Subgroups — #1932

Order 128.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430	C₂	1	1	yes	maximal, characteristic, complemented
2	#421	C₂	1	1	yes	maximal, characteristic
3	C₂×#104	C₂	1	1	yes	maximal, characteristic
4	#413	C₂	1	1	yes	maximal, characteristic
5	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#344	C₂	1	1	yes	maximal, characteristic, complemented
7	#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	#117	C₂²	1	1	yes	characteristic
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	#108	—	2	2	no	—
16	#104	—	2	2	no	—
17	#104	—	2	1	no	—
18	#104	—	2	1	no	—
19	#101	C₄	1	2	yes	complemented
20	#100	C₄	1	2	yes	complemented
21	#97	—	2	1	no	—
22	C₄×C₈	—	2	1	no	—
23	#41	—	2	1	no	—
24	#41	—	2	1	no	—
25	#41	—	2	1	no	—
26	#41	—	2	1	no	—
27	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄	—	2	1	no	—
29	C₂×D₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	—	2	1	no	—
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	M₁₆	—	2	1	no	—
34	M₁₆	—	2	1	no	—
35	M₁₆	—	2	1	no	—
36	M₁₆	—	2	1	no	—
37	M₁₆	—	2	1	no	—
38	M₁₆	C₂×C₄	1	2	yes	—
39	M₁₆	C₂×C₄	1	2	yes	—
40	M₁₆	D₄	1	2	yes	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₂²⋊C₄	—	2	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	2	1	no	—
49	C₄²	D₄	1	1	yes	characteristic
50	C₄²	D₄	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	C₂³	#41	1	1	yes	characteristic
53	Q₈	—	2	1	no	—
54	Q₈	—	2	1	no	—
55	D₄	—	2	1	no	—
56	D₄	—	2	1	no	—
57	D₄	—	4	1	no	—
58	D₄	—	4	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	#41	1	1	yes	characteristic
70	C₂×C₄	#41	1	1	yes	characteristic
71	C₂×C₄	#41	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	2	2	no	—
74	C₈	—	2	2	no	—
75	C₈	—	4	1	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	8	1	no	—
79	C₂²	#117	1	1	yes	characteristic
80	C₂²	#123	1	1	yes	characteristic
81	C₂²	#124	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	1	no	—
86	C₄	C₄×D₄	1	1	yes	characteristic
87	C₄	C₄×D₄	1	1	yes	characteristic
88	C₄	#121	1	1	yes	characteristic
89	C₄	#126	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	—	8	1	no	—
94	C₂	#388	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#388
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #430	C₂	semidirect
5. C₄×M₁₆	C₂	semidirect
6. #344	C₂	semidirect
7. #339	C₂	semidirect
19. #101	C₄	semidirect
20. #100	C₄	semidirect

Maximal subgroups

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

subgroup	index
#430	2
#421	2
C₂×#104	2
#413	2
C₄×M₁₆	2
#344	2
#339	2