Subgroups — #26116459

Order 12032.

592 conjugacy classes of subgroups.

The subgroup lattice has 354 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	#20241615	C₂	1	1	yes	maximal, characteristic
3	#20241268	C₂	1	4	yes	maximal
4	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	#15689029	—	2	1	no	—
8	#15689021	—	2	1	no	—
9	#15689021	C₄	1	2	yes	complemented
10	#15688993	C₄	1	1	yes	characteristic, complemented
11	#15688983	—	2	1	no	—
12	#15688983	C₂²	1	2	yes	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	C₂×C₄₇⋊#95	—	2	1	no	—
15	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
18	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂³×Dic₄₇	—	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	C₄₇⋊#95	—	2	2	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	4	2	no	—
35	C₄₇⋊#95	—	4	2	no	—
36	C₄₇⋊#95	Q₈	1	4	yes	complemented
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	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	4	2	no	—
47	C₂²⋊C₈×C₄₇	—	2	4	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	—	4	1	no	—
52	C₂²×C₄×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	—	4	1	no	—
54	C₂²×C₄×C₄₇	—	4	1	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	4	2	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₂×Dic₉₄	—	4	2	no	—
76	C₂×Dic₉₄	—	4	2	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	4	1	no	—
84	C₄²×C₄₇	—	4	1	no	—
85	C₂²⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	Dic₄₇⋊C₄	—	4	4	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
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	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	#99	1	2	yes	—
120	C₂×C₄×C₄₇	#103	1	2	yes	—
121	C₂×C₄×C₄₇	#103	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
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	4	no	—
134	C₂×Dic₄₇	—	8	2	no	—
135	C₂×Dic₄₇	—	8	2	no	—
136	C₂×Dic₄₇	—	8	2	no	—
137	C₂×Dic₄₇	—	8	2	no	—
138	Dic₉₄	—	8	4	no	—
139	C₈×C₄₇	—	8	2	no	—
140	#9911	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	#391	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
151	C₂²×C₄₇	#410	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
153	C₂²×C₄₇	#418	1	1	yes	characteristic
154	C₂²×C₄₇	#421	1	1	yes	characteristic
155	C₂²×C₄₇	#421	1	1	yes	characteristic
156	C₂²×C₄₇	#476	1	2	yes	—
157	C₂²×C₄₇	#477	1	2	yes	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	4	1	no	—
165	Dic₄₇	—	8	4	no	—
166	Dic₄₇	—	8	2	no	—
167	Dic₄₇	—	8	2	no	—
168	#2857	—	47	1	no	—
169	#1797	D₄₇	1	1	yes	characteristic
170	#1434	—	47	1	no	—
171	#1294	—	47	4	no	—
172	C₂×C₄₇	—	2	2	no	—
173	C₂×C₄₇	—	2	2	no	—
174	C₂×C₄₇	#1510	1	1	yes	characteristic
175	C₂×C₄₇	#1514	1	1	yes	characteristic
176	C₂×C₄₇	#1518	1	1	yes	characteristic
177	C₂×C₄₇	#1520	1	1	yes	characteristic
178	C₂×C₄₇	#1987	1	1	yes	characteristic
179	C₂×C₄₇	#1995	1	1	yes	characteristic
180	C₂×C₄₇	#1996	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	—	47	1	no	—
182	C₂×C₂²⋊Q₈	—	94	1	no	—
183	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄⋊C₈	—	2	1	no	—
185	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
186	#391	—	94	1	no	—
187	#389	—	47	2	no	—
188	#389	—	94	1	no	—
189	#384	—	47	1	no	—
190	#380	—	47	2	no	—
191	#380	—	94	1	no	—
192	C₂×#95	—	47	2	no	—
193	C₂×#95	—	94	1	no	—
