Subgroups — #14300366

Order 1920.

520 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	#11756578	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic
3	#11751335	C₂	1	1	yes	maximal, characteristic
4	#11751100	C₂	1	2	yes	maximal, complemented
5	#11746354	C₂	1	2	yes	maximal, complemented
6	#10600837	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₂²	1	2	yes	—
9	#91837	—	2	4	no	—
10	C₂×#6710	—	2	2	no	—
11	C₂×#6710	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	2	yes	—
13	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₃×C₅×#99	C₄	1	4	yes	—
15	#91114	—	2	2	no	—
16	C₄⋊(C₃⋊#168)	—	2	4	no	—
17	#70044	—	5	1	no	maximal, Hall
18	#66153	—	3	1	no	—
19	C₅×#410	D₃	1	1	yes	characteristic
20	#65523	—	3	1	no	—
21	#65288	—	3	2	no	—
22	#64753	—	3	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₃×C₅×#41	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
26	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	#6710	—	2	4	no	—
34	#6710	—	2	2	no	—
35	#6710	—	2	2	no	—
36	#6710	—	2	2	no	—
37	C₂×C₃⋊#168	—	2	4	no	—
38	C₂×C₃⋊#168	—	2	2	no	—
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₃⋊#168	D₄	1	4	yes	—
42	#5819	—	5	1	no	—
43	C₃×#410	D₅	1	1	yes	characteristic
44	#5234	—	5	1	no	—
45	#4999	—	5	2	no	—
46	#4533	—	5	2	no	—
47	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
48	C₅×#117	C₂×D₃	1	2	yes	—
49	#4094	—	6	4	no	—
50	C₂×#681	—	3	2	no	—
51	C₂×#681	—	3	1	no	—
52	C₂×#681	—	6	2	no	—
53	C₂²×#168	—	3	1	no	—
54	C₅×#99	Dic₃	1	4	yes	—
55	C₂²⋊#168	—	6	2	no	—
56	C₄⋊#168	—	6	4	no	—
57	#1868	—	15	1	no	Sylow
58	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
59	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₅×Q₈	—	2	2	no	—
61	C₃×C₅×D₄	—	2	4	no	—
62	C₃×C₅×D₄	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₃⋊#168	—	2	4	no	—
72	C₃⋊#168	—	4	1	no	—
73	C₃⋊#168	—	4	1	no	—
74	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
75	C₃×#117	C₂×D₅	1	2	yes	—
76	#948	—	10	4	no	—
77	C₂×#208	—	5	2	no	—
78	C₂×#208	—	5	1	no	—
79	C₂×#208	—	10	2	no	—
80	C₂²×C₃⋊C₈	—	5	1	no	—
81	C₃×#99	Dic₅	1	4	yes	—
82	#830	—	10	2	no	—
83	C₄⋊(C₃⋊C₈)	—	10	4	no	—
84	C₅×#41	—	2	2	no	—
85	C₅×#41	—	2	2	no	—
86	C₂×C₅×Q₈	Dic₆	1	1	yes	characteristic
87	C₂×C₅×D₄	Dic₆	1	1	yes	characteristic
88	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
89	C₂²×C₄×C₅	D₁₂	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₂²⋊C₄	C₂×Dic₃	1	4	yes	—
93	C₄²×C₅	—	2	2	no	—
94	#681	—	6	4	no	—
95	#681	—	6	2	no	—
96	#681	—	6	2	no	—
97	#681	—	6	2	no	—
98	C₂×#168	—	3	4	no	—
99	C₂×#168	—	6	4	no	—
100	C₂×#168	—	6	1	no	—
101	C₂×#168	—	6	1	no	—
102	C₂×#168	—	6	2	no	—
103	#568	—	15	1	no	—
104	#424	—	15	2	no	—
105	#410	D₁₅	1	1	yes	characteristic
106	#408	—	15	1	no	—
107	#344	—	15	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	2	no	—
111	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	#120	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
117	C₃×C₄×C₅	#123	1	1	yes	characteristic
118	C₃×#41	—	2	2	no	—
119	C₃×#41	—	2	2	no	—
120	C₂×C₃×Q₈	Dic₁₀	1	1	yes	characteristic
121	C₂×C₃×D₄	Dic₁₀	1	1	yes	characteristic
122	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
123	C₂²×C₃×C₄	D₂₀	1	2	yes	—
124	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₂²⋊C₄	C₂×Dic₅	1	4	yes	—
127	C₃×C₄²	—	2	2	no	—
128	#208	—	10	4	no	—
129	#208	—	10	2	no	—
130	#208	—	10	2	no	—
131	#208	—	10	2	no	—
132	C₂×C₃⋊C₈	—	5	4	no	—
133	C₂×C₃⋊C₈	—	10	4	no	—
134	C₂×C₃⋊C₈	—	10	1	no	—
135	C₂×C₃⋊C₈	—	10	1	no	—
136	C₂×C₃⋊C₈	—	10	2	no	—
137	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
