Finite Group Databasebeta

Subgroups — #12014095

Order 1280.

238 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₅×#2252	C₂	1	1	yes	maximal, characteristic
2	#10602178	C₂	1	1	yes	maximal, characteristic
3	#10602175	C₂	1	1	yes	maximal, characteristic
4	#10602154	C₂	1	1	yes	maximal, characteristic
5	#10602153	C₂	1	1	yes	maximal, characteristic
6	#10599858	C₂	1	1	yes	maximal, characteristic
7	#10597990	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#500	C₂²	1	1	yes	characteristic
10	#65382	—	2	1	no	—
11	#65370	—	2	1	no	—
12	#65370	—	2	1	no	—
13	#65368	—	2	1	no	—
14	#64988	C₂²	1	1	yes	characteristic
15	#64971	C₂²	1	1	yes	characteristic
16	C₅×#366	—	2	2	no	—
17	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	#64723	—	2	2	no	—
19	#64712	—	2	2	no	—
20	#64710	—	2	2	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	C₈×#168	C₂²	1	1	yes	characteristic
23	#12533	—	5	1	no	maximal, Sylow
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₂×Dic₂₀	—	2	1	no	—
27	C₂×Dic₂₀	—	2	1	no	—
28	#4015	—	2	1	no	—
29	#4015	—	2	1	no	—
30	C₂×C₅×C₁₆	D₄	1	2	yes	—
31	C₅×#107	—	2	1	no	—
32	C₅×#107	—	2	1	no	—
33	C₅×#107	—	2	1	no	—
34	C₅×#107	—	2	1	no	—
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	#3967	—	4	2	no	—
37	#3967	—	4	2	no	—
38	#3950	—	2	1	no	—
39	#3950	—	2	1	no	—
40	C₂×#672	D₄	1	2	yes	—
41	#3942	—	4	2	no	—
42	#3940	—	4	2	no	—
43	C₄×#168	D₄	1	1	yes	characteristic
44	C₄×#168	D₄	1	1	yes	characteristic
45	#2254	—	5	1	no	—
46	#2252	D₅	1	1	yes	characteristic
47	#2241	—	5	1	no	—
48	#2241	—	5	1	no	—
49	#1710	—	5	1	no	—
50	#1707	—	5	1	no	—
51	#1312	—	5	1	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×Dic₁₀	—	4	1	no	—
55	C₂×Dic₁₀	—	4	1	no	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	4	1	no	—
59	C₅×C₄⋊C₄	—	4	1	no	—
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	—	4	1	no	—
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₄⋊Dic₅	—	4	1	no	—
64	C₄⋊Dic₅	—	4	1	no	—
65	C₄⋊Dic₅	—	4	1	no	—
66	C₄⋊Dic₅	—	4	1	no	—
67	C₂×#168	D₈	1	2	yes	—
68	C₂×#168	D₈	1	2	yes	—
69	Dic₂₀	—	4	2	no	—
70	Dic₂₀	—	4	2	no	—
71	C₅×C₁₆	—	2	2	no	—
72	#672	—	2	2	no	—
73	#500	—	5	1	no	—
74	#500	C₂×D₅	1	1	yes	characteristic
75	#500	C₂×D₅	1	1	yes	characteristic
76	#498	—	10	1	no	—
77	#494	—	5	1	no	—
78	#487	—	10	1	no	—
79	#487	—	10	1	no	—
80	#487	—	10	1	no	—
81	#367	—	10	2	no	—
82	#366	—	2	2	no	—
83	#358	—	10	2	no	—
84	#358	—	10	2	no	—
85	C₄×C₁₆	—	5	1	no	—
86	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
87	C₈²	—	5	1	no	—
88	C₅×Q₈	—	8	2	no	—
89	C₅×Q₈	—	8	2	no	—
90	C₂×C₄×C₅	—	8	1	no	—
91	C₂×C₄×C₅	—	8	1	no	—
92	C₂×C₄×C₅	#127	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×Dic₅	—	8	1	no	—
