Subgroups — #14331242

Order 1920.

848 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	C₃×D₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11752449	C₂	1	4	yes	maximal
3	#11752370	C₂	1	8	yes	maximal
4	#11752349	C₂	1	2	yes	maximal
5	#10617390	—	3	1	no	maximal, Hall
6	C₂×C₃×Q₈×D₅	C₂²	1	2	yes	—
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	C₃×#4090	C₄	1	2	yes	complemented
11	C₃×D₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃×#4034	C₄	1	4	yes	complemented
13	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
14	C₃×#4015	C₄	1	1	yes	characteristic, complemented
15	#91235	C₂²	1	4	yes	—
16	#91235	C₂²	1	8	yes	—
17	#91235	C₂²	1	8	yes	—
18	#83231	—	5	1	no	maximal, Hall
19	D₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
20	(C₂×AGL(1,𝔽₅))⋊Q₈	—	3	4	no	—
21	#65711	—	3	8	no	—
22	#65690	—	3	2	no	—
23	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	8	yes	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
25	C₃×Q₈×D₅	—	4	2	no	—
26	C₃×Q₈×D₅	—	4	2	no	—
27	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
30	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
31	C₂×C₃×Dic₁₀	C₂×C₄	1	4	yes	—
32	C₄⋊ΓL(1,𝔽₁₆)	—	4	4	no	—
33	C₄×ΓL(1,𝔽₁₆)	—	4	4	no	—
34	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
37	C₃×#683	C₂×C₄	1	8	yes	—
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
41	#5229	—	5	4	no	—
42	#4980	—	5	8	no	—
43	#4937	—	5	2	no	—
44	C₂×Q₈×D₅	C₂×D₃	1	2	yes	—
45	C₂×C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
46	C₂×C₄×AGL(1,𝔽₅)	—	3	4	no	—
47	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
48	#4090	Dic₃	1	2	yes	complemented
49	D₅×C₄⋊C₄	C₂×D₃	1	4	yes	—
50	#4034	Dic₃	1	4	yes	complemented
51	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
52	#4015	Dic₃	1	1	yes	characteristic, complemented
53	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	8	no	—
54	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	4	no	—
55	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	8	no	—
56	#2381	—	15	1	no	Sylow
57	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
58	C₂×ΓL(1,𝔽₁₆)	—	2	8	no	—
59	C₂×ΓL(1,𝔽₁₆)	—	2	16	no	—
60	C₃×C₅×Q₈	—	4	2	no	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
65	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
67	C₃×C₄×D₅	—	2	4	no	—
68	C₃×C₄×D₅	—	2	4	no	—
69	C₃×C₄×D₅	—	2	2	no	—
70	C₃×C₄×D₅	—	2	4	no	—
71	C₃×Dic₁₀	—	4	4	no	—
72	C₃×Dic₁₀	—	4	2	no	—
73	C₂²×C₃×Q₈	—	5	2	no	—
74	C₃×C₄⋊Q₈	—	5	2	no	—
75	C₃×C₄⋊Q₈	—	5	4	no	—
76	C₃×C₄⋊Q₈	—	5	1	no	—
77	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
78	C₂×C₃×C₄⋊C₄	—	5	4	no	—
79	C₂×C₃×C₄²	—	5	1	no	—
80	C₂×C₄⋊Dic₃	—	5	4	no	—
81	C₂×C₄×Dic₃	—	5	4	no	—
82	C₃⋊#95	—	5	4	no	—
83	C₃⋊#95	—	5	8	no	—
84	C₃⋊#95	—	5	8	no	—
85	C₂²×AGL(1,𝔽₅)	—	3	8	no	—
86	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
87	Q₈×D₅	—	4	2	no	—
88	Q₈×D₅	—	4	2	no	—
89	C₂×C₄×D₅	C₂²×D₃	1	4	yes	—
90	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
92	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
93	C₂×Dic₁₀	C₂×Dic₃	1	4	yes	—
94	C₄⋊AGL(1,𝔽₅)	—	12	4	no	—
95	C₄×AGL(1,𝔽₅)	—	12	4	no	—
96	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
97	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
98	C₄⋊Dic₅	C₂×Dic₃	1	4	yes	—
99	#683	C₂×Dic₃	1	8	yes	—
100	C₄×Dic₅	C₂×Dic₃	1	2	yes	—
101	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	—	5	1	no	—
103	#391	—	15	4	no	—
104	#387	—	15	8	no	—
105	#383	—	15	2	no	—
106	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₃×D₅	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₃×D₅	2₋¹⁺⁴	1	4	yes	—
109	ΓL(1,𝔽₁₆)	—	4	8	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	—	2	4	no	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	2	4	no	—
115	C₃×Dic₅	—	2	2	no	—
116	C₂×C₃×Q₈	—	5	4	no	—
117	C₂×C₃×Q₈	—	5	2	no	—
118	C₂×C₃×Q₈	—	20	2	no	—
119	C₂×C₃×Q₈	—	20	2	no	—
120	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
121	C₂²×C₃×C₄	—	5	2	no	—
122	C₂²×C₃×C₄	—	5	1	no	—
123	C₂²×C₃×C₄	—	5	4	no	—
124	C₂²×Dic₃	—	5	8	no	—
125	C₃×C₄⋊C₄	—	5	8	no	—
126	C₃×C₄⋊C₄	—	5	4	no	—
127	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
128	C₃×C₄²	—	5	1	no	—
129	C₃×C₄²	—	5	2	no	—
