Subgroups — #13107121

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×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717273	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717259	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717259	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717258	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717256	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717255	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#1039	C₂	1	1	yes	maximal, characteristic, complemented
10	#11717205	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717203	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717201	C₂	1	1	yes	maximal, characteristic, complemented
13	#11717199	C₂	1	1	yes	maximal, characteristic, complemented
14	#11717199	C₂	1	1	yes	maximal, characteristic, complemented
15	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
17	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic, complemented
24	C₂²×#1039	C₂²	1	1	yes	characteristic
25	C₂²×#1039	C₂²	1	1	yes	characteristic
26	C₂²×#1039	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×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₂²⋊#1039	C₂²	1	1	yes	characteristic
33	C₂²⋊#1039	C₂²	1	1	yes	characteristic
34	C₂²⋊#1039	C₂²	1	1	yes	characteristic
35	C₂²⋊#1039	C₂²	1	1	yes	characteristic
36	C₂²⋊#1039	C₂²	1	1	yes	characteristic
37	C₂²⋊#1039	C₂²	1	1	yes	characteristic
38	C₂²⋊#1039	C₂²	1	1	yes	characteristic
39	C₂²⋊#1039	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊#1039	C₂²	1	1	yes	characteristic, complemented
41	#87808	C₂²	1	1	yes	characteristic, complemented
42	#87808	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
44	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
45	#87806	C₂²	1	1	yes	characteristic, complemented
46	#87806	C₂²	1	1	yes	characteristic, complemented
47	#87806	C₂²	1	1	yes	characteristic, complemented
48	#87806	C₂²	1	1	yes	characteristic, complemented
49	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
50	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
51	#65927	—	5	6	no	maximal
52	C₂³×C₅²	—	2	2	no	—
53	C₂³×C₅²	—	4	1	no	—
54	C₂³×C₅²	—	4	1	no	—
55	C₂³×C₅²	C₂³	1	1	yes	characteristic
56	C₂³×C₅²	C₂³	1	1	yes	characteristic
57	C₂³×C₅²	C₂³	1	1	yes	characteristic
58	C₂³×C₅²	C₂³	1	1	yes	characteristic
59	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
60	C₅²×D₄	—	2	2	no	—
61	C₅²×D₄	—	4	1	no	—
62	C₅²×D₄	—	4	1	no	—
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)	—	2	2	no	—
67	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
68	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
69	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
70	C₂×#1039	—	4	1	no	—
71	C₂×#1039	—	4	1	no	—
72	C₂×#1039	—	4	1	no	—
73	C₂×#1039	C₂³	1	1	yes	characteristic
74	C₂×#1039	C₂³	1	1	yes	characteristic
75	C₂×#1039	C₂³	1	1	yes	characteristic
76	C₂×#1039	C₂³	1	1	yes	characteristic
77	C₂×#1039	C₂³	1	1	yes	characteristic
78	C₂×#1039	C₂³	1	1	yes	characteristic
79	C₂×#1039	C₂³	1	1	yes	characteristic
80	C₄×C₅⋊D₅	—	4	1	no	—
81	#6045	—	4	1	no	—
82	C₅×#120	D₅	1	6	yes	complemented
83	#4099	—	5	6	no	—
84	C₂×C₂²⋊Dic₅	—	5	6	no	—
85	#4085	—	5	6	no	—
86	#4085	—	5	6	no	—
87	#4084	—	5	6	no	—
88	#4082	—	5	6	no	—
89	#4081	—	5	6	no	—
90	D₄×Dic₅	—	5	6	no	—
