Subgroups — #12014617

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64731	C₂	1	2	yes	maximal
2	Dic₅×M₃₂	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₅×#1376	C₂	1	1	yes	maximal, characteristic
5	#10598149	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	C₄²×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	#64731	C₄	1	8	yes	complemented
16	#64722	—	2	2	no	—
17	C₁₆×Dic₅	—	2	2	no	—
18	C₄×#364	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	—	2	2	no	—
23	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×#168	C₂×C₄	1	1	yes	characteristic
26	C₂²×#168	C₂×C₄	1	1	yes	characteristic
27	C₅×M₃₂	—	2	2	no	—
28	C₅×M₃₂	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₁₆	—	2	2	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	C₈×Dic₅	—	2	2	no	—
32	C₈×Dic₅	C₈	1	4	yes	—
33	C₈×Dic₅	C₂×C₄	1	4	yes	—
34	C₈×Dic₅	D₄	1	1	yes	characteristic
35	C₈×Dic₅	D₄	1	1	yes	characteristic
36	C₈×Dic₅	Q₈	1	1	yes	characteristic
37	C₈×Dic₅	Q₈	1	1	yes	characteristic
38	#3944	—	2	2	no	—
39	#3944	C₂×C₄	1	4	yes	—
40	C₂×#672	—	2	2	no	—
41	C₄×#168	C₈	1	4	yes	—
42	C₂×#364	—	5	2	no	—
43	C₄×M₃₂	—	5	2	no	—
44	C₄²×C₈	—	5	1	no	—
45	#1376	—	5	1	no	—
46	#1376	D₅	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	1	no	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅	—	2	1	no	—
50	C₂²×Dic₅	M₁₆	1	2	yes	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	—	2	1	no	—
53	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
55	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	2	yes	—
58	C₄²×C₅	—	2	1	no	—
59	C₄²×C₅	C₂×C₈	1	2	yes	—
60	C₄×Dic₅	—	2	8	no	—
61	C₄×Dic₅	—	2	1	no	—
62	C₄×Dic₅	C₄²	1	2	yes	—
63	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
64	C₄×Dic₅	C₄⋊C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₈	1	2	yes	—
66	C₂×#168	C₄²	1	2	yes	—
67	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
68	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
69	C₂×#168	C₂×C₈	1	2	yes	—
70	C₂×#168	C₂×C₈	1	2	yes	—
71	C₂×#168	C₂×C₈	1	2	yes	—
72	C₂×#168	M₁₆	1	2	yes	—
73	C₅×C₁₆	—	2	4	no	—
74	#672	—	2	4	no	—
75	C₂×M₃₂	—	5	2	no	—
76	C₂×M₃₂	C₂×D₅	1	2	yes	—
77	C₂×C₄×C₈	—	5	1	no	—
78	C₂×C₄×C₈	—	5	1	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
83	C₄³	—	5	1	no	—
84	#364	—	5	8	no	—
85	#346	—	10	2	no	—
86	C₄×C₁₆	—	10	2	no	—
87	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	#117	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂×Dic₅	—	2	4	no	—
97	C₂×Dic₅	—	2	2	no	—
98	C₂×Dic₅	—	2	4	no	—
99	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
100	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	C₄×D₄	1	2	yes	—
103	C₅×C₈	C₄×Q₈	1	2	yes	—
104	#168	C₄×C₈	1	4	yes	—
105	#168	C₄⋊C₈	1	2	yes	—
106	#168	C₄⋊C₈	1	2	yes	—
107	C₂²×C₈	—	5	1	no	—
108	C₂²×C₈	—	5	1	no	—
109	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
111	C₂×C₄²	—	5	1	no	—
112	C₂×C₄²	—	5	1	no	—
113	C₂×C₄²	—	10	2	no	—
114	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	5	4	no	—
117	M₃₂	—	10	2	no	—
118	M₃₂	C₄×D₅	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	10	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	5	4	no	—
123	C₄×C₈	—	5	4	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	5	1	no	—
127	C₄×C₈	—	5	1	no	—
128	C₄×C₈	—	5	4	no	—
129	C₄×C₈	—	10	2	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
132	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂²×C₅	#431	1	1	yes	characteristic
134	C₄×C₅	—	2	4	no	—
135	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₅	#431	1	1	yes	characteristic
139	Dic₅	—	2	4	no	—
140	Dic₅	—	2	2	no	—
141	Dic₅	#364	1	4	yes	complemented
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	5	2	no	—
144	C₂²×C₄	—	10	1	no	—
145	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	1	no	—
150	C₂×C₈	—	5	1	no	—
151	C₂×C₈	—	5	2	no	—
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	5	2	no	—
154	C₂×C₈	—	5	2	no	—
155	C₂×C₈	C₈×D₅	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
157	C₂×C₈	D₄×D₅	1	1	yes	characteristic
158	C₂×C₈	#711	1	2	yes	—
159	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
160	C₄²	—	2	1	no	—
161	C₄²	—	5	1	no	—
162	C₄²	—	5	1	no	—
163	C₄²	—	5	2	no	—
164	C₄²	—	5	2	no	—
165	C₄²	—	10	8	no	—
166	C₄²	—	10	1	no	—
167	C₄²	C₈×D₅	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₁₆	—	10	4	no	—
170	C₂×C₅	—	2	1	no	—
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
173	C₂×C₅	C₂×#364	1	2	yes	—
174	C₂³	D₅×M₁₆	1	1	yes	characteristic
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	5	2	no	—
178	C₂×C₄	—	5	2	no	—
179	C₂×C₄	—	10	4	no	—
180	C₂×C₄	—	10	4	no	—
181	C₂×C₄	—	10	2	no	—
182	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
183	C₂×C₄	#4023	1	1	yes	characteristic
184	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
187	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	5	2	no	—
190	C₈	—	5	2	no	—
191	C₈	—	5	4	no	—
192	C₈	#4027	1	2	yes	—
193	C₈	#4033	1	2	yes	—
194	C₅	C₄×#364	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	#64996	1	1	yes	characteristic
197	C₂²	#65007	1	1	yes	characteristic
198	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	5	4	no	—
201	C₄	—	10	4	no	—
202	C₄	—	10	2	no	—
203	C₄	#64938	1	1	yes	characteristic
204	C₄	C₄×C₈×D₅	1	1	yes	characteristic
205	C₄	#65007	1	1	yes	characteristic
206	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#10599395	1	1	yes	characteristic
210	C₂	D₅×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1376	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1376	D₅
hypocenter	C₅	C₄×#364
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
15. #64731	C₄	semidirect
141. Dic₅	#364	semidirect
194. C₅	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64731	2
Dic₅×M₃₂	2
C₄×C₈×Dic₅	2
C₅×#1376	2
#10598149	2
C₄×#364	5