Subgroups — #26224418

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#125×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244358	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244213	C₂	1	1	yes	maximal, characteristic
5	#20244073	C₂	1	4	yes	maximal, complemented
6	#20243522	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243215	C₂	1	4	yes	maximal, complemented
8	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₇×#490	C₂	1	1	yes	maximal, characteristic
10	C₂×D₄×Dic₄₇	—	2	2	no	—
11	C₂×D₄×Dic₄₇	C₂²	1	2	yes	—
12	C₂×#13093945	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	1	yes	characteristic
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	#15689026	—	2	2	no	—
17	#15689026	C₂²	1	2	yes	—
18	#15689022	C₂²	1	2	yes	—
19	#15689009	—	2	2	no	—
20	#15688993	C₂²	1	1	yes	characteristic
21	#15688989	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×Dic₄₇	—	2	2	no	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
24	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	4	yes	—
26	#15688885	C₂²	1	4	yes	—
27	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#490	C₄	1	8	yes	complemented
30	C₂×#102×C₄₇	C₂²	1	4	yes	—
31	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×Dic₄₇	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	D₄×Dic₄₇	—	2	8	no	—
35	D₄×Dic₄₇	—	4	8	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
41	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
47	#13093945	—	2	8	no	—
48	#13093935	—	2	2	no	—
49	#13093934	—	2	4	no	—
50	C₈×Dic₄₇	—	2	4	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₂²×D₄×C₄₇	—	2	2	no	—
55	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	#127×C₄₇	—	2	2	no	—
58	#127×C₄₇	C₂×C₄	1	4	yes	—
59	#125×C₄₇	—	2	2	no	—
60	#125×C₄₇	C₂×C₄	1	4	yes	complemented
61	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#102×C₄₇	C₂×C₄	1	16	yes	complemented
65	#97×C₄₇	C₂×C₄	1	4	yes	complemented
66	C₂⁴×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	—	2	8	no	—
68	C₂×D₄×C₄₇	—	2	4	no	—
69	C₂×D₄×C₄₇	—	2	4	no	—
70	C₂×D₄×C₄₇	—	2	2	no	—
71	C₂×D₄×C₄₇	—	4	4	no	—
72	C₂×D₄×C₄₇	—	4	4	no	—
73	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	4	8	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
87	C₄⋊C₄×C₄₇	—	2	8	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	#41	1	2	yes	—
92	C₂²⋊Dic₄₇	—	4	4	no	—
93	C₂²⋊Dic₄₇	—	4	8	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	—	4	4	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	2	4	no	—
100	C₄×Dic₄₇	—	4	2	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₄×Dic₄₇	#41	1	4	yes	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	#41	1	4	yes	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	4	2	no	—
109	C₂³×C₄₇	—	4	4	no	—
110	C₂³×C₄₇	—	4	4	no	—
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₄₇	—	2	8	no	—
113	D₄×C₄₇	—	4	16	no	—
114	D₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	8	8	no	—
135	C₂×Dic₄₇	#124	1	2	yes	—
136	C₂×Dic₄₇	#124	1	2	yes	—
137	C₂×Dic₄₇	AGL(1,ℤ/8)	1	4	yes	—
138	C₈×C₄₇	—	2	4	no	—
139	C₄₇⋊C₈	—	2	4	no	—
140	#21656	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	4	8	no	—
142	C₂²×C₄₇	—	4	4	no	—
143	C₂²×C₄₇	—	8	2	no	—
144	C₂²×C₄₇	—	8	2	no	—
145	C₂²×C₄₇	—	8	4	no	—
146	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
148	C₂²×C₄₇	#532	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₂²×C₄₇	#575	1	2	yes	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	4	4	no	—
153	C₄×C₄₇	C₄×#41	1	2	yes	—
154	C₄×C₄₇	#529	1	2	yes	—
155	Dic₄₇	—	2	2	no	—
156	Dic₄₇	—	2	2	no	—
157	Dic₄₇	—	2	4	no	—
158	Dic₄₇	—	8	2	no	—
159	Dic₄₇	—	8	2	no	—
160	C₂×#490	D₄₇	1	1	yes	characteristic
161	C₄×#127	—	47	1	no	—
162	C₄×#125	—	47	1	no	—
163	#1992	—	47	1	no	—
164	#1961	—	47	1	no	—
165	#1934	—	47	4	no	—
166	#1836	—	47	1	no	—
167	#1762	—	47	4	no	—
168	C₄×#97	—	47	1	no	—
169	C₂×C₄₇	—	8	4	no	—
170	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
171	C₂×C₄₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₄₇	#3138	1	1	yes	characteristic
173	C₂×C₄₇	#3142	1	1	yes	characteristic
174	C₂×C₄₇	#3180	1	2	yes	—
175	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
176	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
177	C₂×C₄×D₄	—	47	2	no	—
