Subgroups — #11999424

Order 1280.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10605042	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#10598193	C₂	1	4	yes	maximal
5	C₂×#4015	—	2	1	no	—
6	C₂×#4015	—	2	1	no	—
7	C₂×#4015	C₄	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	#65224	—	2	1	no	—
13	#65224	—	2	1	no	—
14	#65224	C₄	1	2	yes	—
15	#65224	C₂²	1	2	yes	—
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	C₂×C₄×#168	—	2	1	no	—
18	#64744	—	2	1	no	—
19	#9898	—	5	1	no	maximal, Sylow
20	C₂²×Dic₁₀	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₂×C₄⋊Dic₅	—	2	2	no	—
31	C₂×C₄⋊Dic₅	—	2	2	no	—
32	C₂×C₄⋊Dic₅	—	2	2	no	—
33	C₂×C₄⋊Dic₅	—	2	2	no	—
34	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
35	C₂²×#168	—	2	2	no	—
36	#4015	—	4	4	no	—
37	#4015	—	4	2	no	—
38	#4015	—	4	2	no	—
39	#4015	—	4	2	no	—
40	#4015	—	4	2	no	—
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×C₄⋊C₈	—	4	2	no	—
43	C₅×C₄⋊C₈	—	4	2	no	—
44	C₄⋊#168	—	2	2	no	—
45	C₄⋊#168	Q₈	1	4	yes	complemented
46	C₄×#168	—	4	2	no	—
47	#2868	—	5	1	no	—
48	#1842	D₅	1	1	yes	characteristic
49	C₄×C₄⋊C₈	—	5	1	no	—
50	#1296	—	5	4	no	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅	—	2	4	no	—
59	C₂×Dic₁₀	—	4	4	no	—
60	C₂×Dic₁₀	—	4	4	no	—
61	C₂×Dic₁₀	—	4	4	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	2	no	—
65	C₄²×C₅	—	2	2	no	—
66	C₄²×C₅	—	2	2	no	—
67	C₄²×C₅	—	2	2	no	—
68	C₄²×C₅	D₈	1	2	yes	—
69	C₄²×C₅	D₈	1	2	yes	—
70	C₄²×C₅	SD₁₆	1	2	yes	—
71	C₄²×C₅	SD₁₆	1	2	yes	—
72	C₄²×C₅	SD₁₆	1	2	yes	—
73	C₄²×C₅	Q₁₆	1	2	yes	—
74	C₄²×C₅	C₂×Q₈	1	2	yes	—
75	C₄²×C₅	#41	1	2	yes	—
76	C₄⋊Dic₅	—	4	4	no	—
77	C₄⋊Dic₅	—	4	4	no	—
78	C₄⋊Dic₅	—	4	4	no	—
79	C₄⋊Dic₅	—	4	2	no	—
80	C₄⋊Dic₅	—	4	2	no	—
81	C₄⋊Dic₅	—	4	2	no	—
82	C₄⋊Dic₅	—	4	2	no	—
83	C₄⋊Dic₅	—	4	4	no	—
84	C₂×#168	—	2	4	no	—
85	C₂×#168	—	4	2	no	—
86	C₂×#168	—	4	2	no	—
87	C₂×C₄⋊Q₈	—	5	2	no	—
88	C₂×C₄⋊Q₈	—	10	1	no	—
89	C₂×C₄⋊Q₈	—	10	1	no	—
90	C₂×C₄⋊C₈	—	2	1	no	—
91	C₂×C₄⋊C₈	—	2	1	no	—
92	C₂×C₄⋊C₈	—	5	2	no	—
93	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
94	C₂×C₄×C₈	—	10	1	no	—
95	#384	—	5	2	no	—
96	#384	—	5	2	no	—
97	#384	—	10	1	no	—
98	#384	—	10	1	no	—
99	C₄³	C₂×D₅	1	1	yes	characteristic
100	#336	—	10	1	no	—
101	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	#100	1	2	yes	—
109	C₂×C₄×C₅	#101	1	2	yes	—
110	C₂×C₄×C₅	#102	1	2	yes	—
111	C₂×C₄×C₅	#102	1	2	yes	—
