Subgroups — #12014384

Order 1280.

486 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	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic
4	C₂×#64722	C₂	1	2	yes	maximal, complemented
5	#10598295	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	—	2	4	no	—
8	C₂²×C₅×C₁₆	C₄	1	4	yes	complemented
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊#168	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₂²	1	2	yes	—
14	C₂²×#672	C₂²	1	2	yes	—
15	#64770	C₄	1	2	yes	—
16	C₂²⋊#672	C₄	1	4	yes	complemented
17	#64722	—	2	4	no	—
18	#12611	—	5	1	no	maximal, Sylow
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
27	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
28	C₂²×#168	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₁₆	—	2	2	no	—
30	C₂×C₅×C₁₆	—	2	8	no	—
31	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₁₆	D₄	1	4	yes	—
34	#3963	C₈	1	2	yes	—
35	C₂²⋊#168	C₈	1	4	yes	—
36	C₈×Dic₅	—	2	2	no	—
37	C₈×Dic₅	—	2	2	no	—
38	C₂×#672	—	2	2	no	—
39	C₂×#672	—	2	2	no	—
40	C₂×#672	C₂×C₄	1	4	yes	—
41	C₂³×C₁₆	D₅	1	1	yes	characteristic
42	C₂×C₂²⋊C₁₆	—	5	1	no	—
43	C₂×#346	—	5	2	no	—
44	C₈×C₂²⋊C₄	—	5	1	no	—
45	#1374	—	5	2	no	—
46	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	C₄²	1	2	yes	—
50	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₅	C₂×C₈	1	4	yes	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	C₄²	1	2	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₅×C₈	#41	1	2	yes	—
63	C₂²⋊Dic₅	—	2	4	no	—
64	C₄×Dic₅	—	2	2	no	—
65	C₄×Dic₅	—	2	2	no	—
66	C₂×#168	—	2	2	no	—
67	C₂×#168	—	2	2	no	—
68	C₂×#168	C₂×C₈	1	4	yes	—
69	C₅×C₁₆	—	2	2	no	—
70	C₅×C₁₆	C₂²⋊C₄	1	4	yes	complemented
71	#672	—	2	4	no	—
72	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
73	C₂²×C₁₆	—	2	4	no	—
74	C₂²×C₁₆	—	5	2	no	—
75	C₂²×C₁₆	Dic₅	1	4	yes	complemented
76	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
78	C₂×C₂²⋊C₈	—	5	1	no	—
79	C₂×C₄×C₈	—	5	2	no	—
80	C₄×C₂²⋊C₄	—	5	1	no	—
81	C₂²⋊C₁₆	—	5	4	no	—
82	#346	—	10	4	no	—
83	#336	—	5	2	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	2	2	no	—
86	C₂³×C₅	C₄×C₈	1	2	yes	—
87	C₂³×C₅	M₃₂	1	4	yes	—
88	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅	#131	1	2	yes	—
98	C₂×Dic₅	—	2	4	no	—
99	C₂×Dic₅	—	2	4	no	—
100	C₂×Dic₅	—	2	4	no	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₅×C₈	#117	1	1	yes	characteristic
104	C₅×C₈	C₄×D₄	1	2	yes	—
105	#168	—	2	4	no	—
106	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	5	2	no	—
110	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
111	C₂²×C₈	C₄×D₅	1	2	yes	—
112	C₂²×C₈	C₂×Dic₅	1	2	yes	—
113	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	—	5	2	no	—
115	C₂×C₄²	—	5	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	—	5	4	no	—
119	C₂×C₁₆	—	10	2	no	—
120	C₂×C₁₆	—	10	2	no	—
121	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
122	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₅	1	4	yes	—
124	C₂²⋊C₈	—	5	4	no	—
125	C₄×C₈	—	10	2	no	—
126	C₄×C₈	—	10	2	no	—
127	#95	—	5	2	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	—	2	8	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	#346	1	4	yes	complemented
132	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
133	C₂²×C₅	C₈×D₄	1	2	yes	—
134	C₂²×C₅	C₂×M₃₂	1	2	yes	—
135	C₂²×C₅	#504	1	2	yes	—
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₅	#405	1	1	yes	characteristic
139	C₄×C₅	C₈×D₄	1	2	yes	—
140	Dic₅	—	4	4	no	—
141	C₂⁴	C₈×D₅	1	1	yes	characteristic
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	5	4	no	—
145	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
146	C₂²×C₄	C₈×D₅	1	2	yes	—
147	C₂²×C₄	C₄×Dic₅	1	2	yes	—
148	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	5	4	no	—
154	C₂×C₈	—	10	2	no	—
155	C₂×C₈	—	10	2	no	—
156	C₂×C₈	C₄×Dic₅	1	2	yes	—
157	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
158	C₂×C₈	#709	1	2	yes	—
159	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
161	C₂²⋊C₄	—	10	4	no	—
162	C₄²	—	10	2	no	—
163	C₄²	—	10	2	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	10	4	no	—
166	C₁₆	C₂²⋊Dic₅	1	4	yes	complemented
167	C₂×C₅	—	2	4	no	—
168	C₂×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₅	C₂×#346	1	1	yes	characteristic
170	C₂×C₅	#2108	1	1	yes	characteristic
171	C₂×C₅	#2168	1	4	yes	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	#3930	1	4	yes	—
175	C₂³	C₈×Dic₅	1	2	yes	—
176	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	10	4	no	—
182	C₂×C₄	—	10	4	no	—
183	C₂×C₄	—	10	4	no	—
184	C₂×C₄	C₈×Dic₅	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
186	C₂×C₄	#4047	1	2	yes	—
187	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
188	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	—	10	4	no	—
191	C₈	#4072	1	1	yes	characteristic
192	C₈	C₄×C₂²⋊D₅	1	2	yes	—
193	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
194	C₅	#12611	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	#64722	1	4	yes	complemented
199	C₂²	C₂×#3930	1	2	yes	—
200	C₂²	#65190	1	2	yes	—
201	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
202	C₂²	C₈×C₂²⋊D₅	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	20	4	no	—
205	C₄	#65390	1	1	yes	characteristic
206	C₄	C₈×C₂²⋊D₅	1	2	yes	—
207	C₄	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
208	C₂	—	2	4	no	—
209	C₂	C₂×#64722	1	1	yes	characteristic
210	C₂	#10602192	1	1	yes	characteristic
211	C₂	#10602200	1	4	yes	—
212	C₂	C₈×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	#12611
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×#64722	C₂	semidirect
5. #10598295	C₂	semidirect
8. C₂²×C₅×C₁₆	C₄	semidirect
16. C₂²⋊#672	C₄	semidirect
70. C₅×C₁₆	C₂²⋊C₄	semidirect
75. C₂²×C₁₆	Dic₅	semidirect
131. C₂²×C₅	#346	semidirect
166. C₁₆	C₂²⋊Dic₅	semidirect
194. C₅	#12611	semidirect
198. C₂²	#64722	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₈×C₂²⋊Dic₅	2
C₂×C₂²⋊#672	2
C₂×#64722	2
#10598295	2
#12611	5