Subgroups — #12014372

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65411	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#349	C₂	1	1	yes	maximal, characteristic
3	C₂×#64773	C₂	1	1	yes	maximal, characteristic
4	C₂×#64732	C₂	1	2	yes	maximal, complemented
5	#10598301	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂×#4072	C₄	1	1	yes	characteristic
8	C₂³×#168	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	#65411	—	2	4	no	—
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×#3947	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	C₅×#349	C₄	1	4	yes	complemented
17	#64773	C₄	1	4	yes	complemented
18	#64732	—	2	4	no	—
19	#12608	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₅	—	2	1	no	—
28	#4072	—	2	2	no	—
29	#4072	—	2	2	no	—
30	C₂×C₄⋊Dic₅	—	2	1	no	—
31	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×#168	—	2	2	no	—
34	C₂²×#168	—	2	2	no	—
35	C₂²×#168	—	2	2	no	—
36	C₂²×#168	C₂×C₄	1	1	yes	characteristic
37	C₂²×#168	C₂×C₄	1	1	yes	characteristic
38	C₅×M₃₂	—	2	2	no	—
39	C₅×M₃₂	C₂×C₄	1	4	yes	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	#3947	—	2	2	no	—
42	#3947	D₄	1	4	yes	—
43	C₈×Dic₅	—	2	2	no	—
44	C₈×Dic₅	—	2	2	no	—
45	#3944	—	2	2	no	—
46	#3944	C₂×C₄	1	4	yes	—
47	C₂×#672	—	2	2	no	—
48	C₂×#405	—	5	1	no	—
49	C₂×#349	—	5	1	no	—
50	C₂×#349	D₅	1	1	yes	characteristic
51	C₂×#347	—	5	2	no	—
52	#1376	—	5	2	no	—
53	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	C₄²	1	2	yes	—
59	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅	M₁₆	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅	—	2	1	no	—
63	C₂²×Dic₅	—	2	1	no	—
64	C₅×M₁₆	—	2	4	no	—
65	C₅×M₁₆	—	2	4	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	C₄²	1	2	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₅×C₈	#41	1	2	yes	—
73	C₂²⋊Dic₅	—	2	4	no	—
74	C₄⋊Dic₅	—	2	4	no	—
75	C₄×Dic₅	—	2	1	no	—
76	C₄×Dic₅	—	2	1	no	—
77	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
78	C₄×Dic₅	M₁₆	1	2	yes	—
79	C₂×#168	—	2	8	no	—
80	C₂×#168	—	2	1	no	—
81	C₂×#168	—	2	1	no	—
82	C₂×#168	—	2	1	no	—
83	C₂×#168	—	2	1	no	—
84	C₂×#168	C₂²⋊C₄	1	2	yes	—
85	C₂×#168	M₁₆	1	2	yes	—
86	C₅×C₁₆	—	2	4	no	—
87	#672	—	2	4	no	—
88	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
89	C₂³×C₈	—	5	1	no	—
90	C₂×#117	—	5	1	no	—
91	C₂×M₃₂	—	5	2	no	—
92	C₂×M₃₂	C₂×D₅	1	2	yes	—
93	#405	—	10	4	no	—
94	C₂×#97	—	5	1	no	—
95	C₂×#97	—	5	1	no	—
96	C₂×C₄×C₈	—	5	1	no	—
97	C₂×C₄×C₈	—	5	1	no	—
98	#349	—	5	4	no	—
99	#349	Dic₅	1	4	yes	complemented
100	#347	—	10	4	no	—
101	C₂³×C₅	—	2	1	no	—
102	C₂³×C₅	—	2	1	no	—
103	C₂³×C₅	—	4	2	no	—
104	C₂³×C₅	#97	1	2	yes	—
105	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	#97	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#117	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₂×Dic₅	—	2	2	no	—
120	C₂×Dic₅	—	2	2	no	—
121	C₂×Dic₅	—	4	2	no	—
122	C₂×Dic₅	—	4	2	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	C₄×D₄	1	2	yes	—
125	C₅×C₈	C₄×D₄	1	2	yes	—
126	#168	—	2	2	no	—
127	#168	—	2	2	no	—
128	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	C₄×D₅	1	2	yes	—
132	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
133	C₂²×C₈	—	5	1	no	—
134	C₂²×C₈	—	5	1	no	—
135	C₂²×C₈	—	10	2	no	—
136	C₂²×C₈	—	10	2	no	—
137	C₂²×C₈	—	10	2	no	—
138	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
140	#117	—	10	2	no	—
