Subgroups — #14294866

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 212 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	C₄²×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11746371	C₂	1	4	yes	maximal
5	#10600543	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₅	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
11	C₂²×C₃⋊#168	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
13	C₄×C₃⋊#168	C₄	1	4	yes	complemented
14	#69750	—	5	1	no	maximal, Hall
15	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
16	C₄²×Dic₅	—	3	1	no	—
17	C₂×C₄×#168	—	3	1	no	—
18	#64770	—	3	4	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₅	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₅×C₈	—	2	4	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₅	—	2	4	no	—
28	C₄×Dic₁₅	—	2	4	no	—
29	C₄×Dic₁₅	C₈	1	8	yes	complemented
30	C₂×C₃⋊#168	—	2	4	no	—
31	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
33	C₄²×Dic₃	—	5	1	no	—
34	C₂×C₄×C₃⋊C₈	—	5	1	no	—
35	#4550	—	5	4	no	—
36	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄×Dic₅	—	3	2	no	—
39	C₂×C₄×Dic₅	—	3	2	no	—
40	C₂×C₄×Dic₅	—	3	2	no	—
41	C₂²×#168	—	3	2	no	—
42	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
43	C₄×#168	—	3	4	no	—
44	#1744	—	15	1	no	Sylow
45	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
49	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	#41	1	4	yes	—
53	C₂×Dic₁₅	—	2	4	no	—
54	C₂×Dic₁₅	—	2	4	no	—
55	C₂×Dic₁₅	C₂×C₈	1	8	yes	—
56	C₃×C₅×C₈	—	2	4	no	—
57	C₃⋊#168	—	2	4	no	—
58	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
59	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₄×Dic₃	—	5	2	no	—
61	C₂×C₄×Dic₃	—	5	2	no	—
62	C₂×C₄×Dic₃	—	5	2	no	—
63	C₂²×C₃⋊C₈	—	5	2	no	—
64	C₃×C₄×C₈	Dic₅	1	4	yes	complemented
65	C₄×C₃⋊C₈	—	5	4	no	—
66	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₂²×Dic₅	—	3	4	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
72	C₄²×C₅	C₄×D₃	1	2	yes	—
73	C₄²×C₅	C₂×Dic₃	1	2	yes	—
74	C₄×Dic₅	—	3	8	no	—
75	C₄×Dic₅	—	6	4	no	—
76	C₄×Dic₅	—	6	4	no	—
77	C₂×#168	—	3	4	no	—
78	C₂×#168	—	6	4	no	—
79	C₂×C₄×C₈	—	15	1	no	—
80	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
81	C₄³	—	15	1	no	—
82	#336	—	15	4	no	—
83	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
84	C₂²×C₃×C₅	#117	1	2	yes	—
85	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
86	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
87	C₃×C₄×C₅	C₄×C₈	1	4	yes	—
88	C₃×C₄×C₅	#97	1	4	yes	—
89	C₃×C₄×C₅	#117	1	2	yes	—
90	C₃×C₄×C₅	#117	1	2	yes	—
91	Dic₁₅	—	2	8	no	—
92	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
95	C₂²×Dic₃	—	5	4	no	—
96	C₂×C₃×C₈	—	2	4	no	—
97	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	—
98	C₃×C₄²	C₄×D₅	1	2	yes	—
99	C₃×C₄²	C₂×Dic₅	1	2	yes	—
100	C₄×Dic₃	—	5	8	no	—
101	C₄×Dic₃	—	10	4	no	—
102	C₄×Dic₃	—	10	4	no	—
103	C₂×C₃⋊C₈	—	5	4	no	—
104	C₂×C₃⋊C₈	—	10	4	no	—
105	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
106	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
107	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
108	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
109	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
111	C₂×C₄×C₅	#235	1	4	yes	—
112	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
113	C₂×Dic₅	—	3	8	no	—
114	C₂×Dic₅	—	6	4	no	—
115	C₂×Dic₅	—	6	4	no	—
116	C₅×C₈	—	2	4	no	—
117	#168	—	6	4	no	—
118	C₂²×C₈	—	15	2	no	—
119	C₂²×C₈	C₂×D₁₅	1	2	yes	—
120	C₂×C₄²	—	15	2	no	—
