Finite Group Databasebeta

Subgroups — #12013604

Order 1280.

218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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×#2227	C₂	1	1	yes	maximal, characteristic
2	#10602155	C₂	1	1	yes	maximal, characteristic
3	#10601704	C₂	1	1	yes	maximal, characteristic
4	Q₁₆×#168	C₂	1	1	yes	maximal, characteristic
5	#10598293	C₂	1	1	yes	maximal, characteristic
6	#10598056	C₂	1	1	yes	maximal, characteristic
7	#10598046	C₂	1	1	yes	maximal, characteristic
8	C₅×#500	C₂²	1	1	yes	characteristic
9	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	#65381	—	2	1	no	—
11	#65367	—	2	1	no	—
12	Q₈×#168	—	2	1	no	—
13	#64971	C₂²	1	1	yes	characteristic
14	C₅×#366	—	2	1	no	—
15	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#358	—	2	1	no	—
17	#64784	—	2	1	no	—
18	Q₈⋊#168	—	2	1	no	—
19	#64709	—	2	1	no	—
20	C₄⋊#672	C₂²	1	1	yes	characteristic
21	#64678	C₂²	1	1	yes	characteristic
22	C₈×#168	C₂²	1	1	yes	characteristic
23	#12448	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₂×#689	—	4	2	no	—
28	#4015	—	2	1	no	—
29	C₂×C₅×C₁₆	—	2	1	no	—
30	C₅×#107	—	2	1	no	—
31	C₅×#107	—	2	1	no	—
32	C₅×#107	D₄	1	1	yes	characteristic
33	C₅×#103	—	2	1	no	—
34	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
35	#3950	—	2	1	no	—
36	#3950	—	2	1	no	—
37	C₂×#672	—	2	1	no	—
38	#3939	—	4	1	no	—
39	#3939	—	4	1	no	—
40	C₄⋊#168	—	2	1	no	—
41	C₄⋊#168	—	4	1	no	—
42	C₄×#168	—	4	1	no	—
43	C₄×#168	D₄	1	1	yes	characteristic
44	C₄×#168	D₄	1	1	yes	characteristic
45	#2227	—	5	1	no	—
46	#2227	D₅	1	1	yes	characteristic
47	#1713	—	5	1	no	—
48	C₈×Q₁₆	—	5	1	no	—
49	#1370	—	5	1	no	—
50	#1365	—	5	1	no	—
51	#1365	—	5	1	no	—
52	C₂×C₅×Q₈	—	2	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×Dic₁₀	—	4	1	no	—
55	C₅×Q₁₆	—	2	1	no	—
56	C₅×Q₁₆	Q₁₆	1	2	yes	complemented
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	1	no	—
60	C₅×C₄⋊C₄	—	4	1	no	—
61	C₅×C₄⋊C₄	—	4	1	no	—
62	C₅×C₄⋊C₄	—	4	1	no	—
63	C₄²×C₅	—	4	1	no	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	#689	—	8	4	no	—
66	C₄⋊Dic₅	—	4	1	no	—
67	C₄⋊Dic₅	—	4	1	no	—
68	C₂×#168	—	2	1	no	—
69	C₂×#168	—	4	1	no	—
70	C₂×#168	D₈	1	1	yes	characteristic
71	C₂×#168	D₈	1	1	yes	characteristic
72	C₅×C₁₆	—	4	1	no	—
73	#672	—	4	1	no	—
74	#500	—	5	1	no	—
75	#500	—	10	1	no	—
76	#500	C₂×D₅	1	1	yes	characteristic
77	#494	—	10	1	no	—
78	C₈×Q₈	—	10	1	no	—
79	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
80	#366	—	2	1	no	—
81	#366	—	10	1	no	—
82	C₄⋊C₁₆	—	5	1	no	—
83	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
84	#358	—	2	1	no	—
85	#358	—	10	1	no	—
86	#335	—	5	1	no	—
87	Q₈⋊C₈	—	10	1	no	—
88	C₈²	—	5	1	no	—
89	C₅×Q₈	—	2	2	no	—
90	C₅×Q₈	—	4	1	no	—
91	C₅×Q₈	—	8	2	no	—
92	C₂×C₄×C₅	—	4	1	no	—
93	C₂×C₄×C₅	—	4	1	no	—
94	C₂×C₄×C₅	—	8	1	no	—
95	C₂×C₄×C₅	#121	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×Dic₅	—	8	1	no	—
99	Dic₁₀	—	8	2	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
