Subgroups — C₅×#5563 #11755948

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#938	C₂	1	2	yes	maximal, complemented
3	C₅×#930	C₂	1	2	yes	maximal, complemented
4	C₅×#926	C₂	1	2	yes	maximal, complemented
5	C₅×#892	C₂	1	2	yes	maximal, complemented
6	C₅×#885	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#555	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₂×C₅×#67	C₂²	1	4	yes	—
13	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂×C₅×Dic₆	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₅×#217	C₂²	1	4	yes	—
20	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
22	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₄×C₅×Dic₃	C₂²	1	4	yes	—
26	#5563	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×C₄⋊Q₈	—	3	1	no	—
28	C₅×#124	—	3	4	no	—
29	C₅×#121	—	3	2	no	—
30	C₅×#121	—	3	2	no	—
31	C₄×C₅×D₄	—	3	1	no	—
32	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
33	C₅×#117	—	3	2	no	—
34	C₅×#117	—	3	2	no	—
35	C₂³×C₃×C₅	C₂³	1	2	yes	—
36	C₂²×C₅×D₃	C₂³	1	2	yes	—
37	C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₄×C₅	—	2	4	no	—
39	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₅×#67	—	4	4	no	—
44	C₂×C₅×Dic₃	C₂³	1	2	yes	—
45	C₂×C₅×Dic₃	C₂³	1	4	yes	—
46	C₅×D₁₂	—	2	2	no	—
47	C₄×C₅×D₃	—	2	4	no	—
48	C₅×Dic₆	—	4	2	no	—
49	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
50	#938	C₂×C₅	1	2	yes	complemented
51	#930	C₂×C₅	1	2	yes	complemented
52	#926	C₂×C₅	1	2	yes	complemented
53	#892	C₂×C₅	1	2	yes	complemented
54	#885	C₂×C₅	1	4	yes	complemented
55	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
56	#869	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₈	—	3	2	no	—
58	C₂×C₅×D₄	—	3	4	no	—
59	C₂×C₅×D₄	—	3	1	no	—
60	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	3	2	no	—
62	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
63	C₅×C₄⋊C₄	—	3	1	no	—
64	C₅×C₄⋊C₄	—	3	2	no	—
65	C₅×C₄⋊C₄	—	3	2	no	—
66	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₅×C₂²⋊C₄	—	3	4	no	—
68	C₅×C₂²⋊C₄	—	3	4	no	—
69	C₅×C₂²⋊C₄	—	3	2	no	—
70	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
71	C₄²×C₅	—	3	4	no	—
72	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	#555	—	3	1	no	Sylow
74	C₂²×C₃×C₅	—	2	2	no	—
75	C₂²×C₃×C₅	—	2	2	no	—
76	C₂²×C₃×C₅	—	2	2	no	—
77	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×D₃	—	2	2	no	—
79	C₂×C₅×D₃	—	2	2	no	—
80	C₂×C₅×D₃	—	2	2	no	—
81	C₃×C₄×C₅	—	2	2	no	—
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	#41	1	2	yes	—
84	C₃×C₄×C₅	#41	1	2	yes	—
85	C₅×Dic₃	—	2	2	no	—
86	C₅×Dic₃	—	2	4	no	—
87	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
89	C₂×#67	C₂²×C₅	1	4	yes	—
90	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
92	C₂×Dic₆	C₂²×C₅	1	2	yes	—
93	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
95	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
96	#217	C₂²×C₅	1	4	yes	—
97	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
98	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
99	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
100	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
101	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
102	C₄×Dic₃	C₂²×C₅	1	4	yes	—
103	C₂³×C₅	—	3	2	no	—
104	C₂³×C₅	C₂²×D₃	1	2	yes	—
105	C₅×Q₈	—	12	2	no	—
106	C₅×D₄	—	2	2	no	—
107	C₅×D₄	—	6	2	no	—
108	C₅×D₄	—	12	4	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	3	2	no	—
111	C₂×C₄×C₅	—	3	4	no	—
112	C₂×C₄×C₅	—	6	4	no	—
113	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
114	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₄⋊Q₈	—	3	1	no	—
118	#124	—	3	4	no	—
119	#121	—	3	2	no	—
120	#121	—	3	2	no	—
121	C₄×D₄	—	3	1	no	—
122	C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
