Subgroups — #12015224

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64978	C₂	1	1	yes	maximal, characteristic
2	C₂×#64727	C₂	1	2	yes	maximal
3	C₂×#64721	C₂	1	2	yes	maximal, complemented
4	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598148	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#676	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×#4018	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	2	yes	—
14	C₂×#3935	C₄	1	1	yes	characteristic
15	#64978	—	2	4	no	—
16	#64727	—	2	4	no	—
17	#64721	—	2	4	no	—
18	#12784	—	5	1	no	maximal, Sylow
19	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×#685	C₈	1	2	yes	—
24	C₂×#683	C₈	1	2	yes	—
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	C₂×#676	—	2	2	no	—
29	C₂×#676	—	2	2	no	—
30	C₂×#676	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	C₂×C₈×D₅	—	2	2	no	—
33	C₂×C₈×D₅	D₄	1	2	yes	—
34	C₂×C₈×D₅	Q₈	1	2	yes	—
35	#4018	—	2	2	no	—
36	#4018	—	2	2	no	—
37	C₂×C₅×C₁₆	—	2	2	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₄×C₅×C₈	—	2	2	no	—
41	#3947	—	2	2	no	—
42	C₈×Dic₅	D₄	1	4	yes	—
43	C₂×#672	—	2	2	no	—
44	C₂×#672	—	2	2	no	—
45	C₂×#672	—	2	2	no	—
46	#3935	—	2	2	no	—
47	C₂×#405	—	5	1	no	—
48	C₂×C₄⋊C₁₆	—	5	2	no	—
49	C₂×C₂²⋊C₁₆	—	5	2	no	—
50	#1374	—	5	1	no	—
51	#1374	D₅	1	1	yes	characteristic, complemented
52	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅	C₂×C₈	1	2	yes	—
56	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄×D₅	—	2	1	no	—
58	C₂×C₄×D₅	—	2	1	no	—
59	C₂×C₄×D₅	—	2	1	no	—
60	C₂×C₄×D₅	—	2	1	no	—
61	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₅	M₁₆	1	2	yes	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	M₁₆	1	2	yes	—
66	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₅×C₈	#41	1	2	yes	—
70	C₄²×C₅	—	2	1	no	—
71	C₄²×C₅	—	2	1	no	—
72	#685	—	2	4	no	—
73	#683	—	2	4	no	—
74	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
75	C₄×Dic₅	M₁₆	1	2	yes	—
76	C₂×#168	—	2	1	no	—
77	C₂×#168	—	2	1	no	—
78	C₂×#168	—	2	1	no	—
79	C₂×#168	—	2	1	no	—
80	C₂×#168	C₂²⋊C₄	1	2	yes	—
81	C₂×#168	C₄⋊C₄	1	2	yes	—
82	#676	—	2	8	no	—
83	C₈×D₅	—	2	4	no	—
84	C₈×D₅	—	2	4	no	—
85	C₅×C₁₆	—	4	2	no	—
86	#672	—	4	2	no	—
87	C₂²×M₁₆	—	5	1	no	—
88	C₂³×C₈	—	5	1	no	—
89	C₂×#117	—	5	1	no	—
90	C₂²×C₁₆	—	5	2	no	—
91	C₂²×C₁₆	C₂×D₅	1	2	yes	—
92	#405	—	10	4	no	—
93	C₂×#97	—	5	1	no	—
94	C₂×#97	—	5	1	no	—
95	C₂×C₄×C₈	—	5	1	no	—
96	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
97	C₄⋊C₁₆	—	10	4	no	—
98	C₂²⋊C₁₆	—	10	4	no	—
99	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₅	—	2	1	no	—
101	C₂²×D₅	—	2	1	no	—
102	C₂²×D₅	—	2	2	no	—
103	C₂²×D₅	—	2	2	no	—
104	C₂²×D₅	C₄⋊C₈	1	2	yes	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	#117	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₂×Dic₅	—	2	1	no	—
115	C₂×Dic₅	—	2	1	no	—
116	C₂×Dic₅	—	2	2	no	—
117	C₂×Dic₅	—	2	2	no	—
118	C₂×Dic₅	C₂²⋊C₈	1	4	yes	—
119	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
120	C₄×D₅	—	2	2	no	—
121	C₄×D₅	—	2	2	no	—
122	C₄×D₅	—	2	2	no	—
123	C₄×D₅	—	2	2	no	—
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	#121	1	2	yes	—
126	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
127	#168	—	2	2	no	—
128	#168	—	2	2	no	—
129	C₂³×C₄	—	5	1	no	—
130	C₂×M₁₆	—	10	2	no	—
131	C₂×M₁₆	—	10	2	no	—
132	C₂×M₁₆	—	10	2	no	—
133	C₂²×C₈	—	5	1	no	—
134	C₂²×C₈	—	5	1	no	—
135	C₂²×C₈	—	5	2	no	—
136	C₂²×C₈	—	5	2	no	—
137	C₂²×C₈	—	10	2	no	—
138	C₂²×C₈	—	10	2	no	—
139	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
141	#117	—	10	2	no	—
142	#117	—	10	2	no	—
143	C₂×C₄⋊C₄	—	5	2	no	—
144	C₂×C₂²⋊C₄	—	5	2	no	—
145	C₂×C₄²	—	5	1	no	—