102	C₅×C₈	#135	1	1	yes	characteristic
103	#168	—	2	2	no	—
104	#168	—	2	1	no	—
105	#168	—	8	1	no	—
106	#168	Q₃₂	1	2	yes	—
107	C₂×Q₁₆	—	20	2	no	—
108	C₂×Q₁₆	C₂²⋊D₅	1	1	yes	characteristic
109	C₄⋊Q₈	—	2	1	no	—
110	C₄⋊Q₈	—	10	1	no	—
111	C₄×Q₈	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	10	1	no	—
114	#107	—	2	1	no	—
115	#107	—	2	1	no	—
116	#107	—	10	1	no	—
117	#107	—	10	1	no	—
118	#107	—	20	1	no	—
119	#107	—	20	1	no	—
120	#107	C₂²⋊D₅	1	1	yes	characteristic
121	C₄⋊C₈	—	10	1	no	—
122	C₄⋊C₈	—	20	1	no	—
123	#103	—	2	1	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	20	1	no	—
127	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₂²×C₅	#459	1	1	yes	characteristic
129	C₄×C₅	—	2	1	no	—
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	—	8	1	no	—
132	C₄×C₅	—	16	1	no	—
133	C₄×C₅	#467	1	1	yes	characteristic
134	C₄×C₅	#473	1	1	yes	characteristic
135	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
136	C₄×C₅	#510	1	1	yes	characteristic
137	Dic₅	—	16	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	4	1	no	—
140	C₂×Q₈	—	20	1	no	—
141	Q₁₆	—	2	1	no	—
142	Q₁₆	—	40	4	no	—
143	Q₁₆	#689	1	2	yes	complemented
144	C₂×C₈	—	5	1	no	—
145	C₂×C₈	—	5	1	no	—
146	C₂×C₈	—	10	1	no	—
147	C₂×C₈	—	20	1	no	—
148	C₂×C₈	#711	1	1	yes	characteristic
149	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	20	1	no	—
155	C₄⋊C₄	—	20	1	no	—
156	C₄²	—	4	1	no	—
157	C₄²	D₄×D₅	1	1	yes	characteristic
158	C₁₆	—	4	1	no	—
159	C₁₆	—	20	1	no	—
160	C₂×C₅	#1669	1	1	yes	characteristic
161	C₂×C₅	#2209	1	1	yes	characteristic
162	C₂×C₅	#2213	1	1	yes	characteristic
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	8	2	no	—
166	Q₈	—	40	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	40	1	no	—
171	C₂×C₄	D₅×D₈	1	1	yes	characteristic
172	C₂×C₄	#4057	1	1	yes	characteristic
173	C₂×C₄	#4085	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	5	2	no	—
176	C₈	—	10	2	no	—
177	C₈	—	10	1	no	—
178	C₈	—	40	1	no	—
179	C₈	C₂×#689	1	1	yes	characteristic
180	C₈	#4096	1	1	yes	characteristic
181	C₅	#12448	1	1	yes	characteristic, complemented, Sylow
182	C₂²	#65439	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	16	1	no	—
187	C₄	—	80	1	no	—
188	C₄	D₅×Q₃₂	1	1	yes	characteristic
189	C₄	#65207	1	1	yes	characteristic
190	C₄	#65340	1	1	yes	characteristic
191	C₄	#65493	1	1	yes	characteristic
192	C₂	#10601894	1	1	yes	characteristic
193	C₂	#10602290	1	1	yes	characteristic
194	C₂	#10602307	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65439
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2227	C₂
socle, prime radical	C₂²×C₅	#459
hypercenter	#2227	D₅
hypocenter	C₅	#12448
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
56. C₅×Q₁₆	Q₁₆	semidirect
143. Q₁₆	#689	semidirect
181. C₅	#12448	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2227	2
#10602155	2
#10601704	2
Q₁₆×#168	2
#10598293	2
#10598056	2
#10598046	2
#12448	5