Subgroups — #12010594

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic
3	#10600522	C₂	1	1	yes	maximal, characteristic
4	#10598056	C₂	1	4	yes	maximal, complemented
5	C₂×#4015	—	2	1	no	—
6	C₂×C₅×M₃₂	—	2	1	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3950	—	2	1	no	—
9	C₂×#3950	C₄	1	2	yes	—
10	#65214	—	2	1	no	—
11	#64971	—	2	1	no	—
12	#64971	—	2	1	no	—
13	#64971	C₄	1	2	yes	—
14	#64971	C₂²	1	2	yes	—
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
17	#64678	C₂²	1	2	yes	—
18	#64677	—	2	1	no	—
19	#11941	—	5	1	no	maximal, Sylow
20	C₂²×Dic₁₀	—	4	1	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	—	4	1	no	—
26	#4015	—	2	2	no	—
27	#4015	—	2	2	no	—
28	#4015	—	4	1	no	—
29	#4015	—	4	1	no	—
30	C₅×M₃₂	—	4	2	no	—
31	C₂×C₅×C₁₆	—	2	2	no	—
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×C₈	D₄	1	1	yes	characteristic
34	C₄×C₅×C₈	D₄	1	1	yes	characteristic
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂²⋊#168	—	4	1	no	—
37	#3950	—	2	2	no	—
38	#3950	—	2	2	no	—
39	#3950	—	2	2	no	—
40	#3950	C₂×C₄	1	4	yes	—
41	C₄⋊#168	—	2	2	no	—
42	C₄×#168	—	4	1	no	—
43	C₂×#500	—	5	1	no	—
44	#2150	D₅	1	1	yes	characteristic
45	#1564	—	5	1	no	—
46	#1365	—	5	4	no	—
47	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
50	C₂²×Dic₅	—	4	2	no	—
51	C₂×Dic₁₀	—	4	2	no	—
52	C₂×Dic₁₀	—	4	2	no	—
53	C₂×Dic₁₀	—	8	4	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₅×C₈	SD₁₆	1	2	yes	—
58	C₂×C₅×C₈	Q₁₆	1	2	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅	D₈	1	1	yes	characteristic
63	C₄²×C₅	SD₁₆	1	1	yes	characteristic
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	C₄⋊Dic₅	—	2	4	no	—
66	C₄⋊Dic₅	—	4	2	no	—
67	C₄⋊Dic₅	—	4	2	no	—
68	C₄⋊Dic₅	—	4	2	no	—
69	C₂×#168	—	4	2	no	—
70	C₅×C₁₆	—	4	2	no	—
71	C₂×C₄⋊Q₈	—	10	1	no	—
72	C₂×M₃₂	—	2	1	no	—
73	#500	—	5	2	no	—
74	#500	—	5	2	no	—
75	#500	—	10	1	no	—
76	#500	—	10	1	no	—
77	#431	—	10	1	no	—
78	C₂×#107	—	5	2	no	—
79	C₂×#107	—	10	1	no	—
80	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	C₄⋊C₁₆	—	2	1	no	—
82	C₄⋊C₁₆	C₂×D₅	1	2	yes	—
83	#335	—	5	2	no	—
84	#334	—	10	1	no	—
85	C₂³×C₅	#102	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	#101	1	2	yes	—
89	C₂×C₄×C₅	#102	1	1	yes	characteristic
90	C₂×C₄×C₅	#102	1	1	yes	characteristic
91	C₂×C₄×C₅	#102	1	1	yes	characteristic
92	C₂×C₄×C₅	#103	1	2	yes	—
93	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×Dic₅	—	4	4	no	—
97	C₂×Dic₅	—	8	4	no	—
98	Dic₁₀	—	8	4	no	—
99	Dic₁₀	—	8	4	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	1	no	—
102	C₅×C₈	#103	1	2	yes	—
103	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
104	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
105	#168	—	8	2	no	—
106	C₂²×Q₈	—	20	1	no	—
107	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
109	C₄⋊Q₈	—	10	2	no	—
110	C₄⋊Q₈	—	10	2	no	—
111	C₄⋊Q₈	—	20	1	no	—
112	C₄⋊Q₈	—	20	1	no	—
113	C₂×C₄⋊C₄	—	10	2	no	—
114	C₂×C₄⋊C₄	—	20	1	no	—
115	C₂×C₄²	D₂₀	1	1	yes	characteristic
116	M₃₂	—	4	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	#107	—	5	4	no	—
