Subgroups — #14305977

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
2	#11751037	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90967	C₂	1	4	yes	maximal, complemented
5	#10602558	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	2	no	—
7	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
10	C₅×#874	—	2	2	no	—
11	C₅×#874	C₄	1	4	yes	complemented
12	C₂²×C₃×#168	C₂²	1	2	yes	—
13	#91115	—	2	2	no	—
14	C₃×#3935	—	2	2	no	—
15	#90967	—	2	8	no	—
16	#69823	—	5	1	no	maximal, Hall
17	C₂×C₅×#127	—	3	1	no	—
18	C₂×#3964	—	3	4	no	—
19	#65225	—	3	1	no	—
20	C₂×#3935	D₃	1	1	yes	characteristic, complemented
21	C₂³×C₅×D₃	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₅	—	2	2	no	—
25	C₂×C₅×D₁₂	—	2	4	no	—
26	C₂×C₅×D₁₂	—	2	4	no	—
27	C₂×C₅×D₁₂	—	2	8	no	—
28	C₂×C₅×D₁₂	—	2	2	no	—
29	C₂×C₅×D₁₂	—	4	8	no	—
30	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
31	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₄⋊Dic₁₅	—	2	4	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	—	2	2	no	—
36	C₂×C₃×#168	D₄	1	4	yes	complemented
37	C₂×#874	D₅	1	1	yes	characteristic
38	C₂×C₃×#97	—	5	1	no	—
39	C₂×#821	—	5	4	no	—
40	#4936	—	5	1	no	—
41	C₂²×C₅×D₄	—	3	2	no	—
42	C₂²×C₅×D₄	—	6	2	no	—
43	C₅×#127	—	3	4	no	—
44	C₅×#127	—	6	2	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₅	—	3	2	no	—
47	C₂²×#168	C₂×D₃	1	2	yes	—
48	#3964	—	6	8	no	—
49	#3963	—	6	2	no	—
50	#3935	—	2	2	no	—
51	#1961	—	15	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₅×D₃	—	2	4	no	—
54	C₂²×C₅×D₃	—	4	2	no	—
55	C₂²×C₅×D₃	—	4	2	no	—
56	C₂²×C₅×D₃	—	4	8	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×Dic₁₅	—	4	2	no	—
64	C₂×Dic₁₅	—	4	4	no	—
65	C₅×D₁₂	—	2	8	no	—
66	C₅×D₁₂	—	4	16	no	—
67	C₅×D₁₂	—	4	4	no	—
68	C₃×#168	—	2	4	no	—
69	C₂²×D₁₂	—	2	2	no	—
70	C₂²×D₁₂	C₂×D₅	1	2	yes	—
71	C₂²×C₃×C₈	—	5	2	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₂×C₄⋊Dic₃	—	5	2	no	—
74	#874	—	2	2	no	—
75	#874	Dic₅	1	4	yes	complemented
76	C₃×#97	—	10	2	no	—
77	C₃⋊#95	—	10	2	no	—
78	#821	—	10	8	no	—
79	C₂⁴×C₅	—	6	2	no	—
80	C₂×C₅×D₄	—	3	4	no	—
81	C₂×C₅×D₄	—	6	8	no	—
82	C₂×C₅×D₄	—	6	4	no	—
83	C₂×C₅×D₄	—	6	4	no	—
84	C₂×C₅×D₄	—	6	2	no	—
85	C₂×C₅×D₄	—	12	8	no	—
86	C₂²×C₄×C₅	D₁₂	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂²×Dic₅	—	6	2	no	—
89	C₄²×C₅	C₄×D₃	1	2	yes	—
90	C₄²×C₅	#67	1	2	yes	—
91	C₄⋊Dic₅	—	6	4	no	—
92	C₂×#168	—	2	2	no	—
93	C₂×#168	—	2	2	no	—
94	C₂×#168	D₁₂	1	4	yes	complemented
95	C₂×#127	—	3	1	no	—
96	C₂×#102	—	15	4	no	—
97	C₂×#97	—	5	1	no	—
98	#382	—	15	1	no	—
99	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	#124	1	1	yes	characteristic
