Subgroups — #1922

Order 128.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#427	C₂	1	1	yes	maximal, characteristic
2	#424	C₂	1	1	yes	maximal, characteristic
3	#405	C₂	1	1	yes	maximal, characteristic
4	#341	C₂	1	1	yes	maximal, characteristic
5	#339	C₂	1	1	yes	maximal, characteristic
6	#339	C₂	1	1	yes	maximal, characteristic
7	#336	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	#117	C₂²	1	1	yes	characteristic
13	#117	C₂²	1	1	yes	characteristic
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	#107	C₄	1	1	yes	characteristic, complemented
16	#107	C₄	1	1	yes	characteristic, complemented
17	#106	C₄	1	1	yes	characteristic, complemented
18	#106	C₄	1	1	yes	characteristic, complemented
19	C₄⋊C₈	—	2	1	no	—
20	C₄⋊C₈	—	2	1	no	—
21	#97	—	2	1	no	—
22	C₄×C₈	—	2	1	no	—
23	C₂²×C₄	—	2	1	no	—
24	C₂²×C₄	C₂³	1	1	yes	characteristic
25	M₁₆	—	2	1	no	—
26	M₁₆	—	2	1	no	—
27	M₁₆	—	2	1	no	—
28	M₁₆	Q₈	1	2	yes	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈	D₄	1	1	yes	characteristic
38	C₂×C₈	D₄	1	1	yes	characteristic
39	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂²⋊C₄	—	2	1	no	—
44	C₂²⋊C₄	—	2	1	no	—
45	C₄²	—	2	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	2	1	no	—
49	C₂³	#41	1	1	yes	characteristic
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄	#41	1	1	yes	characteristic
61	C₂×C₄	#41	1	1	yes	characteristic
62	C₂×C₄	#41	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	2	2	no	—
66	C₈	—	2	1	no	—
67	C₈	—	2	1	no	—
68	C₈	—	4	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₄×D₄	1	1	yes	characteristic
72	C₂²	#125	1	1	yes	characteristic
73	C₂²	#126	1	1	yes	characteristic
74	C₄	—	4	2	no	—
75	C₄	—	4	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	4	1	no	—
79	C₄	#117	1	1	yes	characteristic
80	C₄	C₄×Q₈	1	1	yes	characteristic
81	C₄	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄	#123	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	2	1	no	—
85	C₂	#387	1	1	yes	characteristic
86	C₂	#443	1	1	yes	characteristic
87	C₂	#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂³	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #107	C₄	semidirect
16. #107	C₄	semidirect
17. #106	C₄	semidirect
18. #106	C₄	semidirect

Maximal subgroups

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

subgroup	index
#427	2
#424	2
#405	2
#341	2
#339	2
#336	2