Subgroups — #14305791

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic
2	#11751036	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×#90973	C₂	1	4	yes	maximal
5	#10602578	—	3	1	no	maximal, Hall
6	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	—	2	2	no	—
9	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
11	C₅×#869	—	2	2	no	—
12	C₅×#869	C₄	1	4	yes	complemented
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	C₃×C₄×#168	—	2	2	no	—
15	#90973	—	2	8	no	—
16	#69771	—	5	1	no	maximal, Hall
17	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
18	C₂×#3967	—	3	4	no	—
19	#65224	—	3	1	no	—
20	C₂×C₄×#168	D₃	1	1	yes	characteristic
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	C₂²×C₅×Dic₃	—	2	2	no	—
25	C₂×C₅×Dic₆	—	2	4	no	—
26	C₂×C₅×Dic₆	—	2	4	no	—
27	C₂×C₅×Dic₆	—	2	2	no	—
28	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₄⋊Dic₁₅	—	2	4	no	—
32	C₄⋊Dic₁₅	—	4	4	no	—
33	C₅×C₄⋊Dic₃	—	2	8	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	—	2	2	no	—
36	C₂×C₃×#168	D₄	1	4	yes	—
37	C₂×#869	D₅	1	1	yes	characteristic
38	C₂×C₃×C₄×C₈	—	5	1	no	—
39	C₂×#816	—	5	4	no	—
40	#4935	—	5	1	no	—
41	C₂²×C₅×Q₈	—	3	2	no	—
42	C₅×C₄⋊Q₈	—	3	4	no	—
43	C₅×C₄⋊Q₈	—	6	2	no	—
44	C₂×C₅×C₄⋊C₄	—	6	2	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₅	—	3	2	no	—
47	C₂×C₄⋊Dic₅	—	6	2	no	—
48	C₂²×#168	C₂×D₃	1	2	yes	—
49	#3967	—	6	8	no	—
50	C₄×#168	—	2	2	no	—
51	#1956	—	15	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
55	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	2	yes	—
60	C₂×Dic₁₅	—	4	2	no	—
61	C₂×Dic₁₅	—	4	4	no	—
62	C₂×C₅×Dic₃	—	2	4	no	—
63	C₂×C₅×Dic₃	—	4	2	no	—
64	C₂×C₅×Dic₃	—	4	2	no	—
65	C₅×Dic₆	—	2	8	no	—
66	C₅×Dic₆	—	4	4	no	—
67	C₃×#168	—	2	4	no	—
68	C₂²×Dic₆	C₂×D₅	1	2	yes	—
69	C₂²×C₃×C₈	—	5	2	no	—
70	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
71	C₂×C₄⋊Dic₃	—	2	2	no	—
72	C₂×C₄⋊Dic₃	—	5	2	no	—
73	C₂×C₄⋊Dic₃	—	10	2	no	—
74	#869	—	2	2	no	—
75	#869	Dic₅	1	4	yes	complemented
76	C₃×C₄×C₈	—	10	2	no	—
77	#816	—	10	8	no	—
78	C₂×C₅×Q₈	—	3	4	no	—
79	C₂×C₅×Q₈	—	6	2	no	—
80	C₂×C₅×Q₈	—	6	4	no	—
81	C₂×C₅×Q₈	—	6	4	no	—
82	C₂²×C₄×C₅	—	6	2	no	—
83	C₂²×C₄×C₅	D₁₂	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂²×Dic₅	—	6	2	no	—
86	C₅×C₄⋊C₄	—	6	8	no	—
87	C₄²×C₅	C₄×D₃	1	2	yes	—
88	C₄²×C₅	#67	1	2	yes	—
89	C₄⋊Dic₅	—	6	4	no	—
90	C₄⋊Dic₅	—	12	4	no	—
91	C₂×#168	—	2	2	no	—
92	C₂×#168	—	2	2	no	—
93	C₂×#168	D₁₂	1	4	yes	—
94	C₂×C₄⋊Q₈	—	3	1	no	—
95	C₂×#103	—	15	4	no	—
96	C₂×C₄×C₈	—	5	1	no	—
97	#384	—	15	1	no	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	#124	1	1	yes	characteristic
100	C₂²×C₃×C₅	#127	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
