Subgroups — C₂×C₃×C₅×#395 #14275421

Order 1920.

1816 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	C₂³×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	3	yes	maximal
3	C₃×C₅×#395	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#95	C₂	1	3	yes	maximal
5	C₂×C₅×#395	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	12	no	—
7	C₂²×C₃×C₅×Q₈	C₂²	1	4	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
10	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	12	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	12	yes	—
13	C₃×C₅×#95	C₂²	1	12	yes	—
14	C₂×C₃×#395	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂³×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	3	yes	—
17	C₅×#395	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₅×#95	C₂×C₃	1	3	yes	—
19	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₃×C₅×Q₈	—	2	32	no	—
21	C₂×C₃×C₅×Q₈	—	2	24	no	—
22	C₂²×C₃×C₄×C₅	—	2	6	no	—
23	C₂²×C₃×C₄×C₅	—	2	12	no	—
24	C₂²×C₃×C₄×C₅	—	2	6	no	—
25	C₂²×C₃×C₄×C₅	—	2	6	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	12	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	12	yes	complemented
28	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	24	no	—
32	C₂³×C₃×Q₈	C₂×C₅	1	1	yes	characteristic
33	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	3	yes	—
34	C₃×#395	C₂×C₅	1	8	yes	direct product factor
35	C₂×C₃×#95	C₂×C₅	1	3	yes	—
36	C₂²×C₅×Q₈	—	2	12	no	—
37	C₂²×C₅×Q₈	C₂²×C₃	1	4	yes	—
38	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
40	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
41	C₂×C₅×C₄⋊C₄	—	2	12	no	—
42	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	12	yes	—
43	C₅×#95	C₂²×C₃	1	12	yes	—
44	C₂×#395	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
46	C₂³×C₃×C₅	C₂×Q₈	1	6	yes	—
47	C₂³×C₃×C₅	#41	1	2	yes	—
48	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	32	no	—
50	C₂×C₃×C₄×C₅	—	2	24	no	—
51	C₂×C₃×C₄×C₅	—	2	24	no	—
52	C₂×C₃×C₄×C₅	—	2	6	no	—
53	C₂×C₃×C₄×C₅	—	2	12	no	—
54	C₂×C₃×C₄×C₅	—	2	6	no	—
55	C₂×C₃×C₄×C₅	—	2	6	no	—
56	C₂×C₃×C₄×C₅	—	4	4	no	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
58	C₂×C₃×C₄×C₅	C₂×Q₈	1	12	yes	complemented
59	C₂²×C₃×Q₈	—	2	12	no	—
60	C₂²×C₃×Q₈	C₂²×C₅	1	4	yes	—
61	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
62	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
63	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
64	C₂×C₃×C₄⋊C₄	—	2	12	no	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	12	yes	—
66	C₃×#95	C₂²×C₅	1	12	yes	—
67	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
68	C₂×C₅×Q₈	—	2	32	no	—
69	C₂×C₅×Q₈	—	2	24	no	—
70	C₂²×C₄×C₅	—	2	6	no	—
71	C₂²×C₄×C₅	—	2	12	no	—
72	C₂²×C₄×C₅	—	2	6	no	—
73	C₂²×C₄×C₅	—	2	6	no	—
74	C₂²×C₄×C₅	C₃×D₄	1	12	yes	—
75	C₂²×C₄×C₅	C₃×Q₈	1	12	yes	complemented
76	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
77	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
78	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
79	C₅×C₄⋊C₄	—	2	24	no	—
80	C₂³×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
81	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	3	yes	—
82	#395	C₂×C₃×C₅	1	8	yes	direct product factor
83	C₂×#95	C₂×C₃×C₅	1	3	yes	—
84	C₂²×C₃×C₅	#120	1	4	yes	—
85	C₂²×C₃×C₅	C₂²⋊Q₈	1	12	yes	—
86	C₂²×C₃×C₅	C₄⋊Q₈	1	12	yes	—
87	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
88	C₂²×C₃×C₅	C₂²×Q₈	1	3	yes	—
89	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
90	C₃×C₄×C₅	—	2	12	no	—
91	C₃×C₄×C₅	—	2	12	no	—
92	C₃×C₄×C₅	—	4	2	no	—
93	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	2	32	no	—
95	C₂×C₃×Q₈	—	2	24	no	—
96	C₂²×C₃×C₄	—	2	6	no	—
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	2	6	no	—
99	C₂²×C₃×C₄	—	2	6	no	—
100	C₂²×C₃×C₄	C₅×D₄	1	12	yes	—
101	C₂²×C₃×C₄	C₅×Q₈	1	12	yes	complemented
102	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
103	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
104	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
105	C₃×C₄⋊C₄	—	2	24	no	—
106	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
107	C₂³×C₅	C₂×C₃×Q₈	1	6	yes	—
108	C₂³×C₅	C₃×#41	1	2	yes	—
109	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
110	C₅×Q₈	—	2	32	no	—
111	C₂×C₄×C₅	—	2	24	no	—
112	C₂×C₄×C₅	—	2	24	no	—
113	C₂×C₄×C₅	—	2	6	no	—
114	C₂×C₄×C₅	—	2	12	no	—
115	C₂×C₄×C₅	—	2	6	no	—
116	C₂×C₄×C₅	—	2	6	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
119	C₂×C₄×C₅	C₂×C₃×Q₈	1	12	yes	complemented
120	C₂²×Q₈	—	2	12	no	—
121	C₂²×Q₈	C₂²×C₃×C₅	1	4	yes	—
122	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
