Finite Group Databasebeta

Subgroups — #12013672

Order 1280.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#10602178	C₂	1	1	yes	maximal, characteristic
2	#10602171	C₂	1	1	yes	maximal, characteristic
3	#10600073	C₂	1	1	yes	maximal, characteristic
4	#10599381	C₂	1	1	yes	maximal, characteristic
5	C₅×#1365	C₂	1	1	yes	maximal, characteristic
6	#10598055	C₂	1	1	yes	maximal, characteristic
7	#10597961	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	#65380	—	2	1	no	—
10	#65138	—	2	1	no	—
11	#64971	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₅×#335	C₂²	1	1	yes	characteristic
15	#64766	—	2	1	no	—
16	#64721	—	2	1	no	—
17	C₁₆×Dic₅	—	2	1	no	—
18	#64710	—	2	1	no	—
19	#64710	—	2	1	no	—
20	#64702	—	2	1	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	#64678	C₂²	1	1	yes	characteristic
23	#12459	—	5	1	no	maximal, Sylow
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₂×Dic₂₀	—	4	1	no	—
26	#4015	—	2	1	no	—
27	C₄×Dic₁₀	D₄	1	1	yes	characteristic
28	C₂×C₅×C₁₆	—	2	1	no	—
29	C₅×#107	—	2	1	no	—
30	C₅×#107	—	2	1	no	—
31	C₅×C₄⋊C₈	—	2	1	no	—
32	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
33	#3950	—	2	1	no	—
34	#3950	—	2	1	no	—
35	#3948	—	4	1	no	—
36	#3946	—	2	1	no	—
37	C₈×Dic₅	—	2	1	no	—
38	C₂×#672	D₄	1	1	yes	characteristic
39	C₂×#672	D₄	1	1	yes	characteristic
40	#3942	—	4	1	no	—
41	#3939	—	4	1	no	—
42	C₄⋊#168	—	2	1	no	—
43	C₄⋊#168	D₄	1	1	yes	characteristic
44	#2254	—	5	1	no	—
45	Q₈×C₁₆	—	5	1	no	—
46	#1676	—	5	1	no	—
47	#1674	—	5	1	no	—
48	#1372	—	5	1	no	—
49	#1365	D₅	1	1	yes	characteristic
50	#1364	—	5	1	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂×Dic₁₀	—	4	1	no	—
53	C₂×Dic₁₀	D₈	1	1	yes	characteristic
54	C₂×C₅×C₈	—	4	1	no	—
55	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	1	yes	characteristic
57	C₅×C₄⋊C₄	—	4	1	no	—
58	C₅×C₄⋊C₄	—	4	1	no	—
59	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
60	C₄⋊Dic₅	—	4	1	no	—
61	C₄⋊Dic₅	—	4	1	no	—
62	C₄⋊Dic₅	D₈	1	1	yes	characteristic
63	#683	—	2	1	no	—
64	C₄×Dic₅	—	2	1	no	—
65	C₂×#168	—	2	1	no	—
66	C₂×#168	—	4	1	no	—
67	Dic₂₀	—	8	2	no	—
68	C₅×C₁₆	—	4	1	no	—
69	#672	—	2	1	no	—
70	#672	Q₁₆	1	2	yes	complemented
71	#500	—	5	1	no	—
72	#500	C₂×D₅	1	1	yes	characteristic
73	#477	—	10	1	no	—
74	#462	—	10	1	no	—
75	C₈×Q₈	—	5	1	no	—
76	#367	—	10	1	no	—
77	#367	—	10	1	no	—
78	C₄⋊C₁₆	—	10	1	no	—
79	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
80	C₄×C₁₆	—	5	1	no	—
81	C₄×C₁₆	—	10	1	no	—
82	#335	—	5	1	no	—
83	#335	C₂×D₅	1	1	yes	characteristic
84	#333	—	10	1	no	—
85	#332	—	10	1	no	—
86	C₅×Q₈	—	16	1	no	—
87	C₂×C₄×C₅	—	8	1	no	—
88	C₂×C₄×C₅	#121	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅	#135	1	1	yes	characteristic
91	C₂×Dic₅	—	2	1	no	—
92	C₂×Dic₅	—	2	1	no	—
93	C₂×Dic₅	—	8	1	no	—
94	Dic₁₀	—	2	1	no	—
95	Dic₁₀	—	8	1	no	—
96	Dic₁₀	—	8	1	no	—
97	Dic₁₀	SD₃₂	1	2	yes	—
98	C₅×C₈	—	2	1	no	—
99	C₅×C₈	—	8	1	no	—
100	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
101	C₅×C₈	#137	1	1	yes	characteristic
