Subgroups — #13106414

Order 1600.

628 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717264	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4092	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6037	C₂	1	1	yes	maximal, characteristic
6	#11717006	C₂	1	2	yes	maximal, complemented
7	#11716997	C₂	1	2	yes	maximal
8	#11716975	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716969	C₂	1	2	yes	maximal, complemented
10	#11716961	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716957	C₂	1	2	yes	maximal, complemented
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×#6045	C₂²	1	1	yes	characteristic
14	C₅×Q₈×D₅	—	2	4	no	—
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
17	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
19	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
20	C₂×#6039	C₂²	1	2	yes	—
21	C₂×#6037	C₂²	1	1	yes	characteristic
22	C₂×D₅×Dic₅	C₂²	1	2	yes	—
23	D₅×Dic₁₀	—	2	4	no	—
24	#87806	C₂²	1	2	yes	—
25	#87806	C₂²	1	2	yes	—
26	C₄×#1039	C₂²	1	1	yes	characteristic
27	C₅×#685	C₂²	1	2	yes	—
28	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₅×#683	C₂²	1	2	yes	—
30	#87781	C₂²	1	1	yes	characteristic
31	#87780	C₂²	1	2	yes	—
32	#87780	C₂²	1	2	yes	—
33	#87778	C₂²	1	1	yes	characteristic
34	#87778	C₂²	1	2	yes	—
35	#87778	C₂²	1	1	yes	characteristic
36	Dic₅²	C₂²	1	2	yes	—
37	Q₈×C₂²⋊D₅	—	5	1	no	maximal
38	#65966	—	5	1	no	maximal
39	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
40	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
41	C₅²×Q₈	—	2	2	no	—
42	C₂×C₄×C₅²	C₂³	1	2	yes	—
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×#1039	C₂³	1	2	yes	—
45	C₂×#1039	C₂³	1	1	yes	characteristic
46	C₂×#1039	C₂³	1	1	yes	characteristic
47	#6045	—	2	2	no	—
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
50	C₂×C₅×Dic₅	C₂³	1	2	yes	—
51	C₂×C₅×Dic₅	C₂³	1	2	yes	—
52	C₄×C₅×D₅	—	2	2	no	—
53	C₄×C₅×D₅	—	2	2	no	—
54	C₄×C₅×D₅	—	2	4	no	—
55	C₅×Dic₁₀	—	2	4	no	—
56	C₅×Dic₁₀	D₄	1	4	yes	complemented
57	#6039	—	2	4	no	—
58	#6037	Q₈	1	4	yes	complemented
59	D₅×Dic₅	—	2	4	no	—
60	C₂²×C₅×Q₈	—	5	1	no	—
61	C₂×Q₈×D₅	—	5	1	no	—
62	C₂×Q₈×D₅	D₅	1	1	yes	characteristic, complemented
63	C₂²×Dic₁₀	—	5	1	no	—
64	C₅×C₂²⋊Q₈	—	5	1	no	—
65	#4092	—	5	2	no	—
66	#4092	D₅	1	1	yes	characteristic, complemented
67	Q₈×Dic₅	—	5	1	no	—
68	#4090	—	5	1	no	—
69	#4090	—	5	1	no	—
70	#4090	—	5	2	no	—
71	#4090	—	10	2	no	—
72	C₄×C₂²⋊D₅	—	5	1	no	—
73	C₄×C₂²⋊D₅	—	5	1	no	—
74	C₄×C₂²⋊D₅	—	5	2	no	—
75	#4070	—	5	1	no	—
76	#4070	—	5	2	no	—
77	#4043	—	5	1	no	—
78	#4042	—	5	2	no	—
79	#4034	—	5	2	no	—
80	#4033	—	5	1	no	—
81	#4027	—	5	2	no	—
82	#4024	—	5	2	no	—