123	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
124	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
125	C₂×C₄⋊C₄	—	2	12	no	—
126	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	12	yes	—
127	#95	C₂²×C₃×C₅	1	12	yes	—
128	C₂×C₃×C₅	#395	1	8	yes	direct product factor
129	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	3	yes	—
131	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	3	yes	—
132	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
133	C₂³×C₃	C₂×C₅×Q₈	1	6	yes	—
134	C₂³×C₃	C₅×#41	1	2	yes	—
135	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
136	C₃×Q₈	—	2	32	no	—
137	C₂×C₃×C₄	—	2	24	no	—
138	C₂×C₃×C₄	—	2	24	no	—
139	C₂×C₃×C₄	—	2	6	no	—
140	C₂×C₃×C₄	—	2	12	no	—
141	C₂×C₃×C₄	—	2	6	no	—
142	C₂×C₃×C₄	—	2	6	no	—
143	C₂×C₃×C₄	—	4	4	no	—
144	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
145	C₂×C₃×C₄	C₂×C₅×Q₈	1	12	yes	complemented
146	C₂²×C₅	C₃×#120	1	4	yes	—
147	C₂²×C₅	C₃×C₂²⋊Q₈	1	12	yes	—
148	C₂²×C₅	C₃×C₄⋊Q₈	1	12	yes	—
149	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
150	C₂²×C₅	C₂²×C₃×Q₈	1	3	yes	—
151	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
152	C₄×C₅	—	2	12	no	—
153	C₄×C₅	—	2	12	no	—
154	C₄×C₅	—	4	2	no	—
155	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
156	C₂×Q₈	—	2	32	no	—
157	C₂×Q₈	—	2	24	no	—
158	C₂²×C₄	—	2	6	no	—
159	C₂²×C₄	—	2	12	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	6	no	—
162	C₂²×C₄	C₃×C₅×D₄	1	12	yes	—
163	C₂²×C₄	C₃×C₅×Q₈	1	12	yes	complemented
164	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
165	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
166	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
167	C₄⋊C₄	—	2	24	no	—
168	C₃×C₅	C₂×#395	1	1	yes	characteristic, direct product factor, Hall
169	C₂²×C₃	C₅×#120	1	4	yes	—
170	C₂²×C₃	C₅×C₂²⋊Q₈	1	12	yes	—
171	C₂²×C₃	C₅×C₄⋊Q₈	1	12	yes	—
172	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
173	C₂²×C₃	C₂²×C₅×Q₈	1	3	yes	—
174	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
175	C₃×C₄	—	2	12	no	—
176	C₃×C₄	—	2	12	no	—
177	C₃×C₄	—	4	2	no	—
178	C₂×C₅	C₃×#395	1	8	yes	direct product factor
179	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
180	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
181	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	3	yes	—
182	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
183	C₂³	C₂×C₃×C₅×Q₈	1	6	yes	—
184	C₂³	C₃×C₅×#41	1	2	yes	—
185	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
186	Q₈	—	2	32	no	—
187	C₂×C₄	—	2	24	no	—
188	C₂×C₄	—	2	6	no	—
189	C₂×C₄	—	2	12	no	—
190	C₂×C₄	—	2	24	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	2	6	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
195	C₂×C₄	C₂×C₃×C₅×Q₈	1	12	yes	complemented
196	C₂×C₃	C₅×#395	1	8	yes	direct product factor
197	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
198	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	3	yes	—
199	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	3	yes	—
200	C₅	C₂×C₃×#395	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	C₃×C₅×#120	1	4	yes	—
202	C₂²	C₃×C₅×C₂²⋊Q₈	1	12	yes	—
203	C₂²	C₃×C₅×C₄⋊Q₈	1	12	yes	—
204	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
205	C₂²	C₂²×C₃×C₅×Q₈	1	3	yes	—
206	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
207	C₄	—	2	12	no	—
208	C₄	—	2	12	no	—
209	C₄	—	4	2	no	—
210	C₃	C₂×C₅×#395	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	C₃×C₅×#395	1	8	yes	direct product factor
212	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic
213	C₂	C₂×C₃×C₅×C₂²⋊Q₈	1	3	yes	—
214	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#395	C₂	direct (N × H)
5. C₂×C₅×#395	C₃	direct (N × H)
14. C₂×C₃×#395	C₅	direct (N × H)
17. C₅×#395	C₂×C₃	direct (N × H)
27. C₂²×C₃×C₄×C₅	Q₈	semidirect
34. C₃×#395	C₂×C₅	direct (N × H)
44. C₂×#395	C₃×C₅	direct (N × H)
58. C₂×C₃×C₄×C₅	C₂×Q₈	semidirect
75. C₂²×C₄×C₅	C₃×Q₈	semidirect
82. #395	C₂×C₃×C₅	direct (N × H)
101. C₂²×C₃×C₄	C₅×Q₈	semidirect
119. C₂×C₄×C₅	C₂×C₃×Q₈	semidirect
128. C₂×C₃×C₅	#395	direct (N × H)
145. C₂×C₃×C₄	C₂×C₅×Q₈	semidirect
163. C₂²×C₄	C₃×C₅×Q₈	semidirect
168. C₃×C₅	C₂×#395	direct (N × H)
178. C₂×C₅	C₃×#395	direct (N × H)
195. C₂×C₄	C₂×C₃×C₅×Q₈	semidirect
196. C₂×C₃	C₅×#395	direct (N × H)
200. C₅	C₂×C₃×#395	direct (N × H)
210. C₃	C₂×C₅×#395	direct (N × H)
211. C₂	C₃×C₅×#395	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×Q₈	2
C₂²×C₃×C₅×C₄⋊C₄	2
C₃×C₅×#395	2
C₂×C₃×C₅×#95	2
C₂×C₅×#395	3
C₂×C₃×#395	5

Subgroups — C2×C3×C5×#395 #14275421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#395 #14275421