Subgroups — #14305777

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91119	C₂	1	4	yes	maximal
3	#11751036	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
5	#10602578	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₅	—	2	2	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
13	C₂²×C₃⋊#168	C₂²	1	2	yes	—
14	#91119	—	2	8	no	—
15	C₄×C₃⋊#168	—	2	2	no	—
16	#71785	—	5	1	no	maximal, Hall
17	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	C₂×#3967	—	3	4	no	—
19	#65224	—	3	1	no	—
20	C₂×C₄×#168	—	3	1	no	—
21	C₂×C₃×C₅×Q₈	—	2	2	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₅	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	—	2	8	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₄⋊Dic₁₅	—	2	4	no	—
33	C₄⋊Dic₁₅	—	4	4	no	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	—	2	2	no	—
36	C₂×C₃⋊#168	D₄	1	4	yes	—
37	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
38	C₂×#835	—	5	4	no	—
39	#4935	—	5	1	no	—
40	C₂×C₄×C₃⋊C₈	—	5	1	no	—
41	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
42	C₅×C₄⋊Q₈	—	2	2	no	—
43	C₅×C₄⋊Q₈	Dic₃	1	4	yes	complemented
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₅	—	3	2	no	—
47	C₂×C₄⋊Dic₅	—	6	2	no	—
48	C₂²×#168	—	3	2	no	—
49	#3967	—	6	8	no	—
50	C₄×#168	—	6	2	no	—
51	#1956	—	15	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₃×C₅×Q₈	—	2	8	no	—
54	C₃×C₅×Q₈	—	4	4	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
59	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
60	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	2	yes	—
65	C₂×Dic₁₅	—	4	2	no	—
66	C₂×Dic₁₅	—	4	4	no	—
67	C₃⋊#168	—	2	4	no	—
68	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
69	C₃×C₄⋊Q₈	—	2	2	no	—
70	C₃×C₄⋊Q₈	Dic₅	1	4	yes	complemented
71	C₂×C₃×C₄⋊C₄	—	2	2	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₂×C₄⋊Dic₃	—	5	2	no	—
74	C₂×C₄⋊Dic₃	—	10	2	no	—
75	C₂²×C₃⋊C₈	—	5	2	no	—
76	#835	—	10	8	no	—
77	C₄×C₃⋊C₈	—	10	2	no	—
78	C₂×C₅×Q₈	—	2	2	no	—
79	C₂×C₅×Q₈	—	2	4	no	—
80	C₂×C₅×Q₈	—	2	4	no	—
81	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	#67	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂²×Dic₅	—	6	2	no	—
86	C₅×C₄⋊C₄	—	2	8	no	—
87	C₄²×C₅	C₂×Dic₃	1	2	yes	—
88	C₄²×C₅	#67	1	2	yes	—
89	C₄⋊Dic₅	—	6	4	no	—
90	C₄⋊Dic₅	—	12	4	no	—
91	C₂×#168	—	3	4	no	—
92	C₂×#168	—	6	2	no	—
93	C₂×#168	—	6	2	no	—
94	C₂×C₄⋊Q₈	D₁₅	1	1	yes	characteristic
95	C₂×#103	—	15	4	no	—
96	C₂×C₄×C₈	—	15	1	no	—
97	#384	—	15	1	no	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	#124	1	1	yes	characteristic
100	C₂²×C₃×C₅	#127	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
102	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
103	C₃×C₄×C₅	—	4	4	no	—
104	C₃×C₄×C₅	#103	1	8	yes	complemented
105	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
106	C₃×C₄×C₅	#121	1	2	yes	—
107	Dic₁₅	—	8	2	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂×C₃×Q₈	—	2	4	no	—
111	C₂×C₃×Q₈	C₂×Dic₅	1	4	yes	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
114	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂²×Dic₃	—	10	2	no	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₃×C₄²	C₂×Dic₅	1	2	yes	—
118	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
119	C₄⋊Dic₃	—	10	4	no	—
120	C₄⋊Dic₃	—	20	4	no	—
121	C₂×C₃⋊C₈	—	5	4	no	—
122	C₂×C₃⋊C₈	—	10	2	no	—
123	C₂×C₃⋊C₈	—	10	2	no	—
124	C₂³×C₅	C₂×#67	1	1	yes	characteristic
125	C₅×Q₈	—	2	8	no	—
126	C₅×Q₈	—	4	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	#215	1	4	yes	—
131	C₂×C₄×C₅	#216	1	4	yes	—
132	C₂×C₄×C₅	#217	1	4	yes	—
133	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
134	C₂×C₄×C₅	#237	1	2	yes	—
135	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
137	C₂×Dic₅	—	12	2	no	—
138	C₂×Dic₅	—	12	4	no	—
139	#168	—	6	4	no	—
140	C₂²×Q₈	C₂×D₁₅	1	2	yes	—
141	C₂²×C₈	—	15	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	Dic₁₅	1	4	yes	complemented