123	#117	—	3	2	no	—
124	#117	—	3	2	no	—
125	C₂×C₃×C₅	—	4	2	no	—
126	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
128	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
129	C₅×D₃	—	4	2	no	—
130	C₂³×C₃	C₂³×C₅	1	2	yes	—
131	C₂²×D₃	C₂³×C₅	1	2	yes	—
132	C₃×D₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
135	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	#67	—	4	4	no	—
139	C₂×Dic₃	C₂³×C₅	1	2	yes	—
140	C₂×Dic₃	C₂³×C₅	1	4	yes	—
141	D₁₂	—	2	2	no	—
142	C₄×D₃	—	2	4	no	—
143	Dic₆	—	4	2	no	—
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	6	2	no	—
148	C₂²×C₅	—	6	2	no	—
149	C₂²×C₅	—	6	2	no	—
150	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	6	2	no	—
154	C₄×C₅	—	6	4	no	—
155	C₄×C₅	#235	1	2	yes	—
156	C₄×C₅	#237	1	2	yes	—
157	C₂×Q₈	—	3	2	no	—
158	C₂×D₄	—	3	4	no	—
159	C₂×D₄	—	3	1	no	—
160	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
161	C₂²×C₄	—	3	2	no	—
162	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
163	C₄⋊C₄	—	3	1	no	—
164	C₄⋊C₄	—	3	2	no	—
165	C₄⋊C₄	—	3	2	no	—
166	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
167	C₂²⋊C₄	—	3	4	no	—
168	C₂²⋊C₄	—	3	4	no	—
169	C₂²⋊C₄	—	3	2	no	—
170	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
171	C₄²	—	3	4	no	—
172	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
173	C₃×C₅	#555	1	1	yes	characteristic, complemented, Hall
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
178	C₂×D₃	—	2	2	no	—
179	C₂×D₃	—	2	2	no	—
180	C₂×D₃	—	2	2	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	C₅×#41	1	2	yes	—
184	C₃×C₄	C₅×#41	1	2	yes	—
185	Dic₃	—	2	2	no	—
186	Dic₃	—	2	4	no	—
187	C₂×C₅	—	4	2	no	—
188	C₂×C₅	—	12	2	no	—
189	C₂×C₅	C₂×#235	1	1	yes	characteristic
190	C₂×C₅	C₂×#237	1	1	yes	characteristic
191	C₂×C₅	#1009	1	1	yes	characteristic
192	C₂³	—	3	2	no	—
193	C₂³	C₂²×C₅×D₃	1	2	yes	—
194	Q₈	—	12	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	6	2	no	—
197	D₄	—	12	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	4	no	—
201	C₂×C₄	—	6	4	no	—
202	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
203	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
206	C₂×C₃	—	4	2	no	—
207	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
208	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
209	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
210	D₃	—	4	2	no	—
211	C₅	#5563	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	6	2	no	—
216	C₂²	—	6	2	no	—
217	C₂²	—	6	2	no	—
218	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	6	2	no	—
222	C₄	—	6	4	no	—
223	C₄	C₅×#235	1	2	yes	—
224	C₄	C₅×#237	1	2	yes	—
225	C₃	C₅×#555	1	1	yes	characteristic, complemented, Sylow
226	C₂	—	4	2	no	—
227	C₂	—	12	2	no	—
228	C₂	C₂×C₅×#235	1	1	yes	characteristic
229	C₂	C₂×C₅×#237	1	1	yes	characteristic
230	C₂	C₅×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×D₄	C₂	semidirect
2. C₅×#938	C₂	semidirect
3. C₅×#930	C₂	semidirect
4. C₅×#926	C₂	semidirect
5. C₅×#892	C₂	semidirect
6. C₅×#885	C₂	semidirect
7. C₄×C₅×D₁₂	C₂	semidirect
8. C₅×#869	C₂	semidirect
26. #5563	C₅	direct (N × H)
32. C₄×C₅×D₄	D₃	semidirect
49. C₃×C₄×D₄	C₂×C₅	semidirect
50. #938	C₂×C₅	semidirect
51. #930	C₂×C₅	semidirect
52. #926	C₂×C₅	semidirect
53. #892	C₂×C₅	semidirect
54. #885	C₂×C₅	semidirect
55. C₄×D₁₂	C₂×C₅	semidirect
56. #869	C₂×C₅	semidirect
122. C₄×D₄	C₅×D₃	semidirect
173. C₃×C₅	#555	semidirect
211. C₅	#5563	direct (N × H)
225. C₃	C₅×#555	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₄	2
C₅×#938	2
C₅×#930	2
C₅×#926	2
C₅×#892	2
C₅×#885	2
C₄×C₅×D₁₂	2
C₅×#869	2
C₅×#555	3
#5563	5

Subgroups — C5×#5563 #11755948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5563 #11755948