Subgroups — C₃×C₅×#398 #11751373

Order 960.

300 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×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#95	C₂	1	2	yes	maximal
5	C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic
6	C₅×#398	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₄	—	2	2	no	—
13	C₃×C₅×C₄⋊C₄	—	2	4	no	—
14	C₃×C₅×C₄⋊C₄	—	2	2	no	—
15	C₃×#398	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	C₂×C₃	1	2	yes	—
18	C₂×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₅×#95	C₂×C₃	1	2	yes	—
20	C₅×#95	C₂×C₃	1	1	yes	characteristic
21	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₄×C₅	—	2	1	no	—
23	C₂×C₃×C₄×C₅	—	2	2	no	—
24	C₂×C₃×C₄×C₅	—	2	1	no	—
25	C₂×C₃×C₄×C₅	—	2	2	no	—
26	C₂×C₃×C₄×C₅	—	2	2	no	—
27	C₂×C₃×C₄×C₅	—	2	2	no	—
28	C₂×C₃×C₄×C₅	—	2	2	no	—
29	C₂×C₃×C₄×C₅	—	2	1	no	—
30	C₂×C₃×C₄×C₅	—	2	2	no	—
31	C₂×C₃×C₄×C₅	—	2	2	no	—
32	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₅	Q₈	1	4	yes	—
35	C₂×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
36	C₂×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×C₄⋊C₄	C₂×C₅	1	2	yes	—
38	C₃×#95	C₂×C₅	1	2	yes	—
39	C₃×#95	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
41	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
45	C₅×C₄⋊C₄	—	2	2	no	—
46	C₅×C₄⋊C₄	—	2	4	no	—
47	C₅×C₄⋊C₄	—	2	2	no	—
48	#398	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
52	C₂²×C₃×C₅	#41	1	1	yes	characteristic
53	C₂²×C₃×C₅	#41	1	2	yes	—
54	C₃×C₄×C₅	—	2	2	no	—
55	C₃×C₄×C₅	—	2	4	no	—
56	C₃×C₄×C₅	—	2	2	no	—
57	C₃×C₄×C₅	—	4	2	no	—
58	C₃×C₄×C₅	—	4	1	no	—
59	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
61	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	1	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	1	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	C₃×D₄	1	2	yes	—
79	C₂×C₄×C₅	C₃×D₄	1	2	yes	—
80	C₂×C₄×C₅	C₃×Q₈	1	4	yes	—
81	C₂×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×C₃×C₅	1	2	yes	—
84	#95	C₂×C₃×C₅	1	2	yes	—
85	#95	C₂×C₃×C₅	1	1	yes	characteristic
86	C₂×C₃×C₅	#121	1	1	yes	characteristic
87	C₂×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₃×C₅	#123	1	1	yes	characteristic
89	C₂×C₃×C₅	#125	1	2	yes	—
90	C₂×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
91	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	C₅×D₄	1	2	yes	—
103	C₂×C₃×C₄	C₅×D₄	1	2	yes	—
104	C₂×C₃×C₄	C₅×Q₈	1	4	yes	—
105	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
106	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
107	C₂²×C₅	C₂×C₃×Q₈	1	2	yes	—
108	C₂²×C₅	C₃×#41	1	1	yes	characteristic
109	C₂²×C₅	C₃×#41	1	2	yes	—
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	—	2	4	no	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	4	2	no	—
114	C₄×C₅	—	4	1	no	—
115	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
116	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	—
117	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
118	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
119	C₂²×C₄	C₂²×C₃×C₅	1	2	yes	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₃×C₅	#398	1	1	yes	characteristic, direct product factor, Hall
124	C₂²×C₃	C₂×C₅×D₄	1	1	yes	characteristic
125	C₂²×C₃	C₂×C₅×D₄	1	1	yes	characteristic
126	C₂²×C₃	C₂×C₅×Q₈	1	2	yes	—
127	C₂²×C₃	C₅×#41	1	1	yes	characteristic
128	C₂²×C₃	C₅×#41	1	2	yes	—
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	4	2	no	—
133	C₃×C₄	—	4	1	no	—
134	C₂×C₅	C₃×#121	1	1	yes	characteristic
135	C₂×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
136	C₂×C₅	C₃×#123	1	1	yes	characteristic
137	C₂×C₅	C₃×#125	1	2	yes	—
138	C₂×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
139	C₂³	C₂³×C₃×C₅	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₃×C₅×D₄	1	2	yes	—
151	C₂×C₄	C₃×C₅×D₄	1	2	yes	—
152	C₂×C₄	C₃×C₅×Q₈	1	4	yes	—
153	C₂×C₃	C₅×#121	1	1	yes	characteristic
154	C₂×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
155	C₂×C₃	C₅×#123	1	1	yes	characteristic
156	C₂×C₃	C₅×#125	1	2	yes	—
157	C₂×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
158	C₅	C₃×#398	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²	C₂×C₃×C₅×D₄	1	1	yes	characteristic
160	C₂²	C₂×C₃×C₅×D₄	1	1	yes	characteristic
161	C₂²	C₂×C₃×C₅×Q₈	1	2	yes	—
162	C₂²	C₃×C₅×#41	1	1	yes	characteristic
163	C₂²	C₃×C₅×#41	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	2	4	no	—
166	C₄	—	2	2	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₃	C₅×#398	1	1	yes	characteristic, direct product factor, Sylow
170	C₂	C₃×C₅×#121	1	1	yes	characteristic
171	C₂	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
172	C₂	C₃×C₅×#123	1	1	yes	characteristic
173	C₂	C₃×C₅×#125	1	2	yes	—
174	C₂	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. C₅×#398	C₃	direct (N × H)
15. C₃×#398	C₅	direct (N × H)
48. #398	C₃×C₅	direct (N × H)
123. C₃×C₅	#398	direct (N × H)
158. C₅	C₃×#398	direct (N × H)
169. C₃	C₅×#398	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×C₅×C₄⋊C₄	2
C₃×C₅×#95	2
C₅×#398	3
C₃×#398	5

Subgroups — C3×C5×#398 #11751373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#398 #11751373