Subgroups — #11986247

Order 1280.

258 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic
2	#10600572	C₂	1	1	yes	maximal, characteristic
3	C₂×#64675	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×M₃₂	—	2	1	no	—
5	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#3951	C₄	1	1	yes	characteristic
7	#65395	C₄	1	1	yes	characteristic
8	#65390	—	2	1	no	—
9	C₂×C₄×#168	—	2	1	no	—
10	C₅×#364	—	2	1	no	—
11	C₅×#364	C₄	1	2	yes	complemented
12	#64743	—	2	1	no	—
13	#64675	—	2	1	no	—
14	#64675	C₄	1	2	yes	complemented
15	#7182	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	#4072	—	2	1	no	—
20	C₂×#681	—	2	1	no	—
21	C₂²×#168	—	2	2	no	—
22	C₅×M₃₂	—	2	2	no	—
23	C₅×M₃₂	—	4	1	no	—
24	C₂×C₅×C₁₆	—	4	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	Q₈	1	1	yes	characteristic
29	#3951	—	2	1	no	—
30	#3951	C₈	1	2	yes	complemented
31	#3950	C₈	1	2	yes	complemented
32	#3949	—	2	1	no	—
33	#3947	—	4	1	no	—
34	C₈×Dic₅	—	4	1	no	—
35	C₄×#168	—	2	2	no	—
36	C₄×#168	—	2	2	no	—
37	C₂×#364	D₅	1	1	yes	characteristic
38	#1772	—	5	1	no	—
39	C₂×#322	—	5	1	no	—
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
49	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
50	C₂×C₅×C₈	D₈	1	1	yes	characteristic
51	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₅×C₈	Q₁₆	1	1	yes	characteristic
54	C₄²×C₅	—	2	1	no	—
55	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²⋊Dic₅	—	4	1	no	—
58	C₄⋊Dic₅	—	2	2	no	—
59	C₄×Dic₅	—	4	1	no	—
60	#681	—	2	2	no	—
61	#681	—	4	1	no	—
62	C₂×#168	—	2	4	no	—
63	C₂×#168	—	4	4	no	—
64	C₂×#168	—	4	1	no	—
65	C₅×C₁₆	—	4	2	no	—
66	C₂×M₃₂	—	2	1	no	—
67	C₂×#108	—	5	1	no	—
68	#427	—	5	1	no	—
69	#405	—	10	1	no	—
70	C₂×C₄×C₈	—	10	1	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#364	—	2	1	no	—
73	#364	Dic₅	1	2	yes	complemented
74	#339	—	10	1	no	—
75	#322	—	5	2	no	—
76	#322	—	10	1	no	—
77	C₂³×C₅	#104	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	#95	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅	#102	1	1	yes	characteristic
85	C₂×C₄×C₅	#103	1	1	yes	characteristic
86	C₂×C₄×C₅	#104	1	1	yes	characteristic
87	C₂×Dic₅	—	4	2	no	—
88	C₅×C₈	C₄×C₈	1	1	yes	characteristic
89	C₅×C₈	#97	1	1	yes	characteristic
90	C₅×C₈	#102	1	1	yes	characteristic
91	C₅×C₈	#103	1	1	yes	characteristic
92	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
93	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
94	C₅×C₈	#106	1	1	yes	characteristic
95	C₅×C₈	#107	1	1	yes	characteristic
96	#168	—	4	4	no	—
97	#168	—	4	2	no	—
98	C₂×M₁₆	—	10	1	no	—
99	C₂²×C₈	—	10	2	no	—
100	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
102	#117	—	10	1	no	—
103	C₂×C₄²	D₂₀	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	4	1	no	—
106	C₂×C₁₆	—	4	1	no	—
107	#108	—	5	2	no	—
108	#108	—	10	1	no	—
109	#107	—	5	2	no	—
110	#106	—	10	1	no	—
111	#97	—	20	1	no	—
112	C₄×C₈	—	10	2	no	—
113	C₄×C₈	—	10	2	no	—
114	C₄×C₈	—	20	1	no	—
115	C₄×C₈	Dic₁₀	1	1	yes	characteristic
