Subgroups — #14313866

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×C₃×#4050	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745897	C₂	1	2	yes	maximal, complemented
5	#11745891	C₂	1	2	yes	maximal, complemented
6	#10601444	—	3	1	no	maximal, Hall
7	C₂×C₃×#709	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#926	—	2	1	no	—
10	C₂×C₃×Dic₂₀	C₂²	1	1	yes	characteristic
11	C₃×#4050	—	2	2	no	—
12	C₃×#4050	C₄	1	4	yes	complemented
13	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×#677	—	2	1	no	—
15	C₂×C₃⋊#672	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	—
17	C₅×#818	C₂²	1	2	yes	—
18	C₅×#817	—	2	1	no	—
19	#72077	—	5	1	no	maximal, Hall
20	C₂×#4050	D₃	1	1	yes	characteristic
21	C₅×#427	—	3	1	no	—
22	C₂²×#672	—	3	1	no	—
23	#64712	—	3	2	no	—
24	#64711	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊D₅	—	4	1	no	—
27	C₃×#709	—	2	4	no	—
28	C₃×#709	—	4	2	no	—
29	C₂×C₃×D₂₀	—	2	1	no	—
30	C₂×C₃×C₄×D₅	—	4	1	no	—
31	C₂×C₃×Dic₁₀	—	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×#217	—	4	1	no	—
37	C₅×C₄⋊Dic₃	—	2	2	no	—
38	C₄×C₅×Dic₃	—	4	1	no	—
39	C₃×Dic₂₀	—	2	1	no	—
40	C₃×Dic₂₀	C₂×C₄	1	2	yes	—
41	C₃×D₄₀	—	2	1	no	—
42	C₃×D₄₀	C₂×C₄	1	2	yes	—
43	C₃×#677	—	2	4	no	—
44	C₃⋊#672	—	2	2	no	—
45	C₂×C₃×#135	—	5	1	no	—
46	#5106	D₅	1	1	yes	characteristic, complemented
47	C₂²×C₃⋊C₁₆	—	5	1	no	—
48	#4564	—	5	2	no	—
49	#4562	—	5	2	no	—
50	C₂×#709	—	2	1	no	—
51	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	C₅×#117	—	6	1	no	—
53	C₂×Dic₂₀	C₂×D₃	1	1	yes	characteristic
54	#4050	—	2	2	no	—
55	#4050	Dic₃	1	4	yes	complemented
56	C₂×D₄₀	C₂×D₃	1	1	yes	characteristic
57	C₂×#677	—	2	1	no	—
58	C₅×#107	—	3	2	no	—
59	C₅×#106	—	6	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×C₃×D₅	—	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	C₂×C₅×Dic₃	—	4	2	no	—
70	C₃×C₂²⋊D₅	—	4	4	no	—
71	C₂×C₃×Dic₅	—	4	1	no	—
72	C₃×D₂₀	—	2	2	no	—
73	C₃×D₂₀	—	4	1	no	—
74	C₃×C₄×D₅	—	4	4	no	—
75	C₃×Dic₁₀	—	2	2	no	—
76	C₃×Dic₁₀	—	4	1	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×C₃×#41	—	10	1	no	—
81	C₃×#135	—	5	4	no	—
82	C₃×#135	—	10	2	no	—
83	C₂×C₃×Q₁₆	—	5	1	no	—
84	C₂×C₃×SD₁₆	—	10	1	no	—
85	C₂×C₃×D₈	—	5	1	no	—
86	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
87	#926	—	2	1	no	—
88	#818	C₂×D₅	1	2	yes	—
89	#817	—	2	1	no	—
90	C₂×C₃⋊C₁₆	—	5	2	no	—
91	C₂×C₃⋊C₁₆	—	10	2	no	—
92	C₂²×C₄×C₅	#67	1	1	yes	characteristic
93	C₂×C₂²⋊D₅	—	4	1	no	—
94	#709	—	2	4	no	—
95	#709	—	4	2	no	—
96	C₂×D₂₀	—	2	1	no	—
97	C₂×C₄×D₅	—	4	1	no	—
98	C₂×Dic₁₀	—	2	1	no	—
99	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
100	C₂×C₅×C₈	#67	1	1	yes	characteristic
