Subgroups — C₃×#10598701 #14278743

Order 1920.

280 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3998	C₂	1	2	yes	maximal
2	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
3	#10598701	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
5	C₂²×C₃×#168	C₄	1	1	yes	characteristic
6	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
7	C₃×#3998	—	2	2	no	—
8	C₃×#3998	C₄	1	4	yes	complemented
9	C₂×C₃×#674	—	2	2	no	—
10	C₃×#3935	—	2	2	no	—
11	C₃×#1377	—	5	1	no	maximal, Hall
12	C₂×#3998	C₂×C₃	1	2	yes	—
13	C₂×#3935	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₃×C₄²×C₅	—	2	1	no	—
17	C₃×C₄²×C₅	—	2	1	no	—
18	C₂×C₃×#168	—	2	1	no	—
19	C₂×C₃×#168	—	2	1	no	—
20	C₂×C₃×#168	C₈	1	2	yes	—
21	C₂×C₃×#168	C₂×C₄	1	2	yes	—
22	C₂×C₃×#168	D₄	1	1	yes	characteristic
23	C₂×C₃×#168	D₄	1	2	yes	—
24	C₂×C₃×#168	Q₈	1	1	yes	characteristic
25	C₃×#674	—	2	4	no	—
26	C₂×C₃×M₃₂	—	5	2	no	—
27	C₂×C₃×#97	—	5	1	no	—
28	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
29	C₂²×#168	C₃×C₄	1	1	yes	characteristic
30	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
31	#3998	—	2	2	no	—
32	#3998	C₃×C₄	1	4	yes	complemented
33	C₂×#674	—	2	2	no	—
34	#3935	—	2	2	no	—
35	#1377	—	5	1	no	Sylow
36	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	—	2	2	no	—
38	C₂×C₃×C₄×C₅	—	4	2	no	—
39	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
45	C₃×#168	—	2	2	no	—
46	C₃×#168	C₂²⋊C₄	1	2	yes	—
47	C₃×#168	C₄⋊C₄	1	2	yes	—
48	C₂²×C₃×C₈	—	5	1	no	—
49	C₂²×C₃×C₈	—	5	1	no	—
50	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₃×M₃₂	—	5	4	no	—
52	C₃×M₃₂	—	10	2	no	—
53	C₂×C₃×C₁₆	—	10	2	no	—
54	C₃×#97	—	10	2	no	—
55	C₂²×C₄×C₅	C₃×C₈	1	2	yes	—
56	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
57	C₄²×C₅	—	2	1	no	—
58	C₄²×C₅	—	2	1	no	—
59	C₂×#168	—	2	1	no	—
60	C₂×#168	—	2	1	no	—
61	C₂×#168	C₃×C₈	1	2	yes	—
62	C₂×#168	C₂×C₃×C₄	1	2	yes	—
63	C₂×#168	C₃×D₄	1	1	yes	characteristic
64	C₂×#168	C₃×D₄	1	2	yes	—
65	C₂×#168	C₃×Q₈	1	1	yes	characteristic
66	#674	—	2	4	no	—
67	C₂×M₃₂	—	5	2	no	—
68	C₂×#97	—	5	1	no	—
69	C₂²×C₃×C₅	—	2	2	no	—
70	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
71	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
72	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
73	C₃×C₄×C₅	—	4	2	no	—
74	C₃×C₄×C₅	#95	1	1	yes	characteristic
75	C₃×C₄×C₅	#97	1	1	yes	characteristic
76	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
77	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₃×C₄	#170	1	2	yes	—
79	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₂×C₃×C₈	—	5	2	no	—
81	C₂×C₃×C₈	—	5	1	no	—
82	C₂×C₃×C₈	—	5	1	no	—
83	C₂×C₃×C₈	—	5	2	no	—
84	C₂×C₃×C₈	—	5	2	no	—
85	C₂×C₃×C₈	—	10	1	no	—
86	C₂×C₃×C₈	—	10	1	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	—	2	1	no	—
89	C₃×C₁₆	—	10	4	no	—
90	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
99	#168	—	2	2	no	—
100	#168	C₃×C₂²⋊C₄	1	2	yes	—
101	#168	C₃×C₄⋊C₄	1	2	yes	—
102	C₂²×C₈	—	5	1	no	—
103	C₂²×C₈	—	5	1	no	—
104	C₂×C₄²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
105	M₃₂	—	5	4	no	—
106	M₃₂	—	10	2	no	—
107	C₂×C₁₆	—	10	2	no	—
108	#97	—	10	2	no	—
109	C₂×C₃×C₅	—	2	1	no	—
110	C₂×C₃×C₅	—	2	1	no	—
