Subgroups — C₂×#540 #3452

Order 128.

630 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#540	C₂	1	16	yes	maximal, direct product factor
2	C₂×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#123	C₂	1	4	yes	maximal, complemented
5	C₂×#121	C₂	1	4	yes	maximal, complemented
6	C₂×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₄	C₂	1	2	yes	maximal, complemented
8	C₂×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄	C₂²	1	2	yes	—
10	C₂²×D₄	C₂²	1	4	yes	—
11	C₂²×D₄	C₂²	1	4	yes	—
12	C₂³×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₄	C₂²	1	4	yes	—
14	#127	C₂²	1	8	yes	complemented
15	#125	C₂²	1	8	yes	complemented
16	#123	C₂²	1	32	yes	complemented
17	#121	C₂²	1	32	yes	complemented
18	#121	C₂²	1	16	yes	complemented
19	C₄×D₄	C₂²	1	16	yes	complemented
20	#117	C₂²	1	8	yes	complemented
21	C₂×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₄²	C₂²	1	1	yes	characteristic
29	C₂⁴	C₂³	1	1	yes	characteristic
30	C₂⁴	C₂³	1	4	yes	—
31	C₂×D₄	—	2	8	no	—
32	C₂×D₄	—	4	8	no	—
33	C₂×D₄	—	4	8	no	—
34	C₂×D₄	C₂³	1	8	yes	—
35	C₂×D₄	C₂³	1	16	yes	—
36	C₂×D₄	C₂³	1	16	yes	—
37	C₂²×C₄	—	2	4	no	—
38	C₂²×C₄	—	4	8	no	—
39	C₂²×C₄	C₂³	1	4	yes	—
40	C₂²×C₄	C₂³	1	2	yes	—
41	C₂²×C₄	C₂³	1	2	yes	—
42	C₂²×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₄	C₂³	1	4	yes	—
44	C₂²×C₄	C₂³	1	16	yes	—
45	C₂²×C₄	C₂³	1	1	yes	characteristic
46	C₄⋊C₄	C₂³	1	8	yes	—
47	C₄⋊C₄	C₂³	1	8	yes	—
48	C₄⋊C₄	C₂³	1	16	yes	—
49	C₂²⋊C₄	C₂³	1	8	yes	—
50	C₂²⋊C₄	C₂³	1	16	yes	—
51	C₂²⋊C₄	C₂³	1	16	yes	—
52	C₄²	C₂³	1	4	yes	—
53	C₄²	C₂³	1	4	yes	—
54	C₂³	—	2	2	no	—
55	C₂³	—	2	8	no	—
56	C₂³	—	2	8	no	—
57	C₂³	—	2	4	no	—
58	C₂³	—	2	8	no	—
59	C₂³	C₂⁴	1	2	yes	—
60	C₂³	C₂⁴	1	8	yes	—
61	C₂³	C₂⁴	1	1	yes	characteristic
62	D₄	—	2	16	no	—
63	D₄	—	4	16	no	—
64	D₄	—	4	16	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	2	8	no	—
70	C₂×C₄	—	4	16	no	—
71	C₂×C₄	#41	1	4	yes	—
72	C₂×C₄	C₂⁴	1	4	yes	—
73	C₂×C₄	C₂⁴	1	4	yes	—
74	C₂×C₄	C₂⁴	1	2	yes	—
75	C₂×C₄	C₂⁴	1	8	yes	—
76	C₂×C₄	C₂⁴	1	2	yes	—
77	C₂²	—	2	2	no	—
78	C₂²	—	2	8	no	—
79	C₂²	—	2	8	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	2	8	no	—
82	C₂²	—	4	4	no	—
83	C₂²	—	4	16	no	—
84	C₂²	C₂×#41	1	2	yes	—
85	C₂²	H(5,𝔽₂)	1	4	yes	—
86	C₂²	C₂⁵	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	2	4	no	—
89	C₄	—	2	8	no	—
90	C₄	—	2	4	no	—
91	C₄	C₂×#41	1	4	yes	—
92	C₂	—	4	2	no	—
93	C₂	—	4	8	no	—
94	C₂	#540	1	4	yes	direct product factor
95	C₂	C₂²×#41	1	1	yes	characteristic
96	C₂	C₂×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #540	C₂	direct (N × H)
2. C₂×#127	C₂	semidirect
3. C₂×#125	C₂	semidirect
4. C₂×#123	C₂	semidirect
5. C₂×#121	C₂	semidirect
6. C₂×#121	C₂	semidirect
7. C₂×C₄×D₄	C₂	semidirect
8. C₂×#117	C₂	semidirect
14. #127	C₂²	semidirect
15. #125	C₂²	semidirect
16. #123	C₂²	semidirect
17. #121	C₂²	semidirect
18. #121	C₂²	semidirect
19. C₄×D₄	C₂²	semidirect
20. #117	C₂²	semidirect
94. C₂	#540	direct (N × H)

Maximal subgroups

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

subgroup	index
#540	2
C₂×#127	2
C₂×#125	2
C₂×#123	2
C₂×#121	2
C₂×C₄×D₄	2
C₂×#117	2

Subgroups — C2×#540 #3452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#540 #3452