Subgroups — #12014307

Order 1280.

338 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₅×M₃₂	C₂	1	2	yes	maximal
2	#10600597	C₂	1	1	yes	maximal, characteristic
3	C₅×#1375	C₂	1	1	yes	maximal, characteristic
4	#10598302	C₂	1	2	yes	maximal
5	#10598154	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×#97	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₄	1	1	yes	characteristic
9	C₂×#3947	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	#65221	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	#64722	C₄	1	4	yes	complemented
16	C₁₆×Dic₅	C₄	1	4	yes	complemented
17	#12593	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×#168	C₂×C₄	1	1	yes	characteristic
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₅×M₃₂	—	2	2	no	—
26	C₅×M₃₂	—	2	2	no	—
27	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
28	C₅×#97	—	2	2	no	—
29	#3963	—	2	2	no	—
30	#3947	—	2	1	no	—
31	#3947	—	2	1	no	—
32	#3947	—	2	2	no	—
33	C₈×Dic₅	C₈	1	4	yes	—
34	C₈×Dic₅	C₂×C₄	1	2	yes	—
35	C₈×Dic₅	C₂×C₄	1	2	yes	—
36	#3944	—	2	2	no	—
37	#3944	—	2	2	no	—
38	C₂×#672	C₂×C₄	1	4	yes	—
39	C₄×#168	C₈	1	4	yes	—
40	C₄×M₃₂	—	5	2	no	—
41	#1726	—	5	1	no	—
42	#1375	—	5	2	no	—
43	#1375	—	5	1	no	—
44	#1375	D₅	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	2	1	no	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	2	1	no	—
48	C₂²×Dic₅	M₁₆	1	2	yes	—
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₅×C₈	#41	1	2	yes	—
54	C₂×C₅×C₈	#41	1	1	yes	characteristic
55	C₂×C₅×C₈	#41	1	1	yes	characteristic
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	C₂×C₈	1	2	yes	—
58	C₄×Dic₅	—	2	1	no	—
59	C₄×Dic₅	C₄²	1	2	yes	—
60	C₄×Dic₅	C₂×C₈	1	2	yes	—
61	C₄×Dic₅	M₁₆	1	2	yes	—
62	C₂×#168	—	2	1	no	—
63	C₂×#168	—	2	1	no	—
64	C₂×#168	C₄²	1	1	yes	characteristic
65	C₂×#168	C₄²	1	1	yes	characteristic
66	C₂×#168	C₂×C₈	1	2	yes	—
67	C₂×#168	C₂×C₈	1	2	yes	—
68	C₂×#168	C₂×C₈	1	2	yes	—
69	C₅×C₁₆	—	2	4	no	—
70	#672	—	2	4	no	—
71	C₂×M₃₂	—	5	2	no	—
72	C₂×M₃₂	C₂×D₅	1	2	yes	—
73	C₂×#97	—	5	1	no	—
74	C₂×#97	—	5	1	no	—
75	C₂×#97	C₂×D₅	1	1	yes	characteristic
76	C₂×C₄×C₈	—	5	1	no	—
77	C₂×C₄×C₈	—	5	1	no	—
78	C₂×C₄×C₈	—	5	1	no	—
79	#376	—	5	1	no	—
80	#346	—	5	4	no	—
81	C₄×C₁₆	—	5	4	no	—
82	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅	#117	1	1	yes	characteristic
87	C₂×C₄×C₅	#117	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₂×Dic₅	—	2	2	no	—
92	C₂×Dic₅	—	4	2	no	—
93	C₂×Dic₅	—	4	2	no	—
94	C₂×Dic₅	#97	1	4	yes	—
95	C₅×C₈	—	2	2	no	—
96	C₅×C₈	#117	1	2	yes	—
97	C₅×C₈	#117	1	2	yes	—
98	#168	—	2	2	no	—
99	#168	C₄×C₈	1	2	yes	—
100	#168	C₄×C₈	1	2	yes	—
101	C₂²×C₈	—	5	1	no	—
102	C₂²×C₈	—	5	1	no	—
103	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
105	C₂×C₄²	—	5	1	no	—
106	C₂×C₄²	—	5	1	no	—
107	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	2	no	—
110	M₃₂	—	10	2	no	—
111	M₃₂	—	10	2	no	—
112	C₂×C₁₆	—	5	4	no	—
113	C₂×C₁₆	C₄×D₅	1	4	yes	—
