Subgroups — #2063

Order 128.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108	C₂	1	1	yes	maximal, characteristic, complemented
2	#428	C₂	1	1	yes	maximal, characteristic, complemented
3	#424	C₂	1	1	yes	maximal, characteristic, complemented
4	#418	C₂	1	1	yes	maximal, characteristic
5	C₂×#102	C₂	1	1	yes	maximal, characteristic
6	C₂×#100	C₂	1	1	yes	maximal, characteristic
7	#343	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	#117	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#108	—	2	2	no	—
16	#107	—	2	1	no	—
17	#106	—	2	1	no	—
18	C₄⋊C₈	—	2	2	no	—
19	#102	—	2	2	no	—
20	#102	—	2	1	no	—
21	#102	—	2	1	no	—
22	#100	—	2	2	no	—
23	C₂⁴	—	2	1	no	—
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	—	2	1	no	—
28	C₂×D₄	—	2	1	no	—
29	C₂×D₄	Q₈	1	2	yes	—
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	C₂³	1	1	yes	characteristic
32	M₁₆	—	2	1	no	—
33	M₁₆	—	2	1	no	—
34	M₁₆	—	2	1	no	—
35	M₁₆	—	2	1	no	—
36	M₁₆	—	2	1	no	—
37	M₁₆	D₄	1	2	yes	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	D₄	1	2	yes	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₄²	—	2	1	no	—
50	C₂³	—	2	2	no	—
51	C₂³	—	4	1	no	—
52	C₂³	—	4	1	no	—
53	C₂³	—	8	1	no	—
54	C₂³	—	8	1	no	—
55	C₂³	#41	1	1	yes	characteristic
56	D₄	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	4	1	no	—
59	D₄	—	4	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	8	1	no	—
64	C₂×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄	#41	1	1	yes	characteristic
69	C₂×C₄	#41	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	2	no	—
72	C₈	—	4	1	no	—
73	C₈	—	4	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	8	2	no	—
79	C₂²	—	8	1	no	—
80	C₂²	—	8	1	no	—
81	C₂²	#121	1	1	yes	characteristic
82	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂²	#126	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	8	1	no	—
87	C₄	#121	1	1	yes	characteristic
88	C₄	#121	1	1	yes	characteristic
89	C₄	C₂²⋊Q₈	1	1	yes	characteristic
90	C₄	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	—	8	2	no	—
94	C₂	#396	1	1	yes	characteristic
95	C₂	#471	1	1	yes	characteristic
96	C₂	#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
1. C₂×#108	C₂	semidirect
2. #428	C₂	semidirect
3. #424	C₂	semidirect
7. #343	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108	2
#428	2
#424	2
#418	2
C₂×#102	2
C₂×#100	2
#343	2