101	C₂×C₅×C₈	#67	1	2	yes	—
102	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	6	2	no	—
104	C₅×C₂²⋊C₄	—	12	1	no	—
105	C₄²×C₅	—	12	1	no	—
106	Dic₂₀	—	2	1	no	—
107	Dic₂₀	C₂×Dic₃	1	2	yes	—
108	D₄₀	—	2	1	no	—
109	D₄₀	C₂×Dic₃	1	2	yes	—
110	#677	—	2	4	no	—
111	#672	—	6	2	no	—
112	C₂×#135	—	5	1	no	—
113	C₂²×C₁₆	—	15	1	no	—
114	#427	—	3	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×C₃×D₅	—	4	2	no	—
121	C₂×C₃×D₅	—	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	—	4	2	no	—
126	C₅×Dic₃	—	8	2	no	—
127	C₃×#41	—	10	4	no	—
128	C₃×#41	—	20	2	no	—
129	C₂×C₃×Q₈	—	10	1	no	—
130	C₂×C₃×D₄	—	10	1	no	—
131	C₂×C₃×D₄	—	20	1	no	—
132	C₂²×C₃×C₄	—	20	1	no	—
133	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
134	C₃×Q₁₆	—	5	2	no	—
135	C₃×Q₁₆	—	10	1	no	—
136	C₃×SD₁₆	—	10	4	no	—
137	C₃×D₈	—	5	2	no	—
138	C₃×D₈	—	10	1	no	—
139	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
140	C₂×C₃×C₈	D₂₀	1	2	yes	—
141	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
143	#217	—	4	1	no	—
144	C₄⋊Dic₃	—	2	2	no	—
145	C₄×Dic₃	—	4	1	no	—
146	C₃⋊C₁₆	—	10	2	no	—
147	C₂³×C₅	#213	1	1	yes	characteristic
148	C₂²×D₅	—	4	1	no	—
149	C₂×C₄×C₅	—	12	2	no	—
150	C₂×C₄×C₅	#213	1	1	yes	characteristic
151	C₂×C₄×C₅	#214	1	2	yes	—
152	C₂×C₄×C₅	#217	1	2	yes	—
153	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
154	C₂²⋊D₅	—	4	4	no	—
155	C₂×Dic₅	—	4	1	no	—
156	D₂₀	—	2	2	no	—
157	D₂₀	—	4	1	no	—
158	C₄×D₅	—	4	4	no	—
159	Dic₁₀	—	2	2	no	—
160	Dic₁₀	—	4	1	no	—
161	C₅×C₈	#217	1	2	yes	—
162	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
163	C₅×C₈	C₂×#67	1	1	yes	characteristic
164	C₂×#41	—	10	1	no	—
165	#135	—	5	4	no	—
166	#135	—	10	2	no	—
167	C₂×Q₁₆	—	5	1	no	—
168	C₂×SD₁₆	—	10	1	no	—
169	C₂×D₈	—	5	1	no	—
170	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
171	#117	—	6	1	no	—
172	C₂×C₁₆	—	15	2	no	—
173	C₂×C₁₆	—	30	2	no	—
174	#107	—	3	2	no	—
175	#106	—	6	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	C₃×D₅	—	8	2	no	—
180	C₂³×C₃	—	20	1	no	—
181	C₂³×C₃	#686	1	1	yes	characteristic
182	C₃×Q₈	—	10	2	no	—
183	C₃×Q₈	—	20	1	no	—
184	C₃×D₄	—	10	2	no	—
185	C₃×D₄	—	20	4	no	—
186	C₃×D₄	—	20	1	no	—
187	C₂×C₃×C₄	—	20	1	no	—
188	C₂×C₃×C₄	—	20	4	no	—
189	C₂×C₃×C₄	#685	1	2	yes	—
190	C₂×C₃×C₄	#686	1	1	yes	characteristic
191	C₂×C₃×C₄	#688	1	2	yes	—
192	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	C₂×Dic₃	—	4	2	no	—
194	C₃×C₈	#685	1	2	yes	—
195	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
196	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	#832	1	2	yes	—
199	C₂²×C₅	C₂×#213	1	1	yes	characteristic
200	C₂×D₅	—	4	2	no	—