83	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
84	C₂×C₅×D₅	—	2	1	no	—
85	C₂×C₅×D₅	—	2	1	no	—
86	C₂×C₅×D₅	—	2	1	no	—
87	C₂×C₅×D₅	—	2	1	no	—
88	C₂×C₅×D₅	C₂×Q₈	1	2	yes	complemented
89	C₂×C₅×D₅	C₂×Q₈	1	2	yes	complemented
90	C₄×C₅²	—	2	2	no	—
91	C₄×C₅²	C₂×D₄	1	2	yes	—
92	#1039	—	2	2	no	—
93	#1039	—	2	1	no	—
94	#1039	C₂×Q₈	1	2	yes	complemented
95	C₅×Dic₅	—	2	1	no	—
96	C₅×Dic₅	—	2	1	no	—
97	C₅×Dic₅	—	2	2	no	—
98	C₅×Dic₅	C₂×D₄	1	4	yes	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	5	1	no	—
101	C₂×C₅×Q₈	—	5	2	no	—
102	C₂×C₅×Q₈	—	10	4	no	—
103	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
104	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	5	1	no	—
106	C₂²×C₄×C₅	—	5	2	no	—
107	C₂²×C₄×C₅	—	5	1	no	—
108	C₂×C₂²⋊D₅	—	5	1	no	—
109	C₂×C₂²⋊D₅	—	5	1	no	—
110	C₂²×Dic₅	—	5	2	no	—
111	Q₈×D₅	—	2	4	no	—
112	Q₈×D₅	—	10	4	no	—
113	C₂×C₄×D₅	—	5	2	no	—
114	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
115	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
116	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
117	C₂×Dic₁₀	—	5	1	no	—
118	C₂×Dic₁₀	—	5	1	no	—
119	C₂×Dic₁₀	—	5	2	no	—
120	C₂×Dic₁₀	—	5	2	no	—
121	C₂×Dic₁₀	—	10	2	no	—
122	C₂×Dic₁₀	—	10	4	no	—
123	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
124	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	5	1	no	—
126	C₅×C₄⋊C₄	—	5	2	no	—
127	C₅×C₂²⋊C₄	—	5	2	no	—
128	C₂²⋊Dic₅	—	5	1	no	—
129	C₂²⋊Dic₅	—	5	1	no	—
130	C₂²⋊Dic₅	—	5	2	no	—
131	#685	—	5	2	no	—
132	#685	—	5	1	no	—
133	#685	—	5	1	no	—
134	#685	C₂×D₅	1	2	yes	—
135	C₄⋊Dic₅	—	5	2	no	—
136	C₄⋊Dic₅	—	5	2	no	—
137	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
138	#683	—	5	2	no	—
139	#683	—	5	2	no	—
140	#683	—	5	2	no	—
141	#683	—	5	2	no	—
142	#683	—	5	2	no	—
143	#683	—	5	1	no	—
144	#683	—	5	2	no	—
145	#683	—	5	1	no	—
146	#683	—	10	4	no	—
147	#683	—	10	4	no	—
148	#683	C₂×D₅	1	2	yes	—
149	C₄×Dic₅	—	5	1	no	—
150	C₄×Dic₅	—	5	1	no	—
151	C₄×Dic₅	—	5	2	no	—
152	C₄×Dic₅	—	5	2	no	—
153	C₄×Dic₅	—	10	2	no	—
154	D₄×Q₈	—	25	1	no	Sylow
155	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
158	C₅×D₅	—	2	2	no	—
159	C₅×D₅	—	2	2	no	—
160	C₂³×C₅	—	5	1	no	—
161	C₂³×C₅	—	5	1	no	—
162	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
163	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
164	C₅×Q₈	—	2	2	no	—
165	C₅×Q₈	—	2	2	no	—
166	C₅×Q₈	—	4	4	no	—
167	C₅×Q₈	—	5	4	no	—
168	C₅×Q₈	—	10	4	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	—	5	2	no	—
172	C₂×C₄×C₅	—	5	2	no	—
173	C₂×C₄×C₅	—	5	1	no	—
174	C₂×C₄×C₅	—	5	1	no	—
175	C₂×C₄×C₅	—	10	4	no	—
176	C₂×C₄×C₅	—	10	2	no	—
177	C₂×C₄×C₅	—	10	2	no	—
