Subgroups — C₂×#361 #2146

Order 128.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈	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	#361	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₄	—	2	1	no	—
6	C₂×D₈	—	2	1	no	—
7	C₂×D₈	—	2	1	no	—
8	C₂×D₈	—	2	1	no	—
9	C₂×D₈	C₄	1	2	yes	—
10	C₂×D₈	C₄	1	2	yes	—
11	C₂×D₈	C₂²	1	2	yes	—
12	C₂×M₁₆	—	2	1	no	—
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	M₃₂	—	2	1	no	—
15	M₃₂	C₂²	1	2	yes	complemented
16	C₂×C₁₆	—	2	1	no	—
17	#108	—	2	1	no	—
18	#108	C₂²	1	2	yes	complemented
19	C₂⁴	—	4	1	no	—
20	C₂×D₄	—	2	2	no	—
21	C₂×D₄	—	2	2	no	—
22	C₂×D₄	—	2	2	no	—
23	C₂×D₄	—	4	1	no	—
24	C₂×D₄	—	4	1	no	—
25	C₂×D₄	—	4	1	no	—
26	C₂²×C₄	D₄	1	1	yes	characteristic
27	D₈	—	2	2	no	—
28	D₈	—	2	2	no	—
29	D₈	—	2	2	no	—
30	D₈	C₂×C₄	1	4	yes	—
31	M₁₆	—	2	2	no	—
32	M₁₆	—	4	1	no	—
33	C₂×C₈	—	4	1	no	—
34	C₂×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈	D₄	1	1	yes	characteristic
37	C₂×C₈	D₄	1	1	yes	characteristic
38	C₂×C₈	D₄	1	1	yes	characteristic
39	C₂×C₈	C₂³	1	1	yes	characteristic
40	C₁₆	—	2	2	no	—
41	C₂³	—	4	2	no	—
42	C₂³	—	4	2	no	—
43	C₂³	—	4	2	no	—
44	C₂³	—	8	4	no	—
45	C₂³	SD₁₆	1	1	yes	characteristic
46	D₄	—	2	4	no	—
47	D₄	—	4	2	no	—
48	D₄	—	4	2	no	—
49	D₄	—	4	2	no	—
50	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄	D₈	1	1	yes	characteristic
53	C₂×C₄	D₈	1	1	yes	characteristic
54	C₂×C₄	SD₁₆	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₈	—	4	2	no	—
57	C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₈	C₂²×C₄	1	1	yes	characteristic
60	C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²	—	2	2	no	—
62	C₂²	—	4	4	no	—
63	C₂²	—	8	4	no	—
64	C₂²	—	8	4	no	—
65	C₂²	—	8	4	no	—
66	C₂²	#102	1	1	yes	characteristic
67	C₂²	#102	1	1	yes	characteristic
68	C₂²	C₂×SD₁₆	1	1	yes	characteristic
69	C₄	#102	1	1	yes	characteristic
70	C₄	#102	1	1	yes	characteristic
71	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄	C₂×D₈	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	8	4	no	—
76	C₂	#361	1	2	yes	direct product factor
77	C₂	C₂×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#102
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #361	C₂	direct (N × H)
15. M₃₂	C₂²	semidirect
18. #108	C₂²	semidirect
76. C₂	#361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₈	2
C₂×M₃₂	2
C₂×#108	2
#361	2

Subgroups — C2×#361 #2146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#361 #2146