96	C₂×Dic₅	—	8	1	no	—
97	Dic₁₀	—	8	1	no	—
98	Dic₁₀	—	8	1	no	—
99	Dic₁₀	—	8	2	no	—
100	C₅×C₈	#137	1	2	yes	—
101	C₅×C₈	#137	1	2	yes	—
102	#168	—	2	2	no	—
103	#168	—	2	2	no	—
104	C₂×Q₁₆	—	10	1	no	—
105	C₂×Q₁₆	—	10	1	no	—
106	C₄⋊Q₈	—	2	1	no	—
107	C₄⋊Q₈	—	2	1	no	—
108	C₄⋊Q₈	—	10	1	no	—
109	C₄⋊Q₈	—	10	1	no	—
110	C₂×C₁₆	—	5	2	no	—
111	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
112	#107	—	2	1	no	—
113	#107	—	2	1	no	—
114	#107	—	2	1	no	—
115	#107	—	2	1	no	—
116	#107	—	10	1	no	—
117	#107	—	10	1	no	—
118	#106	—	20	2	no	—
119	#103	—	20	2	no	—
120	#103	—	20	2	no	—
121	#103	—	20	2	no	—
122	C₄×C₈	—	5	1	no	—
123	C₄×C₈	—	5	1	no	—
124	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
125	C₂²×C₅	#493	1	1	yes	characteristic
126	C₄×C₅	—	16	1	no	—
127	C₄×C₅	—	16	1	no	—
128	C₄×C₅	#497	1	1	yes	characteristic
129	C₄×C₅	#497	1	1	yes	characteristic
130	C₄×C₅	#508	1	2	yes	—
131	C₄×C₅	#510	1	2	yes	—
132	Dic₅	—	16	1	no	—
133	Dic₅	—	16	1	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	20	1	no	—
137	C₂×Q₈	—	20	1	no	—
138	Q₁₆	—	20	2	no	—
139	Q₁₆	—	20	2	no	—
140	C₂×C₈	—	5	2	no	—
141	C₂×C₈	—	5	2	no	—
142	C₂×C₈	D₄×D₅	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	20	1	no	—
149	C₄⋊C₄	—	20	1	no	—
150	C₄⋊C₄	—	20	1	no	—
151	C₄⋊C₄	—	20	1	no	—
152	C₄²	D₄×D₅	1	1	yes	characteristic
153	C₁₆	—	2	2	no	—
154	C₁₆	—	10	2	no	—
155	C₂×C₅	#1722	1	1	yes	characteristic
156	C₂×C₅	#2249	1	1	yes	characteristic
157	C₂×C₅	#2251	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	Q₈	—	8	2	no	—
160	Q₈	—	40	1	no	—
161	Q₈	—	40	2	no	—
162	Q₈	—	40	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	40	1	no	—
166	C₂×C₄	—	40	1	no	—
167	C₂×C₄	D₅×D₈	1	1	yes	characteristic
168	C₂×C₄	D₅×D₈	1	1	yes	characteristic
169	C₂×C₄	#4086	1	1	yes	characteristic
170	C₈	—	10	2	no	—
171	C₈	—	10	2	no	—
172	C₈	#4064	1	2	yes	—
173	C₈	#4088	1	2	yes	—
174	C₅	#12533	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#65440	1	1	yes	characteristic
176	C₄	—	16	1	no	—
177	C₄	—	16	1	no	—
178	C₄	—	80	1	no	—
179	C₄	—	80	1	no	—
180	C₄	#65201	1	2	yes	—
181	C₄	#65207	1	2	yes	—
182	C₄	#65375	1	1	yes	characteristic
183	C₄	#65479	1	1	yes	characteristic
184	C₂	#10602159	1	1	yes	characteristic
185	C₂	#10602258	1	1	yes	characteristic
186	C₂	#10602294	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2252	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#2252	D₅
hypocenter	C₅	#12533
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
174. C₅	#12533	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2252	2
#10602178	2
#10602175	2
#10602154	2
#10602153	2
#10599858	2
#10597990	2
#12533	5