Subgroups — #12014623

Order 1280.

358 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#64731	C₂	1	2	yes	maximal
3	#10600542	C₂	1	1	yes	maximal, characteristic
4	C₄×#3944	C₂	1	1	yes	maximal, characteristic
5	#10598301	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₄	1	1	yes	characteristic
9	C₂×#3947	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	#65221	C₄	1	1	yes	characteristic
14	C₂×#3935	C₄	1	1	yes	characteristic
15	C₅×#346	—	2	1	no	—
16	C₄×C₅×C₁₆	—	2	1	no	—
17	#64731	C₄	1	8	yes	complemented
18	#64681	—	2	1	no	—
19	C₄×#672	—	2	1	no	—
20	#12656	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×#168	C₂×C₄	1	1	yes	characteristic
27	C₂²×#168	C₂×C₄	1	1	yes	characteristic
28	C₅×M₃₂	—	2	2	no	—
29	C₅×M₃₂	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₁₆	—	2	2	no	—
31	C₄×C₅×C₈	D₄	1	1	yes	characteristic
32	C₄×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
34	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
35	#3963	—	2	2	no	—
36	#3947	—	2	2	no	—
37	#3947	C₈	1	4	yes	—
38	C₈×Dic₅	—	2	2	no	—
39	C₈×Dic₅	C₂×C₄	1	4	yes	—
40	#3944	—	2	2	no	—
41	#3944	C₂×C₄	1	4	yes	—
42	C₂×#672	—	2	2	no	—
43	#3935	C₈	1	4	yes	—
44	C₂×#364	—	5	2	no	—
45	C₄×M₃₂	—	5	1	no	—
46	C₄×M₃₂	D₅	1	1	yes	characteristic
47	#1726	—	5	1	no	—
48	#1376	—	5	2	no	—
49	C₂²×C₄×C₅	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅	—	2	1	no	—
52	C₂²×Dic₅	—	2	1	no	—
53	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₅×C₈	M₁₆	1	2	yes	—
57	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
58	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×C₈	#41	1	2	yes	—
61	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₅	C₂×C₈	1	2	yes	—
64	C₄×Dic₅	—	2	1	no	—
65	C₄×Dic₅	—	2	1	no	—
66	C₄×Dic₅	C₄²	1	2	yes	—
67	C₄×Dic₅	C₂×C₈	1	2	yes	—
68	C₂×#168	—	2	1	no	—
69	C₂×#168	—	2	1	no	—
70	C₂×#168	C₄²	1	2	yes	—
71	C₂×#168	C₂×C₈	1	2	yes	—
72	C₂×#168	C₂×C₈	1	2	yes	—
73	C₂×#168	C₂×C₈	1	2	yes	—
74	C₅×C₁₆	—	2	4	no	—
75	#672	—	2	4	no	—
76	C₂×M₃₂	—	5	2	no	—
77	C₂×M₃₂	C₂×D₅	1	2	yes	—
78	C₂×#97	—	5	1	no	—
79	C₂×#97	—	5	1	no	—
80	C₂×#97	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₂×C₄×C₈	—	5	1	no	—
83	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
84	#376	—	5	1	no	—
85	#364	—	5	8	no	—
86	#346	—	2	1	no	—
87	#346	—	10	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	—	10	1	no	—
90	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	—	4	2	no	—
92	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	#117	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₅	—	2	2	no	—
101	C₂×Dic₅	—	2	2	no	—
102	C₂×Dic₅	—	4	2	no	—
103	C₂×Dic₅	—	4	2	no	—
104	C₅×C₈	C₄⋊C₈	1	2	yes	—
105	C₅×C₈	C₄⋊C₈	1	2	yes	—
106	C₅×C₈	C₄×D₄	1	2	yes	—
107	C₅×C₈	C₄×Q₈	1	2	yes	—
108	#168	—	2	2	no	—
109	#168	C₄×C₈	1	4	yes	—
110	C₂²×C₈	—	5	1	no	—
111	C₂²×C₈	—	5	1	no	—
112	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
114	C₂×C₄²	—	5	1	no	—
115	C₂×C₄²	—	5	1	no	—