121	C₂×C₄²	—	15	2	no	—
122	C₂×C₄²	—	15	2	no	—
123	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
124	C₄×C₈	—	15	4	no	—
125	C₄×C₈	Dic₁₅	1	4	yes	complemented
126	C₂×C₃×C₅	#376	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
129	C₂×C₃×C₅	#431	1	4	yes	—
130	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
131	C₂×C₃×C₄	C₈×D₅	1	4	yes	—
132	C₂×C₃×C₄	#676	1	4	yes	—
133	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
134	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
135	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
136	C₂×C₃×C₄	#709	1	4	yes	—
137	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
138	C₂×Dic₃	—	5	8	no	—
139	C₂×Dic₃	—	10	4	no	—
140	C₂×Dic₃	—	10	4	no	—
141	C₃×C₈	—	2	4	no	—
142	C₃⋊C₈	—	10	4	no	—
143	C₂²×C₅	#872	1	2	yes	—
144	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
145	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
146	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
147	C₄×C₅	C₈×Dic₃	1	4	yes	—
148	C₄×C₅	#815	1	4	yes	—
149	C₄×C₅	#872	1	2	yes	—
150	C₄×C₅	#926	1	2	yes	—
151	Dic₅	—	6	8	no	—
152	C₂²×C₄	—	15	4	no	—
153	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	15	4	no	—
158	C₂×C₈	—	30	4	no	—
159	C₂×C₈	C₂×Dic₁₅	1	4	yes	—
160	C₄²	—	15	8	no	—
161	C₄²	—	30	4	no	—
162	C₄²	—	30	4	no	—
163	C₄²	C₄×D₁₅	1	2	yes	—
164	C₄²	C₂×Dic₁₅	1	2	yes	—
165	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
166	C₂²×C₃	#4018	1	2	yes	—
167	C₂²×C₃	C₂×C₈×D₅	1	2	yes	—
168	C₂²×C₃	C₂×#676	1	2	yes	—
169	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
170	C₃×C₄	C₈×Dic₅	1	4	yes	—
171	C₃×C₄	#3947	1	4	yes	—
172	C₃×C₄	#4018	1	2	yes	—
173	C₃×C₄	#4072	1	2	yes	—
174	Dic₃	—	10	8	no	—
175	C₂×C₅	#4685	1	4	yes	—
176	C₂×C₅	#4932	1	1	yes	characteristic
177	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
178	C₂×C₅	C₂×#815	1	1	yes	characteristic
179	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
180	C₂×C₄	—	15	8	no	—
181	C₂×C₄	—	30	4	no	—
182	C₂×C₄	—	30	4	no	—
183	C₂×C₄	C₈×D₁₅	1	4	yes	—
184	C₂×C₄	#6705	1	4	yes	—
185	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
186	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
187	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
188	C₂×C₄	#6817	1	4	yes	—
189	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	30	4	no	—
192	C₂×C₃	#64974	1	4	yes	—
193	C₂×C₃	#65221	1	1	yes	characteristic
194	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
195	C₂×C₃	C₂×#3947	1	1	yes	characteristic
196	C₅	#69750	1	1	yes	characteristic, complemented, Sylow
197	C₂²	#91761	1	2	yes	—
198	C₂²	C₂×C₈×D₁₅	1	2	yes	—
199	C₂²	C₂×#6705	1	2	yes	—
200	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
201	C₄	—	30	8	no	—
202	C₄	C₈×Dic₁₅	1	4	yes	—
203	C₄	#91099	1	4	yes	—
204	C₄	#91761	1	2	yes	—
205	C₄	#91815	1	2	yes	—
206	C₃	#10600543	1	1	yes	characteristic, complemented, Sylow
207	C₂	#11750786	1	4	yes	—
208	C₂	#11751033	1	1	yes	characteristic
209	C₂	C₂×C₈×Dic₁₅	1	1	yes	characteristic
210	C₂	C₂×#91099	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	#1744
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₃×C₄×C₅×C₈	C₄	semidirect
13. C₄×C₃⋊#168	C₄	semidirect
29. C₄×Dic₁₅	C₈	semidirect
42. C₄×C₅×C₈	Dic₃	semidirect
64. C₃×C₄×C₈	Dic₅	semidirect
125. C₄×C₈	Dic₁₅	semidirect
165. C₃×C₅	#1744	semidirect
196. C₅	#69750	semidirect
206. C₃	#10600543	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
C₄²×Dic₁₅	2
C₂×C₄×C₃⋊#168	2
#11746371	2
#10600543	3
#69750	5