Subgroups — C₅×#5567 #11755952

Order 960.

400 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#944	C₂	1	2	yes	maximal
3	C₅×#889	C₂	1	4	yes	maximal
4	C₅×#887	C₂	1	2	yes	maximal
5	C₅×#870	C₂	1	2	yes	maximal
6	C₅×#869	C₂	1	1	yes	maximal, characteristic
7	C₄×C₅×Dic₆	C₂	1	2	yes	maximal
8	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic
9	C₅×#557	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₆	C₂²	1	2	yes	—
12	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₄²×C₅	C₂²	1	2	yes	—
17	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
20	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
21	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
22	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
23	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
24	C₄×C₅×Dic₃	C₂²	1	4	yes	—
25	C₄×C₅×Dic₃	C₂²	1	2	yes	—
26	#5567	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×C₄⋊Q₈	—	3	2	no	—
28	C₅×C₄⋊Q₈	—	3	1	no	—
29	C₅×#125	—	3	4	no	—
30	C₅×#125	—	3	2	no	—
31	C₄×C₅×Q₈	—	3	2	no	—
32	C₄×C₅×Q₈	—	3	2	no	—
33	C₄×C₅×Q₈	—	3	1	no	—
34	C₄×C₅×Q₈	D₃	1	1	yes	characteristic
35	C₃×C₅×Q₈	—	2	2	no	—
36	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
37	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₅×Dic₃	C₂³	1	2	yes	—
42	C₂×C₅×Dic₃	C₂³	1	4	yes	—
43	C₂×C₅×Dic₃	C₂³	1	2	yes	—
44	C₅×Dic₆	—	2	4	no	—
45	C₅×Dic₆	Q₈	1	4	yes	complemented
46	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
47	#944	C₂×C₅	1	2	yes	—
48	#889	C₂×C₅	1	4	yes	—
49	#887	C₂×C₅	1	2	yes	—
50	#870	C₂×C₅	1	2	yes	—
51	#869	C₂×C₅	1	1	yes	characteristic
52	C₄×Dic₆	C₂×C₅	1	2	yes	—
53	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic
54	C₂×C₅×Q₈	—	3	1	no	—
55	C₂×C₅×Q₈	—	3	2	no	—
56	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
57	C₅×C₄⋊C₄	—	3	2	no	—
58	C₅×C₄⋊C₄	—	3	4	no	—
59	C₅×C₄⋊C₄	—	3	1	no	—
60	C₅×C₄⋊C₄	—	3	4	no	—
61	C₅×C₄⋊C₄	—	3	2	no	—
62	C₅×C₄⋊C₄	—	3	4	no	—
63	C₅×C₄⋊C₄	—	3	2	no	—
64	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
66	C₄²×C₅	—	3	4	no	—
67	C₄²×C₅	—	3	2	no	—
68	C₄²×C₅	C₂×D₃	1	2	yes	—
69	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
70	#557	—	3	1	no	Sylow
71	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
72	C₃×C₄×C₅	—	2	2	no	—
73	C₃×C₄×C₅	—	2	2	no	—
74	C₃×C₄×C₅	—	2	1	no	—
75	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
76	C₃×C₄×C₅	#41	1	2	yes	—
77	C₅×Dic₃	—	2	4	no	—
78	C₅×Dic₃	—	2	2	no	—
79	C₅×Dic₃	C₂×Q₈	1	4	yes	—
80	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
81	C₂×Dic₆	C₂²×C₅	1	2	yes	—
82	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
85	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄²	C₂²×C₅	1	2	yes	—
87	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
88	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
89	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
90	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
91	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
92	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
93	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
94	C₄×Dic₃	C₂²×C₅	1	4	yes	—
95	C₄×Dic₃	C₂²×C₅	1	2	yes	—
96	C₅×Q₈	—	2	2	no	—
97	C₅×Q₈	—	3	4	no	—
98	C₅×Q₈	—	6	4	no	—
99	C₂×C₄×C₅	—	3	2	no	—
100	C₂×C₄×C₅	—	3	2	no	—
101	C₂×C₄×C₅	—	3	4	no	—
