Subgroups — #1395

Order 128.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#413	C₂	1	1	yes	maximal, characteristic
2	#410	C₂	1	1	yes	maximal, characteristic
3	#408	C₂	1	1	yes	maximal, characteristic
4	C₂×#41	C₂²	1	1	yes	characteristic
5	C₂×M₁₆	—	2	1	no	—
6	C₂×M₁₆	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₈	C₄	1	1	yes	characteristic, complemented
8	#117	—	2	1	no	—
9	#101	C₄	1	2	yes	complemented
10	#100	—	2	1	no	—
11	#99	—	2	1	no	—
12	#99	C₄	1	2	yes	—
13	C₂²⋊C₈	—	2	1	no	—
14	C₂²⋊C₈	—	2	1	no	—
15	#41	—	4	1	no	—
16	#41	—	4	1	no	—
17	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
18	C₂×D₄	C₂×C₄	1	1	yes	characteristic
19	C₂×D₄	D₄	1	1	yes	characteristic
20	C₂×D₄	Q₈	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	M₁₆	—	2	2	no	—
25	M₁₆	—	4	1	no	—
26	M₁₆	—	4	1	no	—
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	4	1	no	—
30	C₂×C₈	—	4	1	no	—
31	C₄⋊C₄	—	4	1	no	—
32	C₂²⋊C₄	—	2	2	no	—
33	C₄²	—	4	1	no	—
34	C₂³	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³	C₄⋊C₄	1	1	yes	characteristic
36	C₂³	C₄⋊C₄	1	1	yes	characteristic
37	Q₈	—	4	1	no	—
38	D₄	—	4	1	no	—
39	D₄	—	4	1	no	—
40	D₄	—	4	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	2	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	C₄²	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₈	—	4	2	no	—
53	C₈	—	4	1	no	—
54	C₈	—	4	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	4	1	no	—
59	C₂²	—	4	1	no	—
60	C₂²	—	4	1	no	—
61	C₂²	#95	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	—	2	1	no	—
64	C₄	—	2	1	no	—
65	C₄	—	8	2	no	—
66	C₄	#99	1	1	yes	characteristic
67	C₄	#99	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	—	4	1	no	—
71	C₂	—	4	1	no	—
72	C₂	#342	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#99
commutator	C₂×C₄	C₄²
Frattini, prime radical, cosocle	C₂×#41	C₂²
socle	C₂	#342
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
6. C₂×M₁₆	C₄	semidirect
7. C₂²×C₈	C₄	semidirect
9. #101	C₄	semidirect

Maximal subgroups

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

subgroup	index
#413	2
#410	2
#408	2