Subgroups — #14439870

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756229	C₂	1	1	yes	maximal, characteristic
2	C₅×#5470	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
4	#11748458	C₂	1	4	yes	maximal
5	#11746956	C₂	1	4	yes	maximal, complemented
6	#11746955	C₂	1	2	yes	maximal, complemented
7	#11746941	C₂	1	2	yes	maximal, complemented
8	#10615070	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	#91815	C₂²	1	2	yes	—
11	#91813	—	2	2	no	—
12	#91759	C₂²	1	2	yes	—
13	#91758	C₂²	1	2	yes	—
14	C₄×Dic₃₀	—	2	2	no	—
15	C₅×#926	C₂²	1	2	yes	—
16	C₅×#924	—	2	2	no	—
17	C₅×#870	C₂²	1	2	yes	—
18	C₅×#869	C₂²	1	2	yes	—
19	C₄×C₅×Dic₆	—	2	2	no	—
20	C₂²×C₃×#168	C₂²	1	2	yes	—
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	C₃×C₄×#168	C₂²	1	1	yes	characteristic
23	C₃×C₄×#168	C₂²	1	2	yes	—
24	#90988	C₂²	1	8	yes	—
25	#90985	C₂²	1	8	yes	—
26	#78442	—	5	1	no	maximal, Hall
27	C₅×#533	—	3	1	no	—
28	#65804	—	3	1	no	—
29	#65368	—	3	2	no	—
30	#65367	—	3	2	no	—
31	#65353	—	3	4	no	—
32	#65287	—	3	4	no	—
33	C₂×C₄×#168	D₃	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×Dic₃₀	—	2	2	no	—
37	C₂×C₅×Dic₆	—	2	2	no	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6719	—	2	2	no	—
42	C₄⋊Dic₁₅	—	2	2	no	—
43	C₄⋊Dic₁₅	C₂³	1	4	yes	—
44	#6712	—	2	4	no	—
45	C₄×Dic₁₅	—	2	2	no	—
46	C₅×#217	—	2	2	no	—
47	C₅×C₄⋊Dic₃	—	2	2	no	—
48	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
49	C₅×Dic₃⋊C₄	—	2	4	no	—
50	C₄×C₅×Dic₃	—	2	2	no	—
51	C₂×C₃×#168	Q₈	1	8	yes	complemented
52	C₂×C₃×#168	C₂³	1	2	yes	—
53	C₂×C₃×#168	C₂³	1	2	yes	—
54	#5470	—	5	1	no	—
55	#5470	D₅	1	1	yes	characteristic
56	C₂×C₃×C₄×C₈	—	5	1	no	—
57	#5106	—	5	4	no	—
58	#4683	—	5	4	no	—
59	#4682	—	5	2	no	—
60	#4680	—	5	2	no	—
61	C₅×C₄⋊Q₈	—	3	2	no	—
62	C₅×#125	—	3	2	no	—
63	C₅×C₂²⋊Q₈	—	6	2	no	—
64	C₄×C₅×Q₈	—	6	2	no	—
65	C₅×#117	—	3	2	no	—
66	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
67	#4072	—	3	2	no	—
68	#4070	—	6	2	no	—
69	C₂²×#168	C₂×D₃	1	2	yes	—
70	#4016	—	3	2	no	—
71	#4015	—	3	2	no	—
72	C₄×Dic₁₀	—	6	2	no	—
73	#3940	—	3	8	no	—
74	#3939	—	3	8	no	—
75	C₄×#168	C₂×D₃	1	2	yes	—
76	C₄×#168	C₂×D₃	1	1	yes	characteristic
77	C₄×#168	C₂×D₃	1	1	yes	characteristic
78	#3160	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₁₅	—	2	4	no	—
87	Dic₃₀	—	4	4	no	—
88	C₂×C₅×Dic₃	—	2	4	no	—
89	C₅×Dic₆	—	4	4	no	—
90	C₃×#168	C₂×Q₈	1	4	yes	—
91	C₃×#168	C₂×Q₈	1	4	yes	—
92	C₂²×C₃×C₈	—	5	2	no	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	#926	—	5	2	no	—
95	#926	C₂×D₅	1	2	yes	—
96	#924	—	2	2	no	—
97	#924	—	10	2	no	—
98	#870	—	5	2	no	—
99	#870	C₂×D₅	1	2	yes	—
100	#869	—	5	2	no	—
101	#869	C₂×D₅	1	2	yes	—
102	C₄×Dic₆	—	2	2	no	—
103	C₄×Dic₆	—	10	2	no	—
104	C₃×C₄×C₈	—	5	1	no	—
105	C₃×C₄×C₈	—	5	1	no	—
106	C₃×C₄×C₈	—	5	2	no	—
107	#818	—	5	8	no	—
