Subgroups — C₅×#71774 #14284896

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic
3	C₅×#4924	C₂	1	1	yes	maximal, characteristic
4	C₅×#4546	C₂	1	4	yes	maximal, complemented
5	C₅×#1548	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#208	—	2	1	no	—
15	C₅×#830	—	2	1	no	—
16	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₄×C₅×C₃⋊C₈	—	2	1	no	—
20	#71774	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
22	C₅×#431	—	3	1	no	—
23	C₅×#423	—	3	1	no	—
24	C₅×#333	—	3	4	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	4	4	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×#208	—	4	2	no	—
41	C₂×C₅×C₃⋊C₈	—	2	2	no	—
42	C₂×C₅×C₃⋊C₈	—	2	2	no	—
43	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
44	#4925	C₂×C₅	1	1	yes	characteristic
45	#4924	C₂×C₅	1	1	yes	characteristic
46	#4546	C₂×C₅	1	4	yes	complemented
47	C₂²×C₅×Q₈	Dic₃	1	2	yes	—
48	C₂×C₅×M₁₆	—	6	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	Dic₃	1	2	yes	—
52	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
56	C₅×C₄⋊C₈	—	3	2	no	—
57	C₅×C₄⋊C₈	—	3	2	no	—
58	C₅×C₄⋊C₈	—	6	1	no	—
59	C₅×C₂²⋊C₈	—	6	1	no	—
60	C₄×C₅×C₈	—	6	1	no	—
61	#1548	—	3	1	no	Sylow
62	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₃×C₅×Q₈	—	4	4	no	—
64	C₃×C₅×Q₈	—	4	4	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
73	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₅×C₃⋊C₈	—	4	2	no	—
78	C₅×C₃⋊C₈	—	4	2	no	—
79	C₂²×C₃×Q₈	C₄×C₅	1	2	yes	—
80	C₃×C₄⋊Q₈	—	2	1	no	—
81	C₃×C₄⋊Q₈	—	2	1	no	—
82	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
83	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
84	C₂×C₃×C₄⋊C₄	—	2	1	no	—
85	C₂×C₃×C₄⋊C₄	—	2	1	no	—
86	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₂×#208	—	2	1	no	—
88	#830	—	2	1	no	—
89	C₄⋊(C₃⋊C₈)	—	2	1	no	—
90	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
91	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
92	C₄×C₃⋊C₈	—	2	1	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	—	4	4	no	—
95	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
98	C₂²×C₄×C₅	#67	1	1	yes	characteristic
99	C₂²×C₄×C₅	#67	1	1	yes	characteristic
100	C₅×M₁₆	—	12	2	no	—
101	C₂×C₅×C₈	—	6	2	no	—
102	C₂×C₅×C₈	—	6	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
108	C₄²×C₅	#67	1	1	yes	characteristic
109	C₄²×C₅	#67	1	1	yes	characteristic
110	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic
112	#431	—	3	1	no	—
113	#423	—	3	1	no	—
114	#333	—	3	4	no	—
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	—	2	1	no	—
117	C₂²×C₃×C₅	#103	1	2	yes	—
118	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	4	4	no	—
122	C₃×C₄×C₅	#100	1	2	yes	—
123	C₃×C₄×C₅	#103	1	2	yes	—
124	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₃×C₄×C₅	#137	1	2	yes	—
127	C₂×C₃×Q₈	—	2	2	no	—
128	C₂×C₃×Q₈	—	4	4	no	—
129	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
139	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
140	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
141	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
142	#208	—	4	2	no	—
143	C₂×C₃⋊C₈	—	2	2	no	—
144	C₂×C₃⋊C₈	—	2	2	no	—
145	C₂³×C₅	#217	1	1	yes	characteristic
146	C₅×Q₈	—	4	4	no	—
147	C₅×Q₈	—	4	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	4	4	no	—
152	C₂×C₄×C₅	#215	1	2	yes	—
153	C₂×C₄×C₅	#216	1	2	yes	—
154	C₂×C₄×C₅	#217	1	1	yes	characteristic
155	C₂×C₄×C₅	#217	1	1	yes	characteristic
156	C₂×C₄×C₅	#217	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
160	C₅×C₈	—	12	2	no	—
161	C₅×C₈	—	12	2	no	—
162	C₂²×Q₈	C₅×Dic₃	1	2	yes	—
