Subgroups — #14327277

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₅	C₂	1	2	yes	maximal
3	#11751085	C₂	1	6	yes	maximal
4	#11751037	C₂	1	3	yes	maximal
5	C₄×C₄⋊Dic₁₅	C₂	1	3	yes	maximal
6	#10614191	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₄	1	8	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
11	Q₈×Dic₁₅	—	2	8	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	6	yes	—
13	C₂×C₄⋊Dic₁₅	C₂²	1	6	yes	—
14	C₂×C₄×Dic₁₅	C₂²	1	6	yes	—
15	#91115	C₂²	1	1	yes	characteristic
16	#91115	C₂²	1	3	yes	—
17	#91115	C₂²	1	6	yes	—
18	#82460	—	5	1	no	maximal, Hall
19	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic
20	C₂×Q₈×Dic₅	—	3	2	no	—
21	#65274	—	3	6	no	—
22	#65225	—	3	3	no	—
23	C₄×C₄⋊Dic₅	—	3	3	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
26	C₂×C₃×C₅×Q₈	Q₈	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
30	C₂²×Dic₁₅	C₂³	1	6	yes	—
31	C₂²×Dic₁₅	C₂³	1	2	yes	—
32	C₃×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	—
33	C₃×C₄²×C₅	C₂×C₄	1	12	yes	—
34	C₄⋊Dic₁₅	—	2	12	no	—
35	C₄⋊Dic₁₅	—	2	12	no	—
36	C₄×Dic₁₅	—	2	12	no	—
37	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic
38	C₂×Q₈×Dic₃	—	5	2	no	—
39	#4984	—	5	6	no	—
40	#4936	—	5	3	no	—
41	C₄×C₄⋊Dic₃	—	5	3	no	—
42	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
43	C₄×C₅×Q₈	Dic₃	1	8	yes	complemented
44	C₂×C₅×C₄⋊C₄	C₂×D₃	1	3	yes	—
45	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
46	Q₈×Dic₅	—	6	8	no	—
47	C₂×C₄⋊Dic₅	—	3	6	no	—
48	C₂×C₄⋊Dic₅	—	3	6	no	—
49	C₂×C₄×Dic₅	—	3	6	no	—
50	#3963	—	3	1	no	—
51	#3963	—	3	3	no	—
52	#3963	—	3	6	no	—
53	#2351	—	15	1	no	Sylow
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×Q₈	C₄⋊C₄	1	16	yes	complemented
56	C₂×C₃×C₄×C₅	—	2	6	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	6	yes	—
63	C₂×Dic₁₅	—	2	2	no	—
64	C₂×Dic₁₅	—	2	2	no	—
65	C₂×Dic₁₅	—	2	6	no	—
66	C₂×Dic₁₅	—	2	6	no	—
67	C₂×Dic₁₅	—	2	6	no	—
68	C₂×Dic₁₅	#41	1	4	yes	—
69	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
70	C₃×C₄×Q₈	Dic₅	1	8	yes	complemented
71	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
72	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
73	Q₈×Dic₃	—	10	8	no	—
74	C₂×C₄⋊Dic₃	—	5	6	no	—
75	C₂×C₄⋊Dic₃	—	5	6	no	—
76	C₂×C₄×Dic₃	—	5	6	no	—
77	C₃⋊#95	—	5	1	no	—
78	C₃⋊#95	—	5	3	no	—
79	C₃⋊#95	—	5	6	no	—
80	C₂×C₅×Q₈	Dic₆	1	4	yes	—
81	C₂×C₅×Q₈	D₁₂	1	4	yes	—
82	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
83	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
86	C₂²×Dic₅	—	3	2	no	—
87	C₂²×Dic₅	—	3	6	no	—
88	C₅×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
89	C₄²×C₅	C₂×Dic₃	1	12	yes	—
90	C₄⋊Dic₅	—	6	12	no	—
91	C₄⋊Dic₅	—	6	12	no	—
92	C₄×Dic₅	—	6	12	no	—