91	#4031	—	5	6	no	—
92	#4029	—	5	6	no	—
93	#4027	—	5	6	no	—
94	#4025	—	5	6	no	—
95	#4025	—	5	6	no	—
96	#4024	—	5	6	no	—
97	C₂²×C₅²	—	2	1	no	—
98	C₂²×C₅²	—	2	1	no	—
99	C₂²×C₅²	—	2	1	no	—
100	C₂²×C₅²	—	2	1	no	—
101	C₂²×C₅²	—	2	1	no	—
102	C₂²×C₅²	—	2	1	no	—
103	C₂²×C₅²	—	2	1	no	—
104	C₂²×C₅²	—	2	1	no	—
105	C₂²×C₅²	—	2	1	no	—
106	C₂²×C₅²	—	2	1	no	—
107	C₂²×C₅²	—	2	1	no	—
108	C₂²×C₅²	—	4	4	no	—
109	C₂²×C₅²	#41	1	2	yes	—
110	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
111	C₂×C₅⋊D₅	—	2	1	no	—
112	C₂×C₅⋊D₅	—	2	1	no	—
113	C₂×C₅⋊D₅	—	2	1	no	—
114	C₄×C₅²	—	2	1	no	—
115	C₄×C₅²	—	2	1	no	—
116	C₄×C₅²	—	2	1	no	—
117	#1039	—	2	1	no	—
118	#1039	—	2	1	no	—
119	#1039	—	2	1	no	—
120	#1039	—	2	1	no	—
121	#1039	—	2	1	no	—
122	#1039	—	2	1	no	—
123	#1039	—	2	1	no	—
124	C₂⁴×C₅	C₂×D₅	1	6	yes	—
125	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
126	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
127	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
128	C₂×C₂²⋊D₅	—	5	6	no	—
129	C₂×C₂²⋊D₅	—	5	6	no	—
130	C₂×C₂²⋊D₅	—	5	6	no	—
131	C₂×C₂²⋊D₅	—	5	6	no	—
132	C₂²×Dic₅	—	5	6	no	—
133	C₂²×Dic₅	—	5	6	no	—
134	C₂²×Dic₅	—	5	6	no	—
135	C₂×C₄×D₅	—	5	6	no	—
136	C₂×Dic₁₀	—	5	6	no	—
137	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
138	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
139	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
140	C₂²⋊Dic₅	—	5	6	no	—
141	C₂²⋊Dic₅	—	5	6	no	—
142	C₂²⋊Dic₅	—	5	6	no	—
143	C₂²⋊Dic₅	—	5	6	no	—
144	C₂²⋊Dic₅	—	5	6	no	—
145	C₂²⋊Dic₅	—	5	6	no	—
146	C₂²⋊Dic₅	—	5	6	no	—
147	C₂²⋊Dic₅	—	5	6	no	—
148	C₂²⋊Dic₅	—	5	6	no	—
149	#685	—	5	6	no	—
150	#685	—	5	6	no	—
151	C₄⋊Dic₅	—	5	6	no	—
152	C₄⋊Dic₅	—	5	6	no	—
153	#683	—	5	6	no	—
154	#683	—	5	6	no	—
155	#683	—	5	6	no	—
156	#683	—	5	6	no	—
157	C₄×Dic₅	—	5	6	no	—
158	C₄×Dic₅	—	5	6	no	—
159	#538	—	25	1	no	Sylow
160	C₂×C₅²	—	2	2	no	—
161	C₂×C₅²	—	4	1	no	—
162	C₂×C₅²	—	4	1	no	—
163	C₂×C₅²	—	4	1	no	—
164	C₂×C₅²	C₂×#41	1	1	yes	characteristic
165	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
167	C₅⋊D₅	—	4	1	no	—
168	C₂³×C₅	—	2	12	no	—
169	C₂³×C₅	—	4	6	no	—
170	C₂³×C₅	—	4	6	no	—
171	C₂³×C₅	C₂²×D₅	1	6	yes	—
172	C₂³×C₅	C₂²×D₅	1	6	yes	—
173	C₂³×C₅	C₂²×D₅	1	6	yes	—
174	C₂³×C₅	C₂²×D₅	1	6	yes	—
175	C₂²×D₅	—	5	6	no	—
176	C₅×D₄	—	2	12	no	—
177	C₅×D₄	—	4	6	no	—
178	C₅×D₄	—	4	6	no	—
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	—	10	12	no	—
183	C₂²⋊D₅	—	20	6	no	—
184	C₂²⋊D₅	—	20	6	no	—
185	C₂²⋊D₅	—	20	6	no	—
186	C₂×Dic₅	—	5	6	no	—
187	C₂×Dic₅	—	5	6	no	—
188	C₂×Dic₅	—	5	6	no	—
189	C₂×Dic₅	—	5	6	no	—
190	C₂×Dic₅	—	5	6	no	—
191	C₂×Dic₅	—	5	6	no	—
192	C₂×Dic₅	—	5	6	no	—
193	C₂×Dic₅	—	20	6	no	—
194	C₂×Dic₅	—	20	6	no	—
195	C₂×Dic₅	—	20	6	no	—
196	C₄×D₅	—	20	6	no	—
197	Dic₁₀	—	20	6	no	—
198	#126	—	25	1	no	—
199	#126	—	25	1	no	—
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	C₅⋊D₅	1	1	yes	characteristic, complemented
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	—	2	6	no	—
218	C₂²×C₅	—	2	6	no	—
219	C₂²×C₅	—	2	6	no	—
220	C₂²×C₅	—	2	6	no	—
221	C₂²×C₅	—	2	6	no	—
222	C₂²×C₅	—	2	6	no	—
223	C₂²×C₅	—	2	6	no	—
224	C₂²×C₅	—	2	6	no	—
225	C₂²×C₅	—	4	24	no	—
226	C₂²×C₅	#711	1	12	yes	—
227	C₂²×C₅	C₂³×D₅	1	6	yes	—
228	C₂×D₅	—	10	6	no	—
229	C₂×D₅	—	10	6	no	—
230	C₂×D₅	—	10	6	no	—
231	C₄×C₅	—	2	6	no	—
232	C₄×C₅	—	2	6	no	—
