Subgroups — #14314330

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 188 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×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91096	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊#672)	C₂	1	4	yes	maximal, complemented
4	#10600536	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
10	#91096	—	2	4	no	—
11	C₂×C₃⋊#672	C₂²	1	4	yes	—
12	#69743	—	5	1	no	maximal, Hall
13	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
14	C₂×#3944	—	3	2	no	—
15	C₄⋊#672	—	3	4	no	—
16	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	—	2	2	no	—
19	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
22	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
23	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₄²×C₅	C₈	1	1	yes	characteristic
25	C₃×C₄²×C₅	C₈	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₃⋊#672	—	2	4	no	—
28	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
29	C₂×#812	—	5	2	no	—
30	C₄⋊(C₃⋊C₁₆)	—	5	4	no	—
31	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
34	C₄×C₅×C₈	Dic₃	1	2	yes	—
35	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
36	#3944	—	6	4	no	—
37	C₂×#672	—	3	4	no	—
38	#2150	—	15	1	no	Sylow
39	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	—	2	4	no	—
41	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
45	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₅×C₈	—	2	2	no	—
48	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
49	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
50	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₃×C₈	Dic₅	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
53	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
54	C₃×C₄×C₈	Dic₅	1	2	yes	—
55	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
56	#812	—	10	4	no	—
57	C₂×C₃⋊C₁₆	—	5	4	no	—
58	C₂²×C₄×C₅	C₃⋊C₈	1	2	yes	—
59	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	Dic₆	1	2	yes	—
62	C₂×C₅×C₈	D₁₂	1	2	yes	—
63	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
64	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
66	C₄²×C₅	C₃⋊C₈	1	1	yes	characteristic
67	C₄²×C₅	C₃⋊C₈	1	1	yes	characteristic
68	C₄²×C₅	C₂×Dic₃	1	2	yes	—
69	#672	—	6	4	no	—
70	C₂×M₃₂	—	15	2	no	—
71	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
72	C₄⋊C₁₆	—	15	4	no	—
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
75	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
76	C₃×C₄×C₅	—	2	2	no	—
77	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
78	C₃×C₄×C₅	M₃₂	1	4	yes	complemented
79	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
81	C₂²×C₃×C₄	#168	1	2	yes	—
82	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
85	C₂×C₃×C₈	D₂₀	1	2	yes	—
86	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
87	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
89	C₃×C₄²	#168	1	1	yes	characteristic
90	C₃×C₄²	#168	1	1	yes	characteristic
91	C₃×C₄²	C₂×Dic₅	1	2	yes	—
92	C₃⋊C₁₆	—	10	4	no	—
93	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
96	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
98	C₂×C₄×C₅	#208	1	2	yes	—
99	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
100	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
103	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
104	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
105	C₂²×C₈	Dic₁₅	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
107	C₂×C₄²	Dic₁₅	1	1	yes	characteristic
108	M₃₂	—	30	4	no	—
109	C₂×C₁₆	—	15	4	no	—
110	C₄×C₈	Dic₁₅	1	2	yes	—
111	C₄×C₈	C₂×D₁₅	1	2	yes	—
112	C₂×C₃×C₅	—	2	2	no	—
113	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
115	C₂³×C₃	C₂×#168	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	C₂×#168	1	2	yes	—
118	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×#168	1	2	yes	—
120	C₂×C₃×C₄	#681	1	2	yes	—
121	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
122	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	C₄⋊Dic₅	1	2	yes	—
125	C₃×C₈	C₂×Dic₁₀	1	1	yes	characteristic
126	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
129	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
130	C₄×C₅	—	2	2	no	—
131	C₄×C₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
132	C₄×C₅	#812	1	4	yes	complemented
133	C₄×C₅	C₂×#208	1	1	yes	characteristic
134	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
135	C₂²×C₄	C₃⋊#168	1	2	yes	—
136	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	Dic₃₀	1	2	yes	—
139	C₂×C₈	D₆₀	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
141	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
142	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
143	C₄²	C₃⋊#168	1	1	yes	characteristic
144	C₄²	C₃⋊#168	1	1	yes	characteristic
145	C₄²	C₂×Dic₁₅	1	2	yes	—
146	C₁₆	—	30	4	no	—
147	C₃×C₅	#2150	1	1	yes	characteristic, complemented, Hall
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	C₄⋊#168	1	2	yes	—
150	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	C₄⋊#168	1	2	yes	—
153	C₃×C₄	#3944	1	4	yes	complemented
154	C₃×C₄	C₂×#681	1	1	yes	characteristic
155	C₃×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
158	C₂×C₅	C₂×#812	1	2	yes	—
159	C₂³	C₂×C₃⋊#168	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	C₂×C₃⋊#168	1	2	yes	—
162	C₂×C₄	C₂×C₃⋊#168	1	1	yes	characteristic
163	C₂×C₄	C₂×C₃⋊#168	1	2	yes	—
164	C₂×C₄	#6710	1	2	yes	—
165	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
166	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	C₄⋊Dic₁₅	1	2	yes	—
169	C₈	C₂×Dic₃₀	1	1	yes	characteristic
170	C₈	C₂×D₆₀	1	1	yes	characteristic
171	C₂×C₃	—	2	2	no	—
172	C₂×C₃	C₂×C₄⋊#168	1	1	yes	characteristic
173	C₂×C₃	C₂×#3944	1	2	yes	—
174	C₅	#69743	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	C₄⋊(C₃⋊#168)	1	2	yes	—
177	C₂²	C₂²×C₃⋊#168	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	C₄⋊(C₃⋊#168)	1	2	yes	—
180	C₄	#91096	1	4	yes	complemented
181	C₄	C₂×#6710	1	1	yes	characteristic
182	C₄	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
183	C₃	#10600536	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	2	2	no	—
185	C₂	C₂×C₄⋊(C₃⋊#168)	1	1	yes	characteristic
186	C₂	C₂×#91096	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	#2150
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊#672)	C₂	semidirect
78. C₃×C₄×C₅	M₃₂	semidirect
132. C₄×C₅	#812	semidirect
147. C₃×C₅	#2150	semidirect
153. C₃×C₄	#3944	semidirect
174. C₅	#69743	semidirect
180. C₄	#91096	semidirect
183. C₃	#10600536	semidirect

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×C₈	2
C₂×#91096	2
C₄⋊(C₃⋊#672)	2
#10600536	3
#69743	5