Subgroups — #14313868

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745899	C₂	1	2	yes	maximal, complemented
5	#11745893	C₂	1	2	yes	maximal, complemented
6	#10602870	—	3	1	no	maximal, Hall
7	C₂×C₅×#235	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
10	C₅×#904	—	2	2	no	—
11	C₅×#904	C₄	1	4	yes	complemented
12	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₅×#204	—	2	1	no	—
14	C₃×#4072	—	2	1	no	—
15	C₂×C₃⋊#672	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	—
17	C₃×#3950	C₂²	1	2	yes	—
18	C₃×#3949	—	2	1	no	—
19	#70651	—	5	1	no	maximal, Hall
20	C₂×C₅×#135	—	3	1	no	—
21	#65395	D₃	1	1	yes	characteristic, complemented
22	C₂²×#672	—	3	1	no	—
23	#64784	—	3	2	no	—
24	#64782	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂×C₅×#67	—	4	1	no	—
27	C₅×#235	—	2	4	no	—
28	C₅×#235	—	4	2	no	—
29	C₂×C₅×D₁₂	—	2	1	no	—
30	C₂×C₄×C₅×D₃	—	4	1	no	—
31	C₂×C₅×Dic₆	—	2	1	no	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×Dic₁₂	—	2	1	no	—
37	C₅×Dic₁₂	C₂×C₄	1	2	yes	—
38	C₅×D₂₄	—	2	1	no	—
39	C₅×D₂₄	C₂×C₄	1	2	yes	—
40	C₅×#204	—	2	4	no	—
41	C₃×C₂²⋊Dic₅	—	4	1	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄×Dic₅	—	4	1	no	—
44	C₃⋊#672	—	2	2	no	—
45	C₂×#904	D₅	1	1	yes	characteristic
46	C₃×#427	—	5	1	no	—
47	C₂²×C₃⋊C₁₆	—	5	1	no	—
48	#4492	—	5	2	no	—
49	#4491	—	5	2	no	—
50	C₂×C₅×#41	—	6	1	no	—
51	C₅×#135	—	3	4	no	—
52	C₅×#135	—	6	2	no	—
53	C₂×C₅×Q₁₆	—	3	1	no	—
54	C₂×C₅×SD₁₆	—	6	1	no	—
55	C₂×C₅×D₈	—	3	1	no	—
56	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	#4072	—	2	1	no	—
58	#3950	C₂×D₃	1	2	yes	—
59	#3949	—	2	1	no	—
60	C₂×#672	—	3	2	no	—
61	C₂×#672	—	6	2	no	—
62	#2138	—	15	1	no	Sylow
63	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
64	C₂²×C₅×D₃	—	4	1	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₅×#67	—	4	4	no	—
70	C₂×C₅×Dic₃	—	4	1	no	—
71	C₅×D₁₂	—	2	2	no	—
72	C₅×D₁₂	—	4	1	no	—
73	C₄×C₅×D₃	—	4	4	no	—
74	C₅×Dic₆	—	2	2	no	—
75	C₅×Dic₆	—	4	1	no	—
76	C₂×C₃×Dic₅	—	4	2	no	—
77	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×#235	—	2	1	no	—
81	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₃×#117	—	10	1	no	—
83	C₂×Dic₁₂	C₂×D₅	1	1	yes	characteristic
84	#904	—	2	2	no	—
85	#904	Dic₅	1	4	yes	complemented
86	C₂×D₂₄	C₂×D₅	1	1	yes	characteristic
87	C₂×#204	—	2	1	no	—
88	C₃×#107	—	5	2	no	—
89	C₃×#106	—	10	1	no	—
90	C₂×C₃⋊C₁₆	—	5	2	no	—
91	C₂×C₃⋊C₁₆	—	10	2	no	—
92	C₅×#41	—	6	4	no	—
93	C₅×#41	—	12	2	no	—
94	C₂×C₅×Q₈	—	6	1	no	—
95	C₂×C₅×D₄	—	6	1	no	—
96	C₂×C₅×D₄	—	12	1	no	—
97	C₂²×C₄×C₅	—	12	1	no	—
98	C₂²×C₄×C₅	#67	1	1	yes	characteristic
99	C₅×Q₁₆	—	3	2	no	—
100	C₅×Q₁₆	—	6	1	no	—
101	C₅×SD₁₆	—	6	4	no	—
102	C₅×D₈	—	3	2	no	—
103	C₅×D₈	—	6	1	no	—
104	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
105	C₂×C₅×C₈	D₁₂	1	2	yes	—
106	C₂×C₅×C₈	#67	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