233	C₄×C₅	—	2	6	no	—
234	Dic₅	—	10	6	no	—
235	Dic₅	—	10	6	no	—
236	Dic₅	—	10	6	no	—
237	Dic₅	—	10	6	no	—
238	Dic₅	—	10	6	no	—
239	Dic₅	—	10	6	no	—
240	Dic₅	—	10	6	no	—
241	C₂⁴	C₂×C₅⋊D₅	1	1	yes	characteristic
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	C₂×C₅⋊D₅	1	1	yes	characteristic
248	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
249	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
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	—	25	1	no	—
258	C₄⋊C₄	—	25	1	no	—
259	C₄⋊C₄	—	25	1	no	—
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	—	25	1	no	—
276	C₂×C₅	—	2	12	no	—
277	C₂×C₅	—	4	6	no	—
278	C₂×C₅	—	4	6	no	—
279	C₂×C₅	—	4	6	no	—
280	C₂×C₅	C₂×#711	1	6	yes	—
281	C₂×C₅	#4144	1	6	yes	—
282	C₂×C₅	#4144	1	6	yes	—
283	D₅	—	20	6	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	25	1	no	—
288	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
289	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
290	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
291	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
292	Q₈	—	100	1	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	50	2	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	—	25	1	no	—
305	C₂×C₄	—	25	1	no	—
306	C₂×C₄	—	25	1	no	—
307	C₂×C₄	—	100	1	no	—
308	C₂×C₄	—	100	1	no	—
309	C₂×C₄	—	100	1	no	—
310	C₂×C₄	—	100	1	no	—
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	#65927	1	6	yes	complemented
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	4	no	—
327	C₂²	—	50	1	no	—
328	C₂²	—	50	1	no	—
329	C₂²	—	50	1	no	—
330	C₂²	#87902	1	2	yes	—
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	—	50	1	no	—
336	C₄	—	50	1	no	—
337	C₄	—	50	1	no	—
338	C₄	—	50	1	no	—
339	C₄	—	50	1	no	—
340	C₄	—	50	1	no	—
341	C₄	—	50	1	no	—
342	C₂	—	2	2	no	—
343	C₂	—	4	1	no	—
344	C₂	—	4	1	no	—
345	C₂	—	4	1	no	—
346	C₂	—	100	1	no	—
347	C₂	C₂×#87902	1	1	yes	characteristic
348	C₂	#11717517	1	1	yes	characteristic
349	C₂	#11717517	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#120	C₂
hypercenter	#120	C₅⋊D₅
hypocenter	C₅²	#538
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₅²×#120	C₂	semidirect
2. #11717273	C₂	semidirect
3. C₂×C₂²⋊#1039	C₂	semidirect
4. #11717259	C₂	semidirect
5. #11717259	C₂	semidirect
6. #11717258	C₂	semidirect
7. #11717256	C₂	semidirect
8. #11717255	C₂	semidirect
9. D₄×#1039	C₂	semidirect
10. #11717205	C₂	semidirect
11. #11717203	C₂	semidirect
12. #11717201	C₂	semidirect
13. #11717199	C₂	semidirect
14. #11717199	C₂	semidirect
15. #11717198	C₂	semidirect
23. C₂×C₂²⋊(C₅⋊D₅)	C₂²	semidirect
26. C₂²×#1039	C₂²	semidirect
27. C₂×C₄×C₅⋊D₅	C₂²	semidirect
28. C₂×#6045	C₂²	semidirect
39. C₂²⋊#1039	C₂²	semidirect
40. C₂²⋊#1039	C₂²	semidirect
41. #87808	C₂²	semidirect
42. #87808	C₂²	semidirect
43. C₄⋊#1039	C₂²	semidirect
44. C₄⋊#1039	C₂²	semidirect
45. #87806	C₂²	semidirect
46. #87806	C₂²	semidirect
47. #87806	C₂²	semidirect
48. #87806	C₂²	semidirect
49. C₄×#1039	C₂²	semidirect
50. C₄×#1039	C₂²	semidirect
82. C₅×#120	D₅	semidirect
209. #120	C₅⋊D₅	semidirect
213. C₅²	#538	semidirect
314. C₅	#65927	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#120	2
#11717273	2
C₂×C₂²⋊#1039	2
#11717259	2
#11717258	2
#11717256	2
#11717255	2
D₄×#1039	2
#11717205	2
#11717203	2
#11717201	2
#11717199	2
#11717198	2
#65927	5