Subgroups — #12014604

Order 1280.

438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64721	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₅	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₅×#1374	C₂	1	1	yes	maximal, characteristic
5	#10598148	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×C₁₆	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	C₂²×#672	C₂²	1	2	yes	—
13	C₄²×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	#64721	C₄	1	8	yes	complemented
16	C₁₆×Dic₅	—	2	4	no	—
17	C₄×C₄⋊C₁₆	—	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	C₈	1	2	yes	—
22	C₂×C₄×Dic₅	C₈	1	2	yes	—
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	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	—	2	2	no	—
29	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
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	2	yes	—
35	C₈×Dic₅	Q₈	1	2	yes	—
36	C₂×#672	—	2	2	no	—
37	C₂×#672	—	2	2	no	—
38	C₂×#672	C₂×C₄	1	4	yes	—
39	C₄×#168	—	2	2	no	—
40	C₂×C₄⋊C₁₆	—	5	2	no	—
41	C₂×C₄×C₁₆	—	5	2	no	—
42	C₄²×C₈	—	5	1	no	—
43	#1374	—	5	1	no	—
44	#1374	D₅	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₅	C₂×C₈	1	2	yes	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
54	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	2	yes	—
57	C₄²×C₅	—	2	1	no	—
58	C₄²×C₅	—	2	1	no	—
59	C₄×Dic₅	—	2	4	no	—
60	C₄×Dic₅	C₁₆	1	8	yes	complemented
61	C₄×Dic₅	C₄²	1	2	yes	—
62	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₅	C₂×C₈	1	2	yes	—
64	C₄×Dic₅	M₁₆	1	2	yes	—
65	C₂×#168	—	2	1	no	—
66	C₂×#168	—	2	1	no	—
67	C₂×#168	C₄²	1	2	yes	—
68	C₂×#168	C₄⋊C₄	1	2	yes	—
69	C₂×#168	C₂×C₈	1	2	yes	—
70	C₂×#168	M₁₆	1	2	yes	—
71	C₅×C₁₆	—	2	4	no	—
72	#672	—	2	4	no	—
73	C₂²×C₁₆	—	5	2	no	—
74	C₂²×C₁₆	C₂×D₅	1	2	yes	—
75	C₂×C₄×C₈	—	5	1	no	—
76	C₂×C₄×C₈	—	5	1	no	—
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	C₂×D₅	1	1	yes	characteristic
81	C₄³	—	5	1	no	—
82	C₄⋊C₁₆	—	5	8	no	—
83	C₄×C₁₆	—	10	4	no	—
84	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
88	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	#117	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
94	C₂×Dic₅	—	2	2	no	—
95	C₂×Dic₅	C₄×C₈	1	2	yes	—
96	C₂×Dic₅	C₄×C₈	1	2	yes	—
97	C₂×Dic₅	C₄⋊C₈	1	4	yes	—
98	C₂×Dic₅	C₂×C₁₆	1	4	yes	—
99	C₂×Dic₅	C₂×C₁₆	1	4	yes	—
100	C₂×Dic₅	M₃₂	1	4	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	—	2	2	no	—
105	#168	C₄⋊C₈	1	4	yes	—
106	C₂²×C₈	—	5	1	no	—
107	C₂²×C₈	—	5	1	no	—
108	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
110	C₂×C₄²	—	5	2	no	—
111	C₂×C₄²	—	5	1	no	—
112	C₂×C₄²	—	5	1	no	—
113	C₂×C₄²	—	5	2	no	—
114	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	5	4	no	—
118	C₂×C₁₆	—	10	2	no	—
119	C₂×C₁₆	—	10	2	no	—
120	C₂×C₁₆	C₄×D₅	1	4	yes	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	5	4	no	—
123	C₄×C₈	—	5	4	no	—
124	C₄×C₈	—	5	2	no	—
125	C₄×C₈	—	5	2	no	—
126	C₄×C₈	—	10	2	no	—
127	C₄×C₈	—	10	2	no	—
128	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
129	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₅	#431	1	1	yes	characteristic
131	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
132	C₂²×C₅	C₂×M₃₂	1	2	yes	—
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
136	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₅	#431	1	1	yes	characteristic
138	Dic₅	—	2	2	no	—
139	Dic₅	C₄×C₁₆	1	4	yes	complemented
140	Dic₅	C₄⋊C₁₆	1	8	yes	complemented
141	C₂²×C₄	—	5	2	no	—
142	C₂²×C₄	—	5	1	no	—
143	C₂²×C₄	—	5	1	no	—
144	C₂²×C₄	C₈×D₅	1	2	yes	—
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	2	no	—
150	C₂×C₈	—	5	2	no	—
151	C₂×C₈	—	5	2	no	—
152	C₂×C₈	—	10	1	no	—
153	C₂×C₈	—	10	1	no	—
154	C₂×C₈	C₈×D₅	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
156	C₂×C₈	D₄×D₅	1	1	yes	characteristic
157	C₂×C₈	#711	1	2	yes	—
158	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	5	8	no	—
162	C₄²	—	5	2	no	—
163	C₄²	—	5	2	no	—
164	C₄²	—	5	2	no	—
165	C₄²	—	5	2	no	—
166	C₄²	—	10	4	no	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	10	4	no	—
169	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
170	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
171	C₂×C₅	C₄×M₃₂	1	1	yes	characteristic
172	C₂×C₅	C₂×C₄⋊C₁₆	1	2	yes	—
173	C₂×C₅	#2162	1	2	yes	—
174	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	5	4	no	—
178	C₂×C₄	—	5	4	no	—
179	C₂×C₄	—	5	4	no	—
180	C₂×C₄	—	5	4	no	—
181	C₂×C₄	—	5	2	no	—
182	C₂×C₄	—	5	2	no	—
183	C₂×C₄	—	10	2	no	—
184	C₂×C₄	D₅×C₁₆	1	4	yes	—
185	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
186	C₂×C₄	#4023	1	1	yes	characteristic
187	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
189	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
190	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	5	4	no	—
193	C₈	—	10	2	no	—
194	C₈	#4027	1	2	yes	—
195	C₈	#4033	1	2	yes	—
196	C₅	C₄×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
197	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
198	C₂²	#65007	1	1	yes	characteristic
199	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
200	C₂²	C₂×D₅×C₁₆	1	2	yes	—
201	C₂²	D₅×M₃₂	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	5	8	no	—
204	C₄	—	5	4	no	—
205	C₄	—	10	2	no	—
206	C₄	#64938	1	1	yes	characteristic
207	C₄	#64996	1	1	yes	characteristic
208	C₄	#65007	1	1	yes	characteristic
209	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
210	C₂	#10599395	1	1	yes	characteristic
211	C₂	C₄×D₅×C₁₆	1	1	yes	characteristic
212	C₂	#10599864	1	1	yes	characteristic
213	C₂	#10599885	1	2	yes	—
214	C₂	D₅×C₄⋊C₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂×C₅	C₂×C₄×C₁₆
Fitting	C₅×#1374	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1374	D₅
hypocenter	C₅	C₄×C₄⋊C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
15. #64721	C₄	semidirect
60. C₄×Dic₅	C₁₆	semidirect
139. Dic₅	C₄×C₁₆	semidirect
140. Dic₅	C₄⋊C₁₆	semidirect
196. C₅	C₄×C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64721	2
C₂×C₁₆×Dic₅	2
C₄×C₈×Dic₅	2
C₅×#1374	2
#10598148	2
C₄×C₄⋊C₁₆	5