Subgroups — #14418610

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 256 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×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
4	#11748435	C₂	1	2	yes	maximal
5	#11748324	C₂	1	2	yes	maximal
6	#11746852	C₂	1	8	yes	maximal, complemented
7	#10610776	—	3	1	no	maximal, Hall
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91815	C₂²	1	2	yes	—
10	#91813	C₄	1	4	yes	—
11	#91759	C₄	1	2	yes	—
12	#91758	C₄	1	2	yes	—
13	C₄×Dic₃₀	C₂²	1	4	yes	—
14	C₂×C₅×#208	C₂²	1	2	yes	—
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
18	C₃×#3935	C₂²	1	4	yes	—
19	#90983	C₂²	1	8	yes	—
20	#90954	C₂²	1	4	yes	—
21	#90951	C₂²	1	4	yes	—
22	#78480	—	5	1	no	maximal, Hall
23	#65804	—	3	1	no	—
24	C₅×#423	—	3	1	no	—
25	#65411	—	3	2	no	—
26	#65408	—	3	2	no	—
27	C₂×#3935	D₃	1	1	yes	characteristic
28	#65117	—	3	8	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×Dic₃₀	C₂×C₄	1	4	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂³	1	2	yes	—
35	#6719	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
38	#6712	C₂×C₄	1	8	yes	—
39	C₄×Dic₁₅	C₂³	1	4	yes	—
40	C₅×#208	—	2	4	no	—
41	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×#168	Q₈	1	4	yes	complemented
44	C₂×C₃×#168	C₂³	1	2	yes	—
45	C₂×C₃×#168	C₂³	1	2	yes	—
46	#5470	—	5	1	no	—
47	C₂×C₃×#97	—	5	1	no	—
48	#5102	—	5	2	no	—
49	#5101	—	5	2	no	—
50	#4924	D₅	1	1	yes	characteristic
51	#4701	—	5	8	no	—
52	C₂×C₅×M₁₆	—	3	2	no	—
53	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
54	#4072	—	3	2	no	—
55	#4070	—	3	4	no	—
56	C₂²×#168	C₂×D₃	1	1	yes	characteristic
57	C₂²×#168	C₂×D₃	1	1	yes	characteristic
58	#4016	—	3	2	no	—
59	#4015	—	3	2	no	—
60	C₄×Dic₁₀	—	3	4	no	—
61	C₅×C₄⋊C₈	—	3	4	no	—
62	#3947	—	3	4	no	—
63	#3946	—	3	8	no	—
64	C₈×Dic₅	—	3	4	no	—
65	#3935	C₂×D₃	1	4	yes	—
66	#2961	—	15	1	no	Sylow
67	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
73	C₂×C₃×C₄×C₅	#41	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₁₅	C₂²×C₄	1	8	yes	—
76	Dic₃₀	—	4	4	no	—
77	C₃×#168	—	2	2	no	—
78	C₃×#168	C₂×Q₈	1	4	yes	—
79	C₅×C₃⋊C₈	—	2	4	no	—
80	C₂²×C₃×C₈	—	5	1	no	—
81	C₂²×C₃×C₈	—	5	1	no	—
82	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
83	#926	—	5	2	no	—
84	#924	—	5	4	no	—
85	C₂×#208	C₂×D₅	1	2	yes	—
86	#870	—	5	2	no	—
87	#869	—	5	2	no	—
88	C₄×Dic₆	—	5	4	no	—
89	C₃×#97	—	5	4	no	—
90	#815	—	5	4	no	—
91	Dic₃⋊C₈	—	5	8	no	—
92	C₈×Dic₃	—	5	4	no	—
93	C₄⋊(C₃⋊C₈)	C₂×D₅	1	4	yes	—
94	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂×Dic₁₀	—	3	4	no	—
97	C₅×M₁₆	—	6	4	no	—
98	C₂×C₅×C₈	—	3	4	no	—
99	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
100	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
101	C₄²×C₅	C₂²×D₃	1	2	yes	—
102	C₂²⋊Dic₅	—	3	4	no	—
103	C₄⋊Dic₅	—	3	4	no	—
104	C₄⋊Dic₅	—	3	4	no	—
105	#683	—	3	8	no	—
106	C₄×Dic₅	—	3	4	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	Dic₆	1	4	yes	complemented
109	C₂×#168	C₂²×D₃	1	2	yes	—
110	C₂×#168	C₂²×D₃	1	2	yes	—
111	#533	—	15	1	no	—
112	#446	—	15	8	no	—
113	#424	—	15	2	no	—
114	#423	—	3	1	no	—
115	#405	—	15	2	no	—
116	C₂×#97	—	5	1	no	—
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
119	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
123	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
124	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
125	Dic₁₅	—	2	8	no	—
126	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
127	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₂×Dic₆	—	5	4	no	—
129	C₂×C₃×C₈	—	5	4	no	—
130	C₂×C₃×C₈	—	5	2	no	—
131	C₂×C₃×C₈	—	5	2	no	—
132	C₂×C₃×C₈	—	10	2	no	—
133	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
134	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