112	C₂×C₄×C₅	#102	1	2	yes	—
113	C₂×C₄×C₅	#102	1	2	yes	—
114	C₂×C₄×C₅	#103	1	2	yes	—
115	C₂×C₄×C₅	#103	1	2	yes	—
116	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₅	#124	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	#137	1	2	yes	—
127	C₂×Dic₅	—	4	4	no	—
128	C₂×Dic₅	—	4	4	no	—
129	C₂×Dic₅	—	4	4	no	—
130	Dic₁₀	—	8	8	no	—
131	C₅×C₈	—	8	2	no	—
132	#168	—	4	4	no	—
133	C₂²×Q₈	—	10	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	10	2	no	—
136	C₄⋊Q₈	—	20	4	no	—
137	C₄⋊Q₈	—	20	2	no	—
138	C₄⋊Q₈	—	20	2	no	—
139	C₄⋊Q₈	—	20	2	no	—
140	C₄⋊Q₈	—	20	2	no	—
141	C₂×C₄⋊C₄	—	5	4	no	—
142	C₂×C₄⋊C₄	—	10	2	no	—
143	C₂×C₄⋊C₄	—	10	2	no	—
144	C₂×C₄⋊C₄	—	10	2	no	—
145	C₂×C₄⋊C₄	—	10	2	no	—
146	C₂×C₄⋊C₄	—	10	2	no	—
147	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
149	C₂×C₄²	D₂₀	1	1	yes	characteristic
150	C₂×C₄²	D₂₀	1	1	yes	characteristic
151	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	4	2	no	—
156	C₄⋊C₈	—	4	2	no	—
157	C₄⋊C₈	—	5	4	no	—
158	C₄⋊C₈	—	10	2	no	—
159	C₄×C₈	—	20	2	no	—
160	C₂²×C₅	#391	1	1	yes	characteristic
161	C₂²×C₅	C₂×#100	1	1	yes	characteristic
162	C₂²×C₅	C₂×#101	1	1	yes	characteristic
163	C₂²×C₅	C₂×#102	1	1	yes	characteristic
164	C₂²×C₅	C₂×#102	1	1	yes	characteristic
165	C₂²×C₅	C₂×#103	1	1	yes	characteristic
166	C₂²×C₅	#419	1	1	yes	characteristic
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	#331	1	4	yes	—
171	C₄×C₅	#332	1	4	yes	—
172	C₄×C₅	#474	1	2	yes	—
173	C₄×C₅	#475	1	2	yes	—
174	C₄×C₅	#486	1	1	yes	characteristic
175	C₄×C₅	#487	1	1	yes	characteristic
176	C₄×C₅	#498	1	1	yes	characteristic
177	C₄×C₅	#500	1	1	yes	characteristic
178	Dic₅	—	8	4	no	—
179	C₂×Q₈	—	20	4	no	—
180	C₂×Q₈	—	20	4	no	—
181	C₂×Q₈	—	20	4	no	—
182	C₂²×C₄	—	10	4	no	—
183	C₂²×C₄	#685	1	1	yes	characteristic
184	C₂²×C₄	#685	1	1	yes	characteristic
185	C₂²×C₄	#685	1	1	yes	characteristic
186	C₂²×C₄	#685	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	10	4	no	—
194	C₂×C₈	—	20	2	no	—
195	C₂×C₈	—	20	2	no	—
196	C₄⋊C₄	—	20	4	no	—
197	C₄⋊C₄	—	20	4	no	—
198	C₄⋊C₄	—	20	4	no	—
199	C₄⋊C₄	—	20	4	no	—
200	C₄⋊C₄	—	20	2	no	—
201	C₄⋊C₄	—	20	2	no	—
202	C₄⋊C₄	—	20	2	no	—
203	C₄⋊C₄	—	20	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	#677	1	2	yes	—
208	C₄²	D₄₀	1	2	yes	—
209	C₄²	#686	1	2	yes	—
210	C₄²	#687	1	2	yes	—
211	C₄²	#688	1	2	yes	—
212	C₄²	#689	1	2	yes	—
213	C₄²	#711	1	2	yes	—
214	C₄²	Q₈×D₅	1	2	yes	—
