Subgroups — #14307347

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#11751036	C₂	1	1	yes	maximal, characteristic
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic
4	#11746350	C₂	1	4	yes	maximal
5	#10602534	—	3	1	no	maximal, Hall
6	C₃×C₅×#125	—	2	2	no	—
7	C₃×C₅×#125	C₄	1	4	yes	complemented
8	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₅	—	2	2	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
13	C₂²×C₃⋊#168	C₂²	1	2	yes	—
14	#91087	—	2	2	no	—
15	#71741	—	5	1	no	maximal, Hall
16	C₂×C₅×#125	D₃	1	1	yes	characteristic
17	#65224	—	3	1	no	—
18	C₂×#3935	—	3	1	no	—
19	#64749	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₅	—	2	2	no	—
24	C₃×C₅×C₄⋊C₄	—	2	8	no	—
25	C₃×C₅×C₄⋊C₄	—	2	2	no	—
26	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₄⋊Dic₁₅	—	2	4	no	—
30	C₄⋊Dic₁₅	—	4	4	no	—
31	C₂×C₃⋊#168	—	2	2	no	—
32	C₂×C₃⋊#168	—	2	2	no	—
33	C₂×C₃⋊#168	Q₈	1	4	yes	—
34	C₂×C₃×#125	D₅	1	1	yes	characteristic
35	#4935	—	5	1	no	—
36	C₂×#803	—	5	1	no	—
37	#4529	—	5	4	no	—
38	C₅×#125	—	2	2	no	—
39	C₅×#125	Dic₃	1	4	yes	complemented
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₅	—	3	2	no	—
44	C₂×C₄⋊Dic₅	—	6	2	no	—
45	C₂²×#168	—	3	2	no	—
46	#3935	—	6	2	no	—
47	#1992	—	15	1	no	Sylow
48	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	4	no	—
52	C₂×C₃×C₄×C₅	—	4	2	no	—
53	C₂×C₃×C₄×C₅	—	4	2	no	—
54	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	#41	1	2	yes	—
59	C₂×Dic₁₅	—	4	2	no	—
60	C₂×Dic₁₅	—	4	4	no	—
61	C₃⋊#168	—	2	4	no	—
62	C₃×#125	—	2	2	no	—
63	C₃×#125	Dic₅	1	4	yes	complemented
64	C₂×C₃×C₄⋊C₄	—	2	2	no	—
65	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
66	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
67	C₂×C₄⋊Dic₃	—	5	2	no	—
68	C₂×C₄⋊Dic₃	—	10	2	no	—
69	C₂²×C₃⋊C₈	—	5	2	no	—
70	#803	—	10	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	#67	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
74	C₂²×Dic₅	—	6	2	no	—
75	C₅×C₄⋊C₄	—	2	8	no	—
76	C₅×C₄⋊C₄	—	2	2	no	—
77	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
78	C₄²×C₅	C₂×Dic₃	1	2	yes	—
79	C₄²×C₅	#67	1	2	yes	—
80	C₄⋊Dic₅	—	6	4	no	—
81	C₄⋊Dic₅	—	12	4	no	—
82	C₂×#168	—	3	4	no	—
83	C₂×#168	—	6	2	no	—
84	C₂×#168	—	6	2	no	—
85	C₂×#125	D₁₅	1	1	yes	characteristic
86	C₂×#97	—	15	1	no	—
87	#384	—	15	1	no	—
88	#340	—	15	4	no	—
89	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₃×C₅	#124	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
92	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
93	C₂²×C₃×C₅	#137	1	2	yes	—
94	C₃×C₄×C₅	—	2	4	no	—
95	C₃×C₄×C₅	—	4	4	no	—
96	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
97	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
98	Dic₁₅	—	8	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
101	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
102	C₂²×Dic₃	—	10	2	no	—
103	C₃×C₄⋊C₄	—	2	8	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
106	C₃×C₄²	C₂×Dic₅	1	2	yes	—
107	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
108	C₄⋊Dic₃	—	10	4	no	—
109	C₄⋊Dic₃	—	20	4	no	—
110	C₂×C₃⋊C₈	—	5	4	no	—
111	C₂×C₃⋊C₈	—	10	2	no	—
112	C₂×C₃⋊C₈	—	10	2	no	—
113	C₂³×C₅	C₂×#67	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	#217	1	4	yes	—
120	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
121	C₂×C₄×C₅	#239	1	2	yes	—
122	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
124	C₂×Dic₅	—	12	2	no	—
125	C₂×Dic₅	—	12	4	no	—
126	#168	—	6	4	no	—
127	C₂²×C₈	—	15	2	no	—
128	#125	—	2	2	no	—