108	#817	—	5	8	no	—
109	C₂×C₅×Q₈	—	6	2	no	—
110	C₂²×C₄×C₅	D₁₂	1	2	yes	—
111	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₂×Dic₁₀	—	6	2	no	—
113	C₅×C₄⋊C₄	—	3	4	no	—
114	C₅×C₄⋊C₄	—	6	4	no	—
115	C₅×C₄⋊C₄	—	6	2	no	—
116	C₅×C₂²⋊C₄	—	6	2	no	—
117	C₄²×C₅	—	6	2	no	—
118	C₄²×C₅	D₁₂	1	2	yes	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₂²⋊Dic₅	—	6	2	no	—
122	C₄⋊Dic₅	—	3	4	no	—
123	C₄⋊Dic₅	—	6	2	no	—
124	#683	—	6	4	no	—
125	C₄×Dic₅	—	6	2	no	—
126	C₂×#168	Dic₆	1	8	yes	complemented
127	C₂×#168	C₂²×D₃	1	2	yes	—
128	C₂×#168	C₂²×D₃	1	2	yes	—
129	#533	—	3	1	no	—
130	#533	—	15	1	no	—
131	#500	—	15	2	no	—
132	#499	—	15	4	no	—
133	#498	—	15	2	no	—
134	#427	—	15	4	no	—
135	C₂×C₄×C₈	—	5	1	no	—
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
138	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	—	2	2	no	—
140	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
141	C₃×C₄×C₅	#135	1	4	yes	—
142	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
143	Dic₁₅	—	4	4	no	—
144	C₅×Dic₃	—	4	4	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂×Dic₆	—	2	2	no	—
148	C₂×Dic₆	—	10	2	no	—
149	C₂×C₃×C₈	—	5	8	no	—
150	C₂×C₃×C₈	—	5	2	no	—
151	C₂×C₃×C₈	—	5	2	no	—
152	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
153	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
154	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
155	#217	—	2	2	no	—
156	#217	—	10	2	no	—
157	C₄⋊Dic₃	—	2	2	no	—
158	C₄⋊Dic₃	—	5	4	no	—
159	C₄⋊Dic₃	—	10	2	no	—
160	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
161	Dic₃⋊C₄	—	2	4	no	—
162	Dic₃⋊C₄	—	10	4	no	—
163	C₄×Dic₃	—	2	2	no	—
164	C₄×Dic₃	—	10	2	no	—
165	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
166	C₅×Q₈	—	12	4	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	6	4	no	—
169	C₂×C₄×C₅	C₂×Dic₆	1	4	yes	—
170	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
171	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
172	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₂×Dic₅	—	6	4	no	—
175	Dic₁₀	—	12	4	no	—
176	#168	C₂×Dic₆	1	4	yes	—
177	#168	C₂×Dic₆	1	4	yes	—
178	C₂²×C₈	—	5	2	no	—
179	C₄⋊Q₈	—	3	2	no	—
180	C₄⋊Q₈	—	15	2	no	—
181	#125	—	3	2	no	—
182	#125	—	15	2	no	—
183	C₂²⋊Q₈	—	6	2	no	—
184	C₂²⋊Q₈	—	30	2	no	—
185	C₄×Q₈	—	6	2	no	—
186	C₄×Q₈	—	30	2	no	—
187	#117	—	3	2	no	—
188	#117	—	15	2	no	—
189	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
190	#107	—	15	8	no	—
191	#106	—	15	8	no	—
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	2	no	—
194	C₄×C₈	—	5	1	no	—
195	C₂×C₃×C₅	—	2	2	no	—
196	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂×#135	1	2	yes	—
198	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	Q₈×D₅	1	4	yes	—
201	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
