Subgroups — C₃×C₅×#2397 #14275431

Order 1920.

1680 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₄	C₂	1	6	yes	maximal, complemented
2	C₃×C₅×#399	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#392	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2397	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
7	C₂³×C₃×C₄×C₅	C₂²	1	4	yes	—
8	C₃×C₄×C₅×D₄	—	2	24	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
12	C₃×C₅×#95	C₂²	1	4	yes	—
13	C₃×#2397	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₄×C₅×D₄	C₂×C₃	1	6	yes	complemented
15	C₅×#399	C₂×C₃	1	4	yes	complemented
16	C₅×#392	C₂×C₃	1	4	yes	complemented
17	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
19	C₂×C₃×C₅×D₄	—	2	24	no	—
20	C₂×C₃×C₅×D₄	—	2	12	no	—
21	C₂²×C₃×C₄×C₅	—	2	24	no	—
22	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₃×C₅×C₄⋊C₄	—	2	12	no	—
26	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
27	C₃×C₄²×C₅	D₄	1	12	yes	complemented
28	C₂×C₃×C₄×D₄	C₂×C₅	1	6	yes	complemented
29	C₃×#399	C₂×C₅	1	4	yes	complemented
30	C₃×#392	C₂×C₅	1	4	yes	complemented
31	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
32	C₂²×C₅×D₄	C₂²×C₃	1	6	yes	—
33	C₂³×C₄×C₅	C₂²×C₃	1	4	yes	—
34	C₄×C₅×D₄	—	2	24	no	—
35	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
36	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
37	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
38	C₅×#95	C₂²×C₃	1	4	yes	—
39	#2397	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₂³×C₃×C₅	—	2	12	no	—
41	C₂³×C₃×C₅	—	2	12	no	—
42	C₂³×C₃×C₅	#41	1	8	yes	—
43	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
44	C₃×C₅×D₄	—	2	48	no	—
45	C₂×C₃×C₄×C₅	—	2	12	no	—
46	C₂×C₃×C₄×C₅	—	2	12	no	—
47	C₂×C₃×C₄×C₅	—	4	16	no	—
48	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
49	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
50	C₂²×C₃×D₄	C₂²×C₅	1	6	yes	—
51	C₂³×C₃×C₄	C₂²×C₅	1	4	yes	—
52	C₃×C₄×D₄	—	2	24	no	—
53	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
54	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
55	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
56	C₃×#95	C₂²×C₅	1	4	yes	—
57	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
58	C₂×C₅×D₄	—	2	24	no	—
59	C₂×C₅×D₄	—	2	12	no	—
60	C₂²×C₄×C₅	—	2	24	no	—
61	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
62	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
63	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
64	C₅×C₄⋊C₄	—	2	12	no	—
65	C₅×C₂²⋊C₄	—	2	24	no	—
66	C₄²×C₅	C₃×D₄	1	12	yes	complemented
67	C₂×C₄×D₄	C₂×C₃×C₅	1	6	yes	complemented
68	#399	C₂×C₃×C₅	1	4	yes	complemented
69	#392	C₂×C₃×C₅	1	4	yes	complemented
70	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
71	C₂²×C₃×C₅	—	2	24	no	—
72	C₂²×C₃×C₅	—	4	12	no	—
73	C₂²×C₃×C₅	—	4	4	no	—
74	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
75	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
76	C₃×C₄×C₅	—	2	12	no	—
77	C₃×C₄×C₅	—	4	4	no	—
78	C₃×C₄×C₅	#127	1	4	yes	—
79	C₂⁴×C₃	C₂³×C₅	1	4	yes	—
80	C₂×C₃×D₄	—	2	24	no	—
81	C₂×C₃×D₄	—	2	12	no	—
82	C₂²×C₃×C₄	—	2	24	no	—
83	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
84	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
85	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
86	C₃×C₄⋊C₄	—	2	12	no	—
87	C₃×C₂²⋊C₄	—	2	24	no	—
88	C₃×C₄²	C₅×D₄	1	12	yes	complemented
89	C₂³×C₅	—	2	12	no	—
90	C₂³×C₅	—	2	12	no	—
91	C₂³×C₅	C₃×#41	1	8	yes	—
92	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
93	C₅×D₄	—	2	48	no	—
94	C₂×C₄×C₅	—	2	12	no	—
95	C₂×C₄×C₅	—	2	12	no	—
96	C₂×C₄×C₅	—	4	16	no	—
97	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
98	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
