Subgroups — #2897

Order 128.

389 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#520	C₂	1	1	yes	maximal, characteristic, complemented
4	#418	C₂	1	3	yes	maximal, complemented
5	#417	C₂	1	3	yes	maximal, complemented
6	#416	C₂	1	3	yes	maximal, complemented
7	C₂×#102	C₂	1	3	yes	maximal, complemented
8	H(5,𝔽₂)	—	2	4	no	—
9	H(5,𝔽₂)	C₄	1	8	yes	complemented
10	C₂×#41	—	2	1	no	—
11	C₂×#41	C₂²	1	3	yes	—
12	C₂×#41	C₂²	1	1	yes	characteristic
13	C₂²×D₄	—	2	3	no	—
14	C₂²×D₄	C₂²	1	3	yes	—
15	#131	—	2	4	no	—
16	C₂×M₁₆	C₂²	1	3	yes	complemented
17	C₂²×C₈	C₂²	1	3	yes	complemented
18	C₄×Q₈	—	2	1	no	—
19	C₄×D₄	—	2	3	no	—
20	#117	C₂²	1	3	yes	complemented
21	C₂×C₄⋊C₄	C₂²	1	3	yes	complemented
22	#103	C₂²	1	1	yes	characteristic, complemented
23	#103	C₂²	1	3	yes	complemented
24	#102	C₂²	1	3	yes	complemented
25	#102	C₂²	1	6	yes	complemented
26	#102	C₂²	1	3	yes	complemented
27	C₂⁴	—	2	3	no	—
28	#41	—	2	8	no	—
29	#41	—	2	6	no	—
30	#41	C₂×C₄	1	4	yes	—
31	#41	C₂×C₄	1	12	yes	complemented
32	#41	D₄	1	4	yes	—
33	C₂×Q₈	D₄	1	1	yes	characteristic
34	C₂×Q₈	C₂³	1	1	yes	characteristic
35	C₂×D₄	—	2	3	no	—
36	C₂×D₄	—	2	24	no	—
37	C₂×D₄	—	2	3	no	—
38	C₂×D₄	—	2	6	no	—
39	C₂×D₄	C₂×C₄	1	12	yes	complemented
40	C₂×D₄	D₄	1	3	yes	—
41	C₂×D₄	C₂³	1	3	yes	—
42	C₂²×C₄	—	2	3	no	—
43	C₂²×C₄	—	2	3	no	—
44	C₂²×C₄	C₂³	1	3	yes	—
45	M₁₆	—	2	6	no	—
46	C₂×C₈	—	2	6	no	—
47	C₂×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₈	C₂³	1	3	yes	—
49	C₄⋊C₄	—	2	3	no	—
50	C₄⋊C₄	C₂³	1	3	yes	—
51	C₄⋊C₄	C₂³	1	1	yes	characteristic
52	C₂²⋊C₄	—	2	3	no	—
53	C₄²	—	2	3	no	—
54	C₂³	—	2	3	no	—
55	C₂³	—	2	12	no	—
56	C₂³	—	8	3	no	—
57	C₂³	—	8	3	no	—
58	C₂³	C₂×D₄	1	3	yes	—
59	Q₈	—	2	1	no	—
60	Q₈	C₂²⋊C₄	1	4	yes	—
61	Q₈	C₂²×C₄	1	2	yes	—
62	D₄	—	2	12	no	—
63	D₄	—	2	12	no	—
64	D₄	—	2	3	no	—
65	D₄	C₂²⋊C₄	1	12	yes	—
66	D₄	C₂²×C₄	1	6	yes	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	12	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	3	no	—
71	C₂×C₄	—	8	3	no	—
72	C₂×C₄	C₂²×C₄	1	6	yes	—
73	C₂×C₄	C₂×D₄	1	3	yes	—
74	C₂×C₄	C₂×D₄	1	6	yes	—
75	C₂×C₄	C₂⁴	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	3	no	—
78	C₂²	—	2	3	no	—
79	C₂²	—	2	3	no	—
80	C₂²	—	2	6	no	—
81	C₂²	—	4	3	no	—
82	C₂²	—	4	3	no	—
83	C₂²	—	8	12	no	—
84	C₂²	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂²	C₂²×D₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	3	no	—
88	C₄	—	4	1	no	—
89	C₄	C₂×C₂²⋊C₄	1	6	yes	—
90	C₄	C₂³×C₄	1	1	yes	characteristic
91	C₄	C₂²×D₄	1	1	yes	characteristic
92	C₂	—	2	6	no	—
93	C₂	—	4	6	no	—
94	C₂	C₂²×C₂²⋊C₄	1	1	yes	characteristic
95	C₂	#576	1	1	yes	characteristic
96	C₂	#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×H(5,𝔽₂)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#131	C₂	semidirect
3. #520	C₂	semidirect
4. #418	C₂	semidirect
5. #417	C₂	semidirect
6. #416	C₂	semidirect
7. C₂×#102	C₂	semidirect
9. H(5,𝔽₂)	C₄	semidirect
16. C₂×M₁₆	C₂²	semidirect
17. C₂²×C₈	C₂²	semidirect
20. #117	C₂²	semidirect
21. C₂×C₄⋊C₄	C₂²	semidirect
22. #103	C₂²	semidirect
23. #103	C₂²	semidirect
24. #102	C₂²	semidirect
25. #102	C₂²	semidirect
26. #102	C₂²	semidirect
31. #41	C₂×C₄	semidirect
39. C₂×D₄	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)	2
C₂×#131	2
#520	2
#418	2
#417	2
#416	2
C₂×#102	2