202	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
205	C₂×Dic₃	—	2	4	no	—
206	C₂×Dic₃	—	10	4	no	—
207	Dic₆	—	4	4	no	—
208	Dic₆	—	20	4	no	—
209	C₃×C₈	—	5	4	no	—
210	C₃×C₈	—	5	4	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	#869	1	2	yes	—
213	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	12	4	no	—
216	C₄×C₅	#869	1	2	yes	—
217	C₄×C₅	#904	1	4	yes	—
218	C₄×C₅	C₂²×Dic₆	1	2	yes	—
219	Dic₅	—	12	4	no	—
220	C₂×Q₈	—	6	2	no	—
221	C₂×Q₈	—	30	2	no	—
222	C₂²×C₄	#1203	1	2	yes	—
223	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
224	C₂×C₈	—	5	8	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₄⋊C₄	—	3	4	no	—
228	C₄⋊C₄	—	6	4	no	—
229	C₄⋊C₄	—	6	2	no	—
230	C₄⋊C₄	—	15	4	no	—
231	C₄⋊C₄	—	30	4	no	—
232	C₄⋊C₄	—	30	2	no	—
233	C₂²⋊C₄	—	6	2	no	—
234	C₂²⋊C₄	—	30	2	no	—
235	C₄²	—	6	2	no	—
236	C₄²	—	30	2	no	—
237	C₄²	#1203	1	2	yes	—
238	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
239	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
240	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	#4090	1	2	yes	—
243	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	#4090	1	2	yes	—
246	C₃×C₄	#4096	1	4	yes	—
247	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
248	Dic₃	—	4	4	no	—
249	Dic₃	—	20	4	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	C₂×#869	1	1	yes	characteristic
252	C₂×C₅	C₂×#904	1	2	yes	—
253	C₂³	C₂×#1203	1	1	yes	characteristic
254	Q₈	—	12	4	no	—
255	Q₈	—	60	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	6	4	no	—
258	C₂×C₄	—	30	4	no	—
259	C₂×C₄	D₅×Dic₆	1	4	yes	—
260	C₂×C₄	C₂×#1203	1	2	yes	—
261	C₂×C₄	C₂×#1203	1	2	yes	—
262	C₂×C₄	C₂×#1203	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₈	—	5	4	no	—
265	C₈	—	5	4	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	C₂×#4090	1	1	yes	characteristic
268	C₂×C₃	C₂×#4096	1	2	yes	—
269	C₅	#78442	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	#91376	1	2	yes	—
272	C₂²	C₂²×#1203	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	12	4	no	—
275	C₄	—	60	4	no	—
276	C₄	#91303	1	4	yes	—
277	C₄	#91376	1	2	yes	—
278	C₄	C₂×D₅×Dic₆	1	2	yes	—
279	C₃	#10615070	1	1	yes	characteristic, complemented, Sylow
280	C₂	—	2	2	no	—
281	C₂	C₂×#91303	1	2	yes	—
282	C₂	C₂×#91376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1203
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. #11746956	C₂	semidirect
6. #11746955	C₂	semidirect
7. #11746941	C₂	semidirect
51. C₂×C₃×#168	Q₈	semidirect
126. C₂×#168	Dic₆	semidirect
240. C₃×C₅	#3160	semidirect
269. C₅	#78442	semidirect
279. C₃	#10615070	semidirect

Maximal subgroups

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

subgroup	index
#11756229	2
C₅×#5470	2
C₂×C₃×C₄×#168	2
#11748458	2
#11746956	2
#11746955	2
#11746941	2
#10615070	3
#78442	5