Subgroups — #12010598

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610173	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×#64678	C₂	1	1	yes	maximal, characteristic
4	#10598056	C₂	1	2	yes	maximal, complemented
5	#10598055	C₂	1	2	yes	maximal, complemented
6	#65804	—	2	1	no	—
7	C₂×C₅×M₃₂	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#65395	—	2	1	no	—
10	#65395	C₄	1	2	yes	—
11	C₂×C₄⋊#168	—	2	1	no	—
12	#64972	—	2	1	no	—
13	#64972	C₄	1	2	yes	—
14	#64971	C₂²	1	1	yes	characteristic
15	#64971	C₂²	1	1	yes	characteristic
16	#64969	—	2	1	no	—
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
19	#64678	—	2	1	no	—
20	#64678	C₂²	1	2	yes	—
21	#11942	—	5	1	no	maximal, Sylow
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	#4072	—	2	2	no	—
26	#4070	—	4	1	no	—
27	#4070	—	4	1	no	—
28	C₂²×#168	—	4	1	no	—
29	#4016	—	2	2	no	—
30	#4015	—	2	1	no	—
31	#4015	—	2	1	no	—
32	C₄×Dic₁₀	—	4	1	no	—
33	C₄×Dic₁₀	—	4	1	no	—
34	C₅×M₃₂	—	4	2	no	—
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₅×C₈	D₄	1	1	yes	characteristic
38	C₄×C₅×C₈	D₄	1	1	yes	characteristic
39	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
40	#3950	—	2	1	no	—
41	#3950	—	2	1	no	—
42	#3950	C₂×C₄	1	2	yes	—
43	#3950	C₂×C₄	1	2	yes	—
44	#3949	—	2	2	no	—
45	#3949	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₄⋊#168	—	4	1	no	—
48	#3160	—	5	1	no	—
49	#2150	D₅	1	1	yes	characteristic
50	C₂×#335	—	5	1	no	—
51	#1365	—	5	2	no	—
52	#1364	—	5	2	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₂×Dic₁₀	—	4	1	no	—
57	C₂×Dic₁₀	—	4	1	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	SD₁₆	1	2	yes	—
62	C₂×C₅×C₈	Q₁₆	1	2	yes	—
63	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅	D₈	1	1	yes	characteristic
67	C₄²×C₅	SD₁₆	1	1	yes	characteristic
68	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²⋊Dic₅	—	4	2	no	—
70	C₄⋊Dic₅	—	2	2	no	—
71	C₄⋊Dic₅	—	2	2	no	—
72	C₄⋊Dic₅	—	4	1	no	—
73	C₄⋊Dic₅	—	4	1	no	—
74	#683	—	4	2	no	—
75	#683	—	4	2	no	—
76	C₄×Dic₅	—	4	2	no	—
77	C₂×#168	—	4	2	no	—
78	C₂×#168	—	8	2	no	—
79	C₅×C₁₆	—	4	2	no	—
80	#533	—	10	1	no	—
81	C₂×M₃₂	—	2	1	no	—
82	#500	—	5	1	no	—
83	#500	—	5	1	no	—
84	#499	—	5	2	no	—
85	#499	—	10	1	no	—
86	#498	—	10	1	no	—
87	#427	—	5	2	no	—
88	#427	—	10	1	no	—
89	C₂×C₄⋊C₈	—	10	1	no	—
90	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	C₂×D₅	1	2	yes	—
93	#335	—	5	2	no	—
94	#335	—	10	1	no	—
95	C₂³×C₅	#102	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	4	1	no	—
99	C₂×C₄×C₅	#101	1	2	yes	—
100	C₂×C₄×C₅	#102	1	1	yes	characteristic
101	C₂×C₄×C₅	#102	1	1	yes	characteristic
102	C₂×C₄×C₅	#102	1	1	yes	characteristic
103	C₂×C₄×C₅	#103	1	2	yes	—
104	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×Dic₅	—	4	2	no	—
108	C₂×Dic₅	—	4	2	no	—
109	Dic₁₀	—	8	2	no	—
110	Dic₁₀	—	8	1	no	—
111	Dic₁₀	—	8	1	no	—
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	#103	1	2	yes	—
115	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
116	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
117	#168	—	8	2	no	—
118	C₂²×C₈	—	20	1	no	—
119	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
120	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
121	C₄⋊Q₈	—	10	1	no	—