215	C₂×C₅	C₂×#331	1	1	yes	characteristic
216	C₂×C₅	C₂×#332	1	1	yes	characteristic
217	C₂×C₅	#1546	1	1	yes	characteristic
218	C₂×C₅	#1548	1	1	yes	characteristic
219	C₂×C₅	#1988	1	1	yes	characteristic
220	C₂×C₅	#1990	1	1	yes	characteristic
221	C₂×C₅	#1997	1	1	yes	characteristic
222	C₂³	C₂×#685	1	1	yes	characteristic
223	Q₈	—	40	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	20	4	no	—
231	C₂×C₄	—	20	4	no	—
232	C₂×C₄	—	20	4	no	—
233	C₂×C₄	#3941	1	2	yes	—
234	C₂×C₄	#3941	1	2	yes	—
235	C₂×C₄	#3942	1	2	yes	—
236	C₂×C₄	#3942	1	2	yes	—
237	C₂×C₄	#3953	1	2	yes	—
238	C₂×C₄	#3953	1	2	yes	—
239	C₂×C₄	#3955	1	2	yes	—
240	C₂×C₄	#3956	1	2	yes	—
241	C₂×C₄	#4033	1	2	yes	—
242	C₂×C₄	#4043	1	2	yes	—
243	C₂×C₄	C₂×#677	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
245	C₂×C₄	#4055	1	2	yes	—
246	C₂×C₄	C₂×#686	1	1	yes	characteristic
247	C₂×C₄	C₂×#687	1	1	yes	characteristic
248	C₂×C₄	#4082	1	1	yes	characteristic
249	C₂×C₄	C₂×#688	1	1	yes	characteristic
250	C₂×C₄	C₂×#689	1	1	yes	characteristic
251	C₂×C₄	#4090	1	1	yes	characteristic
252	C₈	—	8	2	no	—
253	C₈	—	20	4	no	—
254	C₅	#9898	1	1	yes	characteristic, complemented, Sylow
255	C₂²	C₂×#3941	1	1	yes	characteristic
256	C₂²	C₂×#3942	1	1	yes	characteristic
257	C₂²	#65263	1	1	yes	characteristic
258	C₂²	C₂×#3953	1	1	yes	characteristic
259	C₂²	#65409	1	1	yes	characteristic
260	C₂²	C₂×#3955	1	1	yes	characteristic
261	C₂²	C₂×#3956	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	40	4	no	—
266	C₄	#64705	1	4	yes	—
267	C₄	#64706	1	4	yes	—
268	C₄	#65135	1	2	yes	—
269	C₄	#65140	1	2	yes	—
270	C₄	#65359	1	1	yes	characteristic
271	C₄	#65367	1	1	yes	characteristic
272	C₄	#65368	1	1	yes	characteristic
273	C₄	#65370	1	1	yes	characteristic
274	C₂	#10600375	1	1	yes	characteristic
275	C₂	C₂×#64705	1	1	yes	characteristic
276	C₂	C₂×#64706	1	1	yes	characteristic
277	C₂	#10601253	1	1	yes	characteristic
278	C₂	#10601299	1	1	yes	characteristic
279	C₂	#10601300	1	1	yes	characteristic
280	C₂	#10601317	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1842	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1842	D₅
hypocenter	C₅	#9898
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
45. C₄⋊#168	Q₈	semidirect
254. C₅	#9898	semidirect

Maximal subgroups

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

subgroup	index
#10605042	2
C₅×#1842	2
C₄×C₄⋊#168	2
#10598193	2
#9898	5