111	C₂×C₃×C₅	#336	1	1	yes	characteristic
112	C₂×C₃×C₅	#349	1	2	yes	—
113	C₂³×C₃	C₂×#170	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
117	C₂×C₃×C₄	#700	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₂×C₃×C₄	#704	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₃×C₈	—	5	2	no	—
123	C₃×C₈	—	5	2	no	—
124	C₃×C₈	—	10	2	no	—
125	C₂²×C₅	—	2	2	no	—
126	C₂²×C₅	C₃×C₄×C₈	1	1	yes	characteristic
127	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	C₃×#95	1	1	yes	characteristic
131	C₄×C₅	C₃×#97	1	1	yes	characteristic
132	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₄	C₃×#170	1	2	yes	—
135	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₂×C₈	—	5	2	no	—
137	C₂×C₈	—	5	1	no	—
138	C₂×C₈	—	5	1	no	—
139	C₂×C₈	—	5	2	no	—
140	C₂×C₈	—	5	2	no	—
141	C₂×C₈	—	10	1	no	—
142	C₂×C₈	—	10	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₁₆	—	10	4	no	—
146	C₃×C₅	#1377	1	1	yes	characteristic, complemented, Hall
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	C₄×#170	1	1	yes	characteristic
149	C₂²×C₃	C₄⋊#170	1	1	yes	characteristic
150	C₂²×C₃	C₂²⋊#170	1	1	yes	characteristic
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	#4002	1	1	yes	characteristic
153	C₃×C₄	#4003	1	1	yes	characteristic
154	C₃×C₄	#4004	1	1	yes	characteristic
155	C₃×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	C₃×#336	1	1	yes	characteristic
159	C₂×C₅	C₃×#349	1	2	yes	—
160	C₂³	C₂×C₃×#170	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	C₃×D₅⋊C₈	1	1	yes	characteristic
164	C₂×C₄	C₃×#700	1	1	yes	characteristic
165	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₂×C₄	C₃×#704	1	1	yes	characteristic
168	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
169	C₈	—	5	2	no	—
170	C₈	—	5	2	no	—
171	C₈	—	10	2	no	—
172	C₂×C₃	—	2	1	no	—
173	C₂×C₃	—	2	1	no	—
174	C₂×C₃	#64912	1	2	yes	—
175	C₂×C₃	#64918	1	1	yes	characteristic
176	C₅	C₃×#1377	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	2	no	—
178	C₂²	C₃×C₄×#170	1	1	yes	characteristic
179	C₂²	C₃×C₄⋊#170	1	1	yes	characteristic
180	C₂²	C₃×C₂²⋊#170	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	C₃×#4002	1	1	yes	characteristic
183	C₄	C₃×#4003	1	1	yes	characteristic
184	C₄	C₃×#4004	1	1	yes	characteristic
185	C₄	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
186	C₃	#10598701	1	1	yes	characteristic, direct product factor, Sylow
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₃×#64912	1	2	yes	—
190	C₂	C₃×#64918	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₃×C₄×C₈
Frattini	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₄²×C₅	C₄
socle	C₂²×C₃×C₅	C₂²⋊C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂²×#168	C₂²×C₃
hypercenter	C₂×C₃×C₄²	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1377
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. #10598701	C₃	direct (N × H)
8. C₃×#3998	C₄	semidirect
32. #3998	C₃×C₄	semidirect
146. C₃×C₅	#1377	semidirect
176. C₅	C₃×#1377	semidirect
186. C₃	#10598701	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×#3998	2
C₂×C₃×#3935	2
#10598701	3
C₃×#1377	5

Subgroups — C3×#10598701 #14278743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10598701 #14278743