102	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
103	C₃×C₄×C₅	#103	1	8	yes	complemented
104	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
105	C₃×C₄×C₅	#121	1	2	yes	—
106	Dic₁₅	—	8	2	no	—
107	C₅×Dic₃	—	4	4	no	—
108	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
109	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
110	C₂²×Dic₃	—	2	2	no	—
111	C₂²×Dic₃	—	10	2	no	—
112	C₂×Dic₆	—	2	4	no	—
113	C₂×Dic₆	—	2	4	no	—
114	C₂×Dic₆	—	2	2	no	—
115	C₂×Dic₆	C₂×Dic₅	1	4	yes	—
116	C₂×C₃×C₈	—	5	4	no	—
117	C₂×C₃×C₈	—	10	2	no	—
118	C₂×C₃×C₈	—	10	2	no	—
119	C₃×C₄²	C₂×Dic₅	1	2	yes	—
120	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
121	C₄⋊Dic₃	—	2	8	no	—
122	C₄⋊Dic₃	—	10	4	no	—
123	C₄⋊Dic₃	—	20	4	no	—
124	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
125	C₅×Q₈	—	6	8	no	—
126	C₅×Q₈	—	12	4	no	—
127	C₂×C₄×C₅	—	6	4	no	—
128	C₂×C₄×C₅	—	12	2	no	—
129	C₂×C₄×C₅	—	12	2	no	—
130	C₂×C₄×C₅	#204	1	4	yes	—
131	C₂×C₄×C₅	Dic₁₂	1	4	yes	—
132	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
133	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
135	C₂×C₄×C₅	#235	1	2	yes	—
136	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
137	C₂×Dic₅	—	12	2	no	—
138	C₂×Dic₅	—	12	4	no	—
139	#168	—	2	4	no	—
140	C₂²×Q₈	—	3	2	no	—
141	C₂²×C₈	—	5	2	no	—
142	C₄⋊Q₈	—	3	4	no	—
143	C₄⋊Q₈	—	6	2	no	—
144	C₂×C₄⋊C₄	—	6	2	no	—
145	C₂×C₄⋊C₄	—	15	2	no	—
146	C₂×C₄⋊C₄	—	30	2	no	—
147	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
148	#103	—	30	8	no	—
149	C₄×C₈	—	10	2	no	—
150	C₂×C₃×C₅	#390	1	1	yes	characteristic
151	C₂×C₃×C₅	C₂×#103	1	2	yes	—
152	C₂×C₃×C₅	#487	1	2	yes	—
153	C₂×C₃×C₅	#492	1	1	yes	characteristic
154	C₂×C₃×C₅	#494	1	1	yes	characteristic
155	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
156	C₂×C₃×C₄	#688	1	4	yes	—
157	C₂×C₃×C₄	#689	1	4	yes	—
158	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
159	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
160	C₂×C₃×C₄	#711	1	2	yes	—
161	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
163	C₂×Dic₃	—	2	4	no	—
164	C₂×Dic₃	—	4	2	no	—
165	C₂×Dic₃	—	4	2	no	—
166	C₂×Dic₃	—	20	2	no	—
167	C₂×Dic₃	—	20	4	no	—
168	Dic₆	—	2	8	no	—
169	Dic₆	—	4	4	no	—
170	C₃×C₈	—	10	4	no	—
171	C₂²×C₅	#874	1	1	yes	characteristic
172	C₂²×C₅	#875	1	1	yes	characteristic
173	C₂²×C₅	C₂×#204	1	2	yes	—
174	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
175	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₄×C₅	—	12	4	no	—
177	C₄×C₅	#816	1	8	yes	complemented
178	C₄×C₅	C₄×D₁₂	1	2	yes	—
179	C₄×C₅	#930	1	2	yes	—
180	Dic₅	—	24	2	no	—
181	C₂×Q₈	—	3	4	no	—
182	C₂×Q₈	—	6	2	no	—
183	C₂×Q₈	—	6	4	no	—
184	C₂×Q₈	—	6	4	no	—