93	C₂×C₄×Q₈	—	15	2	no	—
94	C₂×C₄×Q₈	D₁₅	1	1	yes	characteristic
95	#389	—	15	6	no	—
96	#382	—	15	3	no	—
97	C₄×C₄⋊C₄	—	15	3	no	—
98	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
102	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
103	C₃×C₄×C₅	—	2	6	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
106	Dic₁₅	—	2	4	no	—
107	Dic₁₅	—	4	6	no	—
108	C₂×C₃×Q₈	Dic₁₀	1	4	yes	—
109	C₂×C₃×Q₈	D₂₀	1	4	yes	—
110	C₂×C₃×Q₈	C₂×Dic₅	1	4	yes	—
111	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
113	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
114	C₂²×Dic₃	—	5	2	no	—
115	C₂²×Dic₃	—	5	6	no	—
116	C₃×C₄⋊C₄	C₂×Dic₅	1	12	yes	—
117	C₃×C₄²	C₂×Dic₅	1	12	yes	—
118	C₄⋊Dic₃	—	10	12	no	—
119	C₄⋊Dic₃	—	10	12	no	—
120	C₄×Dic₃	—	10	12	no	—
121	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
122	C₅×Q₈	C₄⋊Dic₃	1	16	yes	complemented
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	6	no	—
125	C₂×C₄×C₅	C₂×Dic₆	1	6	yes	—
126	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
127	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
128	C₂×C₄×C₅	C₂²×Dic₃	1	6	yes	—
129	C₂×C₄×C₅	C₂²×Dic₃	1	6	yes	—
130	C₂×Dic₅	—	3	4	no	—
131	C₂×Dic₅	—	6	2	no	—
132	C₂×Dic₅	—	6	2	no	—
133	C₂×Dic₅	—	6	6	no	—
134	C₂×Dic₅	—	6	6	no	—
135	C₂×Dic₅	—	6	6	no	—
136	C₂²×Q₈	C₂×D₁₅	1	1	yes	characteristic
137	C₄×Q₈	—	30	8	no	—
138	C₄×Q₈	Dic₁₅	1	8	yes	complemented
139	C₂×C₄⋊C₄	—	15	6	no	—
140	C₂×C₄⋊C₄	—	15	6	no	—
141	C₂×C₄⋊C₄	C₂×D₁₅	1	3	yes	—
142	C₂×C₄²	—	15	6	no	—
143	C₂×C₄²	C₂×D₁₅	1	3	yes	—
144	#95	—	15	1	no	—
145	#95	—	15	3	no	—
146	#95	—	15	6	no	—
147	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
149	C₂×C₃×C₅	#519	1	1	yes	characteristic
150	C₂×C₃×C₅	#548	1	2	yes	—
151	C₂×C₃×C₅	#557	1	2	yes	—
152	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
153	C₃×Q₈	C₄⋊Dic₅	1	16	yes	complemented
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	6	no	—
156	C₂×C₃×C₄	C₂×Dic₁₀	1	6	yes	—
157	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
158	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
159	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
160	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
161	C₂×Dic₃	—	5	4	no	—
162	C₂×Dic₃	—	10	2	no	—
163	C₂×Dic₃	—	10	2	no	—
164	C₂×Dic₃	—	10	6	no	—
165	C₂×Dic₃	—	10	6	no	—
166	C₂×Dic₃	—	10	6	no	—
167	C₂²×C₅	C₂²×Dic₆	1	1	yes	characteristic
168	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
169	C₂²×C₅	#1007	1	2	yes	—
170	C₂²×C₅	D₃×#41	1	2	yes	—
171	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	6	no	—
174	C₄×C₅	C₂×C₄⋊Dic₃	1	12	yes	—
175	Dic₅	—	6	4	no	—
176	Dic₅	—	12	6	no	—
177	C₂×Q₈	Dic₃₀	1	4	yes	—
178	C₂×Q₈	D₆₀	1	4	yes	—
179	C₂×Q₈	C₂×Dic₁₅	1	4	yes	—
180	C₂²×C₄	—	15	2	no	—
