Finite Group Databasebeta

Subgroups — #12014112

Order 1280.

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

normal subgroups only

This group has 260 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, complemented
2	#10602178	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602174	C₂	1	1	yes	maximal, characteristic
4	#10602154	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602152	C₂	1	1	yes	maximal, characteristic
6	#10599851	C₂	1	1	yes	maximal, characteristic
7	#10597990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#500	C₂²	1	1	yes	characteristic
10	#65382	—	2	1	no	—
11	#65377	—	2	1	no	—
12	#65374	—	2	1	no	—
13	#65370	—	2	1	no	—
14	#64984	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	#64729	—	2	2	no	—
19	#64711	—	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	#12536	—	5	1	no	maximal, Sylow
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₂×#688	—	4	2	no	—
27	C₂×D₄₀	—	2	1	no	—
28	C₂×D₄₀	—	2	1	no	—
29	#4020	—	2	1	no	—
30	#4015	—	2	1	no	—
31	C₂×C₅×C₁₆	D₄	1	2	yes	—
32	C₅×#107	—	2	1	no	—
33	C₅×#107	—	2	1	no	—
34	C₅×#107	—	2	1	no	—
35	C₅×#107	—	2	1	no	—
36	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
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	#3941	—	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, complemented
47	#2240	—	5	1	no	—
48	#2240	—	5	1	no	—
49	#1710	—	5	1	no	—
50	#1706	—	5	1	no	—
51	#1312	—	5	1	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×D₂₀	—	4	1	no	—
55	C₂×D₂₀	—	4	1	no	—
56	C₂×D₂₀	—	4	1	no	—
57	C₂×Dic₁₀	—	4	1	no	—
58	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	—	4	1	no	—
62	C₅×C₄⋊C₄	—	4	1	no	—
63	C₅×C₄⋊C₄	—	4	1	no	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	#688	—	8	4	no	—
66	C₄⋊Dic₅	—	4	1	no	—
67	C₄⋊Dic₅	—	4	1	no	—
68	C₂×#168	D₈	1	2	yes	—
69	C₂×#168	D₈	1	2	yes	—
70	D₄₀	—	4	2	no	—
71	D₄₀	—	4	2	no	—
72	C₅×C₁₆	—	2	2	no	—
73	#672	—	2	2	no	—
74	#500	—	5	1	no	—
75	#500	C₂×D₅	1	1	yes	characteristic
76	#500	C₂×D₅	1	1	yes	characteristic
77	#493	—	5	1	no	—
78	#492	—	10	1	no	—
79	#487	—	10	1	no	—
80	#487	—	10	1	no	—
81	#486	—	10	1	no	—
82	#367	—	10	2	no	—
83	#366	—	2	2	no	—
84	#357	—	10	2	no	—
85	#357	—	10	2	no	—
86	C₄×C₁₆	—	5	1	no	—
87	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
88	C₈²	—	5	1	no	—
89	C₂²×D₅	—	8	1	no	—
90	C₅×Q₈	—	8	2	no	—
91	C₅×Q₈	—	8	2	no	—
92	C₂×C₄×C₅	—	8	1	no	—
93	C₂×C₄×C₅	—	8	1	no	—
94	C₂×C₄×C₅	#127	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
97	C₂×Dic₅	—	8	1	no	—
98	D₂₀	—	8	1	no	—
99	D₂₀	—	8	1	no	—
100	D₂₀	—	8	2	no	—
101	D₂₀	—	8	2	no	—
102	Dic₁₀	—	8	2	no	—
103	C₅×C₈	#137	1	2	yes	—
104	C₅×C₈	#137	1	2	yes	—
105	#168	—	2	2	no	—
106	#168	—	2	2	no	—
107	C₂×SD₁₆	—	20	2	no	—
108	C₂×D₈	—	10	1	no	—
109	C₂×D₈	—	10	1	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	C₄⋊Q₈	—	2	1	no	—
112	C₄⋊Q₈	—	10	1	no	—
113	#127	—	10	1	no	—
114	C₂×C₁₆	—	5	2	no	—
115	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
116	#107	—	2	1	no	—
117	#107	—	2	1	no	—
118	#107	—	2	1	no	—
119	#107	—	2	1	no	—
120	#107	—	10	1	no	—
121	#107	—	10	1	no	—
122	#103	—	20	2	no	—
123	#103	—	20	2	no	—
124	#102	—	20	2	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₂²×C₅	#493	1	1	yes	characteristic
129	C₂×D₅	—	16	1	no	—
130	C₂×D₅	—	16	1	no	—
131	C₂×D₅	—	16	1	no	—
132	C₄×C₅	—	16	1	no	—
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	#508	1	2	yes	—
137	C₄×C₅	#509	1	2	yes	—
138	Dic₅	—	16	1	no	—
139	C₂×Q₈	—	4	1	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	20	1	no	—
142	C₂×D₄	—	20	1	no	—
143	C₂×D₄	—	20	1	no	—
144	C₂×D₄	—	20	1	no	—
145	SD₁₆	—	40	4	no	—
146	D₈	—	20	2	no	—
147	D₈	—	20	2	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	2	no	—
150	C₂×C₈	D₄×D₅	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
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	C₄²	D₄×D₅	1	1	yes	characteristic
159	C₁₆	—	2	2	no	—
160	C₁₆	—	10	2	no	—
161	C₂×C₅	#1722	1	1	yes	characteristic
162	C₂×C₅	#2249	1	1	yes	characteristic
163	C₂×C₅	#2250	1	1	yes	characteristic
164	D₅	—	32	1	no	—
165	C₂³	—	40	1	no	—
166	Q₈	—	8	2	no	—
167	Q₈	—	8	2	no	—
168	Q₈	—	40	2	no	—
169	D₄	—	40	1	no	—
170	D₄	—	40	2	no	—
171	D₄	—	40	2	no	—
172	D₄	—	40	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	40	1	no	—
176	C₂×C₄	D₅×D₈	1	1	yes	characteristic
177	C₂×C₄	D₅×D₈	1	1	yes	characteristic
178	C₂×C₄	#4086	1	1	yes	characteristic
179	C₈	—	10	2	no	—
180	C₈	—	10	2	no	—
181	C₈	#4064	1	2	yes	—
182	C₈	#4088	1	2	yes	—
183	C₅	#12536	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	80	1	no	—
185	C₂²	—	80	1	no	—
186	C₂²	—	80	1	no	—
187	C₂²	#65440	1	1	yes	characteristic
188	C₄	—	16	1	no	—
189	C₄	—	16	1	no	—
190	C₄	—	80	1	no	—
191	C₄	#65200	1	2	yes	—
192	C₄	#65208	1	2	yes	—
193	C₄	#65375	1	1	yes	characteristic
194	C₄	#65479	1	1	yes	characteristic
195	C₂	—	160	1	no	—
196	C₂	#10602159	1	1	yes	characteristic
197	C₂	#10602259	1	1	yes	characteristic
198	C₂	#10602293	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	C₂²×C₅	#493
prime radical	#64984	C₂²
hypercenter	#2252	D₅
hypocenter	C₅	#12536
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
1. C₅×#2252	C₂	semidirect
2. #10602178	C₂	semidirect
4. #10602154	C₂	semidirect
7. #10597990	C₂	semidirect
46. #2252	D₅	semidirect
183. C₅	#12536	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
#10602174	2
#10602154	2
#10602152	2
#10599851	2
#10597990	2
#12536	5