Subgroups — #13107233

Order 1600.

1000 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717218	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717215	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717215	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717214	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717213	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717205	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717204	C₂	1	1	yes	maximal, characteristic, complemented
10	#11717203	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717202	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717202	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²⋊C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
14	#11717195	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₅⋊D₅	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic, complemented
28	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
29	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
31	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
33	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
34	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
35	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
36	C₂²⋊#1039	C₂²	1	1	yes	characteristic
37	#87808	C₂²	1	1	yes	characteristic
38	#87808	C₂²	1	1	yes	characteristic
39	#87808	C₂²	1	1	yes	characteristic
40	#87808	C₂²	1	1	yes	characteristic
41	#87808	C₂²	1	1	yes	characteristic
42	#87808	C₂²	1	1	yes	characteristic
43	#87808	C₂²	1	1	yes	characteristic, complemented
44	#87808	C₂²	1	1	yes	characteristic, complemented
45	#87808	C₂²	1	1	yes	characteristic, complemented
46	#87808	C₂²	1	1	yes	characteristic, complemented
47	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
48	#87806	C₂²	1	1	yes	characteristic, complemented
49	#87806	C₂²	1	1	yes	characteristic, complemented
50	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
51	#66038	—	5	6	no	maximal
52	C₂³×C₅²	C₂³	1	1	yes	characteristic
53	C₂²×C₅⋊D₅	—	2	2	no	—
54	C₂²×C₅⋊D₅	—	4	1	no	—
55	C₂²×C₅⋊D₅	—	4	1	no	—
56	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
57	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
58	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
59	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
62	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
66	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
67	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
68	C₂×#1039	C₂³	1	1	yes	characteristic
69	C₂×#1039	C₂³	1	1	yes	characteristic
70	C₂×#1039	C₂³	1	1	yes	characteristic
71	C₂×#1039	C₂³	1	1	yes	characteristic
72	C₄⋊(C₅⋊D₅)	—	2	2	no	—
73	C₄⋊(C₅⋊D₅)	—	2	2	no	—
74	C₄⋊(C₅⋊D₅)	—	4	1	no	—
75	C₄⋊(C₅⋊D₅)	—	4	1	no	—
76	C₄⋊(C₅⋊D₅)	—	4	1	no	—
