₂²×C₄	C₂²×C₄	1	4	yes	—
52	C₂²×C₄	C₂²×C₄	1	12	yes	—
53	C₂²×C₄	C₂²×C₄	1	12	yes	—
54	C₂²×C₄	C₂²×C₄	1	24	yes	—
55	C₂²×C₄	C₂⁴	1	1	yes	characteristic
56	C₂²×C₄	C₂⁴	1	16	yes	—
57	C₂²×C₄	C₂⁴	1	6	yes	—
58	C₂²×C₄	C₂⁴	1	6	yes	—
59	C₂²×C₄	C₂⁴	1	1	yes	characteristic
60	C₄⋊C₄	C₂²×C₄	1	48	yes	—
61	C₄⋊C₄	C₂²×C₄	1	96	yes	—
62	C₂²⋊C₄	C₂²×C₄	1	32	yes	—
63	C₄²	—	2	16	no	—
64	C₄²	—	4	24	no	—
65	C₄²	C₂²×C₄	1	24	yes	—
66	C₄²	C₂²×C₄	1	24	yes	—
67	C₄²	C₂²×C₄	1	96	yes	—
68	C₂³	—	2	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	C₂³×C₄	1	4	yes	—
71	C₂³	C₂³×C₄	1	2	yes	—
72	C₂³	C₂⁵	1	1	yes	characteristic
73	Q₈	C₂×C₄²	1	64	yes	—
74	C₂×C₄	—	2	16	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	16	no	—
77	C₂×C₄	—	2	12	no	—
78	C₂×C₄	—	2	8	no	—
79	C₂×C₄	—	4	12	no	—
80	C₂×C₄	C₂×C₄²	1	48	yes	—
81	C₂×C₄	C₂³×C₄	1	2	yes	—
82	C₂×C₄	C₂³×C₄	1	32	yes	—
83	C₂×C₄	C₂³×C₄	1	12	yes	—
84	C₂×C₄	C₂³×C₄	1	2	yes	—
85	C₂×C₄	C₂³×C₄	1	12	yes	—
86	C₂×C₄	C₂³×C₄	1	24	yes	—
87	C₂×C₄	2₋¹⁺⁴	1	4	yes	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	2	8	no	—
91	C₂²	C₂²×C₄²	1	4	yes	—
92	C₂²	C₂⁴×C₄	1	1	yes	characteristic
93	C₂²	C₂⁴×C₄	1	2	yes	—
94	C₂²	C₂×2₋¹⁺⁴	1	2	yes	—
95	C₂²	#585	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	2	12	no	—
98	C₄	—	2	32	no	—
99	C₄	C₂²×C₄²	1	24	yes	—
100	C₄	#519	1	4	yes	direct product factor
101	C₂	—	2	4	no	—
102	C₂	C₂³×C₄²	1	1	yes	characteristic
103	C₂	C₂×#519	1	1	yes	characteristic
104	C₂	#3428	1	1	yes	characteristic
105	C₂	C₄×2₋¹⁺⁴	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#519

C₂

yes

maximal

C₂²×C₄×Q₈

C₂

yes

maximal, characteristic

#2276

C₂

yes

maximal, complemented

#2274

C₂

yes

maximal

C₄²×Q₈

C₂

yes

maximal, complemented

C₄×#117

C₂

yes

maximal

C₂³×Q₈

C₂²

yes

characteristic

#519

C₄

yes

direct product factor

C₂×C₄×Q₈

C₄

yes

complemented

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

C₂×#117

C₂²

yes

—

C₂²×C₄⋊C₄

C₂²

yes

—

C₂²×C₄²

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₂²⋊C₄

C₂²

yes

—

C₄×C₂²⋊C₄

C₂²

yes

—

#376

C₂²

yes

—

C₄³

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×Q₈

C₂³

yes

—

C₂³×C₄

C₂³

yes

characteristic

C₂³×C₄

C₂³

yes

—

C₂³×C₄

C₂³

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂×C₄

128

yes

complemented

#117

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

#95

C₂³

yes

—

#95

C₂³

yes

—

C₂⁴

yes

characteristic

C₂×Q₈

C₄²

yes

complemented

C₂×Q₈

C₂²×C₄

yes

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₄⋊C₄

C₂²×C₄

yes

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

C₂²×C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₂³

—

C₂³

—

C₂³

C₂³×C₄

yes

—

C₂³

C₂³×C₄

yes

—

C₂³

C₂⁵

yes

characteristic

Q₈

C₂×C₄²

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

—

C₂×C₄

C₂³×C₄

yes

—

C₂×C₄

C₂³×C₄

yes

—

C₂×C₄

C₂³×C₄

yes

—

C₂×C₄

C₂³×C₄

yes

—

C₂×C₄

C₂³×C₄

yes

—

C₂×C₄

C₂³×C₄

yes

—

C₂×C₄

2₋¹⁺⁴

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×C₄²

yes

—

C₂²

C₂⁴×C₄

yes

characteristic

C₂²

C₂⁴×C₄

yes

—

C₂²

C₂×2₋¹⁺⁴

yes

—

C₂²

#585

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂²×C₄²

yes

—

100

C₄

#519

yes

direct product factor

101

C₂

—

102

C₂

C₂³×C₄²

yes

characteristic

103

C₂

C₂×#519

yes

characteristic

104

C₂

#3428

yes

characteristic

105

C₂

C₄×2₋¹⁺⁴

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂³×C₄²
Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂⁴

commutator

C₂

C₂³×C₄²

Frattini, socle, cosocle

C₂³

C₂⁵

C₂⁴

G ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₄ ◁ G

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
3. #2276	C₂	semidirect
5. C₄²×Q₈	C₂	semidirect
8. #519	C₄	direct (N × H)
9. C₂×C₄×Q₈	C₄	semidirect
31. C₄×Q₈	C₂×C₄	semidirect
46. C₂×Q₈	C₄²	semidirect
100. C₄	#519	direct (N × H)

H ≅ G/N

type

3. #2276

C₂

semidirect

5. C₄²×Q₈

C₂

semidirect

8. #519

C₄

direct (N × H)

9. C₂×C₄×Q₈

C₄

semidirect

31. C₄×Q₈

C₂×C₄

semidirect

46. C₂×Q₈

C₄²

semidirect

100. C₄

#519

direct (N × H)

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

subgroup	index
C₂×#519	2
C₂²×C₄×Q₈	2
#2276	2
#2274	2
C₄²×Q₈	2
C₄×#117	2

subgroup

index

C₂²×C₄×Q₈

C₄²×Q₈

Subgroups — C₄×#519 #34002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#519 #34002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#519 #34002