102	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
103	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
105	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₄⋊Q₈	—	3	2	no	—
108	C₄⋊Q₈	—	3	1	no	—
109	#125	—	3	4	no	—
110	#125	—	3	2	no	—
111	C₄×Q₈	—	3	2	no	—
112	C₄×Q₈	—	3	2	no	—
113	C₄×Q₈	—	3	1	no	—
114	C₄×Q₈	C₅×D₃	1	1	yes	characteristic
115	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
117	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
118	C₃×Q₈	—	2	2	no	—
119	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
120	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
121	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂×Dic₃	C₂³×C₅	1	2	yes	—
125	C₂×Dic₃	C₂³×C₅	1	4	yes	—
126	C₂×Dic₃	C₂³×C₅	1	2	yes	—
127	Dic₆	—	2	4	no	—
128	Dic₆	C₅×Q₈	1	4	yes	complemented
129	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
130	C₄×C₅	—	2	2	no	—
131	C₄×C₅	—	2	2	no	—
132	C₄×C₅	—	2	1	no	—
133	C₄×C₅	—	3	4	no	—
134	C₄×C₅	—	6	4	no	—
135	C₄×C₅	—	6	2	no	—
136	C₄×C₅	#235	1	2	yes	—
137	C₄×C₅	D₃×Q₈	1	2	yes	—
138	C₂×Q₈	—	3	1	no	—
139	C₂×Q₈	—	3	2	no	—
140	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
141	C₄⋊C₄	—	3	2	no	—
142	C₄⋊C₄	—	3	4	no	—
143	C₄⋊C₄	—	3	1	no	—
144	C₄⋊C₄	—	3	4	no	—
145	C₄⋊C₄	—	3	2	no	—
146	C₄⋊C₄	—	3	4	no	—
147	C₄⋊C₄	—	3	2	no	—
148	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
149	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
150	C₄²	—	3	4	no	—
151	C₄²	—	3	2	no	—
152	C₄²	C₂×C₅×D₃	1	2	yes	—
153	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
154	C₃×C₅	#557	1	1	yes	characteristic, complemented, Hall
155	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
160	C₃×C₄	C₅×#41	1	2	yes	—
161	Dic₃	—	2	4	no	—
162	Dic₃	—	2	2	no	—
163	Dic₃	C₂×C₅×Q₈	1	4	yes	—
164	C₂×C₅	C₂×#235	1	1	yes	characteristic
165	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
166	C₂×C₅	#1010	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	Q₈	—	3	4	no	—
169	Q₈	—	6	4	no	—
170	C₂×C₄	—	3	2	no	—
171	C₂×C₄	—	3	2	no	—
172	C₂×C₄	—	3	4	no	—
173	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
174	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
176	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
178	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
179	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
180	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
181	C₅	#5567	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	3	4	no	—
187	C₄	—	6	4	no	—
188	C₄	—	6	2	no	—
189	C₄	C₅×#235	1	2	yes	—
190	C₄	C₅×D₃×Q₈	1	2	yes	—
191	C₃	C₅×#557	1	1	yes	characteristic, complemented, Sylow
192	C₂	C₂×C₅×#235	1	1	yes	characteristic
193	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
194	C₂	C₅×#1010	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×Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂⁴
hypercenter	C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
26. #5567	C₅	direct (N × H)
45. C₅×Dic₆	Q₈	semidirect
128. Dic₆	C₅×Q₈	semidirect
154. C₃×C₅	#557	semidirect
181. C₅	#5567	direct (N × H)
191. C₃	C₅×#557	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×Q₈	2
C₅×#944	2
C₅×#889	2
C₅×#887	2
C₅×#870	2
C₅×#869	2
C₄×C₅×Dic₆	2
C₅×#557	3
#5567	5

Subgroups — C5×#5567 #11755952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5567 #11755952