Subgroups — #2921

Order 128.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117	C₂	1	1	yes	maximal, characteristic
2	#433	C₂	1	8	yes	maximal, complemented
3	#423	C₂	1	2	yes	maximal, complemented
4	#407	C₂	1	2	yes	maximal
5	C₄×M₁₆	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄	C₂²	1	1	yes	characteristic
7	C₂×M₁₆	C₂²	1	4	yes	—
8	#117	C₄	1	4	yes	—
9	#117	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	C₄⋊C₈	C₂²	1	8	yes	complemented
15	C₂²⋊C₈	C₂²	1	8	yes	—
16	#97	C₂²	1	4	yes	complemented
17	C₄×C₈	C₂²	1	4	yes	complemented
18	C₂⁴	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄	—	2	4	no	—
20	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₄	C₂³	1	2	yes	—
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	M₁₆	—	2	8	no	—
27	C₂×C₈	C₂³	1	8	yes	—
28	C₄⋊C₄	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄	C₂×C₄	1	8	yes	—
30	C₄²	C₂×C₄	1	2	yes	—
31	C₄²	C₂×C₄	1	2	yes	—
32	C₄²	C₂³	1	2	yes	—
33	C₄²	C₂³	1	2	yes	—
34	C₂³	—	2	2	no	—
35	C₂³	—	4	2	no	—
36	C₂³	C₂²×C₄	1	2	yes	—
37	C₂³	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₂²×C₄	1	4	yes	—
44	C₂×C₄	C₂²×C₄	1	4	yes	—
45	C₂×C₄	C₂²×C₄	1	2	yes	—
46	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄	#41	1	4	yes	—
48	C₂×C₄	C₂⁴	1	1	yes	characteristic
49	C₈	—	2	8	no	—
50	C₂²	—	2	2	no	—
51	C₂²	—	2	2	no	—
52	C₂²	—	2	4	no	—
53	C₂²	—	4	4	no	—
54	C₂²	#117	1	2	yes	—
55	C₂²	C₂³×C₄	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	2	4	no	—
58	C₄	—	2	4	no	—
59	C₄	#117	1	2	yes	—
60	C₄	C₂×#41	1	2	yes	—
61	C₂	—	2	2	no	—
62	C₂	—	4	2	no	—
63	C₂	C₂×#117	1	1	yes	characteristic
64	C₂	#568	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	C₂⁴	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #433	C₂	semidirect
3. #423	C₂	semidirect
5. C₄×M₁₆	C₂	semidirect
14. C₄⋊C₈	C₂²	semidirect
16. #97	C₂²	semidirect
17. C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117	2
#433	2
#423	2
#407	2
C₄×M₁₆	2