Subgroups — #11986167

Order 1280.

218 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1450	C₂	1	1	yes	maximal, characteristic
2	#10600536	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
4	C₅×#431	C₄	1	1	yes	characteristic
5	C₅×#431	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3944	—	2	1	no	—
8	C₂²×#672	—	2	1	no	—
9	C₅×#322	—	2	1	no	—
10	C₅×C₈²	—	2	1	no	—
11	C₄⋊#672	—	2	1	no	—
12	C₄⋊#672	—	2	1	no	—
13	C₄⋊#672	C₄	1	2	yes	—
14	C₄⋊#672	C₄	1	2	yes	—
15	#7153	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₅×C₄⋊C₈	—	2	1	no	—
20	C₅×C₄⋊C₈	C₈	1	2	yes	—
21	C₅×C₂²⋊C₈	—	2	1	no	—
22	C₅×C₂²⋊C₈	C₈	1	2	yes	—
23	C₄×C₅×C₈	—	2	1	no	—
24	C₄×C₅×C₈	—	2	1	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	#3944	—	4	1	no	—
30	#3944	—	4	1	no	—
31	C₂×#672	—	2	2	no	—
32	C₂×#672	—	2	2	no	—
33	C₂×#672	—	4	1	no	—
34	C₂×#672	—	4	1	no	—
35	#2150	—	5	1	no	—
36	C₂×C₄⋊C₁₆	—	5	1	no	—
37	#1450	D₅	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
41	C₂×C₅×C₈	—	2	1	no	—
42	C₂×C₅×C₈	—	2	1	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	D₈	1	1	yes	characteristic
50	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
51	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₅×C₈	Q₁₆	1	1	yes	characteristic
53	C₄²×C₅	C₄²	1	1	yes	characteristic
54	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₅	M₁₆	1	1	yes	characteristic
57	#672	—	4	2	no	—
58	#672	—	4	2	no	—
59	C₂×M₃₂	—	10	1	no	—
60	C₂²×C₁₆	—	10	1	no	—
61	#431	Dic₅	1	1	yes	characteristic
62	#431	Dic₅	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
64	C₄⋊C₁₆	—	5	2	no	—
65	C₄⋊C₁₆	—	5	2	no	—
66	C₄⋊C₁₆	—	10	1	no	—
67	C₄⋊C₁₆	—	10	1	no	—
68	#322	—	2	1	no	—
69	C₈²	—	2	1	no	—
70	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	4	1	no	—
73	C₂×C₄×C₅	#95	1	1	yes	characteristic
74	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅	#97	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅	#106	1	1	yes	characteristic
80	C₂×C₄×C₅	#107	1	1	yes	characteristic
81	C₂×C₄×C₅	#108	1	1	yes	characteristic
82	C₂×C₄×C₅	#108	1	1	yes	characteristic
83	C₅×C₈	—	2	1	no	—
84	C₅×C₈	—	2	1	no	—
85	C₅×C₈	—	4	2	no	—
86	C₅×C₈	—	4	2	no	—
87	C₅×C₈	#102	1	1	yes	characteristic
88	C₅×C₈	#102	1	1	yes	characteristic
89	C₅×C₈	#103	1	1	yes	characteristic
90	C₅×C₈	#103	1	1	yes	characteristic
91	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
92	C₂²×C₈	D₂₀	1	1	yes	characteristic
93	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
94	M₃₂	—	20	1	no	—
95	M₃₂	—	20	1	no	—
96	C₂×C₁₆	—	10	2	no	—
97	C₂×C₁₆	—	10	2	no	—
98	C₂×C₁₆	—	20	1	no	—
99	C₂×C₁₆	—	20	1	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	#168	1	2	yes	—
102	C₂²⋊C₈	—	2	1	no	—
103	C₂²⋊C₈	#168	1	2	yes	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	C₄×D₅	1	1	yes	characteristic
107	C₄×C₈	C₄×D₅	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
110	C₂²×C₅	—	2	1	no	—
111	C₂²×C₅	—	2	1	no	—
112	C₂²×C₅	#334	1	1	yes	characteristic
113	C₂²×C₅	#335	1	1	yes	characteristic
114	C₂²×C₅	#336	1	1	yes	characteristic
115	C₄×C₅	—	2	1	no	—
116	C₄×C₅	—	2	1	no	—
117	C₄×C₅	—	2	1	no	—
118	C₄×C₅	#334	1	1	yes	characteristic
119	C₄×C₅	#335	1	1	yes	characteristic
120	C₄×C₅	#340	1	1	yes	characteristic
121	C₄×C₅	#341	1	1	yes	characteristic
122	C₄×C₅	#350	1	2	yes	—
123	C₂²×C₄	C₂×#168	1	1	yes	characteristic
124	C₂²×C₄	#681	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	#677	1	1	yes	characteristic
131	C₂×C₈	#677	1	1	yes	characteristic
132	C₂×C₈	D₄₀	1	1	yes	characteristic
133	C₂×C₈	Dic₂₀	1	1	yes	characteristic
134	C₂×C₈	#683	1	1	yes	characteristic
135	C₂×C₈	#683	1	1	yes	characteristic
136	C₂×C₈	#685	1	1	yes	characteristic
137	C₂×C₈	#685	1	1	yes	characteristic
138	C₄²	C₂×#168	1	1	yes	characteristic
139	C₄²	#681	1	1	yes	characteristic
140	C₄²	C₄×Dic₅	1	1	yes	characteristic
141	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
142	C₁₆	—	20	2	no	—
143	C₁₆	—	20	2	no	—
144	C₂×C₅	—	2	1	no	—
145	C₂×C₅	—	2	1	no	—
146	C₂×C₅	#1277	1	1	yes	characteristic
147	C₂×C₅	#1378	1	1	yes	characteristic
148	C₂×C₅	#1379	1	1	yes	characteristic
149	C₂³	C₄⋊#168	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	C₄×#168	1	1	yes	characteristic
153	C₂×C₄	#3935	1	1	yes	characteristic
154	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
155	C₂×C₄	#3949	1	1	yes	characteristic
156	C₂×C₄	#3950	1	1	yes	characteristic
157	C₂×C₄	#3951	1	1	yes	characteristic
158	C₂×C₄	#3951	1	1	yes	characteristic
159	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
160	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
161	C₂×C₄	#3963	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	2	no	—
166	C₈	#3948	1	1	yes	characteristic
167	C₈	#3948	1	1	yes	characteristic
168	C₈	#3953	1	1	yes	characteristic
169	C₈	#3953	1	1	yes	characteristic
170	C₅	#7153	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#64677	1	1	yes	characteristic
174	C₂²	#64678	1	1	yes	characteristic
175	C₂²	#64744	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#64677	1	1	yes	characteristic
180	C₄	#64678	1	1	yes	characteristic
181	C₄	#64771	1	1	yes	characteristic
182	C₄	#64772	1	1	yes	characteristic
183	C₄	#64781	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#10598145	1	1	yes	characteristic
187	C₂	#10598162	1	1	yes	characteristic
188	C₂	#10598163	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂×C₄×C₅	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄⋊C₈
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	#1450	D₅
hypocenter	C₅	#7153
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
170. C₅	#7153	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1450	2
#10600536	2
C₂×C₄⋊#672	2
#7153	5