Subgroups — #14314609

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751207	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#65395	C₂	1	1	yes	maximal, characteristic
4	#11745944	C₂	1	2	yes	maximal, complemented
5	#11745941	C₂	1	2	yes	maximal, complemented
6	#10602197	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	#91815	—	2	1	no	—
9	C₃×#4072	—	2	1	no	—
10	#91102	C₂²	1	2	yes	—
11	#91101	—	2	1	no	—
12	C₂×C₅×C₃⋊C₁₆	C₄	1	4	yes	complemented
13	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₃×#3950	C₂²	1	2	yes	—
16	C₃×#3949	—	2	1	no	—
17	#70648	—	5	1	no	maximal, Hall
18	C₂²×C₅×C₁₆	—	3	1	no	—
19	#65395	—	3	1	no	—
20	#65395	D₃	1	1	yes	characteristic
21	#64725	—	3	2	no	—
22	#64724	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
28	#6719	—	4	1	no	—
29	C₄⋊Dic₁₅	—	2	2	no	—
30	C₄×Dic₁₅	—	4	1	no	—
31	C₃×C₂²⋊Dic₅	—	4	1	no	—
32	C₃×C₄⋊Dic₅	—	2	2	no	—
33	C₃×C₄×Dic₅	—	4	1	no	—
34	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
36	C₃×#427	—	5	1	no	—
37	#5106	—	5	1	no	—
38	C₂²×C₃⋊C₁₆	D₅	1	1	yes	characteristic
39	#4490	—	5	2	no	—
40	#4489	—	5	2	no	—
41	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
42	#4072	—	2	1	no	—
43	#4072	—	6	1	no	—
44	C₂×C₅×C₁₆	—	3	4	no	—
45	C₂×C₅×C₁₆	—	3	1	no	—
46	C₂×C₅×C₁₆	—	3	1	no	—
47	#3950	—	3	2	no	—
48	#3950	C₂×D₃	1	2	yes	—
49	#3949	—	2	1	no	—
50	#3949	—	6	1	no	—
51	#2158	—	15	1	no	Sylow
52	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
54	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₁₅	—	4	2	no	—
58	C₂×C₃×Dic₅	—	4	2	no	—
59	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
60	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
63	C₃×#117	—	10	1	no	—
64	#926	—	10	1	no	—
65	C₃×#107	—	5	2	no	—
66	C₃×#106	—	10	1	no	—
67	#818	—	5	2	no	—
68	#817	—	10	1	no	—
69	C₂×C₃⋊C₁₆	Dic₅	1	4	yes	complemented
70	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
71	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
72	C₂²×C₄×C₅	#67	1	1	yes	characteristic
73	C₂×C₅×C₈	Dic₆	1	2	yes	—
74	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
75	C₂×C₅×C₈	#67	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
77	C₂²⋊Dic₅	—	4	1	no	—
78	C₂²⋊Dic₅	—	12	1	no	—
79	C₄⋊Dic₅	—	2	2	no	—
80	C₄⋊Dic₅	—	6	2	no	—
81	C₄×Dic₅	—	4	1	no	—
82	C₄×Dic₅	—	12	1	no	—
83	C₅×C₁₆	—	3	2	no	—
84	C₅×C₁₆	—	3	2	no	—
85	C₂²×C₁₆	—	3	1	no	—
86	#427	—	5	1	no	—
87	#427	—	15	1	no	—
88	#367	—	15	2	no	—
89	#366	—	15	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	#107	1	2	yes	—
92	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
93	C₃×C₄×C₅	#107	1	2	yes	—
94	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	Dic₁₅	—	8	2	no	—
97	C₃×Dic₅	—	8	2	no	—
98	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
99	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
100	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
102	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	10	2	no	—
104	C₃×C₂²⋊C₄	—	20	1	no	—
105	C₃×C₄²	—	20	1	no	—
106	#217	—	20	1	no	—
107	C₄⋊Dic₃	—	10	2	no	—
108	C₄×Dic₃	—	20	1	no	—
109	C₃⋊C₁₆	C₂×Dic₅	1	2	yes	—
110	C₃⋊C₁₆	C₂×Dic₅	1	2	yes	—
111	C₂³×C₅	#213	1	1	yes	characteristic
112	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
113	C₂×C₄×C₅	#213	1	1	yes	characteristic
114	C₂×C₄×C₅	#216	1	2	yes	—
115	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
116	C₂×Dic₅	—	4	2	no	—
117	C₂×Dic₅	—	12	2	no	—
118	C₅×C₈	Dic₃⋊C₄	1	2	yes	—
