Subgroups — #11720196

Order 864.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₀₈	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₂₇)⋊C₄	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#63509	—	3	1	no	maximal
6	C₂²×D₂₇	C₂²	1	2	yes	—
7	Q₈×C₂₇	—	2	2	no	—
8	C₂×C₄×C₂₇	C₂²	1	2	yes	—
9	C₂×C₄×C₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₇	C₂²	1	2	yes	—
11	D₁₀₈	—	2	2	no	—
12	C₂×Q₈×C₉	D₃	1	1	yes	characteristic
13	C₂×D₃₆	—	3	1	no	—
14	(C₂×D₉)⋊C₄	—	3	4	no	—
15	C₄×Dic₉	—	3	1	no	—
16	C₂²×C₂₇	C₂³	1	1	yes	characteristic
17	C₂×D₂₇	—	2	2	no	—
18	C₂×D₂₇	—	2	4	no	—
19	C₄×C₂₇	—	2	2	no	—
20	C₄×C₂₇	D₄	1	2	yes	—
21	Dic₂₇	—	2	2	no	—
22	#947	—	9	1	no	—
23	C₂²×D₉	—	3	2	no	—
24	Q₈×C₉	—	2	2	no	—
25	C₂×C₄×C₉	C₂×D₃	1	2	yes	—
26	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₉	—	3	2	no	—
28	D₃₆	—	6	2	no	—
29	C₂×C₂₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₂₇	#41	1	2	yes	—
31	D₂₇	—	4	2	no	—
32	C₂×C₃×Q₈	D₉	1	1	yes	characteristic
33	C₂×D₁₂	—	9	1	no	—
34	(C₂×D₃)⋊C₄	—	9	4	no	—
35	C₄×Dic₃	—	9	1	no	—
36	C₂²×C₉	C₂²×D₃	1	1	yes	characteristic
37	C₂×D₉	—	6	4	no	—
38	C₂×D₉	—	6	2	no	—
39	C₄×C₉	—	2	2	no	—
40	C₄×C₉	#67	1	2	yes	—
41	Dic₉	—	6	2	no	—
42	#124	—	27	1	no	Sylow
43	C₂₇	#124	1	1	yes	characteristic, complemented, Sylow
44	C₂²×D₃	—	9	2	no	—
45	C₃×Q₈	—	2	2	no	—
46	C₂×C₃×C₄	C₂×D₉	1	2	yes	—
47	C₂×C₃×C₄	C₂×D₉	1	1	yes	characteristic
48	C₂×Dic₃	—	9	2	no	—
49	D₁₂	—	18	2	no	—
50	C₂×C₉	#239	1	2	yes	—
51	C₂×C₉	C₂×#67	1	1	yes	characteristic
52	D₉	—	12	2	no	—
53	C₂×Q₈	D₂₇	1	1	yes	characteristic, complemented
54	C₂×D₄	—	27	1	no	—
55	C₂²⋊C₄	—	27	4	no	—
56	C₄²	—	27	1	no	—
57	C₂²×C₃	C₂²×D₉	1	1	yes	characteristic
58	C₂×D₃	—	18	4	no	—
59	C₂×D₃	—	18	2	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	#611	1	2	yes	—
62	Dic₃	—	18	2	no	—
63	C₉	#947	1	1	yes	characteristic
64	C₂³	—	27	2	no	—
65	Q₈	—	2	2	no	—
66	D₄	—	54	2	no	—
67	C₂×C₄	—	27	2	no	—
68	C₂×C₄	C₂×D₂₇	1	2	yes	—
69	C₂×C₄	C₂×D₂₇	1	1	yes	characteristic
70	C₂×C₃	#3701	1	2	yes	—
71	C₂×C₃	C₂×#611	1	1	yes	characteristic
72	D₃	—	36	2	no	—
73	C₂²	—	54	4	no	—
74	C₂²	—	54	2	no	—
75	C₂²	C₂²×D₂₇	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	54	2	no	—
78	C₄	#6171	1	2	yes	—
79	C₃	#63509	1	1	yes	characteristic
80	C₂	—	108	2	no	—
81	C₂	#88403	1	2	yes	—
82	C₂	C₂×#6171	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂³
Frattini	C₂²×C₉	C₂²×D₃
Fitting	C₂×Q₈×C₂₇	C₂
socle	C₂²×C₃	C₂²×D₉
prime radical	C₂×D₁₀₈	C₂
hypercenter	C₂×Q₈	D₂₇
hypocenter	C₂₇	#124
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₂₇	C₂	semidirect
3. (C₂×D₂₇)⋊C₄	C₂	semidirect
4. C₄×Dic₂₇	C₂	semidirect
43. C₂₇	#124	semidirect
53. C₂×Q₈	D₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₂₇	2
C₂×D₁₀₈	2
(C₂×D₂₇)⋊C₄	2
C₄×Dic₂₇	2
#63509	3