129	#125	Dic₁₅	1	4	yes	complemented
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	15	2	no	—
132	C₂×C₄⋊C₄	—	30	2	no	—
133	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
134	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
135	#97	—	30	2	no	—
136	C₂×C₃×C₅	#391	1	1	yes	characteristic
137	C₂×C₃×C₅	#417	1	2	yes	—
138	C₂×C₃×C₅	#490	1	1	yes	characteristic
139	C₂×C₃×C₅	#491	1	1	yes	characteristic
140	C₂×C₃×C₅	#501	1	2	yes	—
141	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
148	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
149	C₂×C₃×C₄	#713	1	2	yes	—
150	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
152	C₂×Dic₃	—	20	2	no	—
153	C₂×Dic₃	—	20	4	no	—
154	C₃⋊C₈	—	10	4	no	—
155	C₂²×C₅	#944	1	1	yes	characteristic
156	C₂²×C₅	#947	1	1	yes	characteristic
157	C₂²×C₅	#949	1	2	yes	—
158	C₂²×C₅	#951	1	2	yes	—
159	C₂²×C₅	C₂×#217	1	1	yes	characteristic
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	—	4	4	no	—
162	C₄×C₅	Q₈×Dic₃	1	2	yes	—
163	C₄×C₅	#946	1	2	yes	—
164	Dic₅	—	24	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	30	2	no	—
167	C₂²×C₄	#1220	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
169	C₂×C₈	—	15	4	no	—
170	C₂×C₈	—	30	2	no	—
171	C₂×C₈	—	30	2	no	—
172	C₄⋊C₄	—	2	8	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	30	4	no	—
175	C₄⋊C₄	—	60	4	no	—
176	C₄⋊C₄	C₂×Dic₁₅	1	4	yes	—
177	C₄²	C₂×Dic₁₅	1	2	yes	—
178	C₄²	#1220	1	2	yes	—
179	C₃×C₅	#1992	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	#4090	1	1	yes	characteristic
181	C₂²×C₃	#4093	1	1	yes	characteristic
182	C₂²×C₃	#4095	1	2	yes	—
183	C₂²×C₃	#4097	1	2	yes	—
184	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	Q₈×Dic₅	1	2	yes	—
188	C₃×C₄	#4092	1	2	yes	—
189	Dic₃	—	40	2	no	—
190	C₂×C₅	#5064	1	2	yes	—
191	C₂×C₅	#5067	1	1	yes	characteristic
192	C₂×C₅	#5069	1	1	yes	characteristic
193	C₂×C₅	#5229	1	1	yes	characteristic
194	C₂×C₅	#5232	1	2	yes	—
195	C₂³	C₂×#1220	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	60	2	no	—
202	C₂×C₄	—	60	4	no	—
203	C₂×C₄	#6719	1	4	yes	—
204	C₂×C₄	Q₈×D₁₅	1	2	yes	—
205	C₂×C₄	#6821	1	2	yes	—
206	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
207	C₂×C₄	C₂×#1220	1	1	yes	characteristic
208	C₈	—	30	4	no	—
209	C₂×C₃	#65354	1	2	yes	—
210	C₂×C₃	#65356	1	1	yes	characteristic
211	C₂×C₃	#65358	1	1	yes	characteristic
212	C₂×C₃	#65519	1	1	yes	characteristic
213	C₂×C₃	#65521	1	2	yes	—
214	C₅	#71741	1	1	yes	characteristic, complemented, Sylow
215	C₂²	#91833	1	1	yes	characteristic
216	C₂²	#91836	1	1	yes	characteristic
217	C₂²	#91838	1	2	yes	—
218	C₂²	#91840	1	2	yes	—
219	C₂²	C₂×#6719	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	4	4	no	—
222	C₄	—	120	2	no	—
223	C₄	Q₈×Dic₁₅	1	2	yes	—
224	C₄	#91835	1	2	yes	—
225	C₃	#10602534	1	1	yes	characteristic, complemented, Sylow
226	C₂	#11751165	1	2	yes	—
227	C₂	#11751168	1	1	yes	characteristic
228	C₂	#11751170	1	1	yes	characteristic
229	C₂	#11751330	1	1	yes	characteristic
230	C₂	#11751333	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#125	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#125	D₁₅
hypocenter	C₃×C₅	#1992
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
7. C₃×C₅×#125	C₄	semidirect
39. C₅×#125	Dic₃	semidirect
63. C₃×#125	Dic₅	semidirect
129. #125	Dic₁₅	semidirect
179. C₃×C₅	#1992	semidirect
214. C₅	#71741	semidirect
225. C₃	#10602534	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×#125	2
#11751036	2
C₂×#91087	2
#11746350	2
#10602534	3
#71741	5