Finite Group Databasebeta

Subgroups — #44401601

Order 29056.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300231	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409114	C₂	1	1	yes	maximal, characteristic
4	#28408857	C₂	1	1	yes	maximal, characteristic, complemented
5	#28408834	C₂	1	1	yes	maximal, characteristic
6	#28408807	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₈₁₆	C₂²	1	1	yes	characteristic
9	Q₈×Dic₂₂₇	—	2	1	no	—
10	#20300250	—	2	1	no	—
11	#20300231	C₄	1	4	yes	complemented
12	C₂×C₂₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#20300222	C₂²	1	1	yes	characteristic
14	#20300212	—	2	1	no	—
15	#20300208	—	2	1	no	—
16	C₈×Dic₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
19	#107×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×D₉₀₈	—	2	1	no	—
21	C₂×C₄×D₂₂₇	—	4	1	no	—
22	D₁₈₁₆	—	2	1	no	—
23	D₁₈₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₂₂₇	—	2	1	no	—
26	C₄⋊Dic₂₂₇	—	4	1	no	—
27	(C₂×D₂₂₇)⋊C₄	—	4	1	no	—
28	C₄×Dic₂₂₇	—	4	1	no	—
29	C₄×Dic₂₂₇	D₄	1	1	yes	characteristic
30	C₂₂₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×Q₈×C₂₂₇	—	2	1	no	—
32	Q₁₆×C₂₂₇	—	2	1	no	—
33	Q₁₆×C₂₂₇	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₂₇	—	2	1	no	—
36	C₁₆×C₂₂₇	—	2	1	no	—
37	C₂²×D₂₂₇	—	4	1	no	—
38	D₉₀₈	—	2	2	no	—
39	D₉₀₈	—	4	1	no	—
40	C₂×Dic₂₂₇	—	4	1	no	—
41	C₂×Dic₂₂₇	D₈	1	1	yes	characteristic
42	C₂×Dic₂₂₇	D₈	1	1	yes	characteristic
43	C₄×D₂₂₇	—	8	2	no	—
44	C₂₂₇⋊C₈	—	2	1	no	—
45	Q₈×C₂₂₇	—	2	2	no	—
46	Q₈×C₂₂₇	—	4	1	no	—
47	C₂×C₄×C₂₂₇	—	4	1	no	—
48	C₂×C₄×C₂₂₇	—	4	1	no	—
49	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
50	C₈×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₂₂₇	#41	1	1	yes	characteristic
52	C₂²×C₂₂₇	C₂×D₈	1	1	yes	characteristic
53	C₂×D₂₂₇	—	4	2	no	—
54	C₂×D₂₂₇	—	8	1	no	—
55	C₂×D₂₂₇	—	8	1	no	—
56	C₄×C₂₂₇	—	4	2	no	—
57	C₄×C₂₂₇	—	8	1	no	—
58	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₂₂₇	#135	1	1	yes	characteristic
60	Dic₂₂₇	—	2	1	no	—
61	Dic₂₂₇	—	2	1	no	—
62	Dic₂₂₇	—	8	1	no	—
63	C₂×C₂₂₇	C₄×D₈	1	1	yes	characteristic
64	C₂×C₂₂₇	#509	1	1	yes	characteristic
65	C₂×C₂₂₇	#510	1	1	yes	characteristic
66	D₂₂₇	—	8	2	no	—
67	C₂₂₇	#2175	1	1	yes	characteristic, complemented, Sylow
68	#2175	—	227	1	no	maximal, Sylow
69	C₂×SD₃₂	—	227	1	no	—
70	C₄×Q₁₆	—	227	1	no	—
71	C₄×D₈	—	227	1	no	—
72	#366	—	227	1	no	—
73	#358	D₂₂₇	1	1	yes	characteristic, complemented
74	#357	—	227	1	no	—
75	#346	—	227	1	no	—
76	C₂×Q₁₆	C₂×D₂₂₇	1	1	yes	characteristic
77	C₂×D₈	—	227	1	no	—
78	C₄×Q₈	—	454	1	no	—
79	C₄×D₄	—	454	1	no	—
80	SD₃₂	—	227	4	no	—
81	C₂×C₁₆	—	227	1	no	—
82	C₂×C₁₆	C₂×D₂₂₇	1	1	yes	characteristic
83	#107	—	227	1	no	—
84	#107	C₂×D₂₂₇	1	1	yes	characteristic
85	#103	—	454	1	no	—
86	#102	—	454	1	no	—
87	C₄×C₈	—	227	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	454	1	no	—
90	C₂²×C₄	—	908	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₄×D₂₂₇	1	2	yes	complemented
93	D₈	—	454	1	no	—
94	D₈	—	227	2	no	—
95	C₂×C₈	—	227	1	no	—
96	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	908	1	no	—
99	C₄⋊C₄	—	454	1	no	—
100	C₂²⋊C₄	—	908	1	no	—
101	C₄²	—	908	1	no	—
102	C₄²	—	227	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	227	2	no	—
105	C₂³	—	908	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	908	1	no	—
109	D₄	—	454	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	1816	2	no	—
113	C₂×C₄	—	908	1	no	—
114	C₂×C₄	—	227	1	no	—
115	C₂×C₄	—	227	1	no	—
116	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
117	C₈	—	454	1	no	—
118	C₈	C₂×C₄×D₂₂₇	1	1	yes	characteristic
119	C₈	#16813883	1	1	yes	characteristic
120	C₂²	—	1816	1	no	—
121	C₂²	—	1816	1	no	—
122	C₂²	—	908	2	no	—
123	C₂²	D₈×D₂₂₇	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	8	1	no	—
126	C₄	—	1816	1	no	—
127	C₄	—	454	1	no	—
128	C₄	—	454	1	no	—
129	C₄	(C₂²⋊D₂₂₇)⋊C₄	1	1	yes	characteristic
130	C₄	#20300309	1	1	yes	characteristic
131	C₂	—	1816	2	no	—
132	C₂	#28409112	1	1	yes	characteristic
133	C₂	#28409469	1	1	yes	characteristic
134	C₂	#28409474	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₂₇
commutator	C₈×C₂₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₂₇
Fitting	#358×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×D₈
prime radical	C₂×D₁₈₁₆	C₂²
cosocle	C₂×C₈×C₂₂₇	C₂³
hypercenter	#358	D₂₂₇
hypocenter	C₂₂₇	#2175
LCS	G ▷ C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₂₂₇	C₂	semidirect
4. #28408857	C₂	semidirect
6. #28408807	C₂	semidirect
7. #358×C₂₂₇	C₂	semidirect
11. #20300231	C₄	semidirect
30. C₂₂₇⋊C₁₆	C₂×C₄	semidirect
33. Q₁₆×C₂₂₇	C₂×C₄	semidirect
67. C₂₂₇	#2175	semidirect
73. #358	D₂₂₇	semidirect
92. Q₁₆	C₄×D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300231	2
Q₁₆×Dic₂₂₇	2
#28409114	2
#28408857	2
#28408834	2
#28408807	2
#358×C₂₂₇	2
#2175	227