Subgroups — #1814

Order 128.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#427	C₂	1	1	yes	maximal, characteristic, complemented
4	#421	C₂	1	2	yes	maximal
5	#344	C₂	1	2	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	—	2	1	no	—
8	#131	—	2	1	no	—
9	#131	C₄	1	2	yes	complemented
10	#131	C₄	1	2	yes	complemented
11	C₂×M₁₆	—	2	1	no	—
12	C₂×M₁₆	C₂²	1	2	yes	—
13	C₂×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈	—	2	1	no	—
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	#117	C₂²	1	2	yes	—
17	#108	—	2	1	no	—
18	#108	—	2	1	no	—
19	#107	—	2	1	no	—
20	#106	—	2	1	no	—
21	#104	—	2	4	no	—
22	#41	—	2	1	no	—
23	#41	—	2	1	no	—
24	#41	C₂×C₄	1	2	yes	—
25	#41	D₄	1	2	yes	—
26	C₂×Q₈	Q₈	1	1	yes	characteristic
27	C₂×D₄	—	2	1	no	—
28	C₂×D₄	Q₈	1	1	yes	characteristic
29	C₂²×C₄	—	2	1	no	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	M₁₆	—	2	1	no	—
32	M₁₆	—	2	2	no	—
33	M₁₆	—	2	2	no	—
34	M₁₆	—	2	2	no	—
35	M₁₆	—	2	2	no	—
36	M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈	D₄	1	1	yes	characteristic
44	C₂×C₈	D₄	1	1	yes	characteristic
45	C₂×C₈	D₄	1	1	yes	characteristic
46	C₂×C₈	D₄	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₂²⋊C₄	—	2	2	no	—
49	C₄²	—	2	2	no	—
50	C₂³	—	2	1	no	—
51	C₂³	#41	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	C₄⋊C₄	1	2	yes	—
54	D₄	—	2	2	no	—
55	D₄	—	2	2	no	—
56	D₄	—	2	1	no	—
57	D₄	C₄⋊C₄	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₄	#41	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	4	2	no	—
70	C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	4	1	no	—
76	C₂²	—	4	2	no	—
77	C₂²	#117	1	1	yes	characteristic
78	C₂²	#121	1	1	yes	characteristic
79	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	8	2	no	—
82	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄	#121	1	1	yes	characteristic
85	C₄	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	4	2	no	—
90	C₂	#380	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#380
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
2. C₂×#108	C₂	semidirect
3. #427	C₂	semidirect
5. #344	C₂	semidirect
9. #131	C₄	semidirect
10. #131	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131	2
C₂×#108	2
#427	2
#421	2
#344	2