77	C₄×C₅⋊D₅	—	4	1	no	—
78	C₄×C₅⋊D₅	—	4	1	no	—
79	C₄×C₅⋊D₅	—	4	1	no	—
80	C₄×C₅⋊D₅	—	4	1	no	—
81	#6045	—	4	1	no	—
82	C₅×#126	D₅	1	6	yes	complemented
83	#4044	—	5	6	no	—
84	#4042	—	5	6	no	—
85	#4041	—	5	6	no	—
86	#4041	—	5	6	no	—
87	#4040	—	5	6	no	—
88	#4039	—	5	6	no	—
89	#4031	—	5	6	no	—
90	#4030	—	5	6	no	—
91	#4029	—	5	6	no	—
92	#4028	—	5	6	no	—
93	#4028	—	5	6	no	—
94	D₅×C₂²⋊C₄	—	5	6	no	—
95	#4021	—	5	6	no	—
96	C₄×D₂₀	—	5	6	no	—
97	C₂²×C₅²	—	2	1	no	—
98	C₂²×C₅²	—	2	1	no	—
99	C₂²×C₅²	—	2	1	no	—
100	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
101	C₂×C₅⋊D₅	—	2	1	no	—
102	C₂×C₅⋊D₅	—	2	1	no	—
103	C₂×C₅⋊D₅	—	2	1	no	—
104	C₂×C₅⋊D₅	—	2	1	no	—
105	C₂×C₅⋊D₅	—	2	1	no	—
106	C₂×C₅⋊D₅	—	2	1	no	—
107	C₂×C₅⋊D₅	—	2	1	no	—
108	C₂×C₅⋊D₅	—	2	1	no	—
109	C₂×C₅⋊D₅	—	2	1	no	—
110	C₂×C₅⋊D₅	—	2	1	no	—
111	C₂×C₅⋊D₅	—	2	1	no	—
112	C₂×C₅⋊D₅	—	4	4	no	—
113	C₂×C₅⋊D₅	#41	1	2	yes	—
114	C₄×C₅²	—	2	1	no	—
115	C₄×C₅²	—	2	1	no	—
116	C₄×C₅²	—	2	1	no	—
117	C₄×C₅²	—	2	1	no	—
118	C₄×C₅²	—	2	1	no	—
119	C₄×C₅²	—	2	1	no	—
120	#1039	—	2	1	no	—
121	#1039	—	2	1	no	—
122	#1039	—	2	1	no	—
123	#1039	—	2	1	no	—
124	C₂³×D₅	—	5	6	no	—
125	C₂×C₂²⋊D₅	—	5	6	no	—
126	C₂×C₂²⋊D₅	—	5	6	no	—
127	C₂×D₂₀	—	5	6	no	—
128	C₂×D₂₀	—	5	6	no	—
129	C₂×D₂₀	—	5	6	no	—
130	C₂×D₂₀	—	5	6	no	—
131	C₂×D₂₀	—	5	6	no	—
132	C₂×C₄×D₅	—	5	6	no	—
133	C₂×C₄×D₅	—	5	6	no	—
134	C₂×C₄×D₅	—	5	6	no	—
135	C₂×C₄×D₅	—	5	6	no	—
136	C₂×Dic₁₀	—	5	6	no	—
137	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	complemented
138	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	complemented
139	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	complemented
140	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
141	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
142	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
143	C₄²×C₅	C₂×D₅	1	6	yes	complemented
144	C₂²⋊Dic₅	—	5	6	no	—
145	#685	—	5	6	no	—
146	#685	—	5	6	no	—
147	#685	—	5	6	no	—
148	#685	—	5	6	no	—
149	#685	—	5	6	no	—
150	#685	—	5	6	no	—
151	#685	—	5	6	no	—
152	#685	—	5	6	no	—
153	#685	—	5	6	no	—
154	#685	—	5	6	no	—
155	C₄⋊Dic₅	—	5	6	no	—
156	#683	—	5	6	no	—
157	#683	—	5	6	no	—
158	C₄×Dic₅	—	5	6	no	—
159	#538	—	25	1	no	Sylow
160	C₂×C₅²	—	4	1	no	—
161	C₂×C₅²	C₂×#41	1	1	yes	characteristic
162	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
164	C₅⋊D₅	—	2	2	no	—
165	C₅⋊D₅	—	4	1	no	—
166	C₅⋊D₅	—	4	1	no	—
167	C₅⋊D₅	—	4	1	no	—
168	C₂³×C₅	C₂²×D₅	1	6	yes	—
169	C₂²×D₅	—	5	6	no	—
170	C₂²×D₅	—	5	6	no	—
171	C₂²×D₅	—	5	6	no	—
172	C₂²×D₅	—	5	6	no	—
173	C₂²×D₅	—	10	12	no	—
174	C₂²×D₅	—	20	6	no	—
175	C₂²×D₅	—	20	6	no	—
176	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
177	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
178	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
179	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