163	C₂×M₁₆	—	6	1	no	—
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	—	2	1	no	—
166	C₄⋊Q₈	C₅×Dic₃	1	2	yes	—
167	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
171	C₄⋊C₈	—	3	2	no	—
172	C₄⋊C₈	—	3	2	no	—
173	C₄⋊C₈	—	6	1	no	—
174	C₂²⋊C₈	—	6	1	no	—
175	C₄×C₈	—	6	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
179	C₂×C₃×C₅	#419	1	1	yes	characteristic
180	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₃×Q₈	—	4	4	no	—
182	C₃×Q₈	—	4	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	4	4	no	—
187	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
191	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
192	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₃⋊C₈	—	4	2	no	—
196	C₃⋊C₈	—	4	2	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	#835	1	2	yes	—
200	C₂²×C₅	C₂×#217	1	1	yes	characteristic
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	#833	1	2	yes	—
205	C₄×C₅	#835	1	2	yes	—
206	C₄×C₅	C₂×#215	1	1	yes	characteristic
207	C₄×C₅	#942	1	2	yes	—
208	C₄×C₅	C₂×#216	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	C₂×C₅×Dic₃	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
214	C₂²×C₄	C₅×#67	1	1	yes	characteristic
215	C₂²×C₄	C₅×#67	1	1	yes	characteristic
216	M₁₆	—	12	2	no	—
217	C₂×C₈	—	6	2	no	—
218	C₂×C₈	—	6	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
224	C₄²	C₅×#67	1	1	yes	characteristic
225	C₄²	C₅×#67	1	1	yes	characteristic
226	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
227	C₃×C₅	#1548	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	C₅×#103	1	2	yes	—
231	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	4	no	—
235	C₃×C₄	C₅×#100	1	2	yes	—
236	C₃×C₄	C₅×#103	1	2	yes	—
237	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
238	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
239	C₃×C₄	C₅×#137	1	2	yes	—
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	C₂×#833	1	1	yes	characteristic
242	C₂×C₅	C₂×#835	1	1	yes	characteristic
243	C₂×C₅	#5225	1	1	yes	characteristic
244	C₂³	C₅×#217	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₅×#215	1	2	yes	—
252	C₂×C₄	C₅×#216	1	2	yes	—
253	C₂×C₄	C₅×#217	1	1	yes	characteristic
254	C₂×C₄	C₅×#217	1	1	yes	characteristic
255	C₂×C₄	C₅×#217	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
259	C₈	—	12	2	no	—
260	C₈	—	12	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
263	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
264	C₂×C₃	C₅×#419	1	1	yes	characteristic
265	C₅	#71774	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	C₅×#835	1	2	yes	—
269	C₂²	C₂×C₅×#217	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	4	no	—
273	C₄	C₅×#833	1	2	yes	—
274	C₄	C₅×#835	1	2	yes	—
275	C₄	C₂×C₅×#215	1	1	yes	characteristic
276	C₄	C₅×#942	1	2	yes	—
277	C₄	C₂×C₅×#216	1	1	yes	characteristic
278	C₃	C₅×#1548	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	2	no	—
280	C₂	C₂×C₅×#833	1	1	yes	characteristic
281	C₂	C₂×C₅×#835	1	1	yes	characteristic
282	C₂	C₅×#5225	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₂×C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#1548
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4546	C₂	semidirect
20. #71774	C₅	direct (N × H)
46. #4546	C₂×C₅	semidirect
227. C₃×C₅	#1548	semidirect
265. C₅	#71774	direct (N × H)
278. C₃	C₅×#1548	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₅×#4925	2
C₅×#4924	2
C₅×#4546	2
C₅×#1548	3
#71774	5

Subgroups — C5×#71774 #14284896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71774 #14284896