Subgroups — #1976

Order 128.

142 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	#427	C₂	1	1	yes	maximal, characteristic, complemented
3	#421	C₂	1	2	yes	maximal, complemented
4	#410	C₂	1	2	yes	maximal, complemented
5	#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41	C₂²	1	2	yes	—
7	#135	—	2	1	no	—
8	#135	—	2	1	no	—
9	#135	—	2	1	no	—
10	#135	—	2	1	no	—
11	C₂×Q₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×SD₁₆	C₄	1	2	yes	complemented
13	C₂×D₈	C₄	1	1	yes	characteristic, complemented
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	#117	C₂²	1	2	yes	—
17	#117	C₂²	1	1	yes	characteristic
18	#107	—	2	1	no	—
19	#106	—	2	1	no	—
20	#104	—	2	4	no	—
21	#99	—	2	4	no	—
22	#97	—	2	1	no	—
23	C₄×C₈	—	2	1	no	—
24	#41	—	2	2	no	—
25	#41	—	2	2	no	—
26	#41	—	2	2	no	—
27	#41	—	2	2	no	—
28	C₂×Q₈	C₂×C₄	1	2	yes	—
29	C₂×D₄	—	2	2	no	—
30	C₂×D₄	C₂×C₄	1	2	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	Q₁₆	—	2	1	no	—
34	Q₁₆	—	2	1	no	—
35	SD₁₆	—	2	2	no	—
36	SD₁₆	—	2	2	no	—
37	D₈	—	2	1	no	—
38	D₈	—	2	1	no	—
39	M₁₆	—	2	1	no	—
40	M₁₆	D₄	1	2	yes	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈	D₄	1	1	yes	characteristic
45	C₂×C₈	D₄	1	1	yes	characteristic
46	C₂×C₈	D₄	1	1	yes	characteristic
47	C₂×C₈	D₄	1	1	yes	characteristic
48	C₄⋊C₄	—	2	1	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₂²⋊C₄	—	2	2	no	—
51	C₂²⋊C₄	D₄	1	2	yes	—
52	C₄²	—	2	1	no	—
53	C₄²	—	2	2	no	—
54	C₂³	—	2	2	no	—
55	C₂³	C₂×D₄	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	D₄	—	2	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	4	2	no	—
61	D₄	—	4	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄	#41	1	1	yes	characteristic
72	C₂×C₄	#41	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²	—	4	1	no	—
79	C₂²	—	4	2	no	—
80	C₂²	—	8	2	no	—
81	C₂²	C₄×D₄	1	1	yes	characteristic
82	C₂²	#121	1	1	yes	characteristic
83	C₂²	#127	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	4	2	no	—
86	C₄	—	8	2	no	—
87	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄	C₄×D₄	1	1	yes	characteristic
89	C₄	#121	1	1	yes	characteristic
90	C₄	#124	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	2	no	—
95	C₂	#390	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#390
prime radical	C₂×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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. #427	C₂	semidirect
3. #421	C₂	semidirect
4. #410	C₂	semidirect
5. #405	C₂	semidirect
11. C₂×Q₁₆	C₄	semidirect
12. C₂×SD₁₆	C₄	semidirect
13. 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
#427	2
#421	2
#410	2
#405	2