Subgroups — #1978

Order 128.

158 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×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#418	C₂	1	2	yes	maximal, complemented
4	C₂×#100	C₂	1	2	yes	maximal, complemented
5	#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄	C₂²	1	2	yes	—
7	C₂×D₈	—	2	1	no	—
8	C₂×D₈	—	2	1	no	—
9	C₂×D₈	—	2	2	no	—
10	C₂×D₈	C₄	1	4	yes	complemented
11	C₂×M₁₆	C₂²	1	2	yes	—
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	#117	C₂²	1	1	yes	characteristic
15	#108	—	2	2	no	—
16	#102	—	2	4	no	—
17	#100	—	2	4	no	—
18	#97	—	2	1	no	—
19	C₄×C₈	—	2	1	no	—
20	C₂⁴	—	2	2	no	—
21	C₂×D₄	—	2	2	no	—
22	C₂×D₄	—	2	2	no	—
23	C₂×D₄	—	2	2	no	—
24	C₂×D₄	—	2	2	no	—
25	C₂×D₄	—	2	2	no	—
26	C₂×D₄	C₂×C₄	1	4	yes	—
27	C₂²×C₄	C₂³	1	1	yes	characteristic
28	D₈	—	2	4	no	—
29	D₈	—	2	4	no	—
30	M₁₆	—	2	1	no	—
31	M₁₆	—	2	2	no	—
32	M₁₆	—	2	2	no	—
33	M₁₆	D₄	1	2	yes	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₈	D₄	1	1	yes	characteristic
38	C₂×C₈	D₄	1	1	yes	characteristic
39	C₂×C₈	D₄	1	2	yes	—
40	C₄⋊C₄	D₄	1	2	yes	—
41	C₂²⋊C₄	—	2	1	no	—
42	C₄²	—	2	1	no	—
43	C₂³	—	2	4	no	—
44	C₂³	—	4	2	no	—
45	C₂³	—	4	2	no	—
46	C₂³	—	8	4	no	—
47	C₂³	#41	1	1	yes	characteristic
48	D₄	—	2	4	no	—
49	D₄	—	2	4	no	—
50	D₄	—	4	2	no	—
51	D₄	—	4	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄	#41	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	2	no	—
61	C₈	C₂²⋊C₄	1	2	yes	—
62	C₈	C₂²⋊C₄	1	2	yes	—
63	C₂²	—	2	2	no	—
64	C₂²	—	4	4	no	—
65	C₂²	—	8	4	no	—
66	C₂²	—	8	4	no	—
67	C₂²	—	8	4	no	—
68	C₂²	C₄×D₄	1	1	yes	characteristic
69	C₂²	#121	1	1	yes	characteristic
70	C₂²	#124	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄	C₄×D₄	1	1	yes	characteristic
74	C₄	#121	1	1	yes	characteristic
75	C₄	#127	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	8	4	no	—
79	C₂	#390	1	1	yes	characteristic
80	C₂	#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₂×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
2. C₂×#108	C₂	semidirect
3. #418	C₂	semidirect
4. C₂×#100	C₂	semidirect
5. #405	C₂	semidirect
10. C₂×D₈	C₄	semidirect

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×#108	2
#418	2
C₂×#100	2
#405	2