Subgroups — #26224734

Order 12032.

1398 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×#15689235	C₂	1	4	yes	maximal, complemented
2	#127×Dic₄₇	C₂	1	2	yes	maximal, complemented
3	#20244265	C₂	1	4	yes	maximal, complemented
4	#20244195	C₂	1	2	yes	maximal, complemented
5	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×Dic₄₇	—	2	4	no	—
9	C₂×D₄×Dic₄₇	C₂²	1	4	yes	—
10	#15689235	—	2	16	no	—
11	C₂×#13093956	C₂²	1	4	yes	—
12	C₂×#13093945	C₂²	1	8	yes	—
13	C₂×#13093934	C₂²	1	2	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	#15689026	—	2	4	no	—
17	#15689026	C₂²	1	4	yes	—
18	#15688993	C₂²	1	2	yes	—
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂²×D₈×C₄₇	C₂²	1	4	yes	—
21	C₂×#127×C₄₇	C₂²	1	2	yes	—
22	C₄₇×#493	C₄	1	8	yes	complemented
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
26	D₄×Dic₄₇	—	2	16	no	—
27	D₄×Dic₄₇	—	4	16	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
32	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
35	#13093956	—	2	8	no	—
36	#13093945	—	2	16	no	—
37	#13093934	—	2	4	no	—
38	#13093934	D₄	1	8	yes	complemented
39	C₄×C₄₇⋊C₈	—	2	2	no	—
40	C₂²×D₄×C₄₇	—	2	4	no	—
41	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
42	C₂×D₈×C₄₇	—	2	8	no	—
43	C₂×D₈×C₄₇	—	2	8	no	—
44	C₂×D₈×C₄₇	C₂×C₄	1	16	yes	complemented
45	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
46	#127×C₄₇	—	2	4	no	—
47	#127×C₄₇	C₂×C₄	1	8	yes	complemented
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
50	C₂⁴×C₄₇	—	2	4	no	—
51	C₂×D₄×C₄₇	—	2	16	no	—
52	C₂×D₄×C₄₇	—	2	8	no	—
53	C₂×D₄×C₄₇	—	2	8	no	—
54	C₂×D₄×C₄₇	—	2	4	no	—
55	C₂×D₄×C₄₇	—	4	8	no	—
56	C₂×D₄×C₄₇	—	4	8	no	—
57	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	4	16	no	—
62	C₂²×Dic₄₇	—	4	8	no	—
63	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
64	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
65	D₈×C₄₇	—	2	32	no	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
68	C₂×C₈×C₄₇	#41	1	4	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	#41	1	2	yes	—
71	C₂²⋊Dic₄₇	—	4	16	no	—
72	C₂²⋊Dic₄₇	—	4	8	no	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄⋊Dic₄₇	—	4	8	no	—
75	C₄×Dic₄₇	—	2	8	no	—
76	C₄×Dic₄₇	—	2	2	no	—
77	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	2	2	no	—
80	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
81	C₂³×C₄₇	—	2	8	no	—
82	C₂³×C₄₇	—	4	4	no	—
83	C₂³×C₄₇	—	4	4	no	—
84	C₂³×C₄₇	—	4	8	no	—
85	C₂³×C₄₇	—	4	8	no	—
86	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	D₄×C₄₇	—	2	16	no	—
88	D₄×C₄₇	—	4	8	no	—
89	D₄×C₄₇	—	4	32	no	—
90	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₄₇	—	2	4	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	4	no	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₂×Dic₄₇	—	8	16	no	—
102	C₂×Dic₄₇	#127	1	4	yes	—
103	C₈×C₄₇	C₄×D₄	1	8	yes	complemented
104	C₄₇⋊C₈	—	2	4	no	—
105	#21690	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	4	8	no	—
107	C₂²×C₄₇	—	4	16	no	—
108	C₂²×C₄₇	—	8	4	no	—
109	C₂²×C₄₇	—	8	4	no	—
110	C₂²×C₄₇	—	8	8	no	—
111	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
113	C₂²×C₄₇	#532	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
115	C₄×C₄₇	#442	1	8	yes	complemented
116	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
117	C₄×C₄₇	#532	1	2	yes	—
118	Dic₄₇	—	2	4	no	—
119	Dic₄₇	—	4	2	no	—
120	Dic₄₇	—	8	4	no	—
121	C₂×#493	D₄₇	1	1	yes	characteristic
122	C₂×#442	—	47	4	no	—
123	C₄×#127	—	47	2	no	—
124	#1972	—	47	4	no	—
125	#1957	—	47	2	no	—
126	#1844	—	47	1	no	—
127	C₄×#97	—	47	1	no	—
128	C₂×C₄₇	—	8	8	no	—
129	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
130	C₂×C₄₇	C₂×#442	1	2	yes	—
131	C₂×C₄₇	#3151	1	2	yes	—
132	C₂×C₄₇	#3184	1	2	yes	—
133	C₂²×D₈	C₂×D₄₇	1	4	yes	—
134	C₂×#127	C₂×D₄₇	1	2	yes	—
135	C₂×C₄×D₄	—	47	4	no	—
136	C₂×C₄×D₄	—	94	4	no	—
137	#493	Dic₄₇	1	8	yes	complemented
138	#442	—	94	16	no	—
139	C₂×#106	—	47	4	no	—
140	C₂×#102	—	47	8	no	—
141	C₂×#97	—	47	2	no	—
142	C₂×#97	—	47	2	no	—