130	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₄⋊Dic₃	—	20	4	no	—
132	C₄×Dic₃	—	20	4	no	—
133	C₂²×D₅	C₂³×D₃	1	1	yes	characteristic
134	C₂×AGL(1,𝔽₅)	—	6	8	no	—
135	C₂×AGL(1,𝔽₅)	—	6	16	no	—
136	C₅×Q₈	—	4	2	no	—
137	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
138	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×Dic₃	1	4	yes	—
140	C₂×Dic₅	C₂²×Dic₃	1	2	yes	—
141	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
142	C₂×Dic₅	C₂²×Dic₃	1	4	yes	—
143	C₄×D₅	—	2	4	no	—
144	C₄×D₅	—	2	4	no	—
145	C₄×D₅	—	2	2	no	—
146	C₄×D₅	—	2	4	no	—
147	Dic₁₀	—	4	4	no	—
148	Dic₁₀	—	4	2	no	—
149	C₂²×Q₈	—	5	2	no	—
150	C₄⋊Q₈	—	5	2	no	—
151	C₄⋊Q₈	—	5	4	no	—
152	C₄⋊Q₈	—	5	1	no	—
153	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
154	C₂×C₄⋊C₄	—	5	4	no	—
155	C₂×C₄⋊C₄	—	15	4	no	—
156	C₂×C₄²	—	5	1	no	—
157	C₂×C₄²	—	15	4	no	—
158	#95	—	15	8	no	—
159	#95	—	15	4	no	—
160	#95	—	15	8	no	—
161	C₂×C₃×C₅	#518	1	1	yes	characteristic
162	C₂×C₃×C₅	#519	1	2	yes	—
163	C₃×D₅	#563	1	4	yes	—
164	C₂³×C₃	—	5	1	no	—
165	C₃×Q₈	—	4	2	no	—
166	C₃×Q₈	—	20	2	no	—
167	C₃×Q₈	—	20	4	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	4	no	—
171	C₂×C₃×C₄	—	10	4	no	—
172	C₂×C₃×C₄	—	10	4	no	—
173	C₂×C₃×C₄	—	10	4	no	—
174	C₂×C₃×C₄	—	10	2	no	—
175	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
176	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
178	C₂×Dic₃	—	10	8	no	—
179	C₂×Dic₃	—	10	16	no	—
180	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
181	C₂×D₅	#1004	1	2	yes	—
182	C₂×D₅	#1007	1	4	yes	—
183	AGL(1,𝔽₅)	—	12	8	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	2	2	no	—
187	Dic₅	—	2	1	no	—
188	Dic₅	—	2	4	no	—
189	Dic₅	—	2	2	no	—
190	C₂×Q₈	—	5	4	no	—
191	C₂×Q₈	—	5	2	no	—
192	C₂×Q₈	—	20	2	no	—
193	C₂×Q₈	—	20	2	no	—
194	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
195	C₂²×C₄	—	5	2	no	—
196	C₂²×C₄	—	5	1	no	—
197	C₂²×C₄	—	5	4	no	—
198	C₂²×C₄	—	15	8	no	—
199	C₄⋊C₄	—	5	8	no	—
200	C₄⋊C₄	—	5	4	no	—
201	C₄⋊C₄	—	60	4	no	—
202	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
203	C₄²	—	5	1	no	—
204	C₄²	—	5	2	no	—
205	C₄²	—	60	4	no	—
206	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
207	C₃×C₅	#2381	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	5	2	no	—
209	C₂²×C₃	—	5	4	no	—
210	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	10	4	no	—
215	C₃×C₄	—	10	1	no	—
216	C₃×C₄	—	10	2	no	—
217	Dic₃	—	20	8	no	—
218	C₂×C₅	#5798	1	1	yes	characteristic
219	C₂×C₅	#5812	1	2	yes	—
220	D₅	#5735	1	4	yes	—
221	C₂³	—	5	1	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	20	2	no	—
224	Q₈	—	20	4	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	5	1	no	—
227	C₂×C₄	—	5	4	no	—
228	C₂×C₄	—	10	4	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	4	no	—
232	C₂×C₄	—	30	8	no	—
233	C₂×C₄	—	30	16	no	—
234	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
235	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
236	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	4	yes	—
237	C₂×C₃	—	5	4	no	—
238	C₂×C₃	#66258	1	1	yes	characteristic
239	C₂×C₃	#66262	1	2	yes	—
240	C₅	#83231	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	5	2	no	—
242	C₂²	—	5	4	no	—
243	C₂²	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	10	1	no	—
248	C₄	—	10	4	no	—
249	C₄	—	10	2	no	—
250	C₄	—	60	8	no	—
251	C₃	#10617390	1	1	yes	characteristic, complemented, Sylow
252	C₂	—	5	4	no	—
253	C₂	#11756987	1	1	yes	characteristic
254	C₂	#11756991	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₃×C₅×C₄⋊Q₈	C₄
prime radical, cosocle	C₂²×C₃×D₅	C₂⁴
hypercenter	C₄⋊Q₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2381
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
10. C₃×#4090	C₄	semidirect
12. C₃×#4034	C₄	semidirect
14. C₃×#4015	C₄	semidirect
47. C₅×C₄⋊Q₈	Dic₃	semidirect
48. #4090	Dic₃	semidirect
50. #4034	Dic₃	semidirect
52. #4015	Dic₃	semidirect
77. C₃×C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
153. C₄⋊Q₈	ΓL(1,𝔽₁₆)	semidirect
207. C₃×C₅	#2381	semidirect
240. C₅	#83231	semidirect
251. C₃	#10617390	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₄⋊Q₈	2
#11752449	2
#11752370	2
#11752349	2
#10617390	3
#83231	5