181	C₂²×C₄	—	15	6	no	—
182	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
184	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
185	C₄⋊C₄	—	30	12	no	—
186	C₄⋊C₄	—	30	12	no	—
187	C₄⋊C₄	C₂×Dic₁₅	1	12	yes	—
188	C₄²	—	30	12	no	—
189	C₄²	C₂×Dic₁₅	1	12	yes	—
190	C₃×C₅	#2351	1	1	yes	characteristic, complemented, Hall
191	C₂²×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
192	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
193	C₂²×C₃	#4147	1	2	yes	—
194	C₂²×C₃	D₅×#41	1	2	yes	—
195	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	6	no	—
198	C₃×C₄	C₂×C₄⋊Dic₅	1	12	yes	—
199	Dic₃	—	10	4	no	—
200	Dic₃	—	20	6	no	—
201	C₂×C₅	#5569	1	2	yes	—
202	C₂×C₅	#5576	1	2	yes	—
203	C₂×C₅	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
204	C₂×C₅	#5812	1	1	yes	characteristic
205	C₂×C₅	Dic₃×#41	1	1	yes	characteristic
206	C₂³	C₂³×D₁₅	1	1	yes	characteristic
207	Q₈	C₄⋊Dic₁₅	1	16	yes	complemented
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	6	no	—
210	C₂×C₄	—	15	4	no	—
211	C₂×C₄	—	30	2	no	—
212	C₂×C₄	—	30	2	no	—
213	C₂×C₄	—	30	6	no	—
214	C₂×C₄	—	30	6	no	—
215	C₂×C₄	—	30	6	no	—
216	C₂×C₄	C₂×Dic₃₀	1	6	yes	—
217	C₂×C₄	C₂×D₆₀	1	6	yes	—
218	C₂×C₄	C₂²×Dic₁₅	1	2	yes	—
219	C₂×C₄	C₂²×Dic₁₅	1	6	yes	—
220	C₂×C₄	C₂²×Dic₁₅	1	6	yes	—
221	C₂×C₃	#65903	1	2	yes	—
222	C₂×C₃	#65910	1	2	yes	—
223	C₂×C₃	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
224	C₂×C₃	#66146	1	1	yes	characteristic
225	C₂×C₃	#41×Dic₅	1	1	yes	characteristic
226	C₅	#82460	1	1	yes	characteristic, complemented, Sylow
227	C₂²	C₂²×Dic₃₀	1	1	yes	characteristic
228	C₂²	C₂²×D₆₀	1	1	yes	characteristic
229	C₂²	#92098	1	2	yes	—
230	C₂²	#41×D₁₅	1	2	yes	—
231	C₂²	C₂³×Dic₁₅	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	6	no	—
234	C₄	—	30	4	no	—
235	C₄	—	60	6	no	—
236	C₄	C₂×C₄⋊Dic₁₅	1	12	yes	—
237	C₃	#10614191	1	1	yes	characteristic, complemented, Sylow
238	C₂	#11756328	1	2	yes	—
239	C₂	#11756335	1	2	yes	—
240	C₂	C₂²×C₄⋊Dic₁₅	1	1	yes	characteristic
241	C₂	#11756571	1	1	yes	characteristic
242	C₂	#41×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅
commutator	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₅
hypocenter	C₃×C₅	#2351
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
8. C₃×C₄×C₅×Q₈	C₄	semidirect
43. C₄×C₅×Q₈	Dic₃	semidirect
55. C₃×C₅×Q₈	C₄⋊C₄	semidirect
70. C₃×C₄×Q₈	Dic₅	semidirect
122. C₅×Q₈	C₄⋊Dic₃	semidirect
138. C₄×Q₈	Dic₁₅	semidirect
153. C₃×Q₈	C₄⋊Dic₅	semidirect
190. C₃×C₅	#2351	semidirect
207. Q₈	C₄⋊Dic₁₅	semidirect
226. C₅	#82460	semidirect
237. C₃	#10614191	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×Q₈	2
C₂×Q₈×Dic₁₅	2
#11751085	2
#11751037	2
C₄×C₄⋊Dic₁₅	2
#10614191	3
#82460	5