Subgroups — #2148

Order 128.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	#362	C₂	1	2	yes	maximal, complemented
5	#361	C₂	1	2	yes	maximal, complemented
6	C₂×#41	—	2	1	no	—
7	#135	—	2	1	no	—
8	#135	—	2	1	no	—
9	#135	C₄	1	2	yes	—
10	#135	C₄	1	2	yes	—
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆	—	2	1	no	—
13	C₂×D₈	C₂²	1	1	yes	characteristic
14	C₂×M₁₆	—	2	1	no	—
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	M₃₂	—	2	1	no	—
17	M₃₂	C₂²	1	2	yes	—
18	C₂×C₁₆	—	2	1	no	—
19	#108	—	2	1	no	—
20	#108	C₂²	1	2	yes	—
21	#41	—	2	2	no	—
22	#41	—	2	2	no	—
23	#41	—	4	1	no	—
24	#41	—	4	1	no	—
25	C₂×Q₈	—	2	1	no	—
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	—	4	1	no	—
28	C₂²×C₄	—	4	1	no	—
29	C₂²×C₄	D₄	1	1	yes	characteristic
30	Q₁₆	—	2	1	no	—
31	Q₁₆	C₂×C₄	1	2	yes	—
32	SD₁₆	—	2	2	no	—
33	SD₁₆	—	2	2	no	—
34	D₈	—	2	1	no	—
35	D₈	C₂×C₄	1	2	yes	—
36	M₁₆	—	2	2	no	—
37	M₁₆	—	4	1	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈	D₄	1	1	yes	characteristic
42	C₂×C₈	D₄	1	1	yes	characteristic
43	C₂×C₈	D₄	1	1	yes	characteristic
44	C₂×C₈	C₂³	1	1	yes	characteristic
45	C₁₆	—	2	2	no	—
46	C₂³	—	4	1	no	—
47	C₂³	SD₁₆	1	1	yes	characteristic
48	Q₈	—	2	2	no	—
49	Q₈	—	4	1	no	—
50	D₄	—	2	2	no	—
51	D₄	—	4	2	no	—
52	D₄	—	4	2	no	—
53	D₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄	D₈	1	1	yes	characteristic
60	C₂×C₄	D₈	1	1	yes	characteristic
61	C₂×C₄	SD₁₆	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₈	—	4	2	no	—
64	C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₈	C₂²×C₄	1	1	yes	characteristic
67	C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	8	2	no	—
71	C₂²	#102	1	1	yes	characteristic
72	C₂²	#102	1	1	yes	characteristic
73	C₂²	C₂×SD₁₆	1	1	yes	characteristic
74	C₄	—	4	2	no	—
75	C₄	#102	1	1	yes	characteristic
76	C₄	#102	1	1	yes	characteristic
77	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄	C₂×D₈	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	8	2	no	—
83	C₂	C₂×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#102
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
socle	C₂	C₂×#102
prime radical	C₂×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ 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. C₂×M₃₂	C₂	semidirect
3. C₂×#108	C₂	semidirect
4. #362	C₂	semidirect
5. #361	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135	2
C₂×M₃₂	2
C₂×#108	2
#362	2
#361	2