Finite Group Databasebeta

Subgroups — #12014088

Order 1280.

272 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 79 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×#2240	C₂	1	1	yes	maximal, characteristic
2	#10602177	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602155	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602081	C₂	1	1	yes	maximal, characteristic
5	#10602067	C₂	1	1	yes	maximal, characteristic
6	#10599837	C₂	1	1	yes	maximal, characteristic, complemented
7	#10597990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#493	C₂²	1	1	yes	characteristic
10	#65381	—	2	1	no	—
11	#65367	—	2	1	no	—
12	#65365	—	2	1	no	—
13	#65359	—	2	1	no	—
14	#64971	C₂²	1	1	yes	characteristic
15	#64971	C₂²	1	1	yes	characteristic
16	C₅×#357	—	2	2	no	—
17	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	#64782	—	2	2	no	—
19	#64724	—	2	2	no	—
20	#64709	—	2	2	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	C₈×#168	C₂²	1	1	yes	characteristic
23	#12531	—	5	1	no	maximal, Sylow
24	C₂×C₅×D₈	—	2	1	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×#127	—	2	1	no	—
28	C₂×#689	—	4	2	no	—
29	C₂×#687	—	4	2	no	—
30	#4015	—	2	1	no	—
31	#4015	—	2	1	no	—
32	C₂×C₅×C₁₆	D₄	1	2	yes	—
33	C₅×#107	—	2	1	no	—
34	C₅×#107	—	2	1	no	—
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	#3964	—	4	2	no	—
37	#3950	—	2	1	no	—
38	#3950	—	2	1	no	—
39	#3950	—	2	1	no	—
40	#3950	—	2	1	no	—
41	C₂×#672	D₄	1	2	yes	—
42	#3939	—	4	2	no	—
43	C₄×#168	D₄	1	1	yes	characteristic
44	C₄×#168	D₄	1	1	yes	characteristic
45	#2252	—	5	1	no	—
46	#2252	—	5	1	no	—
47	#2240	—	5	1	no	—
48	#2240	D₅	1	1	yes	characteristic
49	#1713	—	5	1	no	—
50	#1706	—	5	1	no	—
51	#1312	—	5	1	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×D₄	—	4	1	no	—
54	C₂×C₅×D₄	—	4	1	no	—
55	C₂×C₅×D₄	—	4	1	no	—
56	C₂×Dic₁₀	—	4	1	no	—
57	C₂×Dic₁₀	—	4	1	no	—
58	C₅×D₈	—	4	2	no	—
59	C₅×D₈	—	4	2	no	—
60	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	4	1	no	—
63	C₅×C₄⋊C₄	—	4	1	no	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	#689	—	8	4	no	—
66	#687	—	8	4	no	—
67	C₄⋊Dic₅	—	4	1	no	—
68	C₄⋊Dic₅	—	4	1	no	—
69	C₄⋊Dic₅	—	4	1	no	—
70	C₄⋊Dic₅	—	4	1	no	—
71	C₂×#168	D₈	1	2	yes	—
72	C₂×#168	D₈	1	2	yes	—
73	C₅×C₁₆	—	2	2	no	—
74	#672	—	2	2	no	—
75	#500	—	5	1	no	—
76	#500	—	5	1	no	—
77	#500	—	10	1	no	—
78	#500	C₂×D₅	1	1	yes	characteristic
79	#494	—	10	1	no	—
80	#493	C₂×D₅	1	1	yes	characteristic
81	#492	—	10	1	no	—
82	#486	—	10	1	no	—
83	#366	—	10	2	no	—
84	#366	—	10	2	no	—
85	#357	—	2	2	no	—
86	#357	—	10	2	no	—
87	C₄×C₁₆	—	5	1	no	—
88	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
89	C₈²	—	5	1	no	—
90	C₂³×C₅	—	8	1	no	—
91	C₅×Q₈	—	8	2	no	—
92	C₅×D₄	—	8	1	no	—
93	C₅×D₄	—	8	1	no	—
94	C₅×D₄	—	8	2	no	—
95	C₅×D₄	—	8	2	no	—
96	C₂×C₄×C₅	—	8	1	no	—
97	C₂×C₄×C₅	#127	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
100	C₂×Dic₅	—	8	1	no	—
101	C₂×Dic₅	—	8	1	no	—
102	Dic₁₀	—	8	2	no	—
103	Dic₁₀	—	8	2	no	—
104	C₅×C₈	#137	1	2	yes	—
