Subgroups — #11992556

Order 1280.

362 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610176	C₂	1	1	yes	maximal, characteristic
2	C₅×#1848	C₂	1	1	yes	maximal, characteristic
3	#10601302	C₂	1	2	yes	maximal
4	C₂×#64678	C₂	1	1	yes	maximal, characteristic
5	#10598202	C₂	1	2	yes	maximal
6	#65804	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	—	2	1	no	—
8	C₂×C₅×#108	—	2	1	no	—
9	C₅×#423	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#65391	—	2	1	no	—
12	#65390	—	2	1	no	—
13	#65213	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	#64968	—	2	1	no	—
16	#64968	C₄	1	2	yes	—
17	C₈×Dic₁₀	—	2	1	no	—
18	C₈×Dic₁₀	C₄	1	2	yes	—
19	#64706	—	2	2	no	—
20	#64701	—	2	2	no	—
21	#64678	—	2	1	no	—
22	#64678	—	2	1	no	—
23	#8685	—	5	1	no	maximal, Sylow
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	#4072	—	2	1	no	—
29	#4070	—	2	1	no	—
30	#4070	Q₈	1	2	yes	—
31	C₂×#681	—	2	1	no	—
32	C₂²×#168	—	2	2	no	—
33	#4016	—	2	1	no	—
34	#4015	D₄	1	1	yes	characteristic
35	#4015	D₄	1	1	yes	characteristic
36	C₄×Dic₁₀	—	2	1	no	—
37	C₄×Dic₁₀	C₂×C₄	1	2	yes	—
38	C₅×#108	—	4	2	no	—
39	C₅×#108	—	4	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	#3947	—	2	2	no	—
47	#3946	—	2	2	no	—
48	#3946	—	2	2	no	—
49	C₈×Dic₅	—	2	2	no	—
50	C₄⋊#168	—	2	1	no	—
51	C₄⋊#168	—	2	2	no	—
52	C₄⋊#168	—	2	2	no	—
53	C₄⋊#168	C₂×C₄	1	2	yes	—
54	#2960	—	5	1	no	—
55	#1848	D₅	1	1	yes	characteristic
56	C₂×#335	—	5	1	no	—
57	#1550	—	5	2	no	—
58	#1292	—	5	2	no	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×Dic₁₀	—	2	1	no	—
63	C₂×Dic₁₀	C₄⋊C₄	1	2	yes	—
64	C₅×M₁₆	—	4	2	no	—
65	C₅×M₁₆	—	4	2	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	SD₁₆	1	2	yes	—
74	C₂×C₅×C₈	Q₁₆	1	2	yes	—
75	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅	#41	1	1	yes	characteristic
79	C₂²⋊Dic₅	—	2	2	no	—
80	C₄⋊Dic₅	—	2	1	no	—
81	C₄⋊Dic₅	—	2	1	no	—
82	C₄⋊Dic₅	—	2	1	no	—
83	C₄⋊Dic₅	C₄⋊C₄	1	2	yes	—
84	#683	—	2	2	no	—
85	#683	—	2	2	no	—
86	C₄×Dic₅	—	2	2	no	—
87	#681	—	4	1	no	—
88	#681	—	4	1	no	—
89	C₂×#168	—	2	2	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	8	2	no	—
92	#533	—	5	1	no	—
93	#446	—	5	2	no	—
94	#446	—	10	1	no	—
95	C₈×Q₈	—	5	2	no	—
96	C₈×Q₈	—	10	1	no	—
97	C₂×#108	—	2	1	no	—
98	C₂×#108	—	2	1	no	—
99	#424	—	10	1	no	—
100	#423	—	5	1	no	—
101	#423	C₂×D₅	1	2	yes	—
102	C₂×C₄⋊C₈	—	5	2	no	—
103	#405	—	10	1	no	—
104	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	#335	—	10	1	no	—
106	#335	—	10	1	no	—
107	#333	—	10	2	no	—
108	#332	—	10	2	no	—
109	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	4	1	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#117	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	#121	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₅	#137	1	1	yes	characteristic
123	C₂×Dic₅	—	2	2	no	—
124	C₂×Dic₅	—	2	2	no	—
125	Dic₁₀	—	4	1	no	—
126	Dic₁₀	—	4	1	no	—
127	Dic₁₀	—	4	2	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	8	2	no	—
131	C₅×C₈	#103	1	2	yes	—
132	C₅×C₈	#103	1	2	yes	—
133	#168	—	4	2	no	—
134	#168	—	8	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	10	1	no	—
137	C₂²×C₈	—	10	2	no	—
138	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
140	C₄⋊Q₈	—	5	1	no	—
141	C₄⋊Q₈	—	5	1	no	—
142	#125	—	10	1	no	—
143	C₂²⋊Q₈	—	5	2	no	—
144	C₂²⋊Q₈	—	10	1	no	—
145	C₄×Q₈	—	5	2	no	—
146	C₄×Q₈	—	10	1	no	—
147	#117	—	10	1	no	—
148	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
149	#108	—	4	2	no	—
150	#108	—	4	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	5	2	no	—
154	C₄⋊C₈	—	10	2	no	—