102	#168	—	4	1	no	—
103	#168	—	8	1	no	—
104	C₂×Q₁₆	—	20	1	no	—
105	C₄⋊Q₈	—	2	1	no	—
106	C₄⋊Q₈	—	10	1	no	—
107	C₄×Q₈	—	5	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	5	1	no	—
110	C₂×C₁₆	—	5	1	no	—
111	#107	—	2	1	no	—
112	#107	—	2	1	no	—
113	#107	—	10	1	no	—
114	#107	—	10	1	no	—
115	#107	—	20	1	no	—
116	C₄⋊C₈	—	2	1	no	—
117	C₄⋊C₈	—	5	1	no	—
118	C₄⋊C₈	—	10	1	no	—
119	C₄⋊C₈	—	10	1	no	—
120	#103	—	20	1	no	—
121	#103	—	20	1	no	—
122	C₄×C₈	—	10	1	no	—
123	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
124	C₂²×C₅	#466	1	1	yes	characteristic
125	C₄×C₅	—	2	1	no	—
126	C₄×C₅	—	16	1	no	—
127	C₄×C₅	#462	1	1	yes	characteristic
128	C₄×C₅	#473	1	1	yes	characteristic
129	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
130	C₄×C₅	#508	1	1	yes	characteristic
131	Dic₅	—	2	2	no	—
132	Dic₅	—	4	1	no	—
133	Dic₅	—	16	1	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂×Q₈	—	5	1	no	—
136	C₂×Q₈	—	20	1	no	—
137	Q₁₆	—	40	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	10	1	no	—
140	C₂×C₈	—	20	1	no	—
141	C₂×C₈	#709	1	1	yes	characteristic
142	C₂×C₈	D₄×D₅	1	1	yes	characteristic
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	5	1	no	—
146	C₄⋊C₄	—	10	1	no	—
147	C₄⋊C₄	—	20	1	no	—
148	C₄⋊C₄	—	20	1	no	—
149	C₄²	—	10	1	no	—
150	C₄²	D₄×D₅	1	1	yes	characteristic
151	C₁₆	—	4	1	no	—
152	C₁₆	—	5	2	no	—
153	C₁₆	—	10	1	no	—
154	C₂×C₅	#1674	1	1	yes	characteristic
155	C₂×C₅	#2217	1	1	yes	characteristic
156	C₂×C₅	#2223	1	1	yes	characteristic
157	Q₈	—	5	2	no	—
158	Q₈	—	10	1	no	—
159	Q₈	—	16	1	no	—
160	Q₈	—	40	1	no	—
161	Q₈	—	40	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	10	1	no	—
164	C₂×C₄	—	10	1	no	—
165	C₂×C₄	—	40	1	no	—
166	C₂×C₄	#4030	1	1	yes	characteristic
167	C₂×C₄	D₅×D₈	1	1	yes	characteristic
168	C₂×C₄	#4062	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	8	1	no	—
171	C₈	—	20	1	no	—
172	C₈	—	40	1	no	—
173	C₈	#4061	1	1	yes	characteristic
174	C₈	D₅×Q₁₆	1	1	yes	characteristic
175	C₅	#12459	1	1	yes	characteristic, complemented, Sylow
176	C₂²	#65064	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	10	2	no	—
179	C₄	—	16	1	no	—
180	C₄	—	20	1	no	—
181	C₄	—	80	1	no	—
182	C₄	#65044	1	1	yes	characteristic
183	C₄	#65082	1	1	yes	characteristic
184	C₄	D₅×SD₃₂	1	1	yes	characteristic
185	C₄	#65208	1	1	yes	characteristic
186	C₂	#10599260	1	1	yes	characteristic
187	C₂	#10600021	1	1	yes	characteristic
188	C₂	#10600054	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65064
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1365	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#1365	D₅
hypocenter	C₅	#12459
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1365

Complements

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

N	H ≅ G/N	type
70. #672	Q₁₆	semidirect
175. C₅	#12459	semidirect

Maximal subgroups

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

subgroup	index
#10602178	2
#10602171	2
#10600073	2
#10599381	2
C₅×#1365	2
#10598055	2
#10597961	2
#12459	5