141	#117	—	10	2	no	—
142	C₂×C₄⋊C₄	—	10	1	no	—
143	C₂×C₂²⋊C₄	—	10	1	no	—
144	C₂×C₄²	—	5	1	no	—
145	C₂×C₄²	—	5	1	no	—
146	M₃₂	—	2	2	no	—
147	M₃₂	—	5	4	no	—
148	M₃₂	—	10	2	no	—
149	M₃₂	C₂×Dic₅	1	4	yes	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	10	2	no	—
152	#97	—	5	4	no	—
153	#97	—	10	2	no	—
154	C₄×C₈	—	10	2	no	—
155	C₄×C₈	—	10	2	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	—	4	2	no	—
161	C₂²×C₅	C₂×#97	1	1	yes	characteristic
162	C₂²×C₅	#407	1	1	yes	characteristic
163	C₂²×C₅	#432	1	1	yes	characteristic
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
167	C₄×C₅	#407	1	1	yes	characteristic
168	C₄×C₅	#432	1	1	yes	characteristic
169	Dic₅	—	4	2	no	—
170	Dic₅	—	4	2	no	—
171	C₂⁴	#676	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	10	1	no	—
177	C₂²×C₄	—	10	1	no	—
178	C₂²×C₄	#676	1	1	yes	characteristic
179	C₂²×C₄	#676	1	2	yes	—
180	C₂²×C₄	C₄×Dic₅	1	2	yes	—
181	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
182	M₁₆	—	2	4	no	—
183	M₁₆	—	2	4	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	5	2	no	—
187	C₂×C₈	—	5	2	no	—
188	C₂×C₈	—	10	8	no	—
189	C₂×C₈	—	10	1	no	—
190	C₂×C₈	—	10	1	no	—
191	C₂×C₈	—	10	1	no	—
192	C₂×C₈	—	10	1	no	—
193	C₂×C₈	C₄×Dic₅	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
195	C₂×C₈	D₄×D₅	1	2	yes	—
196	C₂×C₈	#711	1	2	yes	—
197	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
198	C₄⋊C₄	—	10	4	no	—
199	C₂²⋊C₄	—	10	4	no	—
200	C₄²	—	5	2	no	—
201	C₄²	—	5	2	no	—
202	C₄²	—	10	1	no	—
203	C₄²	—	10	1	no	—
204	C₁₆	—	2	4	no	—
205	C₁₆	—	10	4	no	—
206	C₂×C₅	—	2	1	no	—
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	—	4	2	no	—
209	C₂×C₅	#1749	1	1	yes	characteristic
210	C₂×C₅	#2166	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	#3947	1	2	yes	—
215	C₂³	C₂×#676	1	1	yes	characteristic
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	10	2	no	—
223	C₂×C₄	—	10	2	no	—
224	C₂×C₄	—	20	2	no	—
225	C₂×C₄	—	20	2	no	—
226	C₂×C₄	#3947	1	2	yes	—
227	C₂×C₄	#4023	1	1	yes	characteristic
228	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×#676	1	1	yes	characteristic
230	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
231	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	10	2	no	—
235	C₈	—	10	2	no	—
236	C₈	#4027	1	2	yes	—
237	C₈	D₄×Dic₅	1	2	yes	—
238	C₅	#12608	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#65008	1	1	yes	characteristic
245	C₂²	#65015	1	1	yes	characteristic
246	C₂²	C₂×#3947	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	20	2	no	—
249	C₄	—	20	2	no	—
250	C₄	#65008	1	1	yes	characteristic
251	C₄	#65015	1	1	yes	characteristic
252	C₄	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
253	C₄	M₁₆×Dic₅	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#10599908	1	2	yes	—
258	C₂	#10600699	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4023
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#349	C₂
socle	C₂²×C₅	#432
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#349	D₅
hypocenter	C₅	#12608
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
4. C₂×#64732	C₂	semidirect
5. #10598301	C₂	semidirect
16. C₅×#349	C₄	semidirect
17. #64773	C₄	semidirect
99. #349	Dic₅	semidirect
238. C₅	#12608	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65411	2
C₂×C₅×#349	2
C₂×#64773	2
C₂×#64732	2
#10598301	2
#12608	5