Subgroups — #14307393

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#11751037	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
4	#11745981	C₂	1	4	yes	maximal
5	#10602581	—	3	1	no	maximal, Hall
6	C₂²×C₅×Dic₆	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	—	2	1	no	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₅×#869	—	2	2	no	—
12	C₅×#869	C₄	1	4	yes	complemented
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	#91115	—	2	1	no	—
15	#91115	—	2	1	no	—
16	C₃×#3935	—	2	2	no	—
17	#69828	—	5	1	no	maximal, Hall
18	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
19	#65225	—	3	1	no	—
20	C₂×#3935	D₃	1	1	yes	characteristic
21	#64749	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₅	—	2	2	no	—
26	C₂²×C₅×Dic₃	—	2	2	no	—
27	C₂×C₅×Dic₆	—	2	4	no	—
28	C₂×C₅×Dic₆	—	4	4	no	—
29	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₄⋊Dic₁₅	—	2	4	no	—
32	C₅×C₄⋊Dic₃	—	2	4	no	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
35	C₂×C₃×#168	—	2	2	no	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	Q₈	1	4	yes	complemented
38	C₂×#869	D₅	1	1	yes	characteristic
39	C₂×C₃×#97	—	5	1	no	—
40	#4936	—	5	1	no	—
41	#4551	—	5	4	no	—
42	C₂²×C₅×Q₈	—	6	1	no	—
43	C₅×C₄⋊Q₈	—	3	4	no	—
44	C₅×C₄⋊Q₈	—	6	2	no	—
45	C₂×C₅×C₄⋊C₄	—	3	2	no	—
46	C₂×C₅×C₄⋊C₄	—	6	1	no	—
47	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂×C₄⋊Dic₅	—	3	2	no	—
49	C₂²×#168	C₂×D₃	1	2	yes	—
50	#3963	—	6	1	no	—
51	#3963	—	6	1	no	—
52	#3935	—	2	2	no	—
53	#1993	—	15	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×Dic₁₅	—	4	2	no	—
64	C₂×Dic₁₅	—	4	2	no	—
65	C₂×Dic₁₅	—	4	2	no	—
66	C₂×C₅×Dic₃	—	2	4	no	—
67	C₂×C₅×Dic₃	—	4	4	no	—
68	C₅×Dic₆	—	4	8	no	—
69	C₃×#168	—	2	4	no	—
70	C₂²×Dic₆	—	2	1	no	—
71	C₂²×C₃×C₈	—	5	2	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₂×C₄⋊Dic₃	—	2	1	no	—
74	C₂×C₄⋊Dic₃	—	5	2	no	—
75	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
76	#869	—	2	2	no	—
77	#869	Dic₅	1	4	yes	complemented
78	C₃×#97	—	10	2	no	—
79	C₃⋊#95	—	10	1	no	—
80	C₃⋊#95	—	10	1	no	—
81	C₂×C₅×Q₈	—	6	4	no	—
82	C₂×C₅×Q₈	—	12	4	no	—
83	C₂²×C₄×C₅	—	6	2	no	—
84	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
85	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂²×Dic₅	—	6	2	no	—
88	C₅×C₄⋊C₄	—	3	4	no	—
89	C₅×C₄⋊C₄	—	6	4	no	—
90	C₅×C₄⋊C₄	—	6	2	no	—
91	C₄²×C₅	C₄×D₃	1	2	yes	—
92	C₄²×C₅	#67	1	2	yes	—
93	C₄⋊Dic₅	—	6	4	no	—
94	C₂×#168	—	2	2	no	—
95	C₂×#168	—	2	2	no	—
96	C₂×#168	Dic₆	1	4	yes	complemented
97	C₂×C₄⋊Q₈	—	3	1	no	—
98	C₂×#97	—	5	1	no	—
99	#382	—	15	1	no	—
100	#340	—	15	4	no	—
101	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	#124	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
105	C₂²×C₃×C₅	#137	1	2	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
109	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
110	Dic₁₅	—	8	2	no	—
111	C₅×Dic₃	—	4	4	no	—
112	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂²×Dic₃	—	2	2	no	—
116	C₂²×Dic₃	—	10	2	no	—
117	C₂×Dic₆	—	2	4	no	—
118	C₂×Dic₆	—	4	4	no	—
119	C₂×C₃×C₈	—	5	4	no	—
120	C₂×C₃×C₈	—	10	2	no	—
121	C₂×C₃×C₈	—	10	2	no	—
122	C₃×C₄²	C₂×Dic₅	1	2	yes	—
123	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
124	C₄⋊Dic₃	—	2	4	no	—
125	C₄⋊Dic₃	—	2	2	no	—
126	C₄⋊Dic₃	—	10	4	no	—
127	C₄⋊Dic₃	C₂×Dic₅	1	4	yes	—
128	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
129	C₅×Q₈	—	12	8	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	6	4	no	—
133	C₂×C₄×C₅	—	12	4	no	—
134	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₄×C₅	#235	1	2	yes	—
139	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
140	C₂×Dic₅	—	12	2	no	—
141	C₂×Dic₅	—	12	2	no	—
142	C₂×Dic₅	—	12	2	no	—
143	#168	—	2	4	no	—