138	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
139	C₅×Q₈	—	2	2	no	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
146	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
149	C₂×C₄×C₅	#237	1	2	yes	—
150	#168	—	6	4	no	—
151	#168	—	12	1	no	—
152	#168	—	12	1	no	—
153	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
154	#131	—	30	4	no	—
155	C₂×M₁₆	—	15	2	no	—
156	C₂×M₁₆	—	15	1	no	—
157	C₂×M₁₆	—	30	2	no	—
158	C₂²×C₈	—	15	1	no	—
159	#117	C₂×D₁₅	1	2	yes	—
160	C₄⋊C₈	—	30	4	no	—
161	#99	Dic₁₅	1	4	yes	—
162	C₂²⋊C₈	—	30	2	no	—
163	C₂×C₃×C₅	—	2	2	no	—
164	C₂×C₃×C₅	—	2	1	no	—
165	C₂×C₃×C₅	—	4	2	no	—
166	C₂×C₃×C₅	#385	1	1	yes	characteristic
167	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
168	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
169	C₃×Q₈	—	2	2	no	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
176	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
178	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
179	C₂×C₃×C₄	#711	1	2	yes	—
180	C₃⋊C₈	—	10	4	no	—
181	C₃⋊C₈	—	20	1	no	—
182	C₃⋊C₈	—	20	1	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
187	C₂²×C₅	D₄×Dic₃	1	2	yes	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	4	4	no	—
190	C₄×C₅	#878	1	2	yes	—
191	C₄×C₅	#882	1	1	yes	characteristic
192	C₄×C₅	#886	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	Dic₃₀	1	1	yes	characteristic
196	C₂×D₄	Dic₃₀	1	1	yes	characteristic
197	C₂×D₄	C₂×Dic₁₅	1	2	yes	—
198	C₂²×C₄	D₆₀	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
200	M₁₆	—	30	4	no	—
201	M₁₆	—	30	2	no	—
202	M₁₆	—	30	2	no	—
203	M₁₆	—	30	2	no	—
204	C₂×C₈	—	15	4	no	—
205	C₂×C₈	—	30	4	no	—
206	C₂×C₈	—	30	1	no	—
207	C₂×C₈	—	30	1	no	—
208	C₂×C₈	—	30	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	C₂×Dic₁₅	1	4	yes	—
211	C₄²	—	2	2	no	—
212	C₃×C₅	#1868	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
217	C₂²×C₃	D₄×Dic₅	1	2	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	#4024	1	2	yes	—
221	C₃×C₄	#4028	1	1	yes	characteristic
222	C₃×C₄	#4032	1	1	yes	characteristic
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	4	2	no	—
226	C₂×C₅	#4950	1	1	yes	characteristic
227	C₂³	C₄⋊Dic₁₅	1	2	yes	—
228	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
236	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
237	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
238	C₂×C₄	D₄×D₁₅	1	2	yes	—
239	C₂×C₄	#6819	1	2	yes	—
240	C₈	—	30	4	no	—
241	C₈	—	60	1	no	—
242	C₈	—	60	1	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	#65239	1	1	yes	characteristic
247	C₅	#70044	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
252	C₂²	D₄×Dic₁₅	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	4	4	no	—
255	C₄	#91767	1	2	yes	—
256	C₄	#91771	1	1	yes	characteristic
257	C₄	#91775	1	1	yes	characteristic
258	C₃	#10600837	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	2	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	#11751051	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91771
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#410	D₁₅
hypocenter	C₃×C₅	#1868
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

Complements

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

N	H ≅ G/N	type
4. #11751100	C₂	semidirect
5. #11746354	C₂	semidirect
212. C₃×C₅	#1868	semidirect
247. C₅	#70044	semidirect
258. C₃	#10600837	semidirect

Maximal subgroups

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

subgroup	index
#11756578	2
C₃×C₅×#410	2
#11751335	2
#11751100	2
#11746354	2
#10600837	3
#70044	5