180	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
181	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
182	C₂²⋊D₅	—	20	6	no	—
183	C₂²⋊D₅	—	20	6	no	—
184	C₂×Dic₅	—	5	6	no	—
185	C₂×Dic₅	—	5	6	no	—
186	C₂×Dic₅	—	5	6	no	—
187	C₂×Dic₅	—	5	6	no	—
188	D₂₀	—	10	12	no	—
189	D₂₀	—	10	12	no	—
190	D₂₀	—	20	6	no	—
191	D₂₀	—	20	6	no	—
192	D₂₀	—	20	6	no	—
193	C₄×D₅	—	20	6	no	—
194	C₄×D₅	—	20	6	no	—
195	C₄×D₅	—	20	6	no	—
196	C₄×D₅	—	20	6	no	—
197	Dic₁₀	—	20	6	no	—
198	#126	—	25	1	no	—
199	#126	C₅⋊D₅	1	1	yes	characteristic, complemented
200	#124	—	25	1	no	—
201	#124	—	25	1	no	—
202	#123	—	25	1	no	—
203	#123	—	25	1	no	—
204	C₂²⋊Q₈	—	25	1	no	—
205	#121	—	25	1	no	—
206	#121	—	25	1	no	—
207	#121	—	25	1	no	—
208	#120	—	25	1	no	—
209	#120	—	25	1	no	—
210	C₄×D₄	—	25	1	no	—
211	C₄×D₄	—	25	1	no	—
212	C₂×C₂²⋊C₄	—	25	1	no	—
213	C₅²	#538	1	1	yes	characteristic, complemented, Sylow
214	C₂²×C₅	—	2	6	no	—
215	C₂²×C₅	—	2	6	no	—
216	C₂²×C₅	—	2	6	no	—
217	C₂²×C₅	C₂³×D₅	1	6	yes	—
218	C₂×D₅	—	5	12	no	—
219	C₂×D₅	—	10	6	no	—
220	C₂×D₅	—	10	6	no	—
221	C₂×D₅	—	10	6	no	—
222	C₂×D₅	—	10	6	no	—
223	C₂×D₅	—	10	6	no	—
224	C₂×D₅	—	10	6	no	—
225	C₂×D₅	—	10	6	no	—
226	C₂×D₅	—	10	6	no	—
227	C₂×D₅	—	10	6	no	—
228	C₂×D₅	—	10	6	no	—
229	C₂×D₅	—	10	6	no	—
230	C₂×D₅	—	20	24	no	—
231	C₄×C₅	—	2	6	no	—
232	C₄×C₅	—	2	6	no	—
233	C₄×C₅	—	2	6	no	—
234	C₄×C₅	—	2	6	no	—
235	C₄×C₅	—	2	6	no	—
236	C₄×C₅	—	2	6	no	—
237	Dic₅	—	10	6	no	—
238	Dic₅	—	10	6	no	—
239	Dic₅	—	10	6	no	—
240	Dic₅	—	10	6	no	—
241	C₂⁴	—	25	1	no	—
242	C₂×Q₈	—	25	1	no	—
243	C₂×D₄	—	25	1	no	—
244	C₂×D₄	—	25	1	no	—
245	C₂×D₄	—	25	1	no	—
246	C₂×D₄	—	25	1	no	—
247	C₂×D₄	—	25	1	no	—
248	C₂×D₄	—	25	1	no	—
249	C₂×D₄	—	25	1	no	—
250	C₂²×C₄	—	25	1	no	—
251	C₂²×C₄	—	25	1	no	—
252	C₂²×C₄	—	25	1	no	—
253	C₂²×C₄	—	25	1	no	—
254	C₄⋊C₄	—	25	1	no	—
255	C₄⋊C₄	—	25	1	no	—
256	C₄⋊C₄	—	25	1	no	—
257	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
258	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
259	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
260	C₂²⋊C₄	—	25	1	no	—
261	C₂²⋊C₄	—	25	1	no	—
262	C₂²⋊C₄	—	25	1	no	—
263	C₂²⋊C₄	—	25	1	no	—
264	C₂²⋊C₄	—	25	1	no	—
265	C₂²⋊C₄	—	25	1	no	—
266	C₂²⋊C₄	—	25	1	no	—
267	C₂²⋊C₄	—	25	1	no	—
268	C₂²⋊C₄	—	25	1	no	—
269	C₂²⋊C₄	—	25	1	no	—
270	C₂²⋊C₄	—	25	1	no	—
271	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
272	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
274	C₄²	—	25	1	no	—
275	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
276	C₂×C₅	—	4	6	no	—
277	C₂×C₅	D₅×#41	1	6	yes	—
278	C₂×C₅	#4149	1	6	yes	—
279	C₂×C₅	#4149	1	6	yes	—
280	D₅	—	10	12	no	—
281	D₅	—	20	6	no	—
282	D₅	—	20	6	no	—
283	D₅	—	20	6	no	—
284	C₂³	—	25	1	no	—
285	C₂³	—	25	1	no	—
286	C₂³	—	25	1	no	—
287	C₂³	—	25	1	no	—
288	C₂³	—	50	2	no	—
289	C₂³	—	100	1	no	—