146	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	10	2	no	—
151	C₂×C₁₆	—	10	2	no	—
152	C₂×C₁₆	—	10	2	no	—
153	#97	—	10	2	no	—
154	#97	—	10	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	5	4	no	—
157	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂²×C₅	#431	1	1	yes	characteristic
160	C₂²×C₅	#504	1	2	yes	—
161	C₂²×C₅	#504	1	2	yes	—
162	C₂×D₅	—	2	2	no	—
163	C₂×D₅	—	2	4	no	—
164	C₂×D₅	—	4	2	no	—
165	C₂×D₅	—	4	1	no	—
166	C₂×D₅	—	4	1	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	#380	1	1	yes	characteristic
169	C₄×C₅	#407	1	1	yes	characteristic
170	C₄×C₅	#423	1	1	yes	characteristic
171	C₄×C₅	#432	1	1	yes	characteristic
172	Dic₅	—	2	2	no	—
173	Dic₅	—	2	4	no	—
174	C₂⁴	—	5	1	no	—
175	C₂²×C₄	—	5	2	no	—
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	—	5	2	no	—
178	C₂²×C₄	—	5	2	no	—
179	C₂²×C₄	—	10	1	no	—
180	C₂²×C₄	—	10	1	no	—
181	C₂²×C₄	—	10	1	no	—
182	C₂²×C₄	—	10	1	no	—
183	C₂²×C₄	C₈×D₅	1	2	yes	—
184	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
185	M₁₆	—	10	8	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	5	2	no	—
189	C₂×C₈	—	5	2	no	—
190	C₂×C₈	—	10	1	no	—
191	C₂×C₈	—	10	1	no	—
192	C₂×C₈	—	10	1	no	—
193	C₂×C₈	—	10	1	no	—
194	C₂×C₈	—	10	4	no	—
195	C₂×C₈	—	10	4	no	—
196	C₂×C₈	#676	1	2	yes	—
197	C₂×C₈	#709	1	2	yes	—
198	C₂×C₈	D₄×D₅	1	1	yes	characteristic
199	C₂×C₈	D₄×D₅	1	2	yes	—
200	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
201	C₄⋊C₄	—	10	4	no	—
202	C₂²⋊C₄	—	10	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	5	2	no	—
206	C₄²	—	5	2	no	—
207	C₁₆	—	4	2	no	—
208	C₁₆	—	20	2	no	—
209	C₂×C₅	#1797	1	1	yes	characteristic
210	C₂×C₅	#2113	1	2	yes	—
211	C₂×C₅	#2151	1	2	yes	—
212	C₂×C₅	#2164	1	1	yes	characteristic
213	C₂×C₅	#2164	1	1	yes	characteristic
214	D₅	—	4	2	no	—
215	C₂³	—	5	2	no	—
216	C₂³	—	10	2	no	—
217	C₂³	—	10	2	no	—
218	C₂³	—	10	1	no	—
219	C₂³	—	10	1	no	—
220	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	5	2	no	—
225	C₂×C₄	—	5	4	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	10	1	no	—
229	C₂×C₄	—	10	1	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	—	10	2	no	—
234	C₂×C₄	#4018	1	1	yes	characteristic
235	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×#676	1	1	yes	characteristic
238	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
239	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	10	2	no	—
242	C₈	—	10	2	no	—
243	C₈	#4030	1	2	yes	—
244	C₈	#4042	1	2	yes	—
245	C₅	#12784	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	10	2	no	—
247	C₂²	—	10	4	no	—
248	C₂²	—	20	1	no	—
249	C₂²	—	20	1	no	—
250	C₂²	—	20	2	no	—
251	C₂²	#64974	1	1	yes	characteristic
252	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
253	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
254	C₂²	#65190	1	2	yes	—
255	C₂²	#65196	1	2	yes	—
256	C₄	—	4	2	no	—
257	C₄	—	10	2	no	—
258	C₄	—	10	4	no	—
259	C₄	#64956	1	1	yes	characteristic
260	C₄	#64994	1	1	yes	characteristic
261	C₄	#65015	1	1	yes	characteristic
262	C₄	#65126	1	1	yes	characteristic
263	C₂	—	20	2	no	—
264	C₂	#10599455	1	1	yes	characteristic
265	C₂	#10599846	1	1	yes	characteristic
266	C₂	#10599867	1	1	yes	characteristic
267	C₂	#10599900	1	2	yes	—
268	C₂	#10600078	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle	C₂³×C₅	C₂²×C₈
Fitting	C₅×#1374	C₂
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1374	D₅
hypocenter	C₅	#12784
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. C₂×#64721	C₂	semidirect
4. C₅×#1374	C₂	semidirect
5. #10598148	C₂	semidirect
51. #1374	D₅	semidirect
245. C₅	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64978	2
C₂×#64727	2
C₂×#64721	2
C₅×#1374	2
#10598148	2
#12784	5