114	#97	—	2	2	no	—
115	#97	—	10	2	no	—
116	#97	—	10	1	no	—
117	#97	—	10	1	no	—
118	C₄×C₈	—	5	4	no	—
119	C₄×C₈	—	5	2	no	—
120	C₄×C₈	—	5	2	no	—
121	C₄×C₈	—	5	4	no	—
122	#95	—	10	2	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	C₂×#97	1	1	yes	characteristic
125	C₂²×C₅	#431	1	1	yes	characteristic
126	C₂²×C₅	#431	1	1	yes	characteristic
127	C₄×C₅	—	4	2	no	—
128	C₄×C₅	#376	1	1	yes	characteristic
129	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
130	C₄×C₅	#431	1	1	yes	characteristic
131	C₄×C₅	#431	1	1	yes	characteristic
132	Dic₅	—	2	4	no	—
133	Dic₅	—	4	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	5	2	no	—
136	C₂²×C₄	—	10	1	no	—
137	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	5	1	no	—
141	C₂×C₈	—	5	1	no	—
142	C₂×C₈	—	5	2	no	—
143	C₂×C₈	—	5	2	no	—
144	C₂×C₈	—	5	2	no	—
145	C₂×C₈	—	10	1	no	—
146	C₂×C₈	—	10	1	no	—
147	C₂×C₈	C₈×D₅	1	2	yes	—
148	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
149	C₂×C₈	#711	1	1	yes	characteristic
150	C₂×C₈	#711	1	2	yes	—
151	C₂×C₈	#713	1	1	yes	characteristic
152	C₄²	—	2	1	no	—
153	C₄²	—	5	2	no	—
154	C₄²	—	5	2	no	—
155	C₄²	—	5	2	no	—
156	C₄²	—	10	1	no	—
157	C₄²	C₈×D₅	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₁₆	—	10	4	no	—
160	C₂×C₅	—	2	1	no	—
161	C₂×C₅	—	2	1	no	—
162	C₂×C₅	#1744	1	1	yes	characteristic
163	C₂×C₅	#2110	1	2	yes	—
164	C₂³	D₅×M₁₆	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	5	4	no	—
168	C₂×C₄	—	10	2	no	—
169	C₂×C₄	—	20	2	no	—
170	C₂×C₄	—	20	2	no	—
171	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
172	C₂×C₄	#4023	1	1	yes	characteristic
173	C₂×C₄	#4038	1	1	yes	characteristic
174	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
175	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
176	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	5	2	no	—
179	C₈	—	5	2	no	—
180	C₈	—	10	2	no	—
181	C₈	#4023	1	2	yes	—
182	C₈	#4038	1	2	yes	—
183	C₅	#12593	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	D₅×#97	1	1	yes	characteristic
186	C₂²	#65007	1	1	yes	characteristic
187	C₂²	#65122	1	1	yes	characteristic
188	C₄	—	4	2	no	—
189	C₄	—	10	4	no	—
190	C₄	—	20	2	no	—
191	C₄	#64937	1	1	yes	characteristic
192	C₄	C₄×C₈×D₅	1	1	yes	characteristic
193	C₄	#65007	1	1	yes	characteristic
194	C₄	#65122	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#10599389	1	1	yes	characteristic
198	C₂	#10599880	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1375	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1375	D₅
hypocenter	C₅	#12593
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #1375

Complements

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

N	H ≅ G/N	type
15. #64722	C₄	semidirect
16. C₁₆×Dic₅	C₄	semidirect
183. C₅	#12593	semidirect

Maximal subgroups

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

subgroup	index
Dic₅×M₃₂	2
#10600597	2
C₅×#1375	2
#10598302	2
#10598154	2
#12593	5