116	C₄×C₈	C₄×D₅	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	D₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₅	Q₈⋊C₈	1	1	yes	characteristic
122	C₂²×C₅	#339	1	1	yes	characteristic
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	D₄⋊C₈	1	1	yes	characteristic
126	C₄×C₅	Q₈⋊C₈	1	1	yes	characteristic
127	C₄×C₅	#336	1	1	yes	characteristic
128	C₄×C₅	#340	1	1	yes	characteristic
129	Dic₅	—	8	2	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	#685	1	1	yes	characteristic
132	M₁₆	—	10	2	no	—
133	M₁₆	—	20	1	no	—
134	C₂×C₈	—	10	4	no	—
135	C₂×C₈	—	20	4	no	—
136	C₂×C₈	—	20	1	no	—
137	C₂×C₈	C₈×D₅	1	1	yes	characteristic
138	C₂×C₈	C₈×D₅	1	1	yes	characteristic
139	C₂×C₈	#676	1	1	yes	characteristic
140	C₂×C₈	#676	1	1	yes	characteristic
141	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
142	C₂×C₈	#683	1	1	yes	characteristic
143	C₂×C₈	#683	1	1	yes	characteristic
144	C₂×C₈	#686	1	1	yes	characteristic
145	C₂×C₈	#687	1	1	yes	characteristic
146	C₂×C₈	#688	1	1	yes	characteristic
147	C₂×C₈	#689	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
149	C₄⋊C₄	—	10	2	no	—
150	C₂²⋊C₄	—	20	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	20	1	no	—
153	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
154	C₄²	#685	1	1	yes	characteristic
155	C₁₆	—	4	2	no	—
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	#1279	1	1	yes	characteristic
159	C₂×C₅	#1336	1	2	yes	—
160	C₂³	#3957	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	20	2	no	—
165	C₂×C₄	#3941	1	1	yes	characteristic
166	C₂×C₄	#3942	1	1	yes	characteristic
167	C₂×C₄	#3952	1	1	yes	characteristic
168	C₂×C₄	#3952	1	1	yes	characteristic
169	C₂×C₄	#3957	1	1	yes	characteristic
170	C₂×C₄	#3963	1	1	yes	characteristic
171	C₈	—	20	4	no	—
172	C₈	—	20	2	no	—
173	C₈	#3939	1	1	yes	characteristic
174	C₈	#3940	1	1	yes	characteristic
175	C₈	C₈×Dic₅	1	1	yes	characteristic
176	C₈	#3946	1	1	yes	characteristic
177	C₈	#3946	1	1	yes	characteristic
178	C₈	#3947	1	1	yes	characteristic
179	C₈	#3964	1	1	yes	characteristic
180	C₈	#3967	1	1	yes	characteristic
181	C₅	#7182	1	1	yes	characteristic, complemented, Sylow
182	C₂²	—	2	2	no	—
183	C₂²	#64703	1	1	yes	characteristic
184	C₂²	#64704	1	1	yes	characteristic
185	C₂²	#64775	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	40	2	no	—
189	C₄	#64703	1	1	yes	characteristic
190	C₄	#64704	1	1	yes	characteristic
191	C₄	#64749	1	1	yes	characteristic
192	C₄	#64770	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#10598060	1	2	yes	—
196	C₂	#10598195	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3941
commutator	C₅×C₈	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₂×C₅×#364	C₂
socle	C₂²×C₅	D₄⋊C₈
prime radical	C₂³×C₅	#104
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	C₂×#364	D₅
hypocenter	C₅	#7182
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
11. C₅×#364	C₄	semidirect
14. #64675	C₄	semidirect
30. #3951	C₈	semidirect
31. #3950	C₈	semidirect
73. #364	Dic₅	semidirect
181. C₅	#7182	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#364	2
#10600572	2
C₂×#64675	2
#7182	5