178	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
179	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
180	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
182	C₂²⋊D₅	—	5	4	no	—
183	C₂²⋊D₅	—	5	4	no	—
184	C₂×Dic₅	—	5	1	no	—
185	C₂×Dic₅	—	5	1	no	—
186	C₂×Dic₅	—	5	1	no	—
187	C₂×Dic₅	—	5	1	no	—
188	C₂×Dic₅	—	5	2	no	—
189	C₂×Dic₅	—	5	2	no	—
190	C₂×Dic₅	—	10	2	no	—
191	C₂×Dic₅	—	10	2	no	—
192	C₂×Dic₅	—	10	4	no	—
193	C₂×Dic₅	—	10	4	no	—
194	C₂×Dic₅	C₂²×D₅	1	2	yes	—
195	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
196	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
197	C₂×Dic₅	C₂²×D₅	1	2	yes	—
198	C₄×D₅	—	2	2	no	—
199	C₄×D₅	—	2	2	no	—
200	C₄×D₅	—	2	4	no	—
201	C₄×D₅	—	10	4	no	—
202	Dic₁₀	—	2	4	no	—
203	Dic₁₀	—	10	4	no	—
204	Dic₁₀	—	10	2	no	—
205	Dic₁₀	—	10	2	no	—
206	Dic₁₀	—	10	4	no	—
207	Dic₁₀	—	20	4	no	—
208	Dic₁₀	C₂²⋊D₅	1	4	yes	complemented
209	C₂²×Q₈	—	5	1	no	—
210	C₂²×Q₈	—	25	1	no	—
211	C₄⋊Q₈	—	25	2	no	—
212	C₄⋊Q₈	—	25	1	no	—
213	C₂²⋊Q₈	—	5	1	no	—
214	C₂²⋊Q₈	—	25	2	no	—
215	C₂²⋊Q₈	—	25	2	no	—
216	C₂²⋊Q₈	—	25	1	no	—
217	C₄×Q₈	—	25	1	no	—
218	C₄×D₄	—	25	2	no	—
219	C₄×D₄	—	25	1	no	—
220	C₅²	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	—	5	2	no	—
223	C₂²×C₅	—	5	2	no	—
224	C₂²×C₅	—	10	1	no	—
225	C₂²×C₅	—	10	1	no	—
226	C₂²×C₅	—	10	1	no	—
227	C₂²×C₅	—	10	1	no	—
228	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
229	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
230	C₂×D₅	—	2	1	no	—
231	C₂×D₅	—	2	1	no	—
232	C₂×D₅	—	2	1	no	—
233	C₂×D₅	—	2	1	no	—
234	C₂×D₅	Q₈×D₅	1	2	yes	complemented
235	C₂×D₅	Q₈×D₅	1	2	yes	complemented
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	—	4	4	no	—
240	C₄×C₅	—	5	4	no	—
241	C₄×C₅	—	10	1	no	—
242	C₄×C₅	—	10	2	no	—
243	C₄×C₅	—	10	1	no	—
244	C₄×C₅	D₄×D₅	1	2	yes	—
245	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
246	Dic₅	—	2	1	no	—
247	Dic₅	—	2	2	no	—
248	Dic₅	—	2	1	no	—
249	Dic₅	—	5	2	no	—
250	Dic₅	—	5	2	no	—
251	Dic₅	—	10	4	no	—
252	Dic₅	—	10	4	no	—
253	Dic₅	—	10	2	no	—
254	Dic₅	—	10	1	no	—
255	Dic₅	—	10	1	no	—
256	Dic₅	—	10	2	no	—
257	Dic₅	—	20	4	no	—
258	Dic₅	C₂×C₂²⋊D₅	1	4	yes	—
259	C₂×Q₈	—	5	1	no	—
260	C₂×Q₈	—	5	2	no	—
261	C₂×Q₈	—	10	4	no	—
262	C₂×Q₈	—	25	2	no	—
263	C₂×Q₈	—	25	1	no	—
264	C₂×Q₈	—	50	4	no	—
265	C₂×Q₈	D₅²	1	1	yes	characteristic
266	C₂×D₄	—	25	1	no	—
267	C₂²×C₄	—	5	1	no	—
268	C₂²×C₄	—	5	2	no	—
269	C₂²×C₄	—	5	1	no	—
270	C₂²×C₄	—	25	2	no	—
271	C₄⋊C₄	—	5	1	no	—
272	C₄⋊C₄	—	5	2	no	—
273	C₄⋊C₄	—	25	1	no	—
274	C₄⋊C₄	—	25	2	no	—
275	C₄⋊C₄	—	25	2	no	—
276	C₄⋊C₄	—	25	2	no	—
277	C₄⋊C₄	—	25	2	no	—
278	C₂²⋊C₄	—	5	2	no	—
279	C₂²⋊C₄	—	25	2	no	—
280	C₂²⋊C₄	—	25	1	no	—
281	C₂²⋊C₄	—	25	1	no	—
