Subgroups — #11998673

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 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×#1846	C₂	1	1	yes	maximal, characteristic
3	#10600517	C₂	1	1	yes	maximal, characteristic
4	#10600339	C₂	1	4	yes	maximal
5	#65804	C₂²	1	2	yes	—
6	C₂×C₅×#107	—	2	1	no	—
7	C₂×C₅×#106	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#382	C₂²	1	2	yes	—
10	#65395	—	2	1	no	—
11	#65395	—	2	1	no	—
12	#65214	C₂²	1	2	yes	—
13	#64972	—	2	1	no	—
14	#64972	—	2	1	no	—
15	#64971	C₄	1	2	yes	complemented
16	#64969	C₄	1	2	yes	complemented
17	#64675	—	2	1	no	—
18	C₈×#168	—	2	1	no	—
19	#64672	—	2	8	no	—
20	#9755	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	#4072	—	2	2	no	—
26	#4070	—	2	4	no	—
27	#4016	—	2	2	no	—
28	#4015	C₂×C₄	1	4	yes	—
29	C₄×Dic₁₀	—	2	4	no	—
30	C₅×#107	—	4	2	no	—
31	C₅×#106	—	4	2	no	—
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×C₈	D₄	1	1	yes	characteristic
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	C₅×#95	—	2	4	no	—
37	C₂²⋊#168	D₄	1	4	yes	—
38	#3950	—	2	2	no	—
39	#3950	—	2	2	no	—
40	#3949	—	2	2	no	—
41	#3949	—	2	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄×#168	—	2	2	no	—
44	#3160	—	5	1	no	—
45	#1846	D₅	1	1	yes	characteristic
46	#1518	—	5	4	no	—
47	#1450	—	5	1	no	—
48	C₂²×C₄×C₅	—	4	2	no	—
49	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₀	—	2	4	no	—
52	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	SD₁₆	1	2	yes	—
57	C₂×C₅×C₈	SD₁₆	1	2	yes	—
58	C₂×C₅×C₈	Q₁₆	1	2	yes	—
59	C₂×C₅×C₈	Q₁₆	1	2	yes	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₅×C₄⋊C₄	—	4	2	no	—
62	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅	#41	1	2	yes	—
65	C₂²⋊Dic₅	—	4	2	no	—
66	C₄⋊Dic₅	—	2	4	no	—
67	C₄⋊Dic₅	—	2	4	no	—
68	#683	—	4	4	no	—
69	C₄×Dic₅	—	4	2	no	—
70	C₂×#168	—	2	4	no	—
71	#533	—	5	2	no	—
72	#500	—	5	2	no	—
73	#499	—	10	1	no	—
74	#499	—	10	1	no	—
75	#498	—	5	2	no	—
76	#431	—	5	2	no	—
77	#427	—	10	1	no	—
78	#427	—	10	1	no	—
79	C₂×#107	—	2	1	no	—
80	C₂×#106	—	2	1	no	—
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	#382	C₂×D₅	1	2	yes	—
83	#328	—	10	8	no	—
84	#322	—	10	1	no	—
85	C₈²	—	10	1	no	—
86	C₂³×C₅	#127	1	1	yes	characteristic
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	8	4	no	—
89	C₂×C₄×C₅	—	8	2	no	—
90	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	#124	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×Dic₅	—	4	4	no	—
99	Dic₁₀	—	8	4	no	—
100	C₅×C₈	#103	1	2	yes	—
101	C₅×C₈	#103	1	2	yes	—
102	C₅×C₈	#103	1	2	yes	—
103	C₅×C₈	#103	1	2	yes	—
104	#168	—	4	4	no	—
105	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
107	C₄⋊Q₈	—	5	4	no	—
108	#125	—	10	2	no	—
109	C₂²⋊Q₈	—	10	4	no	—
110	C₄×Q₈	—	10	4	no	—
111	#117	—	10	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	#107	—	4	2	no	—
115	#107	—	10	2	no	—
116	#107	—	10	2	no	—
117	#106	—	4	2	no	—
118	#106	—	10	2	no	—