122	C₄⋊Q₈	—	10	1	no	—
123	#125	—	10	2	no	—
124	C₂²⋊Q₈	—	20	1	no	—
125	C₂²⋊Q₈	—	20	1	no	—
126	C₄×Q₈	—	20	1	no	—
127	C₄×Q₈	—	20	1	no	—
128	#117	—	10	2	no	—
129	C₂×C₄²	D₂₀	1	1	yes	characteristic
130	M₃₂	—	4	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	#107	—	5	2	no	—
133	#107	—	5	2	no	—
134	#107	—	10	1	no	—
135	#107	—	10	1	no	—
136	#106	—	10	2	no	—
137	#106	—	10	2	no	—
138	C₄⋊C₈	—	10	2	no	—
139	C₄⋊C₈	—	20	1	no	—
140	C₄×C₈	C₄×D₅	1	1	yes	characteristic
141	C₄×C₈	D₂₀	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	#332	1	2	yes	—
147	C₂²×C₅	C₂×#102	1	1	yes	characteristic
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	#332	1	2	yes	—
152	C₄×C₅	C₂×#101	1	1	yes	characteristic
153	C₄×C₅	C₂×#103	1	1	yes	characteristic
154	C₄×C₅	#509	1	1	yes	characteristic
155	C₄×C₅	#510	1	1	yes	characteristic
156	Dic₅	—	8	2	no	—
157	Dic₅	—	8	2	no	—
158	C₂×Q₈	—	20	1	no	—
159	C₂×Q₈	—	20	1	no	—
160	C₂²×C₄	#677	1	1	yes	characteristic
161	C₂²×C₄	D₄₀	1	1	yes	characteristic
162	C₂²×C₄	#685	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	20	2	no	—
165	C₂×C₈	—	40	2	no	—
166	C₂×C₈	#685	1	1	yes	characteristic
167	C₂×C₈	#685	1	1	yes	characteristic
168	C₂×C₈	#687	1	2	yes	—
169	C₂×C₈	#689	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
172	C₄⋊C₄	—	10	2	no	—
173	C₄⋊C₄	—	10	2	no	—
174	C₄⋊C₄	—	20	2	no	—
175	C₄⋊C₄	—	20	2	no	—
176	C₄⋊C₄	—	20	1	no	—
177	C₄⋊C₄	—	20	1	no	—
178	C₂²⋊C₄	—	20	2	no	—
179	C₄²	—	20	2	no	—
180	C₄²	#677	1	1	yes	characteristic
181	C₄²	D₄₀	1	1	yes	characteristic
182	C₄²	#685	1	1	yes	characteristic
183	C₄²	C₂×D₂₀	1	1	yes	characteristic
184	C₁₆	—	4	2	no	—
185	C₂×C₅	—	2	2	no	—
186	C₂×C₅	C₂×#332	1	1	yes	characteristic
187	C₂×C₅	#2139	1	1	yes	characteristic
188	C₂×C₅	#2148	1	1	yes	characteristic
189	C₂³	#3953	1	1	yes	characteristic
190	Q₈	—	40	2	no	—
191	Q₈	—	40	1	no	—
192	Q₈	—	40	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	20	2	no	—
197	C₂×C₄	—	20	2	no	—
198	C₂×C₄	#3942	1	2	yes	—
199	C₂×C₄	#3953	1	1	yes	characteristic
200	C₂×C₄	#3953	1	1	yes	characteristic
201	C₂×C₄	#3953	1	1	yes	characteristic
202	C₂×C₄	#3956	1	2	yes	—
203	C₂×C₄	C₂×#677	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
205	C₂×C₄	C₂×#685	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	40	2	no	—
209	C₈	#3942	1	2	yes	—
210	C₈	C₂×#687	1	1	yes	characteristic
211	C₈	C₂×#689	1	1	yes	characteristic
212	C₅	#11942	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#64706	1	2	yes	—
216	C₂²	C₂×#3953	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	40	2	no	—
221	C₄	—	40	2	no	—
222	C₄	#64706	1	2	yes	—
223	C₄	#65197	1	1	yes	characteristic
224	C₄	#65198	1	1	yes	characteristic
225	C₄	C₂×#3942	1	1	yes	characteristic
226	C₄	C₂×#3956	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#64706	1	1	yes	characteristic
229	C₂	#10602232	1	1	yes	characteristic
230	C₂	#10602235	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	#11942
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
5. #10598055	C₂	semidirect
212. C₅	#11942	semidirect

Maximal subgroups

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

subgroup	index
#10610173	2
C₅×#2150	2
C₂×#64678	2
#10598056	2
#10598055	2
#11942	5