185	C₂²×C₄	—	6	2	no	—
186	C₂²×C₄	—	30	2	no	—
187	C₂²×C₄	#1203	1	2	yes	—
188	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
189	C₂×C₈	—	5	4	no	—
190	C₂×C₈	—	10	2	no	—
191	C₂×C₈	—	10	2	no	—
192	C₄⋊C₄	—	6	8	no	—
193	C₄⋊C₄	—	30	4	no	—
194	C₄⋊C₄	—	60	4	no	—
195	C₄²	D₃×Dic₅	1	2	yes	—
196	C₄²	#1201	1	2	yes	—
197	C₃×C₅	#1956	1	1	yes	characteristic, complemented, Hall
198	C₂²×C₃	#4082	1	1	yes	characteristic
199	C₂²×C₃	#4086	1	1	yes	characteristic
200	C₂²×C₃	C₂×#688	1	2	yes	—
201	C₂²×C₃	C₂×#689	1	2	yes	—
202	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
203	C₃×C₄	#3967	1	8	yes	complemented
204	C₃×C₄	D₄×Dic₅	1	2	yes	—
205	C₃×C₄	#4084	1	2	yes	—
206	Dic₃	—	4	4	no	—
207	Dic₃	—	40	2	no	—
208	C₂×C₅	#4681	1	2	yes	—
209	C₂×C₅	#4693	1	1	yes	characteristic
210	C₂×C₅	#4699	1	1	yes	characteristic
211	C₂×C₅	#4939	1	1	yes	characteristic
212	C₂×C₅	C₂×#816	1	2	yes	—
213	C₂³	C₂×#1203	1	1	yes	characteristic
214	Q₈	—	6	8	no	—
215	Q₈	—	12	4	no	—
216	C₂×C₄	—	6	4	no	—
217	C₂×C₄	—	12	2	no	—
218	C₂×C₄	—	12	2	no	—
219	C₂×C₄	—	60	2	no	—
220	C₂×C₄	—	60	4	no	—
221	C₂×C₄	#6660	1	4	yes	—
222	C₂×C₄	#6663	1	4	yes	—
223	C₂×C₄	(C₂×D₃)⋊Dic₅	1	4	yes	—
224	C₂×C₄	#6772	1	2	yes	—
225	C₂×C₄	D₅×D₁₂	1	2	yes	—
226	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
227	C₂×C₄	C₂×#1201	1	1	yes	characteristic
228	C₈	—	10	4	no	—
229	C₂×C₃	#65370	1	2	yes	—
230	C₂×C₃	#65374	1	1	yes	characteristic
231	C₂×C₃	#65381	1	1	yes	characteristic
232	C₂×C₃	#65510	1	1	yes	characteristic
233	C₂×C₃	C₂×#3967	1	2	yes	—
234	C₅	#69771	1	1	yes	characteristic, complemented, Sylow
235	C₂²	C₂×#6660	1	2	yes	—
236	C₂²	C₂×#6663	1	2	yes	—
237	C₂²	#91373	1	1	yes	characteristic
238	C₂²	#91451	1	1	yes	characteristic
239	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
240	C₄	—	12	4	no	—
241	C₄	—	120	2	no	—
242	C₄	#90973	1	8	yes	complemented
243	C₄	Dic₅×D₁₂	1	2	yes	—
244	C₄	#91449	1	2	yes	—
245	C₃	#10602578	1	1	yes	characteristic, complemented, Sylow
246	C₂	#11746929	1	2	yes	—
247	C₂	#11747067	1	1	yes	characteristic
248	C₂	#11747096	1	1	yes	characteristic
249	C₂	C₂×#90973	1	2	yes	—
250	C₂	#11748764	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1203
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1956
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
12. C₅×#869	C₄	semidirect
75. #869	Dic₅	semidirect
103. C₃×C₄×C₅	#103	semidirect
177. C₄×C₅	#816	semidirect
197. C₃×C₅	#1956	semidirect
203. C₃×C₄	#3967	semidirect
234. C₅	#69771	semidirect
242. C₄	#90973	semidirect
245. C₃	#10602578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#869	2
#11751036	2
C₂×C₃×C₄×#168	2
C₂×#90973	2
#10602578	3
#69771	5