Subgroups — #1979

Order 128.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	1	yes	maximal, characteristic
3	#419	C₂	1	2	yes	maximal
4	C₂×#101	C₂	1	2	yes	maximal
5	#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈	C₂²	1	2	yes	—
7	C₂×Q₁₆	—	2	1	no	—
8	C₂×Q₁₆	—	2	1	no	—
9	C₂×Q₁₆	—	2	2	no	—
10	C₂×Q₁₆	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	#103	—	2	4	no	—
17	#101	—	2	4	no	—
18	#97	—	2	1	no	—
19	C₄×C₈	—	2	1	no	—
20	C₂×Q₈	—	2	2	no	—
21	C₂×Q₈	—	2	2	no	—
22	C₂×Q₈	—	2	2	no	—
23	C₂×Q₈	—	2	2	no	—
24	C₂×Q₈	—	2	2	no	—
25	C₂×Q₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄	—	2	2	no	—
27	C₂²×C₄	C₂³	1	1	yes	characteristic
28	Q₁₆	—	2	4	no	—
29	Q₁₆	—	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	#41	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	2	4	no	—
46	Q₈	—	4	2	no	—
47	Q₈	—	4	2	no	—
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	4	2	no	—
51	C₂×C₄	—	4	2	no	—
52	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄	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	#41	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	4	2	no	—
60	C₈	C₂²⋊C₄	1	2	yes	—
61	C₈	C₂²⋊C₄	1	2	yes	—
62	C₂²	—	2	2	no	—
63	C₂²	C₄×D₄	1	1	yes	characteristic
64	C₂²	#121	1	1	yes	characteristic
65	C₂²	#124	1	1	yes	characteristic
66	C₄	—	4	4	no	—
67	C₄	—	4	2	no	—
68	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄	C₄×D₄	1	1	yes	characteristic
70	C₄	#121	1	1	yes	characteristic
71	C₄	#127	1	1	yes	characteristic
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	#390	1	1	yes	characteristic
75	C₂	#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂³	#41
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
10. C₂×Q₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆	2
C₂×#108	2
#419	2
C₂×#101	2
#405	2