194	C₂×#95	—	94	1	no	—
195	#336	—	2	1	no	—
196	C₄₇	#9911	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	94	1	no	—
198	C₂³×C₄	—	94	2	no	—
199	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
200	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	—	2	2	no	—
203	C₄⋊Q₈	—	188	2	no	—
204	C₂²⋊Q₈	—	188	4	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	94	1	no	—
209	C₂×C₄⋊C₄	—	94	1	no	—
210	C₂×C₄⋊C₄	—	94	1	no	—
211	C₂×C₂²⋊C₄	—	94	2	no	—
212	C₂×C₄²	—	4	2	no	—
213	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₄⋊C₈	—	4	2	no	—
218	C₂²⋊C₈	—	2	4	no	—
219	#95	—	47	4	no	—
220	#95	—	94	2	no	—
221	#95	—	94	2	no	—
222	#95	—	94	2	no	—
223	#95	—	188	2	no	—
224	#95	—	188	2	no	—
225	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
226	C₂×Q₈	—	188	2	no	—
227	C₂×Q₈	—	188	2	no	—
228	C₂×Q₈	—	188	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	94	2	no	—
237	C₂²×C₄	—	94	2	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	—	188	2	no	—
242	C₂²×C₄	—	188	2	no	—
243	C₂²×C₄	—	188	2	no	—
244	C₂²×C₄	—	188	2	no	—
245	C₂²×C₄	#10625325	1	2	yes	—
246	C₂²×C₄	Dic₁₈₈	1	2	yes	—
247	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	#10625352	1	2	yes	—
253	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₄⋊C₄	—	188	2	no	—
259	C₄⋊C₄	—	188	2	no	—
260	C₄⋊C₄	—	188	4	no	—
261	C₄⋊C₄	—	188	2	no	—
262	C₄⋊C₄	—	188	2	no	—
263	C₄⋊C₄	—	188	2	no	—
264	C₂²⋊C₄	—	188	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
274	C₂³	#13094004	1	2	yes	—
275	C₂³	C₂×#10625325	1	1	yes	characteristic
276	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
277	C₂³	#13094052	1	2	yes	—
278	Q₈	—	376	4	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	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	376	2	no	—
298	C₂×C₄	—	376	2	no	—
299	C₂×C₄	—	376	2	no	—
300	C₂×C₄	—	376	2	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	4	no	—
310	C₂×C₄	#13093938	1	2	yes	—
311	C₂×C₄	#13093949	1	2	yes	—
312	C₂×C₄	#13093949	1	2	yes	—
313	C₂×C₄	#13093989	1	2	yes	—
314	C₂×C₄	#13094011	1	1	yes	characteristic
315	C₂×C₄	#13094022	1	1	yes	characteristic
316	C₈	—	8	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	#15689028	1	1	yes	characteristic
326	C₂²	C₂×#13093938	1	1	yes	characteristic
327	C₂²	#15689096	1	1	yes	characteristic
328	C₂²	C₂×#13093949	1	1	yes	characteristic
329	C₂²	#15689129	1	1	yes	characteristic
330	C₂²	#15689139	1	1	yes	characteristic
331	C₂²	#15689139	1	1	yes	characteristic
332	C₂²	#15689414	1	2	yes	—
333	C₂²	#15689419	1	2	yes	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	376	4	no	—
342	C₄	—	376	2	no	—
343	C₄	—	376	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	2	2	no	—
346	C₂	#20241918	1	1	yes	characteristic
347	C₂	#20241936	1	1	yes	characteristic
348	C₂	#20241953	1	1	yes	characteristic
349	C₂	#20241961	1	1	yes	characteristic
350	C₂	#20244335	1	1	yes	characteristic
351	C₂	#20244368	1	1	yes	characteristic
352	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×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#9911
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
9. #15689021	C₄	semidirect
10. #15688993	C₄	semidirect
36. C₄₇⋊#95	Q₈	semidirect
196. C₄₇	#9911	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
#20241615	2
#20241268	2
C₄₇×#1797	2
#9911	47