Subgroups — #26220514

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245853	C₂	1	2	yes	maximal, complemented
2	#20244265	C₂	1	2	yes	maximal
3	#20244127	C₂	1	2	yes	maximal, complemented
4	#20244087	C₂	1	4	yes	maximal, complemented
5	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093945	C₂	1	2	yes	maximal, complemented
7	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	—	2	2	no	—
10	C₂×#13093956	C₂²	1	2	yes	—
11	C₂×#13093945	C₂²	1	4	yes	—
12	C₂×#13093934	C₂²	1	2	yes	—
13	C₂×#13093934	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	#15689026	C₂²	1	4	yes	—
16	#15689022	C₂²	1	4	yes	—
17	#15689016	—	2	4	no	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
20	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688886	C₂²	1	2	yes	—
22	#15688885	C₂²	1	4	yes	—
23	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
25	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	—	4	8	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
34	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093956	—	2	4	no	—
40	#13093945	—	2	8	no	—
41	#13093934	—	2	4	no	—
42	#13093934	—	2	4	no	—
43	C₄×C₄₇⋊C₈	—	2	2	no	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₇	—	2	2	no	—
48	C₂×D₈×C₄₇	—	2	4	no	—
49	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₄₇	—	2	4	no	—
53	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#107×C₄₇	C₂×C₄	1	4	yes	—
58	#102×C₄₇	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	—	2	2	no	—
61	C₂×D₄×C₄₇	—	2	4	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	8	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
68	C₂²×C₄×C₄₇	—	4	4	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
74	D₈×C₄₇	—	2	8	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	#41	1	4	yes	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²⋊C₄×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	#41	1	2	yes	—
84	C₂²⋊Dic₄₇	—	4	8	no	—
85	C₂²⋊Dic₄₇	—	4	8	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	4	8	no	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	2	2	no	—
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	4	2	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	#41	1	4	yes	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	—	4	4	no	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	D₄×C₄₇	—	2	4	no	—
104	D₄×C₄₇	—	2	4	no	—
105	D₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	16	no	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	#117	1	4	yes	complemented
126	C₄₇⋊C₈	—	2	4	no	—
127	#21277	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	4	4	no	—
129	C₂²×C₄₇	—	4	4	no	—
130	C₂²×C₄₇	—	4	4	no	—
131	C₂²×C₄₇	—	8	4	no	—
132	C₂²×C₄₇	—	8	4	no	—
133	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₄₇	#532	1	2	yes	—
135	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
136	C₂²×C₄₇	#575	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	#442	1	4	yes	—
140	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
141	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
142	C₄×C₄₇	#529	1	2	yes	—
143	Dic₄₇	—	4	4	no	—
144	Dic₄₇	—	8	4	no	—
145	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
146	#2293	—	47	2	no	—
147	#1972	—	47	2	no	—
148	#1943	—	47	2	no	—
149	#1936	—	47	4	no	—
150	C₄×#106	—	47	1	no	—
151	C₄×#102	—	47	2	no	—
152	C₄×#97	—	47	1	no	—
153	C₂×C₄₇	—	8	4	no	—
154	C₂×C₄₇	#2293	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
156	C₂×C₄₇	#2943	1	1	yes	characteristic
157	C₂×C₄₇	#3171	1	4	yes	—
158	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
160	C₂×#117	—	94	2	no	—
161	C₄×D₈	Dic₄₇	1	8	yes	complemented
162	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#106	—	47	2	no	—
164	C₂×#102	—	47	4	no	—
165	C₂×#102	C₂×D₄₇	1	2	yes	—
166	C₂×#97	—	47	2	no	—
167	C₂×#97	—	47	2	no	—
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
170	#390	—	47	4	no	—
171	#389	—	47	4	no	—
172	#388	—	94	4	no	—
173	C₄×C₄⋊C₄	—	47	2	no	—
174	C₄×C₂²⋊C₄	—	94	2	no	—
175	C₄³	—	47	1	no	—
176	#340	—	47	2	no	—
177	#340	—	47	4	no	—
178	C₄₇	#21277	1	1	yes	characteristic, complemented, Sylow
179	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
180	C₂³×C₄	—	2	2	no	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₄×D₄	—	2	4	no	—
187	C₄×D₄	C₂×Dic₄₇	1	8	yes	—
188	#117	—	188	8	no	—
189	C₂×C₄⋊C₄	—	47	4	no	—
190	C₂×C₄⋊C₄	—	94	4	no	—
191	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	94	4	no	—
194	C₂×C₂²⋊C₄	—	94	4	no	—
195	C₂×C₄²	—	47	2	no	—
196	C₂×C₄²	—	47	2	no	—
197	C₂×C₄²	—	47	2	no	—
198	C₂×C₄²	—	94	2	no	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#107	C₂×Dic₄₇	1	4	yes	—
201	#106	—	94	4	no	—
202	#102	—	94	8	no	—
203	#102	C₂×Dic₄₇	1	8	yes	—
204	#97	—	94	4	no	—
205	#97	—	94	4	no	—
206	C₄×C₈	—	94	2	no	—
207	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
208	#95	—	94	4	no	—
209	#95	—	94	2	no	—
210	C₂⁴	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	47	4	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	D₈	—	2	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	47	4	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	#10625350	1	4	yes	—
230	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
231	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₄⋊C₄	—	188	8	no	—
236	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	188	8	no	—
239	C₂²⋊C₄	—	188	8	no	—
240	C₄²	—	94	4	no	—
241	C₄²	—	94	2	no	—
242	C₄²	—	94	2	no	—
243	C₄²	—	188	2	no	—
244	C₄²	—	188	4	no	—
245	C₄²	#10625352	1	2	yes	—
246	C₄²	C₂²×Dic₄₇	1	2	yes	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	4	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	94	4	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	94	4	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
268	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
269	C₂×C₄	#13094048	1	4	yes	—
270	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
271	C₂×C₄	#41×D₄₇	1	2	yes	—
272	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
273	C₂×C₄	C₂×#10625350	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	94	4	no	—
276	C₈	#13093983	1	4	yes	complemented
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
283	C₂²	#15689670	1	2	yes	—
284	C₂²	C₂×#13094048	1	2	yes	—
285	C₂²	#15689920	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	376	4	no	—
289	C₄	—	188	4	no	—
290	C₄	#15689235	1	4	yes	—
291	C₄	C₂×#13093983	1	1	yes	characteristic
292	C₄	#41×Dic₄₇	1	1	yes	characteristic
293	C₄	#15689655	1	2	yes	—
294	C₂	—	8	4	no	—
295	C₂	#20245853	1	1	yes	characteristic
296	C₂	C₂×#15689235	1	1	yes	characteristic
297	C₂	#20252189	1	1	yes	characteristic
298	C₂	#20253981	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#21277
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20245853	C₂	semidirect
3. #20244127	C₂	semidirect
4. #20244087	C₂	semidirect
5. C₄×#13093956	C₂	semidirect
6. C₄×#13093945	C₂	semidirect
7. C₄×#13093934	C₂	semidirect
25. C₄×D₈×C₄₇	C₄	semidirect
125. C₈×C₄₇	#117	semidirect
161. C₄×D₈	Dic₄₇	semidirect
178. C₄₇	#21277	semidirect
276. C₈	#13093983	semidirect

Maximal subgroups

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

subgroup	index
#20245853	2
#20244265	2
#20244127	2
#20244087	2
C₄×#13093956	2
C₄×#13093945	2
C₄×#13093934	2
C₂×C₄×D₈×C₄₇	2
#21277	47