Subgroups — #1529

Order 128.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529	C₂	1	1	yes	maximal, characteristic
2	#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#330	C₂	1	2	yes	maximal, complemented
5	#329	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆	—	2	1	no	—
7	#126	—	2	1	no	—
8	#125	C₂²	1	1	yes	characteristic
9	#124	C₂²	1	1	yes	characteristic
10	#123	—	2	1	no	—
11	C₂²⋊Q₈	C₄	1	1	yes	characteristic
12	#121	C₄	1	1	yes	characteristic
13	C₄×Q₈	C₄	1	1	yes	characteristic
14	C₄×D₄	—	2	1	no	—
15	C₄×D₄	C₄	1	1	yes	characteristic
16	#117	—	2	1	no	—
17	C₂×C₄²	C₂²	1	1	yes	characteristic
18	C₄⋊C₈	—	2	1	no	—
19	C₂²⋊C₈	—	2	1	no	—
20	#97	C₂²	1	2	yes	—
21	#97	C₂²	1	2	yes	—
22	C₄×C₈	—	2	1	no	—
23	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₄	D₄	1	1	yes	characteristic
30	M₁₆	—	2	4	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄	—	2	1	no	—
40	C₂²⋊C₄	—	2	1	no	—
41	C₂²⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₂²⋊C₄	—	2	1	no	—
44	C₄²	—	2	1	no	—
45	C₄²	C₂×C₄	1	1	yes	characteristic
46	C₄²	D₄	1	1	yes	characteristic
47	C₄²	D₄	1	1	yes	characteristic
48	C₄²	C₂³	1	1	yes	characteristic
49	C₂³	—	2	1	no	—
50	C₂³	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈	—	4	1	no	—
52	D₄	—	4	1	no	—
53	D₄	—	4	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₈	—	2	4	no	—
68	C₈	—	4	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	4	1	no	—
84	C₄	#104	1	2	yes	—
85	C₂	—	4	1	no	—
86	C₂	—	8	1	no	—
87	C₂	#413	1	1	yes	characteristic
88	C₂	C₂×#104	1	1	yes	characteristic
89	C₂	#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
prime radical	#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #432	C₂	semidirect
3. C₄×M₁₆	C₂	semidirect
4. #330	C₂	semidirect
5. #329	C₂	semidirect

Maximal subgroups

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

subgroup	index
#529	2
#432	2
C₄×M₁₆	2
#330	2
#329	2