119	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
120	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
122	#117	—	10	1	no	—
123	#117	—	30	1	no	—
124	C₂×C₁₆	—	3	4	no	—
125	C₂×C₁₆	—	3	1	no	—
126	C₂×C₁₆	—	3	1	no	—
127	#107	—	5	2	no	—
128	#107	—	15	2	no	—
129	#106	—	10	1	no	—
130	#106	—	30	1	no	—
131	C₂×C₃×C₅	—	2	2	no	—
132	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
133	C₂×C₃×C₅	#508	1	2	yes	—
134	C₂³×C₃	D₄₀	1	1	yes	characteristic
135	C₂×C₃×C₄	—	20	2	no	—
136	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
137	C₂×C₃×C₄	Dic₂₀	1	2	yes	—
138	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
139	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
140	C₂×Dic₃	—	20	2	no	—
141	C₃×C₈	C₄⋊Dic₅	1	2	yes	—
142	C₃×C₈	C₂×Dic₁₀	1	1	yes	characteristic
143	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	#807	1	2	yes	—
146	C₂²×C₅	C₂×#213	1	1	yes	characteristic
147	C₄×C₅	#807	1	2	yes	—
148	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
149	C₄×C₅	C₂×#216	1	1	yes	characteristic
150	Dic₅	—	8	2	no	—
151	Dic₅	—	24	2	no	—
152	C₂²×C₄	#1202	1	1	yes	characteristic
153	C₂×C₈	D₃×Dic₅	1	2	yes	—
154	C₂×C₈	#1202	1	1	yes	characteristic
155	C₂×C₈	#1204	1	2	yes	—
156	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
157	C₄⋊C₄	—	10	2	no	—
158	C₄⋊C₄	—	30	2	no	—
159	C₂²⋊C₄	—	20	1	no	—
160	C₂²⋊C₄	—	60	1	no	—
161	C₄²	—	20	1	no	—
162	C₄²	—	60	1	no	—
163	C₁₆	—	3	2	no	—
164	C₁₆	—	3	2	no	—
165	C₃×C₅	#2158	1	1	yes	characteristic, complemented, Hall
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	#3950	1	2	yes	—
168	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
169	C₃×C₄	—	40	2	no	—
170	C₃×C₄	#3950	1	2	yes	—
171	C₃×C₄	C₂×Dic₂₀	1	1	yes	characteristic
172	C₃×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
173	Dic₃	—	40	2	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	C₂×#807	1	1	yes	characteristic
176	C₂×C₅	#5194	1	2	yes	—
177	C₂³	#6653	1	1	yes	characteristic
178	C₂×C₄	—	20	2	no	—
179	C₂×C₄	—	60	2	no	—
180	C₂×C₄	#6653	1	1	yes	characteristic
181	C₂×C₄	#6662	1	2	yes	—
182	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
183	C₂×C₄	C₂×#1202	1	1	yes	characteristic
184	C₈	Dic₃⋊Dic₅	1	2	yes	—
185	C₈	C₂×D₃×Dic₅	1	1	yes	characteristic
186	C₈	C₂×#1204	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	#65193	1	2	yes	—
189	C₂×C₃	C₂×#3950	1	1	yes	characteristic
190	C₅	#70648	1	1	yes	characteristic, complemented, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	#90990	1	2	yes	—
193	C₂²	C₂×#6653	1	1	yes	characteristic
194	C₄	—	40	2	no	—
195	C₄	—	120	2	no	—
196	C₄	#90990	1	2	yes	—
197	C₄	C₂×#6662	1	1	yes	characteristic
198	C₄	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
199	C₃	#10602197	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	2	2	no	—
201	C₂	#11747542	1	2	yes	—
202	C₂	C₂×#90990	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6653
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2158
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #11745944	C₂	semidirect
5. #11745941	C₂	semidirect
12. C₂×C₅×C₃⋊C₁₆	C₄	semidirect
69. C₂×C₃⋊C₁₆	Dic₅	semidirect
165. C₃×C₅	#2158	semidirect
190. C₅	#70648	semidirect
199. C₃	#10602197	semidirect

Maximal subgroups

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

subgroup	index
#11751207	2
C₂²×C₅×C₃⋊C₁₆	2
C₃×#65395	2
#11745944	2
#11745941	2
#10602197	3
#70648	5