178	C₂×C₄×D₄	—	94	2	no	—
179	#490	Dic₄₇	1	8	yes	complemented
180	C₂×#102	—	47	4	no	—
181	C₂×#102	C₂×D₄₇	1	4	yes	—
182	C₂×#97	—	47	2	no	—
183	C₂×#97	—	47	1	no	—
184	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
185	C₂×C₄×C₈	—	47	2	no	—
186	#390	—	47	2	no	—
187	#390	—	94	2	no	—
188	#389	—	47	2	no	—
189	#387	—	94	2	no	—
190	#384	—	47	1	no	—
191	#382	—	47	1	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₄×C₄⋊C₄	—	94	2	no	—
194	C₄³	—	47	1	no	—
195	#340	—	47	4	no	—
196	#340	—	47	4	no	—
197	C₄₇	#21656	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
200	C₂³×C₄	—	94	2	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
203	#127	—	2	2	no	—
204	#127	C₂×Dic₄₇	1	4	yes	—
205	#125	—	2	2	no	—
206	#125	C₂×Dic₄₇	1	4	yes	complemented
207	C₄×D₄	—	94	8	no	—
208	C₄×D₄	—	188	8	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	—	47	2	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
215	C₂×C₂²⋊C₄	—	94	2	no	—
216	C₂×C₂²⋊C₄	—	94	4	no	—
217	C₂×C₄²	—	47	1	no	—
218	C₂×C₄²	—	47	2	no	—
219	C₂×C₄²	—	47	1	no	—
220	C₂×C₄²	—	47	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#102	—	94	8	no	—
224	#102	C₂×Dic₄₇	1	16	yes	complemented
225	#97	—	94	4	no	—
226	#97	—	94	2	no	—
227	#97	C₂×Dic₄₇	1	4	yes	complemented
228	C₄×C₈	—	94	4	no	—
229	#95	—	94	2	no	—
230	#95	—	94	2	no	—
231	#95	—	94	2	no	—
232	C₂⁴	—	2	2	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	47	1	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	1	no	—
244	C₂²×C₄	—	94	2	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	188	4	no	—
247	C₂²×C₄	—	188	8	no	—
248	C₂²×C₄	D₄×D₄₇	1	2	yes	—
249	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	47	4	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	94	4	no	—
259	C₄⋊C₄	—	94	4	no	—
260	C₄⋊C₄	—	188	4	no	—
261	C₄⋊C₄	—	188	4	no	—
262	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
263	C₂²⋊C₄	—	188	4	no	—
264	C₂²⋊C₄	—	188	8	no	—
265	C₄²	—	47	4	no	—
266	C₄²	—	94	4	no	—
267	C₄²	—	94	2	no	—
268	C₄²	—	188	2	no	—
269	C₄²	—	188	4	no	—
270	C₄²	#10625352	1	2	yes	—
271	C₄²	C₂²×Dic₄₇	1	2	yes	—
272	C₂³	—	2	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
278	D₄	—	2	8	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	47	2	no	—
287	C₂×C₄	—	47	2	no	—
288	C₂×C₄	—	47	4	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	2	no	—
291	C₂×C₄	—	94	2	no	—
292	C₂×C₄	—	94	2	no	—
293	C₂×C₄	—	376	8	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	—	188	2	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
300	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
301	C₂×C₄	#41×D₄₇	1	2	yes	—
302	C₂×C₄	#41×D₄₇	1	2	yes	—
303	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	94	4	no	—
306	C₂²	—	4	8	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	4	no	—
311	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
312	C₂²	#124×D₄₇	1	1	yes	characteristic
313	C₂²	#15689672	1	1	yes	characteristic
314	C₂²	AGL(1,ℤ/8)×D₄₇	1	2	yes	—
315	C₂²	#15689919	1	2	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	94	2	no	—
319	C₄	—	94	2	no	—
320	C₄	—	94	4	no	—
321	C₄	—	376	2	no	—
322	C₄	—	376	2	no	—
323	C₄	#41×Dic₄₇	1	2	yes	—
324	C₄	#15689652	1	2	yes	—
325	C₂	—	8	4	no	—
326	C₂	#124×Dic₄₇	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×Dic₄₇	1	2	yes	—
328	C₂	#20253761	1	1	yes	characteristic
329	C₂	#20253787	1	1	yes	characteristic
330	C₂	#20254085	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#490	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#490	D₄₇
hypocenter	C₄₇	#21656
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#490

Complements

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

N	H ≅ G/N	type
2. #125×Dic₄₇	C₂	semidirect
3. #20244358	C₂	semidirect
5. #20244073	C₂	semidirect
6. #20243522	C₂	semidirect
7. #20243215	C₂	semidirect
8. #97×Dic₄₇	C₂	semidirect
29. C₄₇×#490	C₄	semidirect
60. #125×C₄₇	C₂×C₄	semidirect
64. #102×C₄₇	C₂×C₄	semidirect
65. #97×C₄₇	C₂×C₄	semidirect
179. #490	Dic₄₇	semidirect
197. C₄₇	#21656	semidirect
206. #125	C₂×Dic₄₇	semidirect
224. #102	C₂×Dic₄₇	semidirect
227. #97	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#127×Dic₄₇	2
#125×Dic₄₇	2
#20244358	2
#20244213	2
#20244073	2
#20243522	2
#20243215	2
#97×Dic₄₇	2
C₂×C₄₇×#490	2
#21656	47