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	15	2	no	—
146	C₂×C₄⋊C₄	—	30	2	no	—
147	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
148	#103	—	30	8	no	—
149	C₄×C₈	—	30	2	no	—
150	C₂×C₃×C₅	#390	1	1	yes	characteristic
151	C₂×C₃×C₅	C₂×#103	1	2	yes	—
152	C₂×C₃×C₅	#487	1	2	yes	—
153	C₂×C₃×C₅	#492	1	1	yes	characteristic
154	C₂×C₃×C₅	#494	1	1	yes	characteristic
155	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
156	C₃×Q₈	—	2	8	no	—
157	C₃×Q₈	—	4	4	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	#688	1	4	yes	—
162	C₂×C₃×C₄	#689	1	4	yes	—
163	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
164	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
165	C₂×C₃×C₄	#711	1	2	yes	—
166	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₂×Dic₃	—	20	2	no	—
169	C₂×Dic₃	—	20	4	no	—
170	C₃⋊C₈	—	10	4	no	—
171	C₂²×C₅	#936	1	1	yes	characteristic
172	C₂²×C₅	#940	1	1	yes	characteristic
173	C₂²×C₅	C₂×#215	1	2	yes	—
174	C₂²×C₅	C₂×#216	1	2	yes	—
175	C₂²×C₅	C₂×#217	1	1	yes	characteristic
176	C₄×C₅	—	4	4	no	—
177	C₄×C₅	#835	1	8	yes	complemented
178	C₄×C₅	D₄×Dic₃	1	2	yes	—
179	C₄×C₅	#938	1	2	yes	—
180	Dic₅	—	24	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	C₂×Dic₁₅	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	30	2	no	—
187	C₂²×C₄	#1220	1	2	yes	—
188	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
189	C₂×C₈	—	15	4	no	—
190	C₂×C₈	—	30	2	no	—
191	C₂×C₈	—	30	2	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	30	4	no	—
194	C₄⋊C₄	—	60	4	no	—
195	C₄²	C₂×Dic₁₅	1	2	yes	—
196	C₄²	#1220	1	2	yes	—
197	C₃×C₅	#1956	1	1	yes	characteristic, complemented, Hall
198	C₂²×C₃	#4082	1	1	yes	characteristic
199	C₂²×C₃	#4086	1	1	yes	characteristic
200	C₂²×C₃	C₂×#688	1	2	yes	—
201	C₂²×C₃	C₂×#689	1	2	yes	—
202	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
203	C₃×C₄	—	4	4	no	—
204	C₃×C₄	#3967	1	8	yes	complemented
205	C₃×C₄	D₄×Dic₅	1	2	yes	—
206	C₃×C₄	#4084	1	2	yes	—
207	Dic₃	—	40	2	no	—
208	C₂×C₅	#5079	1	2	yes	—
209	C₂×C₅	#5085	1	1	yes	characteristic
210	C₂×C₅	#5092	1	1	yes	characteristic
211	C₂×C₅	#5221	1	1	yes	characteristic
212	C₂×C₅	C₂×#835	1	2	yes	—
213	C₂³	C₂×#1220	1	1	yes	characteristic
214	Q₈	—	2	8	no	—
215	Q₈	—	4	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	60	2	no	—
220	C₂×C₄	—	60	4	no	—
221	C₂×C₄	#6717	1	4	yes	—
222	C₂×C₄	#6718	1	4	yes	—
223	C₂×C₄	#6719	1	4	yes	—
224	C₂×C₄	D₄×D₁₅	1	2	yes	—
225	C₂×C₄	#6819	1	2	yes	—
226	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
227	C₂×C₄	C₂×#1220	1	1	yes	characteristic
228	C₈	—	30	4	no	—
229	C₂×C₃	#65370	1	2	yes	—
230	C₂×C₃	#65374	1	1	yes	characteristic
231	C₂×C₃	#65381	1	1	yes	characteristic
232	C₂×C₃	#65510	1	1	yes	characteristic
233	C₂×C₃	C₂×#3967	1	2	yes	—
234	C₅	#71785	1	1	yes	characteristic, complemented, Sylow
235	C₂²	#91825	1	1	yes	characteristic
236	C₂²	#91829	1	1	yes	characteristic
237	C₂²	C₂×#6717	1	2	yes	—
238	C₂²	C₂×#6718	1	2	yes	—
239	C₂²	C₂×#6719	1	1	yes	characteristic
240	C₄	—	4	4	no	—
241	C₄	—	120	2	no	—
242	C₄	#91119	1	8	yes	complemented
243	C₄	D₄×Dic₁₅	1	2	yes	—
244	C₄	#91827	1	2	yes	—
245	C₃	#10602578	1	1	yes	characteristic, complemented, Sylow
246	C₂	#11751180	1	2	yes	—
247	C₂	#11751186	1	1	yes	characteristic
248	C₂	#11751193	1	1	yes	characteristic
249	C₂	#11751322	1	1	yes	characteristic
250	C₂	C₂×#91119	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	#1956
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
43. C₅×C₄⋊Q₈	Dic₃	semidirect
70. C₃×C₄⋊Q₈	Dic₅	semidirect
104. C₃×C₄×C₅	#103	semidirect
143. C₄⋊Q₈	Dic₁₅	semidirect
177. C₄×C₅	#835	semidirect
197. C₃×C₅	#1956	semidirect
204. C₃×C₄	#3967	semidirect
234. C₅	#71785	semidirect
242. C₄	#91119	semidirect
245. C₃	#10602578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×#91119	2
#11751036	2
C₂×C₄×C₃⋊#168	2
#10602578	3
#71785	5