Subgroups — #2194

Order 128.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#576	C₂	1	1	yes	maximal, characteristic
2	#510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#473	C₂	1	1	yes	maximal, characteristic, complemented
5	#361	C₂	1	1	yes	maximal, characteristic, complemented
6	#359	C₂	1	1	yes	maximal, characteristic, complemented
7	#350	C₂	1	1	yes	maximal, characteristic, complemented
8	H(5,𝔽₂)	—	2	1	no	—
9	#137	—	2	1	no	—
10	AGL(1,ℤ/8)	—	2	2	no	—
11	AGL(1,ℤ/8)	—	2	1	no	—
12	#135	—	2	1	no	—
13	#135	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₈	—	2	1	no	—
16	C₂×D₈	C₂²	1	1	yes	characteristic
17	#131	C₂²	1	1	yes	characteristic
18	Q₃₂	—	2	1	no	—
19	SD₃₂	—	2	2	no	—
20	SD₃₂	—	2	1	no	—
21	M₃₂	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₁₆	C₂²	1	1	yes	characteristic, complemented
23	#108	C₂²	1	1	yes	characteristic, complemented
24	#101	—	2	1	no	—
25	#41	—	2	2	no	—
26	#41	—	2	1	no	—
27	#41	—	4	1	no	—
28	#41	D₄	1	1	yes	characteristic
29	C₂×Q₈	—	2	1	no	—
30	C₂×D₄	—	2	2	no	—
31	C₂×D₄	—	2	1	no	—
32	C₂×D₄	—	4	2	no	—
33	C₂×D₄	—	4	1	no	—
34	Q₁₆	—	2	1	no	—
35	Q₁₆	—	2	1	no	—
36	Q₁₆	—	4	1	no	—
37	Q₁₆	D₄	1	1	yes	characteristic
38	SD₁₆	—	2	2	no	—
39	SD₁₆	—	2	1	no	—
40	SD₁₆	—	4	1	no	—
41	D₈	—	2	2	no	—
42	D₈	—	2	1	no	—
43	D₈	D₄	1	1	yes	characteristic
44	D₈	D₄	1	2	yes	—
45	M₁₆	—	2	1	no	—
46	M₁₆	—	2	1	no	—
47	M₁₆	D₄	1	1	yes	characteristic
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	C₂³	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	—	2	1	no	—
52	C₂³	—	4	2	no	—
53	C₂³	—	4	1	no	—
54	Q₈	—	2	1	no	—
55	Q₈	—	4	1	no	—
56	Q₈	—	4	1	no	—
57	Q₈	D₈	1	1	yes	characteristic
58	D₄	—	2	1	no	—
59	D₄	—	2	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	4	2	no	—
62	D₄	—	4	2	no	—
63	D₄	—	4	1	no	—
64	D₄	—	4	1	no	—
65	D₄	D₈	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	C₂×D₄	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	4	1	no	—
73	C₈	C₂×D₄	1	1	yes	characteristic
74	C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²	—	2	1	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	16	1	no	—
79	C₂²	—	16	2	no	—
80	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	8	1	no	—
84	C₄	#120	1	1	yes	characteristic
85	C₄	C₂×D₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	4	1	no	—
88	C₂	—	8	2	no	—
89	C₂	—	8	1	no	—
90	C₂	#447	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #510	C₂	semidirect
3. C₂×SD₃₂	C₂	semidirect
4. #473	C₂	semidirect
5. #361	C₂	semidirect
6. #359	C₂	semidirect
7. #350	C₂	semidirect
14. C₂×Q₁₆	C₂²	semidirect
21. M₃₂	C₂²	semidirect
22. C₂×C₁₆	C₂²	semidirect
23. #108	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#576	2
#510	2
C₂×SD₃₂	2
#473	2
#361	2
#359	2
#350	2