155	C₄⋊C₈	—	10	2	no	—
156	C₄⋊C₈	—	10	1	no	—
157	C₄⋊C₈	—	10	2	no	—
158	C₄⋊C₈	—	10	2	no	—
159	#97	—	10	2	no	—
160	C₄×C₈	—	10	2	no	—
161	C₄×C₈	C₄×D₅	1	1	yes	characteristic
162	C₄×C₈	C₄×D₅	1	1	yes	characteristic
163	C₄×C₈	D₂₀	1	1	yes	characteristic
164	C₄×C₈	D₂₀	1	1	yes	characteristic
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	#380	1	1	yes	characteristic
168	C₂²×C₅	#475	1	1	yes	characteristic
169	C₂²×C₅	#477	1	1	yes	characteristic
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	2	1	no	—
173	C₄×C₅	C₂×#103	1	1	yes	characteristic
174	C₄×C₅	#417	1	1	yes	characteristic
175	C₄×C₅	#460	1	1	yes	characteristic
176	C₄×C₅	#462	1	1	yes	characteristic
177	C₄×C₅	#472	1	1	yes	characteristic
178	C₄×C₅	#473	1	1	yes	characteristic
179	Dic₅	—	4	2	no	—
180	Dic₅	—	4	2	no	—
181	C₂×Q₈	—	5	2	no	—
182	C₂×Q₈	—	10	1	no	—
183	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
184	C₂²×C₄	#713	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	20	1	no	—
189	M₁₆	—	20	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	10	2	no	—
194	C₂×C₈	—	20	2	no	—
195	C₂×C₈	—	40	2	no	—
196	C₂×C₈	#685	1	1	yes	characteristic
197	C₂×C₈	#685	1	1	yes	characteristic
198	C₂×C₈	#685	1	1	yes	characteristic
199	C₂×C₈	#685	1	1	yes	characteristic
200	C₂×C₈	#687	1	2	yes	—
201	C₂×C₈	#689	1	2	yes	—
202	C₄⋊C₄	—	5	2	no	—
203	C₄⋊C₄	—	10	2	no	—
204	C₄⋊C₄	—	10	1	no	—
205	C₄⋊C₄	—	10	2	no	—
206	C₄⋊C₄	—	10	1	no	—
207	C₄⋊C₄	—	10	1	no	—
208	C₂²⋊C₄	—	10	2	no	—
209	C₄²	—	10	2	no	—
210	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
211	C₄²	C₂×D₂₀	1	1	yes	characteristic
212	C₄²	D₄×D₅	1	1	yes	characteristic
213	C₄²	#711	1	1	yes	characteristic
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	#1803	1	1	yes	characteristic
217	C₂×C₅	#1813	1	1	yes	characteristic
218	C₂×C₅	#1816	1	1	yes	characteristic
219	C₂³	#4092	1	1	yes	characteristic
220	Q₈	—	20	2	no	—
221	Q₈	—	20	1	no	—
222	Q₈	—	20	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#4038	1	1	yes	characteristic
230	C₂×C₄	#4041	1	1	yes	characteristic
231	C₂×C₄	#4043	1	1	yes	characteristic
232	C₂×C₄	C₂×#685	1	1	yes	characteristic
233	C₂×C₄	#4078	1	1	yes	characteristic
234	C₂×C₄	C₂×#687	1	1	yes	characteristic
235	C₂×C₄	#4084	1	1	yes	characteristic
236	C₂×C₄	#4088	1	1	yes	characteristic
237	C₂×C₄	C₂×#689	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	—	20	2	no	—
242	C₈	—	40	2	no	—
243	C₈	#3942	1	2	yes	—
244	C₈	#3942	1	2	yes	—
245	C₅	#8685	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#65276	1	1	yes	characteristic
249	C₂²	#65380	1	1	yes	characteristic
250	C₂²	#65383	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	20	2	no	—
255	C₄	—	20	2	no	—
256	C₄	#65185	1	1	yes	characteristic
257	C₄	#65186	1	1	yes	characteristic
258	C₄	#65255	1	1	yes	characteristic
259	C₄	C₂×#3942	1	1	yes	characteristic
260	C₄	#65364	1	1	yes	characteristic
261	C₄	#65379	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	#10600381	1	1	yes	characteristic
265	C₂	#10601511	1	1	yes	characteristic
266	C₂	#10601514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65276
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1848	C₂
socle	C₂²×C₅	#380
prime radical	C₂³×C₅	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1848	D₅
hypocenter	C₅	#8685
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
245. C₅	#8685	semidirect

Maximal subgroups

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

subgroup	index
#10610176	2
C₅×#1848	2
#10601302	2
C₂×#64678	2
#10598202	2
#8685	5