108	C₂²⋊Dic₅	—	4	1	no	—
109	C₄⋊Dic₅	—	2	2	no	—
110	C₄×Dic₅	—	4	1	no	—
111	#672	—	6	2	no	—
112	C₂×#135	—	3	1	no	—
113	C₂²×C₁₆	—	15	1	no	—
114	#427	—	5	1	no	—
115	#358	—	15	2	no	—
116	#357	—	15	2	no	—
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	#102	1	2	yes	—
119	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₅×D₃	—	4	2	no	—
121	C₂×C₅×D₃	—	8	2	no	—
122	C₃×C₄×C₅	#102	1	2	yes	—
123	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₃×Dic₅	—	8	2	no	—
126	C₅×Dic₃	—	4	2	no	—
127	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
128	C₂×#67	—	4	1	no	—
129	#235	—	2	4	no	—
130	#235	—	4	2	no	—
131	C₂×D₁₂	—	2	1	no	—
132	C₂×C₄×D₃	—	4	1	no	—
133	C₂×Dic₆	—	2	1	no	—
134	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
135	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
137	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	10	2	no	—
139	C₃×C₂²⋊C₄	—	20	1	no	—
140	C₃×C₄²	—	20	1	no	—
141	Dic₁₂	—	2	1	no	—
142	Dic₁₂	C₂×Dic₅	1	2	yes	—
143	D₂₄	—	2	1	no	—
144	D₂₄	C₂×Dic₅	1	2	yes	—
145	#204	—	2	4	no	—
146	C₃⋊C₁₆	—	10	2	no	—
147	C₂³×C₅	—	12	1	no	—
148	C₂³×C₅	#213	1	1	yes	characteristic
149	C₅×Q₈	—	6	2	no	—
150	C₅×Q₈	—	12	1	no	—
151	C₅×D₄	—	6	2	no	—
152	C₅×D₄	—	12	4	no	—
153	C₅×D₄	—	12	1	no	—
154	C₂×C₄×C₅	—	12	1	no	—
155	C₂×C₄×C₅	—	12	4	no	—
156	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
157	C₂×C₄×C₅	#213	1	1	yes	characteristic
158	C₂×C₄×C₅	#215	1	2	yes	—
159	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
160	C₂×Dic₅	—	4	2	no	—
161	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
162	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
163	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
164	C₂×#41	—	6	1	no	—
165	#135	—	3	4	no	—
166	#135	—	6	2	no	—
167	C₂×Q₁₆	—	3	1	no	—
168	C₂×SD₁₆	—	6	1	no	—
169	C₂×D₈	—	3	1	no	—
170	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
171	#117	—	10	1	no	—
172	C₂×C₁₆	—	15	2	no	—
173	C₂×C₁₆	—	30	2	no	—
174	#107	—	5	2	no	—
175	#106	—	10	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
178	C₂×C₃×C₅	#508	1	2	yes	—
179	C₅×D₃	—	8	2	no	—
180	C₂³×C₃	#686	1	1	yes	characteristic
181	C₂²×D₃	—	4	1	no	—
182	C₂×C₃×C₄	—	20	2	no	—
183	C₂×C₃×C₄	#686	1	1	yes	characteristic
184	C₂×C₃×C₄	#687	1	2	yes	—
185	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
186	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	#67	—	4	4	no	—
188	C₂×Dic₃	—	4	1	no	—
189	D₁₂	—	2	2	no	—
190	D₁₂	—	4	1	no	—
191	C₄×D₃	—	4	4	no	—
192	Dic₆	—	2	2	no	—
193	Dic₆	—	4	1	no	—
194	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
195	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
196	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	12	2	no	—
199	C₂²×C₅	—	24	2	no	—
200	C₂²×C₅	#809	1	2	yes	—
201	C₂²×C₅	C₂×#213	1	1	yes	characteristic
202	C₄×C₅	—	12	2	no	—