135	C₃×C₄²	C₂²×D₅	1	2	yes	—
136	#217	—	5	4	no	—
137	C₄⋊Dic₃	—	5	4	no	—
138	C₄⋊Dic₃	—	5	4	no	—
139	Dic₃⋊C₄	—	5	8	no	—
140	C₄×Dic₃	—	5	4	no	—
141	#208	—	2	4	no	—
142	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
143	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	#235	1	2	yes	—
150	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
151	C₂×Dic₅	—	3	8	no	—
152	Dic₁₀	—	12	4	no	—
153	C₅×C₈	—	6	4	no	—
154	#168	—	2	2	no	—
155	#168	C₂×Dic₆	1	4	yes	—
156	C₂×M₁₆	—	3	2	no	—
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	—	5	1	no	—
159	C₄⋊Q₈	—	15	2	no	—
160	#125	—	15	2	no	—
161	C₂²⋊Q₈	—	15	4	no	—
162	C₄×Q₈	—	15	4	no	—
163	#117	—	15	2	no	—
164	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
165	C₄⋊C₈	—	3	4	no	—
166	C₄⋊C₈	—	15	8	no	—
167	#97	—	5	4	no	—
168	#97	—	15	4	no	—
169	C₄×C₈	—	15	4	no	—
170	C₂×C₃×C₅	—	2	2	no	—
171	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
172	C₂×C₃×C₅	#568	1	2	yes	—
173	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	#711	1	2	yes	—
179	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
180	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
181	C₂×Dic₃	—	5	8	no	—
182	Dic₆	—	20	4	no	—
183	C₃×C₈	—	5	4	no	—
184	C₃×C₈	—	10	2	no	—
185	C₃⋊C₈	—	2	4	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	C₄×Dic₆	1	2	yes	—
188	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	C₄×Dic₆	1	2	yes	—
192	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
193	C₄×C₅	C₂×#235	1	1	yes	characteristic
194	Dic₅	—	6	8	no	—
195	C₂×Q₈	—	15	4	no	—
196	C₂²×C₄	#1200	1	2	yes	—
197	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
198	M₁₆	—	6	4	no	—
199	C₂×C₈	—	3	4	no	—
200	C₂×C₈	—	5	4	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	10	2	no	—
204	C₄⋊C₄	—	15	4	no	—
205	C₄⋊C₄	—	15	8	no	—
206	C₄⋊C₄	—	15	4	no	—
207	C₂²⋊C₄	—	15	4	no	—
208	C₄²	—	15	4	no	—
209	C₄²	#1200	1	1	yes	characteristic
210	C₄²	#1200	1	1	yes	characteristic
211	C₄²	C₂×D₃×D₅	1	2	yes	—
212	C₃×C₅	#2961	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	#4033	1	2	yes	—
215	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	#4033	1	2	yes	—
219	C₃×C₄	C₂×#711	1	1	yes	characteristic
220	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
221	Dic₃	—	10	8	no	—
222	C₂×C₅	—	2	2	no	—
223	C₂×C₅	C₂×C₄×Dic₆	1	1	yes	characteristic
224	C₂×C₅	#5745	1	2	yes	—
225	C₂³	C₂×#1200	1	1	yes	characteristic
226	Q₈	—	60	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	15	8	no	—
229	C₂×C₄	D₅×Dic₆	1	2	yes	—
230	C₂×C₄	#6772	1	2	yes	—
231	C₂×C₄	C₂×#1200	1	2	yes	—
232	C₂×C₄	C₂×#1200	1	2	yes	—
233	C₂×C₄	C₂×#1200	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
235	C₈	—	5	4	no	—
236	C₈	—	6	4	no	—
237	C₈	—	10	2	no	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	C₂×#4033	1	1	yes	characteristic
240	C₂×C₃	#66079	1	2	yes	—
241	C₅	#78480	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	2	2	no	—
243	C₂²	#91333	1	2	yes	—
244	C₂²	C₂²×#1200	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	30	8	no	—
248	C₄	#91333	1	2	yes	—
249	C₄	C₂×D₅×Dic₆	1	1	yes	characteristic
250	C₄	C₂×#6772	1	1	yes	characteristic
251	C₃	#10610776	1	1	yes	characteristic, complemented, Sylow
252	C₂	—	2	2	no	—
253	C₂	#11754130	1	2	yes	—
254	C₂	C₂×#91333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2961
LCS	G ▷ C₂²×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
6. #11746852	C₂	semidirect
43. C₂×C₃×#168	Q₈	semidirect
108. C₂×#168	Dic₆	semidirect
212. C₃×C₅	#2961	semidirect
241. C₅	#78480	semidirect
251. C₃	#10610776	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×#4924	2
C₂×C₃×#3935	2
#11748435	2
#11748324	2
#11746852	2
#10610776	3
#78480	5