Subgroups — #12015738

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#64724	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#10601400	C₂	1	2	yes	maximal
5	C₂×#64710	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	C₂²	1	2	yes	—
8	C₂×#3950	C₂²	1	2	yes	—
9	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	#65266	—	2	2	no	—
12	C₂×#3940	—	2	2	no	—
13	C₅×#366	—	2	2	no	—
14	#64724	—	2	2	no	—
15	C₁₆×Dic₅	C₄	1	4	yes	complemented
16	#64710	—	2	4	no	—
17	#12907	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄⋊Dic₅	—	2	2	no	—
21	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
22	C₂²×#168	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	D₄	1	2	yes	—
25	C₂×C₅×C₁₆	Q₈	1	2	yes	—
26	C₅×#107	—	2	2	no	—
27	C₅×#107	—	2	2	no	—
28	#3963	—	4	4	no	—
29	#3950	—	2	2	no	—
30	#3950	—	2	2	no	—
31	C₈×Dic₅	C₂×C₄	1	2	yes	—
32	C₈×Dic₅	Q₈	1	2	yes	—
33	C₂×#672	C₂×C₄	1	2	yes	—
34	C₂×#672	D₄	1	2	yes	—
35	C₂×#672	Q₈	1	2	yes	—
36	#3940	—	4	4	no	—
37	C₂×#367	—	5	2	no	—
38	C₂×#366	—	5	1	no	—
39	C₂×#366	D₅	1	1	yes	characteristic
40	C₂×C₄×C₁₆	—	5	1	no	—
41	#1846	—	5	2	no	—
42	C₂²×C₄×C₅	—	4	2	no	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅	—	4	2	no	—
45	C₂²×Dic₅	D₈	1	2	yes	—
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₅×C₄⋊C₄	—	4	2	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄⋊Dic₅	—	4	2	no	—
56	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₅	Q₁₆	1	2	yes	—
58	C₂×#168	C₄⋊C₄	1	2	yes	—
59	C₂×#168	D₈	1	2	yes	—
60	C₂×#168	Q₁₆	1	2	yes	—
61	C₅×C₁₆	C₄⋊C₄	1	4	yes	—
62	#672	C₄⋊C₄	1	4	yes	—
63	C₂²×C₁₆	—	5	1	no	—
64	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
65	C₂×#107	—	5	2	no	—
66	C₂×#107	C₂×D₅	1	2	yes	—
67	C₂×#106	—	10	2	no	—
68	C₂×C₄×C₈	—	5	1	no	—
69	#382	—	10	2	no	—
70	#367	—	10	4	no	—
71	#366	—	2	2	no	—
72	#366	—	10	2	no	—
73	C₄×C₁₆	—	5	4	no	—
74	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	8	2	no	—
76	C₂×C₄×C₅	—	8	4	no	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	#127	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×Dic₅	—	2	2	no	—
84	C₂×Dic₅	—	2	2	no	—
85	C₂×Dic₅	—	8	2	no	—
86	C₂×Dic₅	—	8	4	no	—
87	C₂×Dic₅	#107	1	4	yes	—
88	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
91	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
92	#168	#107	1	4	yes	—
93	C₂²×C₈	—	5	1	no	—
94	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	10	2	no	—
97	C₂×C₄²	—	5	1	no	—
98	C₂×C₁₆	—	5	2	no	—
99	C₂×C₁₆	—	5	2	no	—
100	C₂×C₁₆	—	5	2	no	—
101	C₂×C₁₆	Dic₁₀	1	2	yes	—
102	C₂×C₁₆	C₄×D₅	1	2	yes	—
103	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
104	#107	—	2	2	no	—
105	#107	—	2	2	no	—
106	#107	—	10	2	no	—
107	#107	—	10	2	no	—
108	#106	—	20	4	no	—
109	C₄×C₈	—	5	2	no	—
110	C₄×C₈	—	5	2	no	—
111	#95	—	20	4	no	—
112	C₂²×C₅	C₂×#107	1	1	yes	characteristic
113	C₂²×C₅	#493	1	1	yes	characteristic
114	C₂²×C₅	#500	1	1	yes	characteristic
115	C₂²×C₅	#508	1	2	yes	—
116	C₂²×C₅	#508	1	2	yes	—
117	C₄×C₅	—	16	2	no	—
118	C₄×C₅	#384	1	1	yes	characteristic
119	C₄×C₅	C₂×#107	1	1	yes	characteristic
120	C₄×C₅	#494	1	1	yes	characteristic
121	C₄×C₅	#500	1	1	yes	characteristic
122	Dic₅	—	2	4	no	—
123	Dic₅	—	16	2	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	—	5	2	no	—
126	C₂²×C₄	—	20	2	no	—
127	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
128	C₂×C₈	—	5	2	no	—
129	C₂×C₈	—	5	2	no	—
130	C₂×C₈	—	5	2	no	—
131	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
133	C₂×C₈	D₄×D₅	1	1	yes	characteristic
134	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	20	2	no	—
140	C₄⋊C₄	—	20	2	no	—
141	C₄²	—	5	2	no	—
142	C₄²	—	5	2	no	—
143	C₁₆	—	5	4	no	—
144	C₁₆	#683	1	4	yes	—
145	C₂×C₅	#1845	1	1	yes	characteristic
146	C₂×C₅	#2158	1	2	yes	—
147	C₂×C₅	#2249	1	1	yes	characteristic
148	C₂×C₅	#2249	1	1	yes	characteristic
149	C₂×C₅	#2253	1	2	yes	—
150	C₂³	D₅×D₈	1	1	yes	characteristic
151	C₂×C₄	—	5	4	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	—	10	2	no	—
155	C₂×C₄	—	10	2	no	—
156	C₂×C₄	—	40	2	no	—
157	C₂×C₄	—	40	4	no	—
158	C₂×C₄	#4034	1	1	yes	characteristic
159	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄	D₅×D₈	1	1	yes	characteristic
161	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
162	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
163	C₂×C₄	#4086	1	1	yes	characteristic
164	C₈	—	5	4	no	—
165	C₈	#4034	1	1	yes	characteristic
166	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
167	C₈	C₂×#683	1	1	yes	characteristic
168	C₈	#4090	1	1	yes	characteristic
169	C₅	#12907	1	1	yes	characteristic, complemented, Sylow
170	C₂²	#65163	1	1	yes	characteristic
171	C₂²	D₅×#107	1	1	yes	characteristic
172	C₂²	#65201	1	2	yes	—
173	C₂²	#65208	1	2	yes	—
174	C₂²	#65440	1	1	yes	characteristic
175	C₄	—	10	4	no	—
176	C₄	—	16	2	no	—
177	C₄	—	80	2	no	—
178	C₄	#65163	1	1	yes	characteristic
179	C₄	D₅×#107	1	1	yes	characteristic
180	C₄	#65265	1	1	yes	characteristic
181	C₄	#65470	1	1	yes	characteristic
182	C₂	#10600114	1	2	yes	—
183	C₂	#10600118	1	2	yes	—
184	C₂	#10601399	1	1	yes	characteristic
185	C₂	#10602258	1	1	yes	characteristic
186	C₂	#10602293	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₈
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#366	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#366	D₅
hypocenter	C₅	#12907
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
15. C₁₆×Dic₅	C₄	semidirect
169. C₅	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#366	2
C₂×#64724	2
C₂×C₁₆×Dic₅	2
#10601400	2
C₂×#64710	2
#12907	5