Subgroups — C₃×C₅×#554 #11756643

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#125	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×C₅×D₄	C₂	1	2	yes	maximal
7	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₅×#554	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₅	C₂²	1	4	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	C₂²	1	2	yes	—
25	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₃×#554	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#125	C₂×C₃	1	2	yes	complemented
29	C₅×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
30	C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
31	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₄×C₅×D₄	C₂×C₃	1	2	yes	—
33	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
35	C₂³×C₃×C₅	C₂³	1	2	yes	—
36	C₃×C₅×Q₈	—	2	2	no	—
37	C₃×C₅×Q₈	—	4	2	no	—
38	C₃×C₅×D₄	Q₈	1	4	yes	complemented
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	—	4	4	no	—
41	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
44	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×#125	C₂×C₅	1	2	yes	complemented
50	C₃×C₂²⋊Q₈	C₂×C₅	1	4	yes	complemented
51	C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	complemented
52	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×C₄×D₄	C₂×C₅	1	2	yes	—
54	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
55	C₂×C₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
58	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₃	1	4	yes	—
61	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
63	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
69	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
70	C₄²×C₅	C₂²×C₃	1	2	yes	—
71	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
72	#554	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	—	2	2	no	—
75	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	—
76	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
77	C₃×C₄×C₅	—	2	2	no	—
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	—	2	2	no	—
80	C₃×C₄×C₅	—	2	4	no	—
81	C₃×C₄×C₅	—	2	1	no	—
82	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
83	C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
86	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×C₅	1	4	yes	—
89	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
91	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
94	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
96	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
97	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
98	C₃×C₄²	C₂²×C₅	1	2	yes	—
99	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	C₂³×C₅	C₂³×C₃	1	2	yes	—
101	C₅×Q₈	—	2	2	no	—
102	C₅×Q₈	—	4	2	no	—
103	C₅×D₄	C₃×Q₈	1	4	yes	complemented
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
107	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
109	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
111	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
114	#125	C₂×C₃×C₅	1	2	yes	complemented
115	C₂²⋊Q₈	C₂×C₃×C₅	1	4	yes	complemented
116	C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	complemented
117	C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
118	C₄×D₄	C₂×C₃×C₅	1	2	yes	—
119	C₄×D₄	C₂×C₃×C₅	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
121	C₂×C₃×C₅	—	2	4	no	—
122	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
124	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂³×C₃	C₂³×C₅	1	2	yes	—
126	C₃×Q₈	—	2	2	no	—
127	C₃×Q₈	—	4	2	no	—
128	C₃×D₄	C₅×Q₈	1	4	yes	complemented
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	4	4	no	—
131	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
132	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
134	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
135	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
136	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	C₂×C₃×Q₈	1	4	yes	—
141	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	—	2	4	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
148	C₄×C₅	C₃×#41	1	2	yes	—
149	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
151	C₂×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	—
153	C₂²×C₄	C₂²×C₃×C₅	1	4	yes	—
154	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
155	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
156	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
157	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
159	C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
160	C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
161	C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
162	C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
163	C₄²	C₂²×C₃×C₅	1	2	yes	—
164	C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
165	C₃×C₅	#554	1	1	yes	characteristic, direct product factor, Hall
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	C₂×C₅×Q₈	1	4	yes	—
169	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
176	C₃×C₄	C₅×#41	1	2	yes	—
177	C₂×C₅	—	2	4	no	—
178	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
179	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
180	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂³	C₂³×C₃×C₅	1	2	yes	—
182	Q₈	—	2	2	no	—
183	Q₈	—	4	2	no	—
184	D₄	C₃×C₅×Q₈	1	4	yes	complemented
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
188	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
189	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
190	C₂×C₄	C₂³×C₃×C₅	1	2	yes	—
191	C₂×C₄	C₂³×C₃×C₅	1	4	yes	—
192	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
193	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
196	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
197	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
198	C₅	C₃×#554	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	C₂×C₃×C₅×Q₈	1	4	yes	—
202	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	4	no	—
207	C₄	—	2	1	no	—
208	C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
209	C₄	C₃×C₅×#41	1	2	yes	—
210	C₃	C₅×#554	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	4	no	—
212	C₂	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
213	C₂	C₂×C₃×C₅×#41	1	1	yes	characteristic
214	C₂	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₃×C₅×#125	C₂	semidirect
3. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
4. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
5. C₃×C₄×C₅×Q₈	C₂	semidirect
8. C₂×C₃×C₅×C₄⋊C₄	C₂	semidirect
9. C₅×#554	C₃	direct (N × H)
26. C₃×#554	C₅	direct (N × H)
27. C₅×C₄⋊Q₈	C₂×C₃	semidirect
28. C₅×#125	C₂×C₃	semidirect
29. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
30. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
31. C₄×C₅×Q₈	C₂×C₃	semidirect
34. C₂×C₅×C₄⋊C₄	C₂×C₃	semidirect
38. C₃×C₅×D₄	Q₈	semidirect
48. C₃×C₄⋊Q₈	C₂×C₅	semidirect
49. C₃×#125	C₂×C₅	semidirect
50. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
51. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
52. C₃×C₄×Q₈	C₂×C₅	semidirect
55. C₂×C₃×C₄⋊C₄	C₂×C₅	semidirect
72. #554	C₃×C₅	direct (N × H)
103. C₅×D₄	C₃×Q₈	semidirect
113. C₄⋊Q₈	C₂×C₃×C₅	semidirect
114. #125	C₂×C₃×C₅	semidirect
115. C₂²⋊Q₈	C₂×C₃×C₅	semidirect
116. C₂²⋊Q₈	C₂×C₃×C₅	semidirect
117. C₄×Q₈	C₂×C₃×C₅	semidirect
120. C₂×C₄⋊C₄	C₂×C₃×C₅	semidirect
128. C₃×D₄	C₅×Q₈	semidirect
165. C₃×C₅	#554	direct (N × H)
184. D₄	C₃×C₅×Q₈	semidirect
198. C₅	C₃×#554	direct (N × H)
210. C₃	C₅×#554	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
C₃×C₅×#125	2
C₃×C₅×C₂²⋊Q₈	2
C₃×C₄×C₅×Q₈	2
C₃×C₄×C₅×D₄	2
C₂×C₃×C₅×C₄⋊C₄	2
C₅×#554	3
C₃×#554	5

Subgroups — C3×C5×#554 #11756643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#554 #11756643