Subgroups — #2341

Order 128.

436 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈	C₂	1	2	yes	maximal, complemented
3	#390	C₂	1	6	yes	maximal, complemented
4	#380	C₂	1	3	yes	maximal, complemented
5	C₄×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₂×#41	—	2	8	no	—
7	C₂×#41	C₄	1	8	yes	complemented
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×D₄	C₂²	1	3	yes	—
10	C₂³×C₄	C₂²	1	3	yes	—
11	C₄×Q₈	—	2	8	no	—
12	C₂×C₄⋊C₄	C₂²	1	6	yes	—
13	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
14	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₂×C₄²	C₂²	1	6	yes	—
16	#95	C₂²	1	1	yes	characteristic
17	#95	C₂²	1	3	yes	—
18	C₂⁴	C₂³	1	3	yes	—
19	#41	—	2	32	no	—
20	C₂×Q₈	C₂×C₄	1	4	yes	—
21	C₂×Q₈	D₄	1	8	yes	—
22	C₂×D₄	—	2	12	no	—
23	C₂×D₄	C₂×C₄	1	12	yes	—
24	C₂²×C₄	—	2	12	no	—
25	C₂²×C₄	C₂×C₄	1	12	yes	—
26	C₂²×C₄	C₂³	1	6	yes	—
27	C₂²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₄	C₂³	1	2	yes	—
29	C₂²×C₄	C₂³	1	3	yes	—
30	C₄⋊C₄	—	2	12	no	—
31	C₂²⋊C₄	—	2	12	no	—
32	C₂²⋊C₄	—	4	6	no	—
33	C₄²	—	2	12	no	—
34	C₂³	—	2	6	no	—
35	C₂³	—	2	6	no	—
36	C₂³	—	2	3	no	—
37	C₂³	—	2	3	no	—
38	C₂³	C₂²×C₄	1	6	yes	—
39	C₂³	C₂⁴	1	1	yes	characteristic
40	Q₈	C₂²⋊C₄	1	16	yes	complemented
41	D₄	—	2	24	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	24	no	—
46	C₂×C₄	—	2	6	no	—
47	C₂×C₄	—	2	6	no	—
48	C₂×C₄	—	2	6	no	—
49	C₂×C₄	C₂²×C₄	1	2	yes	—
50	C₂×C₄	C₂²×C₄	1	6	yes	—
51	C₂×C₄	C₂×D₄	1	12	yes	—
52	C₂×C₄	#41	1	4	yes	complemented
53	C₂²	—	2	6	no	—
54	C₂²	—	2	6	no	—
55	C₂²	—	2	6	no	—
56	C₂²	—	4	6	no	—
57	C₂²	—	4	6	no	—
58	C₂²	C₂³×C₄	1	1	yes	characteristic
59	C₂²	C₂²×D₄	1	2	yes	—
60	C₂²	C₂×#41	1	2	yes	—
61	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	4	no	—
65	C₄	—	4	6	no	—
66	C₄	C₂×C₂²⋊C₄	1	12	yes	—
67	C₂	—	4	6	no	—
68	C₂	C₂²×C₂²⋊C₄	1	1	yes	characteristic
69	C₂	C₄×#41	1	1	yes	characteristic
70	C₂	#520	1	1	yes	characteristic
71	C₂	#548	1	2	yes	—
72	C₂	#550	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈	C₂	semidirect
3. #390	C₂	semidirect
4. #380	C₂	semidirect
5. C₄×C₂²⋊C₄	C₂	semidirect
7. C₂×#41	C₄	semidirect
40. Q₈	C₂²⋊C₄	semidirect
52. C₂×C₄	#41	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41	2
C₂×C₄×Q₈	2
#390	2
#380	2
C₄×C₂²⋊C₄	2