290	C₂³	—	100	1	no	—
291	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
292	Q₈	—	100	1	no	—
293	D₄	—	50	2	no	—
294	D₄	—	50	2	no	—
295	D₄	—	100	1	no	—
296	D₄	—	100	1	no	—
297	D₄	—	100	1	no	—
298	D₄	—	100	1	no	—
299	D₄	—	100	1	no	—
300	C₂×C₄	—	25	1	no	—
301	C₂×C₄	—	25	1	no	—
302	C₂×C₄	—	25	1	no	—
303	C₂×C₄	—	25	1	no	—
304	C₂×C₄	—	100	1	no	—
305	C₂×C₄	—	100	1	no	—
306	C₂×C₄	—	100	1	no	—
307	C₂×C₄	—	100	1	no	—
308	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
314	C₅	#66038	1	6	yes	complemented
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	25	2	no	—
319	C₂²	—	50	1	no	—
320	C₂²	—	50	1	no	—
321	C₂²	—	50	1	no	—
322	C₂²	—	50	1	no	—
323	C₂²	—	50	1	no	—
324	C₂²	—	50	1	no	—
325	C₂²	—	50	1	no	—
326	C₂²	—	50	1	no	—
327	C₂²	—	50	1	no	—
328	C₂²	—	50	1	no	—
329	C₂²	—	50	1	no	—
330	C₂²	—	100	4	no	—
331	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	50	1	no	—
339	C₄	—	50	1	no	—
340	C₄	—	50	1	no	—
341	C₄	—	50	1	no	—
342	C₂	—	4	1	no	—
343	C₂	—	50	2	no	—
344	C₂	—	100	1	no	—
345	C₂	—	100	1	no	—
346	C₂	—	100	1	no	—
347	C₂	#41×C₅⋊D₅	1	1	yes	characteristic
348	C₂	#11717522	1	1	yes	characteristic
349	C₂	#11717522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#126	C₂
hypercenter	#126	C₅⋊D₅
hypocenter	C₅²	#538
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. C₅²×#126	C₂	semidirect
2. #11717218	C₂	semidirect
3. #11717216	C₂	semidirect
4. #11717215	C₂	semidirect
5. #11717215	C₂	semidirect
6. #11717214	C₂	semidirect
7. #11717213	C₂	semidirect
8. #11717205	C₂	semidirect
9. #11717204	C₂	semidirect
10. #11717203	C₂	semidirect
11. #11717202	C₂	semidirect
12. #11717202	C₂	semidirect
13. C₂²⋊C₄×C₅⋊D₅	C₂	semidirect
14. #11717195	C₂	semidirect
15. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
23. C₂×C₄⋊(C₅⋊D₅)	C₂²	semidirect
26. C₂×C₄×C₅⋊D₅	C₂²	semidirect
27. C₂×C₄×C₅⋊D₅	C₂²	semidirect
28. C₂×#6045	C₂²	semidirect
29. C₅²×C₄⋊C₄	C₂²	semidirect
30. C₅²×C₄⋊C₄	C₂²	semidirect
31. C₅²×C₄⋊C₄	C₂²	semidirect
35. C₄²×C₅²	C₂²	semidirect
43. #87808	C₂²	semidirect
44. #87808	C₂²	semidirect
45. #87808	C₂²	semidirect
46. #87808	C₂²	semidirect
47. C₄⋊#1039	C₂²	semidirect
48. #87806	C₂²	semidirect
49. #87806	C₂²	semidirect
50. C₄×#1039	C₂²	semidirect
82. C₅×#126	D₅	semidirect
137. C₅×C₄⋊C₄	C₂×D₅	semidirect
138. C₅×C₄⋊C₄	C₂×D₅	semidirect
139. C₅×C₄⋊C₄	C₂×D₅	semidirect
143. C₄²×C₅	C₂×D₅	semidirect
199. #126	C₅⋊D₅	semidirect
213. C₅²	#538	semidirect
257. C₄⋊C₄	C₂×C₅⋊D₅	semidirect
258. C₄⋊C₄	C₂×C₅⋊D₅	semidirect
259. C₄⋊C₄	C₂×C₅⋊D₅	semidirect
275. C₄²	C₂×C₅⋊D₅	semidirect
314. C₅	#66038	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#126	2
#11717218	2
#11717216	2
#11717215	2
#11717214	2
#11717213	2
#11717205	2
#11717204	2
#11717203	2
#11717202	2
C₂²⋊C₄×C₅⋊D₅	2
#11717195	2
C₄×C₄⋊(C₅⋊D₅)	2
#66038	5