282	C₄²	—	25	2	no	—
283	C₄²	—	25	1	no	—
284	C₂×C₅	—	2	2	no	—
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	2	2	no	—
287	C₂×C₅	—	10	2	no	—
288	C₂×C₅	—	10	2	no	—
289	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
290	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
291	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
292	C₂×C₅	#4147	1	1	yes	characteristic
293	C₂×C₅	#4150	1	1	yes	characteristic
294	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
295	D₅	—	2	2	no	—
296	D₅	—	2	2	no	—
297	C₂³	—	5	1	no	—
298	C₂³	—	5	1	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	5	4	no	—
301	Q₈	—	10	4	no	—
302	Q₈	—	50	4	no	—
303	Q₈	—	50	2	no	—
304	D₄	—	25	4	no	—
305	C₂×C₄	—	5	2	no	—
306	C₂×C₄	—	5	2	no	—
307	C₂×C₄	—	5	1	no	—
308	C₂×C₄	—	5	1	no	—
309	C₂×C₄	—	10	4	no	—
310	C₂×C₄	—	10	2	no	—
311	C₂×C₄	—	10	2	no	—
312	C₂×C₄	—	25	1	no	—
313	C₂×C₄	—	25	1	no	—
314	C₂×C₄	—	25	2	no	—
315	C₂×C₄	—	50	4	no	—
316	C₂×C₄	C₂×D₅²	1	2	yes	—
317	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
318	C₅	—	2	2	no	—
319	C₅	#65966	1	1	yes	characteristic, complemented
320	C₅	Q₈×C₂²⋊D₅	1	1	yes	characteristic, complemented
321	C₂²	—	5	2	no	—
322	C₂²	—	5	2	no	—
323	C₂²	—	10	1	no	—
324	C₂²	—	10	1	no	—
325	C₂²	—	10	1	no	—
326	C₂²	—	10	1	no	—
327	C₂²	C₂²×D₅²	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	5	4	no	—
330	C₄	—	10	2	no	—
331	C₄	—	10	1	no	—
332	C₄	—	10	1	no	—
333	C₄	—	25	2	no	—
334	C₄	—	50	1	no	—
335	C₄	—	50	2	no	—
336	C₄	D₅×C₂²⋊D₅	1	2	yes	—
337	C₂	—	10	2	no	—
338	C₂	—	10	2	no	—
339	C₂	#11717480	1	1	yes	characteristic
340	C₂	Q₈×D₅²	1	1	yes	characteristic
341	C₂	C₂×D₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₂×C₅²×Q₈	C₂²
prime radical	C₂×#6037	C₂²
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	D₄×Q₈
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×Q₈×D₅	C₂	semidirect
2. C₂×D₅×Dic₁₀	C₂	semidirect
3. #11717264	C₂	semidirect
4. C₅×#4092	C₂	semidirect
6. #11717006	C₂	semidirect
8. #11716975	C₂	semidirect
9. #11716969	C₂	semidirect
10. #11716961	C₂	semidirect
11. #11716957	C₂	semidirect
56. C₅×Dic₁₀	D₄	semidirect
58. #6037	Q₈	semidirect
62. C₂×Q₈×D₅	D₅	semidirect
66. #4092	D₅	semidirect
88. C₂×C₅×D₅	C₂×Q₈	semidirect
89. C₂×C₅×D₅	C₂×Q₈	semidirect
94. #1039	C₂×Q₈	semidirect
208. Dic₁₀	C₂²⋊D₅	semidirect
220. C₅²	D₄×Q₈	semidirect
234. C₂×D₅	Q₈×D₅	semidirect
235. C₂×D₅	Q₈×D₅	semidirect
319. C₅	#65966	semidirect
320. C₅	Q₈×C₂²⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Q₈×D₅	2
C₂×D₅×Dic₁₀	2
#11717264	2
C₅×#4092	2
C₄×#6037	2
#11717006	2
#11716997	2
#11716975	2
#11716969	2
#11716961	2
#11716957	2
Q₈×C₂²⋊D₅	5
#65966	5