201	C₂×D₅	—	8	2	no	—
202	C₄×C₅	—	24	2	no	—
203	C₄×C₅	#832	1	2	yes	—
204	C₄×C₅	C₂×#214	1	1	yes	characteristic
205	C₄×C₅	C₂×#217	1	1	yes	characteristic
206	Dic₅	—	4	2	no	—
207	#41	—	10	4	no	—
208	#41	—	20	2	no	—
209	C₂×Q₈	—	10	1	no	—
210	C₂×D₄	—	10	1	no	—
211	C₂×D₄	—	20	1	no	—
212	C₂²×C₄	—	20	1	no	—
213	C₂²×C₄	#1201	1	1	yes	characteristic
214	Q₁₆	—	5	2	no	—
215	Q₁₆	—	10	1	no	—
216	SD₁₆	—	10	4	no	—
217	D₈	—	5	2	no	—
218	D₈	—	10	1	no	—
219	C₂×C₈	D₅×Dic₃	1	2	yes	—
220	C₂×C₈	#1201	1	1	yes	characteristic
221	C₂×C₈	#1202	1	2	yes	—
222	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
223	C₄⋊C₄	—	6	2	no	—
224	C₂²⋊C₄	—	12	1	no	—
225	C₄²	—	12	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	—	20	2	no	—
230	C₂²×C₃	—	40	2	no	—
231	C₂²×C₃	#3941	1	2	yes	—
232	C₂²×C₃	C₂×#686	1	1	yes	characteristic
233	C₃×C₄	—	20	2	no	—
234	C₃×C₄	#3941	1	2	yes	—
235	C₃×C₄	C₂×#685	1	1	yes	characteristic
236	C₃×C₄	C₂×#688	1	1	yes	characteristic
237	Dic₃	—	8	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	#5194	1	2	yes	—
240	C₂×C₅	C₂×#832	1	1	yes	characteristic
241	D₅	—	8	2	no	—
242	C₂³	—	20	1	no	—
243	C₂³	#6652	1	1	yes	characteristic
244	Q₈	—	10	2	no	—
245	Q₈	—	20	1	no	—
246	D₄	—	10	2	no	—
247	D₄	—	20	4	no	—
248	D₄	—	20	1	no	—
249	C₂×C₄	—	12	2	no	—
250	C₂×C₄	—	20	1	no	—
251	C₂×C₄	—	20	4	no	—
252	C₂×C₄	#6652	1	1	yes	characteristic
253	C₂×C₄	#6655	1	2	yes	—
254	C₂×C₄	#6665	1	2	yes	—
255	C₂×C₄	C₂×#1201	1	1	yes	characteristic
256	C₈	#6665	1	2	yes	—
257	C₈	C₂×D₅×Dic₃	1	1	yes	characteristic
258	C₈	C₂×#1202	1	1	yes	characteristic
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	40	2	no	—
261	C₂×C₃	C₂×#3941	1	1	yes	characteristic
262	C₂×C₃	#65483	1	2	yes	—
263	C₅	#72077	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	20	2	no	—
266	C₂²	—	40	2	no	—
267	C₂²	#90964	1	2	yes	—
268	C₂²	C₂×#6652	1	1	yes	characteristic
269	C₄	—	20	2	no	—
270	C₄	—	24	2	no	—
271	C₄	#90964	1	2	yes	—
272	C₄	C₂×#6655	1	1	yes	characteristic
273	C₄	C₂×#6665	1	1	yes	characteristic
274	C₃	#10601444	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	40	2	no	—
277	C₂	#11747538	1	2	yes	—
278	C₂	C₂×#90964	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×C₃×#4050	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. C₅×#5106	C₂	semidirect
4. #11745897	C₂	semidirect
5. #11745891	C₂	semidirect
12. C₃×#4050	C₄	semidirect
46. #5106	D₅	semidirect
55. #4050	Dic₃	semidirect
227. C₃×C₅	#2138	semidirect
263. C₅	#72077	semidirect
274. C₃	#10601444	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4050	2
C₂²×C₃⋊#672	2
C₅×#5106	2
#11745897	2
#11745891	2
#10601444	3
#72077	5