Subgroups — #14418579

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	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	#11751203	C₂	1	2	yes	maximal
4	#11751202	C₂	1	2	yes	maximal
5	#11751025	C₂	1	1	yes	maximal, characteristic
6	#11750802	C₂	1	8	yes	maximal, complemented
7	#10610211	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#91815	C₂²	1	2	yes	—
12	#91813	C₄	1	4	yes	—
13	C₂×#6710	C₂²	1	2	yes	—
14	#91759	C₄	1	2	yes	—
15	#91758	C₄	1	2	yes	—
16	C₄×Dic₃₀	C₂²	1	4	yes	—
17	C₃×C₅×#97	C₂²	1	4	yes	—
18	#91099	C₂²	1	4	yes	—
19	#91098	C₂²	1	8	yes	—
20	C₈×Dic₁₅	C₂²	1	4	yes	—
21	C₄⋊(C₃⋊#168)	C₂²	1	4	yes	—
22	#78480	—	5	1	no	maximal, Hall
23	#65804	—	3	1	no	—
24	C₂×C₅×#97	D₃	1	1	yes	characteristic
25	#65391	—	3	2	no	—
26	#65390	—	3	2	no	—
27	#65213	—	3	1	no	—
28	#64990	—	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	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
34	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂³	1	2	yes	—
39	#6719	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
42	#6712	C₂×C₄	1	8	yes	—
43	C₄×Dic₁₅	C₂³	1	4	yes	—
44	#6710	—	2	4	no	—
45	C₂×C₃⋊#168	C₂³	1	4	yes	—
46	#5470	—	5	1	no	—
47	C₂×C₃×#97	D₅	1	1	yes	characteristic
48	#5102	—	5	2	no	—
49	#5101	—	5	2	no	—
50	#4924	—	5	1	no	—
51	#4701	—	5	8	no	—
52	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4072	—	3	2	no	—
56	#4070	—	3	4	no	—
57	C₂×#681	—	3	2	no	—
58	#4016	—	3	2	no	—
59	#4015	—	3	2	no	—
60	C₄×Dic₁₀	—	3	4	no	—
61	C₅×#97	C₂×D₃	1	4	yes	—
62	#3947	—	3	4	no	—
63	#3946	—	3	8	no	—
64	C₈×Dic₅	—	3	4	no	—
65	C₄⋊#168	—	3	4	no	—
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×C₅×C₈	—	2	2	no	—
78	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
79	C₃⋊#168	—	2	4	no	—
80	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
83	#926	—	5	2	no	—
84	#924	—	5	4	no	—
85	C₂×#208	—	5	2	no	—
86	#870	—	5	2	no	—
87	#869	—	5	2	no	—
88	C₄×Dic₆	—	5	4	no	—
89	C₃×#97	C₂×D₅	1	4	yes	—
90	#815	—	5	4	no	—
91	Dic₃⋊C₈	—	5	8	no	—
92	C₈×Dic₃	—	5	4	no	—
93	C₄⋊(C₃⋊C₈)	—	5	4	no	—
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	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	Dic₆	1	4	yes	complemented
99	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
100	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
101	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
102	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
103	C₄²×C₅	C₂²×D₃	1	2	yes	—
104	C₂²⋊Dic₅	—	3	4	no	—
105	C₄⋊Dic₅	—	3	4	no	—
106	C₄⋊Dic₅	—	3	4	no	—
107	#683	—	3	8	no	—
108	C₄×Dic₅	—	3	4	no	—
109	#681	—	6	4	no	—
110	C₂×#168	—	3	4	no	—
111	#533	—	15	1	no	—
112	#446	—	15	8	no	—
113	#424	—	15	2	no	—
114	#423	—	15	1	no	—
115	#405	—	15	2	no	—
116	C₂×#97	D₁₅	1	1	yes	characteristic
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	—	2	2	no	—
130	C₂×C₃×C₈	Dic₁₀	1	4	yes	complemented
131	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
132	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
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	—	10	4	no	—
142	C₂×C₃⋊C₈	—	5	4	no	—
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	—	2	2	no	—
154	C₅×C₈	C₂×Dic₆	1	4	yes	—
155	#168	—	6	4	no	—
156	C₂×M₁₆	—	15	2	no	—
157	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
158	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
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	C₂×D₁₅	1	1	yes	characteristic
165	C₄⋊C₈	—	15	8	no	—
166	C₄⋊C₈	—	15	4	no	—
167	#97	—	15	4	no	—
168	#97	C₂×D₁₅	1	4	yes	—
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×Dic₁₀	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
179	C₂×C₃×C₄	#709	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	—	2	2	no	—
184	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
185	C₃⋊C₈	—	10	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	C₄×D₁₅	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
198	M₁₆	—	30	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	15	4	no	—
201	C₂×C₈	Dic₃₀	1	4	yes	complemented
202	C₂×C₈	C₂²×D₁₅	1	2	yes	—
203	C₂×C₈	C₂²×D₁₅	1	2	yes	—
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	C₄×D₁₅	1	1	yes	characteristic
210	C₄²	C₄×D₁₅	1	1	yes	characteristic
211	C₄²	C₂²×D₁₅	1	2	yes	—
212	C₃×C₅	#2961	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	C₄×Dic₁₀	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	C₄×Dic₁₀	1	2	yes	—
219	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
220	C₃×C₄	C₂×#709	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×C₄×D₁₅	1	1	yes	characteristic
226	Q₈	—	60	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	15	8	no	—
229	C₂×C₄	C₂×Dic₃₀	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
233	C₂×C₄	#6817	1	2	yes	—
234	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	30	4	no	—
237	C₈	C₂×Dic₃₀	1	4	yes	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	C₂×C₄×Dic₁₀	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	C₄×Dic₃₀	1	2	yes	—
244	C₂²	C₂²×C₄×D₁₅	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	30	8	no	—
248	C₄	C₄×Dic₃₀	1	2	yes	—
249	C₄	C₂²×Dic₃₀	1	1	yes	characteristic
250	C₄	C₂×#6817	1	1	yes	characteristic
251	C₃	#10610211	1	1	yes	characteristic, complemented, Sylow
252	C₂	—	2	2	no	—
253	C₂	C₂×C₄×Dic₃₀	1	1	yes	characteristic
254	C₂	#11756504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₅×#97	C₂
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×#97	D₁₅
hypocenter	C₃×C₅	#2961
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
6. #11750802	C₂	semidirect
33. C₂×C₃×C₅×C₈	Q₈	semidirect
98. C₂×C₅×C₈	Dic₆	semidirect
130. C₂×C₃×C₈	Dic₁₀	semidirect
201. C₂×C₈	Dic₃₀	semidirect
212. C₃×C₅	#2961	semidirect
241. C₅	#78480	semidirect
251. C₃	#10610211	semidirect

Maximal subgroups

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

subgroup	index
#11756229	2
C₂×C₃×C₅×#97	2
#11751203	2
#11751202	2
#11751025	2
#11750802	2
#10610211	3
#78480	5