Finite Group Databasebeta

Subgroups — #12013662

Order 1280.

192 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 71 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×#1676	C₂	1	1	yes	maximal, characteristic
2	#10602177	C₂	1	1	yes	maximal, characteristic
3	Q₈×#672	C₂	1	1	yes	maximal, characteristic
4	#10601738	C₂	1	1	yes	maximal, characteristic
5	#10598293	C₂	1	1	yes	maximal, characteristic
6	#10597967	C₂	1	1	yes	maximal, characteristic
7	#10597960	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₅×#477	—	2	1	no	—
10	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
11	#65310	—	2	1	no	—
12	#64971	C₂²	1	1	yes	characteristic
13	C₅×#335	C₂²	1	1	yes	characteristic
14	C₅×#333	—	2	1	no	—
15	#64709	—	2	1	no	—
16	#64709	—	2	1	no	—
17	#64701	—	2	1	no	—
18	C₄⋊#672	—	2	1	no	—
19	C₄⋊#672	C₂²	1	1	yes	characteristic
20	C₄×#672	—	2	1	no	—
21	C₄×#672	C₂²	1	1	yes	characteristic
22	#64678	C₂²	1	1	yes	characteristic
23	#12458	—	5	1	no	maximal, Sylow
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
26	#4015	—	2	1	no	—
27	C₅×#107	—	2	1	no	—
28	C₅×#107	—	2	1	no	—
29	C₅×#107	—	4	1	no	—
30	C₅×C₄⋊C₈	—	2	1	no	—
31	C₅×C₄⋊C₈	—	2	1	no	—
32	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
33	C₅×#103	—	4	1	no	—
34	C₄×C₅×C₈	—	2	1	no	—
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	#3967	—	4	1	no	—
37	#3950	—	2	1	no	—
38	#3950	—	2	1	no	—
39	C₂×#672	—	2	1	no	—
40	C₂×#672	D₄	1	1	yes	characteristic
41	C₂×#672	D₄	1	1	yes	characteristic
42	#3939	—	4	1	no	—
43	C₄⋊#168	—	2	1	no	—
44	#2252	—	5	1	no	—
45	Q₈×C₁₆	—	5	1	no	—
46	#1676	—	5	1	no	—
47	#1676	D₅	1	1	yes	characteristic
48	#1371	—	5	1	no	—
49	#1365	—	5	1	no	—
50	#1365	—	5	1	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
53	C₂×Dic₁₀	—	4	1	no	—
54	C₂×C₅×C₈	—	2	1	no	—
55	C₂×C₅×C₈	—	4	1	no	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	2	1	no	—
59	C₅×C₄⋊C₄	—	4	1	no	—
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
62	C₄²×C₅	—	2	1	no	—
63	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₅	—	4	1	no	—
65	C₄⋊Dic₅	—	4	1	no	—
66	C₂×#168	—	4	1	no	—
67	#672	—	2	1	no	—
68	#672	—	4	1	no	—
69	#672	Q₁₆	1	2	yes	—
70	#500	—	5	1	no	—
71	#500	C₂×D₅	1	1	yes	characteristic
72	#477	—	2	1	no	—
73	#477	—	10	1	no	—
74	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
75	#366	—	10	1	no	—
76	#366	—	10	1	no	—
77	C₄⋊C₁₆	—	5	1	no	—
78	C₄⋊C₁₆	—	10	1	no	—
79	C₄×C₁₆	—	5	1	no	—
80	C₄×C₁₆	—	10	1	no	—
81	#335	—	5	1	no	—
82	#335	C₂×D₅	1	1	yes	characteristic
83	#333	—	2	1	no	—
84	#333	—	10	1	no	—
85	C₅×Q₈	—	2	1	no	—
86	C₅×Q₈	—	16	1	no	—
87	C₅×Q₈	Q₃₂	1	2	yes	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	8	1	no	—
91	C₂×C₄×C₅	#121	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅	#135	1	1	yes	characteristic
94	C₂×Dic₅	—	8	1	no	—
95	Dic₁₀	—	16	1	no	—
96	C₅×C₈	—	2	1	no	—
97	C₅×C₈	—	4	1	no	—
98	C₅×C₈	—	8	1	no	—
99	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
100	C₅×C₈	#137	1	1	yes	characteristic
101	#168	—	8	1	no	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	—	10	1	no	—
104	C₄×Q₈	C₂²⋊D₅	1	1	yes	characteristic
105	C₂×C₁₆	—	5	1	no	—
106	C₂×C₁₆	—	5	1	no	—
107	C₂×C₁₆	—	10	1	no	—
108	#107	—	2	1	no	—
109	#107	—	2	1	no	—
110	#107	—	4	1	no	—
111	#107	—	10	1	no	—
112	#107	—	10	1	no	—
113	#107	—	20	1	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	10	1	no	—
117	C₄⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
118	#103	—	4	1	no	—
119	#103	—	20	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
122	C₂²×C₅	#466	1	1	yes	characteristic
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	1	no	—
125	C₄×C₅	—	4	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×Q₃₂	1	1	yes	characteristic
130	C₄×C₅	#508	1	1	yes	characteristic
131	Dic₅	—	16	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	20	1	no	—
134	C₂×Q₈	#686	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	20	1	no	—
138	C₂×C₈	D₄×D₅	1	1	yes	characteristic
139	C₂×C₈	#711	1	1	yes	characteristic
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	20	1	no	—
144	C₄⋊C₄	—	20	1	no	—
145	C₄⋊C₄	#686	1	1	yes	characteristic
146	C₄²	—	2	1	no	—
147	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
148	C₁₆	—	5	2	no	—
149	C₁₆	—	10	1	no	—
150	C₁₆	—	20	1	no	—
151	C₂×C₅	#1674	1	1	yes	characteristic
152	C₂×C₅	#2216	1	1	yes	characteristic
153	C₂×C₅	#2221	1	1	yes	characteristic
154	Q₈	—	2	1	no	—
155	Q₈	—	16	1	no	—
156	Q₈	—	80	1	no	—
157	Q₈	#3961	1	2	yes	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	40	1	no	—
162	C₂×C₄	C₂×#686	1	1	yes	characteristic
163	C₂×C₄	#4085	1	1	yes	characteristic
164	C₂×C₄	#4096	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	4	1	no	—
167	C₈	—	8	1	no	—
168	C₈	—	40	1	no	—
169	C₈	D₅×Q₁₆	1	1	yes	characteristic
170	C₈	#4064	1	1	yes	characteristic
171	C₅	#12458	1	1	yes	characteristic, complemented, Sylow
172	C₂²	#65327	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	16	1	no	—
177	C₄	—	80	1	no	—
178	C₄	C₂×#3961	1	1	yes	characteristic
179	C₄	#65471	1	1	yes	characteristic
180	C₄	#65483	1	1	yes	characteristic
181	C₄	#65493	1	1	yes	characteristic
182	C₂	#10601846	1	1	yes	characteristic
183	C₂	#10601874	1	1	yes	characteristic
184	C₂	#10601885	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65327
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1676	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#1676	D₅
hypocenter	C₅	#12458
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #1676

Complements

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

N	H ≅ G/N	type
171. C₅	#12458	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1676	2
#10602177	2
Q₈×#672	2
#10601738	2
#10598293	2
#10597967	2
#10597960	2
#12458	5