119	#107	—	10	2	no	—
120	#107	—	10	2	no	—
121	#107	—	10	2	no	—
122	C₄⋊C₈	—	10	2	no	—
123	C₂²⋊C₈	—	20	1	no	—
124	C₄×C₈	—	20	1	no	—
125	C₄×C₈	C₄×D₅	1	1	yes	characteristic
126	C₄×C₈	D₂₀	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
129	C₂²×C₅	—	2	1	no	—
130	C₂²×C₅	—	2	1	no	—
131	C₂²×C₅	#332	1	2	yes	—
132	C₂²×C₅	C₂×#102	1	1	yes	characteristic
133	C₄×C₅	—	2	1	no	—
134	C₄×C₅	—	2	1	no	—
135	C₄×C₅	#332	1	2	yes	—
136	C₄×C₅	#361	1	2	yes	—
137	C₄×C₅	C₂×#101	1	1	yes	characteristic
138	C₄×C₅	C₂×#103	1	1	yes	characteristic
139	C₄×C₅	#510	1	2	yes	—
140	Dic₅	—	8	4	no	—
141	C₂×Q₈	—	20	2	no	—
142	C₂×Q₈	—	20	2	no	—
143	C₂×Q₈	—	40	4	no	—
144	C₂²×C₄	—	20	2	no	—
145	C₂²×C₄	#677	1	1	yes	characteristic
146	C₂²×C₄	D₄₀	1	1	yes	characteristic
147	C₂²×C₄	#685	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	20	2	no	—
150	C₂×C₈	#685	1	1	yes	characteristic
151	C₂×C₈	#685	1	1	yes	characteristic
152	C₂×C₈	#687	1	2	yes	—
153	C₂×C₈	#689	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
156	C₄⋊C₄	—	10	4	no	—
157	C₄⋊C₄	—	20	2	no	—
158	C₄⋊C₄	—	20	2	no	—
159	C₄⋊C₄	—	20	2	no	—
160	C₄²	#677	1	1	yes	characteristic
161	C₄²	D₄₀	1	1	yes	characteristic
162	C₄²	#685	1	1	yes	characteristic
163	C₄²	C₂×D₂₀	1	1	yes	characteristic
164	C₁₆	—	4	2	no	—
165	C₂×C₅	—	2	2	no	—
166	C₂×C₅	C₂×#332	1	1	yes	characteristic
167	C₂×C₅	#2141	1	1	yes	characteristic
168	C₂×C₅	C₂×#361	1	1	yes	characteristic
169	C₂³	#3953	1	1	yes	characteristic
170	Q₈	—	40	4	no	—
171	Q₈	—	40	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	20	4	no	—
175	C₂×C₄	—	40	4	no	—
176	C₂×C₄	#3942	1	2	yes	—
177	C₂×C₄	#3953	1	1	yes	characteristic
178	C₂×C₄	#3953	1	1	yes	characteristic
179	C₂×C₄	#3953	1	1	yes	characteristic
180	C₂×C₄	#3956	1	2	yes	—
181	C₂×C₄	C₂×#677	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
183	C₂×C₄	C₂×#685	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	40	2	no	—
187	C₈	#3942	1	2	yes	—
188	C₈	C₂×#687	1	1	yes	characteristic
189	C₈	C₂×#689	1	1	yes	characteristic
190	C₅	#11941	1	1	yes	characteristic, complemented, Sylow
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#64706	1	2	yes	—
194	C₂²	C₂×#3953	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	40	4	no	—
198	C₄	#64706	1	2	yes	—
199	C₄	#64736	1	2	yes	—
200	C₄	#65198	1	2	yes	—
201	C₄	C₂×#3942	1	1	yes	characteristic
202	C₄	C₂×#3956	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#64706	1	1	yes	characteristic
205	C₂	C₂×#64736	1	1	yes	characteristic
206	C₂	#10602238	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3953
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2150	C₂
socle	C₂²×C₅	C₂×#102
prime radical	C₂³×C₅	#102
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#2150	D₅
hypocenter	C₅	#11941
LCS	G ▷ C₂×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
4. #10598056	C₂	semidirect
190. C₅	#11941	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64971	2
C₅×#2150	2
#10600522	2
#10598056	2
#11941	5