116	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	5	4	no	—
119	M₃₂	—	10	2	no	—
120	M₃₂	C₄×D₅	1	4	yes	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	10	2	no	—
123	#97	—	5	4	no	—
124	#97	—	5	4	no	—
125	#97	—	10	2	no	—
126	C₄×C₈	—	5	4	no	—
127	C₄×C₈	—	10	2	no	—
128	C₄×C₈	Dic₁₀	1	1	yes	characteristic
129	C₄×C₈	Dic₁₀	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
132	#95	—	10	2	no	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
135	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₅	#431	1	1	yes	characteristic
137	C₄×C₅	—	2	2	no	—
138	C₄×C₅	—	2	2	no	—
139	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₄×C₅	#431	1	1	yes	characteristic
143	Dic₅	—	4	2	no	—
144	Dic₅	—	4	2	no	—
145	C₂²×C₄	—	10	1	no	—
146	C₂²×C₄	—	10	1	no	—
147	C₂²×C₄	#676	1	2	yes	—
148	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
149	C₂×C₈	—	5	2	no	—
150	C₂×C₈	—	5	2	no	—
151	C₂×C₈	—	5	2	no	—
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	10	1	no	—
154	C₂×C₈	—	10	1	no	—
155	C₂×C₈	C₈×D₅	1	2	yes	—
156	C₂×C₈	#676	1	2	yes	—
157	C₂×C₈	#683	1	1	yes	characteristic
158	C₂×C₈	#683	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
161	C₂×C₈	#709	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
163	C₄²	—	5	2	no	—
164	C₄²	—	5	2	no	—
165	C₄²	—	10	1	no	—
166	C₄²	—	10	1	no	—
167	C₄²	C₈×D₅	1	2	yes	—
168	C₄²	#683	1	1	yes	characteristic
169	C₄²	#683	1	1	yes	characteristic
170	C₁₆	—	2	4	no	—
171	C₁₆	—	10	4	no	—
172	C₂×C₅	—	2	1	no	—
173	C₂×C₅	—	2	1	no	—
174	C₂×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic
175	C₂×C₅	#2153	1	2	yes	—
176	C₂³	C₂×#676	1	1	yes	characteristic
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	10	2	no	—
179	C₂×C₄	—	10	2	no	—
180	C₂×C₄	—	20	2	no	—
181	C₂×C₄	—	20	2	no	—
182	C₂×C₄	#3946	1	2	yes	—
183	C₂×C₄	#3946	1	2	yes	—
184	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
185	C₂×C₄	#4018	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
188	C₂×C₄	C₂×#676	1	1	yes	characteristic
189	C₂×C₄	C₂×#683	1	1	yes	characteristic
190	C₈	—	5	4	no	—
191	C₈	—	10	2	no	—
192	C₈	#3946	1	2	yes	—
193	C₈	#3946	1	2	yes	—
194	C₈	C₄×Dic₁₀	1	2	yes	—
195	C₈	C₄×C₂²⋊D₅	1	2	yes	—
196	C₅	#12656	1	1	yes	characteristic, complemented, Sylow
197	C₂²	—	2	2	no	—
198	C₂²	#64974	1	1	yes	characteristic
199	C₂²	C₄×#676	1	1	yes	characteristic
200	C₂²	C₂×#3946	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	20	2	no	—
204	C₄	—	20	2	no	—
205	C₄	C₄×C₈×D₅	1	1	yes	characteristic
206	C₄	#64974	1	1	yes	characteristic
207	C₄	C₄×#683	1	1	yes	characteristic
208	C₄	C₂×#3946	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄×#3946	1	1	yes	characteristic
212	C₂	#10602216	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#12656
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
17. #64731	C₄	semidirect
196. C₅	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×M₃₂	2
C₂×#64731	2
#10600542	2
C₄×#3944	2
#10598301	2
#12656	5