Subgroups — #26116364

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	#20243150	C₂	1	1	yes	maximal, characteristic
3	#20241267	C₂	1	4	yes	maximal
4	C₄₇×#1434	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#13093935	—	2	1	no	—
9	#15689029	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	C₄	1	2	yes	complemented
12	#15688993	C₄	1	1	yes	characteristic, complemented
13	#15688983	—	2	1	no	—
14	#15688983	C₂²	1	2	yes	—
15	#15688878	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#95×C₄₇	—	2	1	no	—
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	4	2	no	—
23	#13094003	—	4	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₂²⋊(C₄₇⋊C₈)	Q₈	1	4	yes	complemented
33	#13093935	—	4	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₇	—	2	2	no	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	#95×C₄₇	—	2	4	no	—
46	#95×C₄₇	—	2	4	no	—
47	#95×C₄₇	—	4	2	no	—
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	1	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂×Dic₉₄	—	4	2	no	—
75	C₂×Dic₉₄	—	4	2	no	—
76	C₂×Dic₉₄	—	4	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₂²⋊Dic₄₇	—	4	4	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄⋊Dic₄₇	—	4	2	no	—
84	C₄⋊Dic₄₇	—	4	2	no	—
85	C₄⋊Dic₄₇	—	4	2	no	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	Dic₄₇⋊C₄	—	4	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	#124	1	1	yes	characteristic
97	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
99	C₂³×C₄₇	#137	1	2	yes	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	8	2	no	—
122	C₂×C₄×C₄₇	—	8	2	no	—
123	C₂×C₄×C₄₇	—	8	2	no	—
124	C₂×C₄×C₄₇	—	8	2	no	—
125	C₂×C₄×C₄₇	#99	1	2	yes	—
126	C₂×C₄×C₄₇	#104	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	Dic₉₄	—	8	4	no	—
136	C₄₇⋊C₈	—	4	4	no	—
137	#9889	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	#391	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
148	C₂²×C₄₇	#410	1	1	yes	characteristic
149	C₂²×C₄₇	#418	1	1	yes	characteristic
150	C₂²×C₄₇	#419	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
152	C₂²×C₄₇	#421	1	1	yes	characteristic
153	C₂²×C₄₇	#489	1	2	yes	—
154	C₂²×C₄₇	#501	1	2	yes	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	4	1	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	—	8	4	no	—
162	Dic₄₇	—	8	2	no	—
163	Dic₄₇	—	8	2	no	—
164	#2857	—	47	1	no	—
165	#1749	—	47	1	no	—
166	#1434	D₄₇	1	1	yes	characteristic
167	#1294	—	47	4	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	#1505	1	1	yes	characteristic
171	C₂×C₄₇	#1508	1	1	yes	characteristic
172	C₂×C₄₇	#1510	1	1	yes	characteristic
173	C₂×C₄₇	#1515	1	1	yes	characteristic
174	C₂×C₄₇	#1986	1	1	yes	characteristic
175	C₂×C₄₇	#1993	1	1	yes	characteristic
176	C₂×C₄₇	#1996	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	—	47	1	no	—
178	C₂×C₂²⋊Q₈	—	94	1	no	—
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	C₂×C₂²⋊C₈	—	47	2	no	—
181	C₂×#97	—	94	1	no	—
182	#391	—	94	1	no	—
183	#389	—	47	2	no	—
184	#389	—	94	1	no	—
185	#384	—	47	1	no	—
186	#380	—	47	2	no	—
187	#380	—	94	1	no	—
188	C₂×#95	—	2	1	no	—
189	C₂×#95	—	2	1	no	—
190	C₂×#95	C₂×D₄₇	1	2	yes	—
191	#336	—	94	1	no	—
192	C₄₇	#9889	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₈	—	94	1	no	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₈	—	94	2	no	—
199	C₄⋊Q₈	—	188	2	no	—
200	C₂²⋊Q₈	—	188	4	no	—
201	C₂×C₄⋊C₄	—	47	2	no	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	—	94	1	no	—
205	C₂×C₄⋊C₄	—	94	1	no	—
206	C₂×C₄⋊C₄	—	94	1	no	—
207	C₂×C₂²⋊C₄	—	94	2	no	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
212	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
213	C₂²⋊C₈	—	47	4	no	—
214	C₂²⋊C₈	—	94	2	no	—
215	#97	—	188	2	no	—
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	#95	—	4	2	no	—
219	#95	—	94	2	no	—
220	#95	—	94	2	no	—
221	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂×Q₈	—	188	2	no	—
223	C₂×Q₈	—	188	2	no	—
224	C₂×Q₈	—	188	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	94	2	no	—
241	C₂²×C₄	—	94	2	no	—
242	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂²×C₄	#10625352	1	2	yes	—
249	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	188	2	no	—
252	C₂×C₈	—	188	2	no	—
253	C₄⋊C₄	—	188	2	no	—
254	C₄⋊C₄	—	188	2	no	—
255	C₄⋊C₄	—	188	4	no	—
256	C₄⋊C₄	—	188	2	no	—
257	C₄⋊C₄	—	188	2	no	—
258	C₄⋊C₄	—	188	2	no	—
259	C₂²⋊C₄	—	188	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂³	#13094011	1	1	yes	characteristic
270	C₂³	#13094022	1	1	yes	characteristic
271	C₂³	#13094049	1	2	yes	—
272	C₂³	#13094054	1	2	yes	—
273	Q₈	—	376	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	#13093938	1	2	yes	—
306	C₂×C₄	#13093952	1	2	yes	—
307	C₂×C₄	#13093989	1	2	yes	—
308	C₂×C₄	#13094004	1	2	yes	—
309	C₈	—	188	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	#15689028	1	1	yes	characteristic
319	C₂²	C₂×#13093938	1	1	yes	characteristic
320	C₂²	#15689096	1	1	yes	characteristic
321	C₂²	#15689127	1	1	yes	characteristic
322	C₂²	#15689131	1	1	yes	characteristic
323	C₂²	C₂×#13093952	1	1	yes	characteristic
324	C₂²	#15689141	1	1	yes	characteristic
325	C₂²	#15689482	1	2	yes	—
326	C₂²	#15689533	1	2	yes	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	8	4	no	—
334	C₄	—	376	2	no	—
335	C₄	—	376	2	no	—
336	C₂	—	2	2	no	—
337	C₂	—	2	2	no	—
338	C₂	#20241895	1	1	yes	characteristic
339	C₂	#20241907	1	1	yes	characteristic
340	C₂	#20241916	1	1	yes	characteristic
341	C₂	#20241938	1	1	yes	characteristic
342	C₂	#20244331	1	1	yes	characteristic
343	C₂	#20244360	1	1	yes	characteristic
344	C₂	#20244372	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1434	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1434	D₄₇
hypocenter	C₄₇	#9889
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
11. #15689021	C₄	semidirect
12. #15688993	C₄	semidirect
32. C₂²⋊(C₄₇⋊C₈)	Q₈	semidirect
192. C₄₇	#9889	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
#20243150	2
#20241267	2
C₄₇×#1434	2
#9889	47