Subgroups — #14278702

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×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#91096	C₂	1	2	yes	maximal
3	#10598154	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₃×C₅×#97	—	2	2	no	—
8	#91096	—	2	2	no	—
9	#91096	—	2	2	no	—
10	C₂×C₃⋊#672	C₄	1	4	yes	complemented
11	#67663	—	5	1	no	maximal, Hall
12	C₂×C₅×#97	D₃	1	1	yes	characteristic
13	C₂×#3944	—	3	2	no	—
14	C₂²×C₃×C₄×C₅	—	2	1	no	—
15	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₅×C₈	—	2	1	no	—
17	C₂×C₃×C₅×C₈	—	2	1	no	—
18	C₂×C₃×C₅×C₈	C₈	1	2	yes	—
19	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
22	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
23	C₃×C₄²×C₅	—	2	1	no	—
24	C₃×C₄²×C₅	C₈	1	2	yes	—
25	C₃⋊#672	—	2	4	no	—
26	C₂×C₃×#97	D₅	1	1	yes	characteristic
27	C₂×#812	—	5	2	no	—
28	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
30	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
31	C₅×#97	—	2	2	no	—
32	#3944	—	6	2	no	—
33	#3944	—	6	2	no	—
34	C₂×#672	—	3	4	no	—
35	#1375	—	15	1	no	Sylow
36	C₂³×C₃×C₅	M₁₆	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	C₂×C₈	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
45	C₃×C₅×C₈	—	2	2	no	—
46	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
47	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
48	C₂²×C₃×C₈	Dic₅	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
50	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
51	C₃×#97	—	2	2	no	—
52	#812	—	10	2	no	—
53	#812	—	10	2	no	—
54	C₂×C₃⋊C₁₆	—	5	4	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	C₃⋊C₈	1	2	yes	—
60	C₂×C₅×C₈	Dic₆	1	1	yes	characteristic
61	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
62	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
63	C₂×C₅×C₈	#67	1	2	yes	—
64	C₄²×C₅	—	2	1	no	—
65	C₄²×C₅	C₃⋊C₈	1	2	yes	—
66	#672	—	6	4	no	—
67	C₂×M₃₂	—	15	2	no	—
68	C₂×#97	D₁₅	1	1	yes	characteristic
69	C₂²×C₃×C₅	—	2	2	no	—
70	C₂²×C₃×C₅	#97	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	C₄×C₈	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	—	2	1	no	—
79	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	—	2	1	no	—
82	C₂×C₃×C₈	#168	1	2	yes	—
83	C₂×C₃×C₈	Dic₁₀	1	1	yes	characteristic
84	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
85	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	#168	1	2	yes	—
89	C₃⋊C₁₆	—	10	4	no	—
90	C₂³×C₅	#208	1	1	yes	characteristic
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	#208	1	1	yes	characteristic
96	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
97	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
98	C₂×C₄×C₅	#217	1	1	yes	characteristic
99	C₅×C₈	—	2	2	no	—
100	C₅×C₈	Dic₃⋊C₄	1	2	yes	—
101	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
102	C₂²×C₈	Dic₁₅	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
104	C₂×C₄²	Dic₁₅	1	1	yes	characteristic
105	M₃₂	—	30	2	no	—
106	M₃₂	—	30	2	no	—
107	C₂×C₁₆	—	15	4	no	—
108	#97	—	2	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	#347	1	2	yes	—
113	C₂³×C₃	#681	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
118	C₂×C₃×C₄	#681	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
120	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
122	C₃×C₈	—	2	2	no	—
123	C₃×C₈	#683	1	2	yes	—
124	C₃×C₈	#685	1	2	yes	—
125	C₂²×C₅	—	2	2	no	—
126	C₂²×C₅	#803	1	1	yes	characteristic
127	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
128	C₂²×C₅	#830	1	1	yes	characteristic
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	C₄×C₃⋊C₈	1	1	yes	characteristic
131	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
132	C₄×C₅	#830	1	1	yes	characteristic
133	C₄×C₅	C₃⋊#95	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₃⋊#168	1	2	yes	—
139	C₂×C₈	Dic₃₀	1	1	yes	characteristic
140	C₂×C₈	C₄×D₁₅	1	2	yes	—
141	C₂×C₈	D₆₀	1	1	yes	characteristic
142	C₂×C₈	#1220	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	C₃⋊#168	1	2	yes	—
145	C₁₆	—	30	4	no	—
146	C₃×C₅	#1375	1	1	yes	characteristic, complemented, Hall
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	#3935	1	1	yes	characteristic
149	C₂²×C₃	C₄⋊#168	1	1	yes	characteristic
150	C₂²×C₃	C₂²⋊#168	1	1	yes	characteristic
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	C₄×#168	1	1	yes	characteristic
153	C₃×C₄	C₄⋊#168	1	1	yes	characteristic
154	C₃×C₄	C₂²⋊#168	1	1	yes	characteristic
155	C₃×C₄	#3963	1	1	yes	characteristic
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	#4466	1	2	yes	—
159	C₂×C₅	#4524	1	1	yes	characteristic
160	C₂³	#6710	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	C₂×C₃⋊#168	1	1	yes	characteristic
164	C₂×C₄	C₂×C₃⋊#168	1	1	yes	characteristic
165	C₂×C₄	#6710	1	1	yes	characteristic
166	C₂×C₄	C₄×Dic₁₅	1	1	yes	characteristic
167	C₂×C₄	C₄⋊Dic₁₅	1	1	yes	characteristic
168	C₂×C₄	#6719	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	#6712	1	2	yes	—
171	C₈	#6714	1	2	yes	—
172	C₂×C₃	—	2	1	no	—
173	C₂×C₃	—	2	1	no	—
174	C₂×C₃	#64686	1	2	yes	—
175	C₂×C₃	#64744	1	1	yes	characteristic
176	C₅	#67663	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	2	no	—
178	C₂²	#91087	1	1	yes	characteristic
179	C₂²	C₄⋊(C₃⋊#168)	1	1	yes	characteristic
180	C₂²	#91114	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	C₄×C₃⋊#168	1	1	yes	characteristic
183	C₄	C₄⋊(C₃⋊#168)	1	1	yes	characteristic
184	C₄	#91114	1	1	yes	characteristic
185	C₄	#91115	1	1	yes	characteristic
186	C₃	#10598154	1	1	yes	characteristic, complemented, Sylow
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#11746287	1	2	yes	—
190	C₂	#11746345	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₅
commutator	C₃×C₄×C₅	C₄×C₈
Frattini	C₂²×C₈	C₂×D₁₅
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	#97
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂²×C₃×C₅×C₈	C₂²
hypercenter	C₂×#97	D₁₅
hypocenter	C₃×C₅	#1375
LCS	G ▷ C₃×C₄×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
10. C₂×C₃⋊#672	C₄	semidirect
146. C₃×C₅	#1375	semidirect
176. C₅	#67663	semidirect
186. C₃	#10598154	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#97	2
C₂×#91096	2
#10598154	3
#67663	5