99	C₂²×D₄	C₂²×C₃×C₅	1	6	yes	—
100	C₂³×C₄	C₂²×C₃×C₅	1	4	yes	—
101	C₄×D₄	—	2	24	no	—
102	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
104	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
105	#95	C₂²×C₃×C₅	1	4	yes	—
106	C₂×C₃×C₅	—	4	8	no	—
107	C₂×C₃×C₅	#525	1	6	yes	—
108	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
109	C₂³×C₃	—	2	12	no	—
110	C₂³×C₃	—	2	12	no	—
111	C₂³×C₃	C₅×#41	1	8	yes	—
112	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
113	C₃×D₄	—	2	48	no	—
114	C₂×C₃×C₄	—	2	12	no	—
115	C₂×C₃×C₄	—	2	12	no	—
116	C₂×C₃×C₄	—	4	16	no	—
117	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
118	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
119	C₂²×C₅	—	2	24	no	—
120	C₂²×C₅	—	4	12	no	—
121	C₂²×C₅	—	4	4	no	—
122	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
123	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
124	C₄×C₅	—	2	12	no	—
125	C₄×C₅	—	4	4	no	—
126	C₄×C₅	C₃×#127	1	4	yes	—
127	C₂⁴	C₂³×C₃×C₅	1	4	yes	—
128	C₂×D₄	—	2	24	no	—
129	C₂×D₄	—	2	12	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
132	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
133	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
134	C₄⋊C₄	—	2	12	no	—
135	C₂²⋊C₄	—	2	24	no	—
136	C₄²	C₃×C₅×D₄	1	12	yes	complemented
137	C₃×C₅	#2397	1	1	yes	characteristic, direct product factor, Hall
138	C₂²×C₃	—	2	24	no	—
139	C₂²×C₃	—	4	12	no	—
140	C₂²×C₃	—	4	4	no	—
141	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
142	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
143	C₃×C₄	—	2	12	no	—
144	C₃×C₄	—	4	4	no	—
145	C₃×C₄	C₅×#127	1	4	yes	—
146	C₂×C₅	—	4	8	no	—
147	C₂×C₅	C₃×#525	1	6	yes	—
148	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
149	C₂³	—	2	12	no	—
150	C₂³	—	2	12	no	—
151	C₂³	C₃×C₅×#41	1	8	yes	—
152	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
153	D₄	—	2	48	no	—
154	C₂×C₄	—	2	12	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	—	4	16	no	—
157	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
158	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
159	C₂×C₃	—	4	8	no	—
160	C₂×C₃	C₅×#525	1	6	yes	—
161	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
162	C₅	C₃×#2397	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	—	2	24	no	—
164	C₂²	—	4	12	no	—
165	C₂²	—	4	4	no	—
166	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
167	C₂²	C₂×C₃×C₅×#41	1	4	yes	—
168	C₄	—	2	12	no	—
169	C₄	—	4	4	no	—
170	C₄	C₃×C₅×#127	1	4	yes	—
171	C₃	C₅×#2397	1	1	yes	characteristic, direct product factor, Sylow
172	C₂	—	4	8	no	—
173	C₂	C₃×C₅×#525	1	6	yes	—
174	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
socle	C₂³×C₃×C₅	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×D₄	C₂	semidirect
2. C₃×C₅×#399	C₂	semidirect
3. C₃×C₅×#392	C₂	semidirect
4. C₃×C₅×#376	C₂	semidirect
5. C₅×#2397	C₃	direct (N × H)
13. C₃×#2397	C₅	direct (N × H)
14. C₂×C₄×C₅×D₄	C₂×C₃	semidirect
15. C₅×#399	C₂×C₃	semidirect
16. C₅×#392	C₂×C₃	semidirect
17. C₅×#376	C₂×C₃	semidirect
27. C₃×C₄²×C₅	D₄	semidirect
28. C₂×C₃×C₄×D₄	C₂×C₅	semidirect
29. C₃×#399	C₂×C₅	semidirect
30. C₃×#392	C₂×C₅	semidirect
31. C₃×#376	C₂×C₅	semidirect
39. #2397	C₃×C₅	direct (N × H)
66. C₄²×C₅	C₃×D₄	semidirect
67. C₂×C₄×D₄	C₂×C₃×C₅	semidirect
68. #399	C₂×C₃×C₅	semidirect
69. #392	C₂×C₃×C₅	semidirect
70. #376	C₂×C₃×C₅	semidirect
88. C₃×C₄²	C₅×D₄	semidirect
136. C₄²	C₃×C₅×D₄	semidirect
137. C₃×C₅	#2397	direct (N × H)
162. C₅	C₃×#2397	direct (N × H)
171. C₃	C₅×#2397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₄	2
C₃×C₅×#399	2
C₃×C₅×#392	2
C₃×C₅×#376	2
C₅×#2397	3
C₃×#2397	5

Subgroups — C3×C5×#2397 #14275431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2397 #14275431