Subgroups — #26233974

Order 12032.

866 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
2	#20246255	C₂	1	2	yes	maximal
3	#20244126	C₂	1	2	yes	maximal, complemented
4	#20243981	C₂	1	4	yes	maximal, complemented
5	#20243475	C₂	1	2	yes	maximal
6	#20243413	C₂	1	1	yes	maximal, characteristic
7	#20243285	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093980	—	2	2	no	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×#13093958	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093956	C₂²	1	2	yes	—
16	#15689021	C₂²	1	4	yes	—
17	#15689008	—	2	4	no	—
18	#15688984	—	2	2	no	—
19	#15688983	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
21	#15688886	C₂²	1	2	yes	—
22	#15688886	C₂²	1	4	yes	—
23	#15688886	C₂²	1	4	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#336	C₂²	1	2	yes	—
29	C₂³×Dic₄₇	—	2	2	no	—
30	#13093980	—	4	8	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
37	C₂×Dic₄₇⋊C₄	—	2	4	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	4	no	—
40	#13093958	C₂×C₄	1	8	yes	—
41	#13093958	C₂×C₄	1	4	yes	—
42	#13093958	C₂×C₄	1	4	yes	—
43	#13093956	—	2	4	no	—
44	C₄₇⋊#95	—	2	4	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	—	2	4	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂²⋊C₈×C₄₇	—	2	2	no	—
56	#97×C₄₇	—	2	4	no	—
57	#95×C₄₇	D₄	1	2	yes	—
58	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	8	no	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	4	yes	—
80	C₂²⋊C₄×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₂²⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	2	8	no	—
85	C₄⋊Dic₄₇	—	2	2	no	—
86	C₄⋊Dic₄₇	—	2	2	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
90	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
91	Dic₄₇⋊C₄	—	4	8	no	—
92	C₄×Dic₄₇	—	4	8	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	C₄×D₄	1	2	yes	—
96	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
97	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	16	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	—	4	2	no	—
122	C₈×C₄₇	#117	1	4	yes	complemented
123	#22617	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	8	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	#442	1	4	yes	—
128	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₇	#547	1	2	yes	—
130	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₂²×C₄₇	#578	1	2	yes	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	4	4	no	—
134	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
135	C₄×C₄₇	#519	1	1	yes	characteristic
136	C₄×C₄₇	#552	1	2	yes	—
137	Dic₄₇	—	4	4	no	—
138	Dic₄₇	—	4	4	no	—
139	Dic₄₇	—	8	4	no	—
140	C₂²×#107	—	47	1	no	—
141	#2344	—	47	2	no	—
142	#1943	—	47	2	no	—
143	#1921	—	47	4	no	—
144	#1823	—	47	2	no	—
145	#1807	—	47	1	no	—
146	#1776	—	47	2	no	—
147	#1752	D₄₇	1	1	yes	characteristic
148	C₂×C₄₇	—	2	4	no	—
149	C₂×C₄₇	#2344	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
151	C₂×C₄₇	#2950	1	1	yes	characteristic
152	C₂×C₄₇	#3343	1	4	yes	—
153	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×#117	—	94	2	no	—
155	C₂²×C₄⋊C₄	—	47	2	no	—
156	C₂×#107	—	47	8	no	—
157	C₂×#107	—	47	2	no	—
158	C₂×#107	—	47	1	no	—
159	C₂×#107	—	47	1	no	—
160	C₂×#106	—	47	2	no	—
161	C₂×C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#97	C₂×D₄₇	1	2	yes	—
163	#389	—	47	4	no	—
164	#387	—	94	4	no	—
165	#381	—	94	2	no	—
166	#380	—	47	2	no	—
167	C₄×C₄⋊C₄	—	47	4	no	—
168	C₄×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
169	#340	—	47	2	no	—
170	#340	—	47	4	no	—
171	#340	—	47	4	no	—
172	#336	C₂×D₄₇	1	2	yes	—
173	C₄₇	#22617	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	94	2	no	—
175	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	C₄×D₄₇	1	4	yes	—
178	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	#117	—	188	8	no	—
182	C₂×C₄⋊C₄	—	47	8	no	—
183	C₂×C₄⋊C₄	—	47	2	no	—
184	C₂×C₄⋊C₄	—	47	2	no	—
185	C₂×C₄⋊C₄	—	47	4	no	—
186	C₂×C₄⋊C₄	—	94	2	no	—
187	C₂×C₄⋊C₄	—	94	4	no	—
188	C₂×C₄⋊C₄	—	94	4	no	—
189	C₂×C₂²⋊C₄	—	94	2	no	—
190	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
191	C₂×C₄²	—	94	4	no	—
192	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
193	#107	—	47	8	no	—
194	#107	—	47	4	no	—
195	#107	—	47	4	no	—
196	#106	—	94	4	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	4	no	—
199	#95	—	94	4	no	—
200	#95	—	94	2	no	—
201	#95	—	94	4	no	—
202	#95	D₁₈₈	1	2	yes	—
203	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	94	8	no	—
207	C₂²×C₄	—	94	2	no	—
208	C₂²×C₄	—	94	2	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	188	4	no	—
211	C₂²×C₄	—	188	4	no	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
214	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
215	C₂²×C₄	#10625350	1	2	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
224	C₂×C₈	#10625352	1	4	yes	—
225	C₄⋊C₄	—	47	4	no	—
226	C₄⋊C₄	—	47	4	no	—
227	C₄⋊C₄	—	94	8	no	—
228	C₄⋊C₄	—	94	2	no	—
229	C₄⋊C₄	—	94	2	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₄⋊C₄	—	94	4	no	—
232	C₄⋊C₄	—	188	8	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	188	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	188	8	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₄×D₁₈₈	1	2	yes	—
241	C₂³	#13094052	1	4	yes	—
242	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	94	4	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	—	188	16	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	—	188	2	no	—
258	C₂×C₄	—	188	2	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
261	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#10625352	1	2	yes	—
263	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
264	C₂×C₄	#13094071	1	2	yes	—
265	C₈	—	2	2	no	—
266	C₈	—	4	2	no	—
267	C₈	#13093980	1	4	yes	complemented
268	C₂²	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	#15689236	1	4	yes	—
272	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
273	C₂²	#15689777	1	2	yes	—
274	C₂²	C₂×#13094052	1	2	yes	—
275	C₂²	#15689940	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	376	4	no	—
279	C₄	—	188	4	no	—
280	C₄	—	188	4	no	—
281	C₄	C₂×#13093980	1	1	yes	characteristic
282	C₄	#15689600	1	1	yes	characteristic
283	C₄	#15689813	1	2	yes	—
284	C₂	—	2	4	no	—
285	C₂	#20246255	1	1	yes	characteristic
286	C₂	C₂×#15689236	1	1	yes	characteristic
287	C₂	#20252257	1	1	yes	characteristic
288	C₂	#20255830	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1752	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1752	D₄₇
hypocenter	C₄₇	#22617
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ #1752

Complements

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

N	H ≅ G/N	type
3. #20244126	C₂	semidirect
4. #20243981	C₂	semidirect
7. #20243285	C₂	semidirect
11. C₂×#13093958	C₄	semidirect
122. C₈×C₄₇	#117	semidirect
173. C₄₇	#22617	semidirect
267. C₈	#13093980	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093958	2
#20246255	2
#20244126	2
#20243981	2
#20243475	2
#20243413	2
#20243285	2
C₄₇×#1752	2
#22617	47