119	#106	—	10	2	no	—
120	C₄⋊C₈	—	10	2	no	—
121	C₂²⋊C₈	—	5	4	no	—
122	C₄×C₈	—	10	2	no	—
123	C₄×C₈	C₄×D₅	1	1	yes	characteristic
124	C₄×C₈	C₄×D₅	1	1	yes	characteristic
125	C₄×C₈	D₂₀	1	1	yes	characteristic
126	C₄×C₈	D₂₀	1	1	yes	characteristic
127	#95	—	2	4	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	#390	1	1	yes	characteristic
130	C₂²×C₅	#492	1	1	yes	characteristic
131	C₂²×C₅	#494	1	1	yes	characteristic
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	16	2	no	—
135	C₄×C₅	C₂×#103	1	1	yes	characteristic
136	C₄×C₅	C₂×#103	1	1	yes	characteristic
137	C₄×C₅	#487	1	1	yes	characteristic
138	C₄×C₅	#487	1	1	yes	characteristic
139	Dic₅	—	8	4	no	—
140	C₂×Q₈	—	10	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₄×D₅	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₈	—	10	4	no	—
145	C₂×C₈	#685	1	1	yes	characteristic
146	C₂×C₈	#685	1	1	yes	characteristic
147	C₂×C₈	#685	1	1	yes	characteristic
148	C₂×C₈	#685	1	1	yes	characteristic
149	C₂×C₈	#687	1	2	yes	—
150	C₂×C₈	#687	1	2	yes	—
151	C₂×C₈	#689	1	2	yes	—
152	C₂×C₈	#689	1	2	yes	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	10	4	no	—
156	C₄⋊C₄	—	10	4	no	—
157	C₄⋊C₄	—	20	4	no	—
158	C₂²⋊C₄	—	20	2	no	—
159	C₄²	—	20	2	no	—
160	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
161	C₄²	C₂×D₂₀	1	1	yes	characteristic
162	C₄²	#713	1	2	yes	—
163	C₂×C₅	—	2	1	no	—
164	C₂×C₅	—	2	1	no	—
165	C₂×C₅	#1956	1	1	yes	characteristic
166	C₂×C₅	#1976	1	2	yes	—
167	C₂³	#4086	1	1	yes	characteristic
168	Q₈	—	40	4	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	20	4	no	—
173	C₂×C₄	#4039	1	2	yes	—
174	C₂×C₄	#4041	1	2	yes	—
175	C₂×C₄	C₂×#685	1	1	yes	characteristic
176	C₂×C₄	C₂×#687	1	1	yes	characteristic
177	C₂×C₄	C₂×#687	1	1	yes	characteristic
178	C₂×C₄	C₂×#689	1	1	yes	characteristic
179	C₂×C₄	C₂×#689	1	1	yes	characteristic
180	C₂×C₄	#4093	1	1	yes	characteristic
181	C₈	—	20	4	no	—
182	C₈	#3942	1	2	yes	—
183	C₈	#3942	1	2	yes	—
184	C₈	#3942	1	2	yes	—
185	C₈	#3942	1	2	yes	—
186	C₅	#9755	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	2	2	no	—
188	C₂²	#65277	1	1	yes	characteristic
189	C₂²	#65365	1	1	yes	characteristic
190	C₂²	#65381	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	16	2	no	—
194	C₄	—	40	4	no	—
195	C₄	C₂×#3942	1	1	yes	characteristic
196	C₄	C₂×#3942	1	1	yes	characteristic
197	C₄	#65382	1	1	yes	characteristic
198	C₄	#65382	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#10600372	1	1	yes	characteristic
202	C₂	#10601451	1	2	yes	—

	subgroup	G/H
center	C₂²	#65277
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1846	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1846	D₅
hypocenter	C₅	#9755
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
15. #64971	C₄	semidirect
16. #64969	C₄	semidirect
186. C₅	#9755	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×#1846	2
#10600517	2
#10600339	2
#9755	5