144	C₂²×Q₈	—	6	1	no	—
145	C₂²×C₈	—	5	2	no	—
146	C₄⋊Q₈	—	3	4	no	—
147	C₄⋊Q₈	—	6	2	no	—
148	C₂×C₄⋊C₄	—	3	2	no	—
149	C₂×C₄⋊C₄	—	6	1	no	—
150	C₂×C₄⋊C₄	—	15	2	no	—
151	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
152	#97	—	10	2	no	—
153	#95	—	30	1	no	—
154	#95	—	30	1	no	—
155	C₂×C₃×C₅	#391	1	1	yes	characteristic
156	C₂×C₃×C₅	#418	1	1	yes	characteristic
157	C₂×C₃×C₅	#419	1	1	yes	characteristic
158	C₂×C₃×C₅	#489	1	2	yes	—
159	C₂×C₃×C₅	#501	1	2	yes	—
160	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
164	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
165	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
166	C₂×C₃×C₄	#713	1	2	yes	—
167	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
169	C₂×Dic₃	—	2	4	no	—
170	C₂×Dic₃	—	4	4	no	—
171	C₂×Dic₃	—	20	2	no	—
172	C₂×Dic₃	—	20	2	no	—
173	C₂×Dic₃	—	20	2	no	—
174	Dic₆	—	4	8	no	—
175	C₃×C₈	—	10	4	no	—
176	C₂²×C₅	#869	1	1	yes	characteristic
177	C₂²×C₅	#875	1	1	yes	characteristic
178	C₂²×C₅	#908	1	2	yes	—
179	C₂²×C₅	#909	1	2	yes	—
180	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	12	4	no	—
184	C₄×C₅	C₄×Dic₆	1	2	yes	—
185	C₄×C₅	#924	1	2	yes	—
186	Dic₅	—	24	2	no	—
187	C₂×Q₈	—	6	4	no	—
188	C₂×Q₈	—	12	4	no	—
189	C₂²×C₄	—	6	2	no	—
190	C₂²×C₄	—	30	2	no	—
191	C₂²×C₄	#1203	1	1	yes	characteristic
192	C₂²×C₄	#1203	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
194	C₂×C₈	—	5	4	no	—
195	C₂×C₈	—	10	2	no	—
196	C₂×C₈	—	10	2	no	—
197	C₄⋊C₄	—	3	4	no	—
198	C₄⋊C₄	—	6	4	no	—
199	C₄⋊C₄	—	6	2	no	—
200	C₄⋊C₄	—	30	4	no	—
201	C₄²	D₃×Dic₅	1	2	yes	—
202	C₄²	#1201	1	2	yes	—
203	C₃×C₅	#1993	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	#4078	1	2	yes	—
205	C₂²×C₃	#4088	1	2	yes	—
206	C₂²×C₃	#4090	1	1	yes	characteristic
207	C₂²×C₃	#4093	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	Q₈×Dic₅	1	2	yes	—
212	C₃×C₄	#4092	1	2	yes	—
213	Dic₃	—	4	4	no	—
214	Dic₃	—	40	2	no	—
215	C₂×C₅	#4702	1	2	yes	—
216	C₂×C₅	#4714	1	2	yes	—
217	C₂×C₅	#4929	1	1	yes	characteristic
218	C₂×C₅	#5120	1	1	yes	characteristic
219	C₂×C₅	#5131	1	1	yes	characteristic
220	C₂³	C₂×#1203	1	1	yes	characteristic
221	Q₈	—	12	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	6	4	no	—
225	C₂×C₄	—	12	4	no	—
226	C₂×C₄	—	60	2	no	—
227	C₂×C₄	—	60	2	no	—
228	C₂×C₄	—	60	2	no	—
229	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
230	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
231	C₂×C₄	D₅×Dic₆	1	2	yes	—
232	C₂×C₄	#6773	1	2	yes	—
233	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
234	C₂×C₄	C₂×#1201	1	1	yes	characteristic
235	C₈	—	10	4	no	—
236	C₂×C₃	#65369	1	2	yes	—
237	C₂×C₃	#65378	1	2	yes	—
238	C₂×C₃	#65504	1	1	yes	characteristic
239	C₂×C₃	#65514	1	1	yes	characteristic
240	C₂×C₃	#65519	1	1	yes	characteristic
241	C₅	#69828	1	1	yes	characteristic, complemented, Sylow
242	C₂²	#91304	1	2	yes	—
243	C₂²	#91321	1	2	yes	—
244	C₂²	#91375	1	1	yes	characteristic
245	C₂²	#91376	1	1	yes	characteristic
246	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	12	4	no	—
250	C₄	—	120	2	no	—
251	C₄	Dic₅×Dic₆	1	2	yes	—
252	C₄	#91359	1	2	yes	—
253	C₃	#10602581	1	1	yes	characteristic, complemented, Sylow
254	C₂	#11746914	1	2	yes	—
255	C₂	#11746957	1	2	yes	—
256	C₂	#11748389	1	1	yes	characteristic
257	C₂	#11748409	1	1	yes	characteristic
258	C₂	#11748589	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	#1993
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
37. C₂×C₃×#168	Q₈	semidirect
77. #869	Dic₅	semidirect
96. C₂×#168	Dic₆	semidirect
203. C₃×C₅	#1993	semidirect
241. C₅	#69828	semidirect
253. C₃	#10602581	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
#11751037	2
C₂×C₃×#3935	2
#11745981	2
#10602581	3
#69828	5