Subgroups — #1975

Order 128.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#104	C₂	1	2	yes	maximal, complemented
4	#408	C₂	1	2	yes	maximal, complemented
5	C₂×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41	C₂²	1	2	yes	—
7	#135	—	2	2	no	—
8	#135	—	2	2	no	—
9	C₂×Q₁₆	C₄	1	1	yes	characteristic, complemented
10	C₂×SD₁₆	C₄	1	2	yes	complemented
11	C₂×D₈	C₄	1	1	yes	characteristic, complemented
12	C₂×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	#108	—	2	2	no	—
17	#104	—	2	4	no	—
18	C₂²⋊C₈	—	2	4	no	—
19	#97	—	2	2	no	—
20	#41	—	2	2	no	—
21	#41	—	2	2	no	—
22	#41	—	2	2	no	—
23	#41	—	2	2	no	—
24	C₂×Q₈	C₂×C₄	1	2	yes	—
25	C₂×D₄	—	2	2	no	—
26	C₂×D₄	C₂×C₄	1	2	yes	—
27	C₂²×C₄	—	2	1	no	—
28	C₂²×C₄	—	2	2	no	—
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	Q₁₆	—	2	2	no	—
31	SD₁₆	—	2	4	no	—
32	D₈	—	2	2	no	—
33	M₁₆	—	2	2	no	—
34	M₁₆	—	2	2	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈	D₄	1	2	yes	—
41	C₂×C₈	D₄	1	2	yes	—
42	C₂×C₈	D₄	1	2	yes	—
43	C₄²	—	2	1	no	—
44	C₄²	D₄	1	2	yes	—
45	C₂³	—	2	2	no	—
46	C₂³	#41	1	1	yes	characteristic
47	Q₈	—	2	2	no	—
48	Q₈	—	2	2	no	—
49	D₄	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	—	4	2	no	—
52	D₄	—	4	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄	#41	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	4	2	no	—
66	C₈	C₂²⋊C₄	1	2	yes	—
67	C₈	C₂²⋊C₄	1	2	yes	—
68	C₂²	—	2	2	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	4	no	—
71	C₂²	C₄×D₄	1	1	yes	characteristic
72	C₂²	#121	1	1	yes	characteristic
73	C₂²	#124	1	1	yes	characteristic
74	C₄	—	4	2	no	—
75	C₄	—	4	2	no	—
76	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄	C₄×D₄	1	1	yes	characteristic
78	C₄	#121	1	1	yes	characteristic
79	C₄	#127	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	8	2	no	—
83	C₂	#390	1	1	yes	characteristic
84	C₂	#444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂×#135	C₂
LCS	G ▷ C₂×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
2. C₂×#108	C₂	semidirect
3. C₂×#104	C₂	semidirect
4. #408	C₂	semidirect
5. C₂×#97	C₂	semidirect
9. C₂×Q₁₆	C₄	semidirect
10. C₂×SD₁₆	C₄	semidirect
11. 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
C₂×#135	2
C₂×#108	2
C₂×#104	2
#408	2
C₂×#97	2