143	C₂×C₄×C₈	—	47	1	no	—
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	#390	—	47	4	no	—
146	#390	—	94	4	no	—
147	#384	—	47	2	no	—
148	C₄³	—	47	1	no	—
149	C₄₇	#21690	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	2	4	no	—
151	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
152	C₂³×C₄	—	94	4	no	—
153	C₂×D₈	—	2	8	no	—
154	C₂×D₈	—	2	8	no	—
155	C₂×D₈	C₂×Dic₄₇	1	16	yes	complemented
156	C₂²×C₈	—	47	2	no	—
157	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
158	#127	—	2	4	no	—
159	#127	C₂×Dic₄₇	1	8	yes	complemented
160	C₄×D₄	—	94	16	no	—
161	C₄×D₄	—	188	16	no	—
162	C₂×C₄⋊C₄	—	47	4	no	—
163	C₂×C₄⋊C₄	—	94	4	no	—
164	C₂×C₂²⋊C₄	—	94	4	no	—
165	C₂×C₂²⋊C₄	—	94	8	no	—
166	C₂×C₄²	—	47	1	no	—
167	C₂×C₄²	—	47	4	no	—
168	C₂×C₄²	—	47	1	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#106	—	94	8	no	—
171	#102	—	94	16	no	—
172	#97	—	47	8	no	—
173	#97	—	94	4	no	—
174	C₄×C₈	—	94	2	no	—
175	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
176	C₂⁴	—	2	4	no	—
177	C₂×D₄	—	2	16	no	—
178	C₂×D₄	—	2	8	no	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	8	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂×D₄	C₂²×Dic₄₇	1	8	yes	—
184	C₂²×C₄	—	47	2	no	—
185	C₂²×C₄	—	47	2	no	—
186	C₂²×C₄	—	94	4	no	—
187	C₂²×C₄	—	188	8	no	—
188	C₂²×C₄	—	188	16	no	—
189	C₂²×C₄	D₄×D₄₇	1	2	yes	—
190	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
191	D₈	—	2	32	no	—
192	C₂×C₈	—	47	4	no	—
193	C₂×C₈	—	94	2	no	—
194	C₂×C₈	—	94	2	no	—
195	C₂×C₈	D₄×D₄₇	1	4	yes	—
196	C₂×C₈	#10625352	1	4	yes	—
197	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
198	C₄⋊C₄	—	94	8	no	—
199	C₄⋊C₄	—	188	8	no	—
200	C₂²⋊C₄	—	188	8	no	—
201	C₂²⋊C₄	—	188	16	no	—
202	C₄²	—	47	4	no	—
203	C₄²	—	94	8	no	—
204	C₄²	—	94	2	no	—
205	C₄²	#10625352	1	2	yes	—
206	C₄²	C₂²×Dic₄₇	1	2	yes	—
207	C₂³	—	2	8	no	—
208	C₂³	—	4	8	no	—
209	C₂³	—	4	8	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	4	no	—
212	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
213	D₄	—	2	16	no	—
214	D₄	—	4	32	no	—
215	D₄	—	4	8	no	—
216	C₂×C₄	—	47	4	no	—
217	C₂×C₄	—	94	4	no	—
218	C₂×C₄	—	94	4	no	—
219	C₂×C₄	—	94	2	no	—
220	C₂×C₄	—	376	16	no	—
221	C₂×C₄	—	188	8	no	—
222	C₂×C₄	—	188	4	no	—
223	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
224	C₂×C₄	#13094048	1	8	yes	—
225	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
227	C₂×C₄	C₂×#10625352	1	2	yes	—
228	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
229	C₈	—	94	4	no	—
230	C₈	D₄×Dic₄₇	1	8	yes	complemented
231	C₂²	—	4	16	no	—
232	C₂²	—	4	8	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	8	no	—
236	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
237	C₂²	#127×D₄₇	1	1	yes	characteristic
238	C₂²	#15689671	1	1	yes	characteristic
239	C₂²	C₂×#13094048	1	4	yes	—
240	C₄	—	94	4	no	—
241	C₄	—	376	4	no	—
242	C₄	—	188	2	no	—
243	C₄	#15689235	1	8	yes	complemented
244	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
245	C₄	#15689671	1	2	yes	—
246	C₂	—	8	8	no	—
247	C₂	#127×Dic₄₇	1	1	yes	characteristic
248	C₂	C₂×#15689235	1	2	yes	—
249	C₂	#20253839	1	2	yes	—
250	C₂	#20254124	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	#21690
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#15689235	C₂	semidirect
2. #127×Dic₄₇	C₂	semidirect
3. #20244265	C₂	semidirect
4. #20244195	C₂	semidirect
5. #20243545	C₂	semidirect
6. C₄×#13093934	C₂	semidirect
22. C₄₇×#493	C₄	semidirect
38. #13093934	D₄	semidirect
44. C₂×D₈×C₄₇	C₂×C₄	semidirect
47. #127×C₄₇	C₂×C₄	semidirect
49. C₄×C₈×C₄₇	C₂×C₄	semidirect
103. C₈×C₄₇	C₄×D₄	semidirect
115. C₄×C₄₇	#442	semidirect
137. #493	Dic₄₇	semidirect
149. C₄₇	#21690	semidirect
155. C₂×D₈	C₂×Dic₄₇	semidirect
159. #127	C₂×Dic₄₇	semidirect
175. C₄×C₈	C₂×Dic₄₇	semidirect
230. C₈	D₄×Dic₄₇	semidirect
243. C₄	#15689235	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689235	2
#127×Dic₄₇	2
#20244265	2
#20244195	2
#20243545	2
C₄×#13093934	2
C₂×C₄₇×#493	2
#21690	47