105	C₅×C₈	#137	1	2	yes	—
106	#168	—	2	2	no	—
107	#168	D₁₆	1	4	yes	—
108	C₂×Q₁₆	—	20	2	no	—
109	C₂×SD₁₆	—	20	2	no	—
110	C₂×D₈	—	2	1	no	—
111	C₂×D₈	—	2	1	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	10	1	no	—
114	C₄⋊Q₈	—	10	1	no	—
115	#127	—	2	1	no	—
116	C₂×C₁₆	—	5	2	no	—
117	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
118	#107	—	2	1	no	—
119	#107	—	2	1	no	—
120	#107	—	10	1	no	—
121	#107	—	10	1	no	—
122	#107	—	10	1	no	—
123	#107	—	10	1	no	—
124	#107	—	20	2	no	—
125	#102	—	20	2	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	—	5	1	no	—
128	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
129	C₂²×C₅	—	16	1	no	—
130	C₂²×C₅	—	16	1	no	—
131	C₂²×C₅	—	16	1	no	—
132	C₂²×C₅	#493	1	1	yes	characteristic
133	C₄×C₅	—	16	1	no	—
134	C₄×C₅	#497	1	1	yes	characteristic
135	C₄×C₅	#497	1	1	yes	characteristic
136	C₄×C₅	C₂×D₁₆	1	2	yes	—
137	C₄×C₅	#510	1	2	yes	—
138	Dic₅	—	16	1	no	—
139	Dic₅	—	16	1	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	20	1	no	—
142	C₂×Q₈	—	20	1	no	—
143	C₂×D₄	—	4	1	no	—
144	C₂×D₄	—	4	1	no	—
145	C₂×D₄	—	4	1	no	—
146	Q₁₆	—	40	4	no	—
147	SD₁₆	—	40	4	no	—
148	D₈	—	4	2	no	—
149	D₈	—	4	2	no	—
150	C₂×C₈	—	5	2	no	—
151	C₂×C₈	—	5	2	no	—
152	C₂×C₈	D₄×D₅	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	20	1	no	—
157	C₄⋊C₄	—	20	1	no	—
158	C₄⋊C₄	—	20	1	no	—
159	C₄⋊C₄	—	20	1	no	—
160	C₄²	D₄×D₅	1	1	yes	characteristic
161	C₁₆	—	2	2	no	—
162	C₁₆	—	10	2	no	—
163	C₂×C₅	—	32	1	no	—
164	C₂×C₅	#1722	1	1	yes	characteristic
165	C₂×C₅	#2246	1	1	yes	characteristic
166	C₂×C₅	#2251	1	1	yes	characteristic
167	C₂³	—	8	1	no	—
168	Q₈	—	8	2	no	—
169	Q₈	—	40	2	no	—
170	Q₈	—	40	2	no	—
171	D₄	—	8	1	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	1	no	—
174	D₄	—	8	2	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	40	1	no	—
177	C₂×C₄	—	40	1	no	—
178	C₂×C₄	D₅×D₈	1	1	yes	characteristic
179	C₂×C₄	D₅×D₈	1	1	yes	characteristic
180	C₂×C₄	#4086	1	1	yes	characteristic
181	C₈	—	5	4	no	—
182	C₈	—	10	2	no	—
183	C₈	#4061	1	2	yes	—
184	C₈	#4097	1	2	yes	—
185	C₅	#12531	1	1	yes	characteristic, complemented, Sylow
186	C₂²	—	16	1	no	—
187	C₂²	—	16	1	no	—
188	C₂²	—	16	1	no	—
189	C₂²	#65440	1	1	yes	characteristic
190	C₄	—	16	1	no	—
191	C₄	—	80	1	no	—
192	C₄	—	80	1	no	—
193	C₄	D₅×D₁₆	1	2	yes	—
194	C₄	#65204	1	2	yes	—
195	C₄	#65349	1	1	yes	characteristic
196	C₄	#65456	1	1	yes	characteristic
197	C₂	—	32	1	no	—
198	C₂	#10601988	1	1	yes	characteristic
199	C₂	#10602257	1	1	yes	characteristic
200	C₂	#10602275	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2240	C₂
socle	C₂²×C₅	#493
prime radical	C₅×#493	C₂²
hypercenter	#2240	D₅
hypocenter	C₅	#12531
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
2. #10602177	C₂	semidirect
3. #10602155	C₂	semidirect
6. #10599837	C₂	semidirect
7. #10597990	C₂	semidirect
185. C₅	#12531	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2240	2
#10602177	2
#10602155	2
#10602081	2
#10602067	2
#10599837	2
#10597990	2
#12531	5