Subgroups — #1765

Order 128.

146 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×#104	C₂	1	1	yes	maximal, characteristic
3	#416	C₂	1	1	yes	maximal, characteristic
4	#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	#341	C₂	1	1	yes	maximal, characteristic, complemented
7	#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	1	yes	characteristic
9	#131	—	2	2	no	—
10	#131	C₄	1	4	yes	complemented
11	C₂×M₁₆	—	2	1	no	—
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈	—	2	1	no	—
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₈	C₂²	1	1	yes	characteristic
17	#117	C₂²	1	1	yes	characteristic
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#104	C₄	1	4	yes	complemented
20	#103	C₄	1	2	yes	complemented
21	#102	C₄	1	2	yes	complemented
22	#97	—	2	2	no	—
23	#97	—	2	1	no	—
24	C₄×C₈	—	2	1	no	—
25	#41	—	2	1	no	—
26	#41	—	2	1	no	—
27	#41	C₂×C₄	1	2	yes	—
28	#41	C₂×C₄	1	2	yes	—
29	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄	—	2	1	no	—
31	C₂×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	M₁₆	—	2	1	no	—
36	M₁₆	—	2	1	no	—
37	M₁₆	—	2	4	no	—
38	M₁₆	C₂×C₄	1	2	yes	—
39	M₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	C₂×C₄	1	2	yes	—
46	C₂×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₈	D₄	1	2	yes	—
48	C₂×C₈	D₄	1	1	yes	characteristic
49	C₂×C₈	D₄	1	1	yes	characteristic
50	C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄	—	2	1	no	—
52	C₄²	—	2	1	no	—
53	C₄²	—	2	1	no	—
54	C₄²	C₂×C₄	1	2	yes	—
55	C₂³	—	2	1	no	—
56	C₂³	#41	1	1	yes	characteristic
57	Q₈	—	2	1	no	—
58	Q₈	C₄²	1	2	yes	—
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	1	no	—
62	D₄	C₄²	1	2	yes	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄	#41	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	C₂²⋊C₄	1	2	yes	—
79	C₈	C₂²⋊C₄	1	2	yes	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	#117	1	1	yes	characteristic
84	C₂²	C₄×D₄	1	1	yes	characteristic
85	C₂²	C₄×D₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	2	no	—
88	C₄	—	4	2	no	—
89	C₄	C₂×C₄²	1	1	yes	characteristic
90	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₄	C₄×D₄	1	1	yes	characteristic
92	C₄	C₄×D₄	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	—	2	1	no	—
95	C₂	—	4	2	no	—
96	C₂	C₄×C₂²⋊C₄	1	1	yes	characteristic
97	C₂	#444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #405	C₂	semidirect
5. C₂×#97	C₂	semidirect
6. #341	C₂	semidirect
7. #339	C₂	semidirect
10. #131	C₄	semidirect
19. #104	C₄	semidirect
20. #103	C₄	semidirect
21. #102	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×#104	2
#416	2
#405	2
C₂×#97	2
#341	2
#339	2