Subgroups — #12015699

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#1374	C₂	1	1	yes	maximal, characteristic
3	#10598295	C₂	1	2	yes	maximal
4	#10598295	C₂	1	2	yes	maximal
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	#12894	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
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	—	2	2	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	C₈×Dic₅	—	2	2	no	—
32	C₈×Dic₅	—	2	2	no	—
33	C₈×Dic₅	—	2	1	no	—
34	C₈×Dic₅	—	2	1	no	—
35	C₈×Dic₅	—	2	2	no	—
36	C₂×#672	—	2	2	no	—
37	C₂×#672	—	2	2	no	—
38	C₂×#672	—	2	2	no	—
39	C₄×#168	—	2	2	no	—
40	C₄²×C₈	—	5	1	no	—
41	#1374	—	5	2	no	—
42	#1374	—	5	2	no	—
43	#1374	—	5	1	no	—
44	#1374	D₅	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
52	C₂×C₅×C₈	—	2	1	no	—
53	C₂×C₅×C₈	—	2	1	no	—
54	C₂×C₅×C₈	M₁₆	1	2	yes	—
55	C₂×C₅×C₈	#41	1	2	yes	—
56	C₂×C₅×C₈	#41	1	2	yes	—
57	C₂×C₅×C₈	#41	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	1	yes	characteristic
59	C₄²×C₅	—	2	1	no	—
60	C₄²×C₅	—	2	1	no	—
61	C₄×Dic₅	—	2	2	no	—
62	C₄×Dic₅	—	2	1	no	—
63	C₄×Dic₅	—	2	1	no	—
64	C₄×Dic₅	—	2	1	no	—
65	C₄×Dic₅	—	2	1	no	—
66	C₄×Dic₅	—	2	2	no	—
67	C₄×Dic₅	—	2	2	no	—
68	C₄×Dic₅	C₁₆	1	4	yes	complemented
69	C₂×#168	—	2	1	no	—
70	C₂×#168	—	2	1	no	—
71	C₂×#168	—	2	1	no	—
72	C₂×#168	—	2	1	no	—
73	C₂×#168	M₁₆	1	2	yes	—
74	C₂×#168	M₁₆	1	2	yes	—
75	C₅×C₁₆	—	4	2	no	—
76	#672	—	4	2	no	—
77	C₂²×C₁₆	—	5	2	no	—
78	C₂²×C₁₆	C₂×D₅	1	2	yes	—
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	—	5	1	no	—
83	C₂×C₄×C₈	—	5	1	no	—
84	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
85	C₄³	—	5	1	no	—
86	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅	—	2	1	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
92	C₂×C₄×C₅	M₃₂	1	2	yes	—
93	C₂×C₄×C₅	#117	1	1	yes	characteristic
94	C₂×C₄×C₅	#117	1	1	yes	characteristic
95	C₂×C₄×C₅	#117	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
99	C₂×Dic₅	—	2	1	no	—
100	C₂×Dic₅	—	2	1	no	—
101	C₂×Dic₅	—	2	1	no	—
102	C₂×Dic₅	—	2	1	no	—
103	C₂×Dic₅	—	2	1	no	—
104	C₂×Dic₅	—	2	1	no	—
105	C₂×Dic₅	—	2	1	no	—
106	C₂×Dic₅	—	2	1	no	—
107	C₂×Dic₅	C₂×C₁₆	1	2	yes	—
108	C₂×Dic₅	C₂×C₁₆	1	2	yes	—
109	C₂×Dic₅	M₃₂	1	2	yes	—
110	C₂×Dic₅	M₃₂	1	2	yes	—
111	C₅×C₈	—	2	2	no	—
112	C₅×C₈	#126	1	2	yes	—
113	C₅×C₈	#126	1	2	yes	—
114	#168	—	2	2	no	—
115	#168	—	2	2	no	—
116	C₂²×C₈	—	5	1	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
120	C₂×C₄²	—	5	1	no	—
121	C₂×C₄²	—	5	1	no	—
122	C₂×C₄²	—	5	1	no	—
123	C₂×C₄²	—	5	1	no	—
124	C₂×C₄²	—	5	1	no	—
125	C₂×C₄²	—	5	1	no	—
126	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	10	2	no	—
131	C₂×C₁₆	—	10	2	no	—
132	C₂×C₁₆	—	10	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	10	2	no	—
135	C₄×C₈	—	10	2	no	—
136	C₄×C₈	—	10	2	no	—
137	C₄×C₈	—	10	2	no	—
138	C₄×C₈	—	10	1	no	—
139	C₄×C₈	—	10	1	no	—
140	C₂²×C₅	#431	1	1	yes	characteristic
141	C₂²×C₅	#431	1	1	yes	characteristic
142	C₂²×C₅	#431	1	1	yes	characteristic
143	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
144	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
145	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
146	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	#383	1	1	yes	characteristic