101	C₂²×C₃×C₅	#127	1	1	yes	characteristic
102	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
103	C₂×C₅×D₃	—	4	4	no	—
104	C₂×C₅×D₃	—	4	8	no	—
105	C₂×C₅×D₃	—	8	4	no	—
106	C₂×C₅×D₃	—	8	4	no	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₃×C₄×C₅	#121	1	2	yes	—
110	Dic₁₅	—	8	2	no	—
111	C₂³×D₃	—	2	2	no	—
112	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
113	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂²×Dic₃	—	10	2	no	—
115	C₂×D₁₂	—	2	4	no	—
116	C₂×D₁₂	—	2	4	no	—
117	C₂×D₁₂	—	2	8	no	—
118	C₂×D₁₂	—	2	2	no	—
119	C₂×D₁₂	—	4	8	no	—
120	C₂×D₁₂	C₂×Dic₅	1	4	yes	—
121	C₂×C₃×C₈	—	5	4	no	—
122	C₂×C₃×C₈	—	10	2	no	—
123	C₂×C₃×C₈	—	10	2	no	—
124	C₃×C₄²	C₂×Dic₅	1	2	yes	—
125	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
126	C₄⋊Dic₃	—	10	4	no	—
127	C₂³×C₅	—	6	4	no	—
128	C₂³×C₅	—	12	2	no	—
129	C₂³×C₅	—	12	2	no	—
130	C₂³×C₅	—	12	8	no	—
131	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
132	C₅×D₄	—	6	8	no	—
133	C₅×D₄	—	12	16	no	—
134	C₅×D₄	—	12	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
137	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
139	C₂×C₄×C₅	#235	1	2	yes	—
140	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
141	C₂×Dic₅	—	12	2	no	—
142	C₂×Dic₅	—	12	4	no	—
143	#168	—	2	4	no	—
144	C₂²×D₄	—	3	2	no	—
145	C₂²×D₄	—	6	2	no	—
146	C₂²×C₈	—	5	2	no	—
147	#127	—	3	4	no	—
148	#127	—	6	2	no	—
149	C₂×C₄⋊C₄	—	15	2	no	—
150	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
151	#102	—	30	8	no	—
152	#97	—	10	2	no	—
153	#95	—	30	2	no	—
154	C₂×C₃×C₅	#390	1	1	yes	characteristic
155	C₂×C₃×C₅	#418	1	2	yes	—
156	C₂×C₃×C₅	#490	1	2	yes	—
157	C₂×C₃×C₅	#496	1	2	yes	—
158	C₅×D₃	—	8	4	no	—
159	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
160	C₂²×D₃	—	2	4	no	—
161	C₂²×D₃	—	4	8	no	—
162	C₂²×D₃	—	4	2	no	—
163	C₂²×D₃	—	4	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
166	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
167	C₂×C₃×C₄	#711	1	2	yes	—
168	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
170	C₂×Dic₃	—	20	4	no	—
171	C₂×Dic₃	—	20	2	no	—
172	D₁₂	—	2	8	no	—
173	D₁₂	—	4	16	no	—
174	D₁₂	—	4	4	no	—
175	C₃×C₈	—	10	4	no	—
176	C₂²×C₅	—	12	8	no	—
177	C₂²×C₅	—	12	4	no	—
178	C₂²×C₅	—	24	4	no	—
179	C₂²×C₅	—	24	4	no	—
180	C₂²×C₅	#874	1	1	yes	characteristic
181	C₂²×C₅	#875	1	1	yes	characteristic
182	C₂²×C₅	#908	1	4	yes	—
183	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	C₄×D₁₂	1	2	yes	—
186	C₄×C₅	#930	1	2	yes	—
187	Dic₅	—	24	2	no	—
188	C₂⁴	—	6	2	no	—
189	C₂×D₄	—	3	4	no	—
190	C₂×D₄	—	6	8	no	—
191	C₂×D₄	—	6	4	no	—
192	C₂×D₄	—	6	4	no	—
193	C₂×D₄	—	6	2	no	—