203	C₄×C₅	#809	1	2	yes	—
204	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₄×C₅	C₂×#215	1	1	yes	characteristic
206	Dic₅	—	8	2	no	—
207	#41	—	6	4	no	—
208	#41	—	12	2	no	—
209	C₂×Q₈	—	6	1	no	—
210	C₂×D₄	—	6	1	no	—
211	C₂×D₄	—	12	1	no	—
212	C₂²×C₄	—	12	1	no	—
213	C₂²×C₄	#1201	1	1	yes	characteristic
214	Q₁₆	—	3	2	no	—
215	Q₁₆	—	6	1	no	—
216	SD₁₆	—	6	4	no	—
217	D₈	—	3	2	no	—
218	D₈	—	6	1	no	—
219	C₂×C₈	D₃×Dic₅	1	2	yes	—
220	C₂×C₈	#1201	1	1	yes	characteristic
221	C₂×C₈	#1203	1	2	yes	—
222	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
223	C₄⋊C₄	—	10	2	no	—
224	C₂²⋊C₄	—	20	1	no	—
225	C₄²	—	20	1	no	—
226	C₁₆	—	30	2	no	—
227	C₃×C₅	#2138	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	#3964	1	2	yes	—
230	C₂²×C₃	C₂×#686	1	1	yes	characteristic
231	C₂×D₃	—	4	2	no	—
232	C₂×D₃	—	8	2	no	—
233	C₃×C₄	—	40	2	no	—
234	C₃×C₄	#3964	1	2	yes	—
235	C₃×C₄	C₂×#687	1	1	yes	characteristic
236	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
237	Dic₃	—	4	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	24	2	no	—
240	C₂×C₅	C₂×#809	1	1	yes	characteristic
241	C₂×C₅	#5194	1	2	yes	—
242	C₂³	—	12	1	no	—
243	C₂³	#6652	1	1	yes	characteristic
244	Q₈	—	6	2	no	—
245	Q₈	—	12	1	no	—
246	D₄	—	6	2	no	—
247	D₄	—	12	4	no	—
248	D₄	—	12	1	no	—
249	C₂×C₄	—	12	1	no	—
250	C₂×C₄	—	12	4	no	—
251	C₂×C₄	—	20	2	no	—
252	C₂×C₄	#6652	1	1	yes	characteristic
253	C₂×C₄	#6656	1	2	yes	—
254	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
255	C₂×C₄	C₂×#1201	1	1	yes	characteristic
256	C₈	(C₂×D₃)⋊Dic₅	1	2	yes	—
257	C₈	C₂×D₃×Dic₅	1	1	yes	characteristic
258	C₈	C₂×#1203	1	1	yes	characteristic
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	#65483	1	2	yes	—
261	C₂×C₃	C₂×#3964	1	1	yes	characteristic
262	D₃	—	8	2	no	—
263	C₅	#70651	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	12	2	no	—
266	C₂²	—	24	2	no	—
267	C₂²	#90966	1	2	yes	—
268	C₂²	C₂×#6652	1	1	yes	characteristic
269	C₄	—	12	2	no	—
270	C₄	—	40	2	no	—
271	C₄	#90966	1	2	yes	—
272	C₄	C₂×#6656	1	1	yes	characteristic
273	C₄	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
274	C₃	#10602870	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	24	2	no	—
277	C₂	#11747538	1	2	yes	—
278	C₂	C₂×#90966	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6652
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×#904	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2138
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊#672	C₂	semidirect
3. C₃×#65395	C₂	semidirect
4. #11745899	C₂	semidirect
5. #11745893	C₂	semidirect
11. C₅×#904	C₄	semidirect
21. #65395	D₃	semidirect
85. #904	Dic₅	semidirect
227. C₃×C₅	#2138	semidirect
263. C₅	#70651	semidirect
274. C₃	#10602870	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#904	2
C₂²×C₃⋊#672	2
C₃×#65395	2
#11745899	2
#11745893	2
#10602870	3
#70651	5