150	C₄×C₅	#432	1	1	yes	characteristic
151	C₄×C₅	#432	1	1	yes	characteristic
152	C₄×C₅	#432	1	1	yes	characteristic
153	Dic₅	—	2	2	no	—
154	Dic₅	—	2	2	no	—
155	Dic₅	—	2	2	no	—
156	Dic₅	—	2	2	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	5	1	no	—
159	C₂²×C₄	—	5	1	no	—
160	C₂²×C₄	—	5	1	no	—
161	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
162	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	5	2	no	—
167	C₂×C₈	—	5	2	no	—
168	C₂×C₈	—	10	1	no	—
169	C₂×C₈	—	10	1	no	—
170	C₂×C₈	—	10	1	no	—
171	C₂×C₈	—	10	1	no	—
172	C₂×C₈	#676	1	2	yes	—
173	C₂×C₈	#709	1	2	yes	—
174	C₂×C₈	#711	1	1	yes	characteristic
175	C₂×C₈	#711	1	2	yes	—
176	C₂×C₈	#713	1	1	yes	characteristic
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	5	4	no	—
180	C₄²	—	10	2	no	—
181	C₄²	—	10	2	no	—
182	C₄²	—	10	2	no	—
183	C₄²	—	10	1	no	—
184	C₄²	—	10	1	no	—
185	C₄²	—	10	1	no	—
186	C₄²	—	10	1	no	—
187	C₁₆	—	4	2	no	—
188	C₁₆	—	20	2	no	—
189	C₂×C₅	#1839	1	1	yes	characteristic
190	C₂×C₅	#2162	1	1	yes	characteristic
191	C₂×C₅	#2162	1	1	yes	characteristic
192	C₂×C₅	#2162	1	1	yes	characteristic
193	C₂×C₅	#2163	1	1	yes	characteristic
194	C₂×C₅	#2163	1	1	yes	characteristic
195	C₂×C₅	#2163	1	1	yes	characteristic
196	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	5	2	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	5	2	no	—
204	C₂×C₄	—	5	2	no	—
205	C₂×C₄	—	10	1	no	—
206	C₂×C₄	—	10	1	no	—
207	C₂×C₄	—	10	1	no	—
208	C₂×C₄	—	10	1	no	—
209	C₂×C₄	—	10	1	no	—
210	C₂×C₄	—	10	1	no	—
211	C₂×C₄	—	10	1	no	—
212	C₂×C₄	—	10	1	no	—
213	C₂×C₄	D₅×C₁₆	1	2	yes	—
214	C₂×C₄	#3930	1	2	yes	—
215	C₂×C₄	#4018	1	1	yes	characteristic
216	C₂×C₄	#4023	1	1	yes	characteristic
217	C₂×C₄	#4038	1	1	yes	characteristic
218	C₂×C₄	C₂×#676	1	1	yes	characteristic
219	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
220	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	10	2	no	—
223	C₈	—	10	2	no	—
224	C₈	#4025	1	2	yes	—
225	C₈	#4044	1	2	yes	—
226	C₅	#12894	1	1	yes	characteristic, complemented, Sylow
227	C₂²	#64974	1	1	yes	characteristic
228	C₂²	#65007	1	1	yes	characteristic
229	C₂²	#65122	1	1	yes	characteristic
230	C₂²	C₂×D₅×C₁₆	1	1	yes	characteristic
231	C₂²	C₂×#3930	1	1	yes	characteristic
232	C₂²	D₅×M₃₂	1	1	yes	characteristic
233	C₂²	D₅×M₃₂	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	10	2	no	—
237	C₄	—	10	2	no	—
238	C₄	—	10	2	no	—
239	C₄	—	10	2	no	—
240	C₄	#64940	1	1	yes	characteristic
241	C₄	#64994	1	1	yes	characteristic
242	C₄	#65008	1	1	yes	characteristic
243	C₄	#65124	1	1	yes	characteristic
244	C₂	#10599400	1	1	yes	characteristic
245	C₂	#10599845	1	1	yes	characteristic
246	C₂	#10599866	1	1	yes	characteristic
247	C₂	#10599885	1	1	yes	characteristic
248	C₂	#10599886	1	1	yes	characteristic
249	C₂	#10600067	1	1	yes	characteristic
250	C₂	#10600068	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₁₆
Fitting	C₅×#1374	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1374	D₅
hypocenter	C₅	#12894
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
68. C₄×Dic₅	C₁₆	semidirect
226. C₅	#12894	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₅	2
C₅×#1374	2
#10598295	2
#10598148	2
#12894	5