194	C₂×D₄	—	12	8	no	—
195	C₂²×C₄	—	30	2	no	—
196	C₂²×C₄	#1203	1	2	yes	—
197	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
198	C₂×C₈	—	5	4	no	—
199	C₂×C₈	—	10	2	no	—
200	C₂×C₈	—	10	2	no	—
201	C₄⋊C₄	—	30	4	no	—
202	C₄²	D₃×Dic₅	1	2	yes	—
203	C₄²	#1201	1	2	yes	—
204	C₃×C₅	#1961	1	1	yes	characteristic, complemented, Hall
205	C₂²×C₃	#4078	1	4	yes	—
206	C₂²×C₃	#4082	1	1	yes	characteristic
207	C₂²×C₃	#4086	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
209	C₂×D₃	—	4	8	no	—
210	C₂×D₃	—	4	4	no	—
211	C₂×D₃	—	8	4	no	—
212	C₂×D₃	—	8	4	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	D₄×Dic₅	1	2	yes	—
215	C₃×C₄	#4084	1	2	yes	—
216	Dic₃	—	40	2	no	—
217	C₂×C₅	—	24	4	no	—
218	C₂×C₅	#4712	1	2	yes	—
219	C₂×C₅	#4713	1	2	yes	—
220	C₂×C₅	#4939	1	1	yes	characteristic
221	C₂×C₅	#5131	1	2	yes	—
222	C₂³	—	6	4	no	—
223	C₂³	—	12	2	no	—
224	C₂³	—	12	8	no	—
225	C₂³	—	12	2	no	—
226	C₂³	C₂×#1203	1	1	yes	characteristic
227	D₄	—	6	8	no	—
228	D₄	—	12	16	no	—
229	D₄	—	12	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	60	2	no	—
232	C₂×C₄	—	60	4	no	—
233	C₂×C₄	(C₂×D₃)⋊Dic₅	1	4	yes	—
234	C₂×C₄	#6772	1	2	yes	—
235	C₂×C₄	D₅×D₁₂	1	2	yes	—
236	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
237	C₂×C₄	C₂×#1201	1	1	yes	characteristic
238	C₈	—	10	4	no	—
239	C₂×C₃	#65360	1	2	yes	—
240	C₂×C₃	#65363	1	2	yes	—
241	C₂×C₃	#65504	1	2	yes	—
242	C₂×C₃	#65510	1	1	yes	characteristic
243	D₃	—	8	4	no	—
244	C₅	#69823	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	12	8	no	—
246	C₂²	—	12	4	no	—
247	C₂²	—	24	4	no	—
248	C₂²	—	24	4	no	—
249	C₂²	#91304	1	4	yes	—
250	C₂²	#91373	1	1	yes	characteristic
251	C₂²	#91451	1	1	yes	characteristic
252	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	120	2	no	—
255	C₄	Dic₅×D₁₂	1	2	yes	—
256	C₄	#91449	1	2	yes	—
257	C₃	#10602558	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	24	4	no	—
259	C₂	#11746908	1	2	yes	—
260	C₂	#11747031	1	2	yes	—
261	C₂	#11748389	1	2	yes	—
262	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	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#874	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1961
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
2. #11751037	C₂	semidirect
3. C₂×C₃×#3935	C₂	semidirect
4. C₂×#90967	C₂	semidirect
11. C₅×#874	C₄	semidirect
20. C₂×#3935	D₃	semidirect
36. C₂×C₃×#168	D₄	semidirect
75. #874	Dic₅	semidirect
94. C₂×#168	D₁₂	semidirect
204. C₃×C₅	#1961	semidirect
244. C₅	#69823	semidirect
257. C₃	#10602558	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#874	2
#11751037	2
C₂×C₃×#3935	2
C₂×#90967	2
#10602558	3
#69823	5