Subgroups — #26106352

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689178	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689075	C₂	1	1	yes	maximal, characteristic
3	#20245781	C₂	1	1	yes	maximal, characteristic
4	#20241270	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241270	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241269	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1295	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	—	2	1	no	—
9	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	#15689178	—	2	2	no	—
11	#15689178	C₄	1	4	yes	complemented
12	C₂×Dic₄₇⋊C₈	—	2	1	no	—
13	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
14	#15689075	—	2	2	no	—
15	#15689075	C₄	1	4	yes	complemented
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	—	2	1	no	—
19	#15688990	—	2	1	no	—
20	#15688989	C₂²	1	1	yes	characteristic
21	#15688985	—	2	1	no	—
22	#15688980	C₄	1	2	yes	—
23	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
24	#15688970	—	2	1	no	—
25	#15688968	—	2	1	no	—
26	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#382	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
32	C₂×#10625324	—	2	8	no	—
33	C₂×#10625324	—	2	4	no	—
34	C₂²×C₄₇⋊C₈	—	2	1	no	—
35	C₂²×C₄₇⋊C₈	—	2	1	no	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
38	C₂×C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₄²×D₄₇	C₂×C₄	1	2	yes	—
43	C₄²×D₄₇	C₂×C₄	1	2	yes	—
44	C₄²×D₄₇	D₄	1	2	yes	—
45	C₄²×D₄₇	Q₈	1	2	yes	—
46	Dic₄₇⋊C₈	—	2	4	no	—
47	(C₂×D₄₇)⋊C₈	—	2	4	no	—
48	#13093935	—	2	1	no	—
49	#13093935	—	2	1	no	—
50	#13093935	C₂×C₄	1	2	yes	—
51	#13093935	C₂×C₄	1	2	yes	—
52	C₈×Dic₄₇	—	2	4	no	—
53	C₄₇⋊#95	—	2	1	no	—
54	C₄₇⋊#95	—	2	1	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#97×C₄₇	—	2	1	no	—
62	#97×C₄₇	—	2	1	no	—
63	#97×C₄₇	C₂×C₄	1	2	yes	—
64	#97×C₄₇	C₂×C₄	1	2	yes	—
65	#95×C₄₇	—	2	1	no	—
66	#95×C₄₇	—	2	1	no	—
67	C₂³×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₄₇	—	2	4	no	—
77	C₂×C₄×D₄₇	—	4	1	no	—
78	C₂×C₄×D₄₇	—	4	1	no	—
79	C₂×C₄×D₄₇	—	4	2	no	—
80	C₂×C₄×D₄₇	C₄²	1	2	yes	—
81	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	4	2	no	—
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
95	(C₂×D₄₇)⋊C₄	—	4	4	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄×Dic₄₇	—	4	4	no	—
98	C₄×Dic₄₇	C₄²	1	2	yes	—
99	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
100	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
101	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
103	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	4	2	no	—
108	#10625324	—	2	16	no	—
109	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×D₄₇	—	2	1	no	—
111	C₂²×D₄₇	—	2	1	no	—
112	C₂²×D₄₇	—	2	2	no	—
113	C₂²×D₄₇	—	2	2	no	—
114	C₂²×D₄₇	#95	1	2	yes	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	1	no	—
129	C₂×Dic₄₇	—	2	1	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	#95	1	2	yes	—
135	C₂×Dic₄₇	#95	1	2	yes	—
136	C₂×Dic₄₇	#95	1	2	yes	—
137	C₂×Dic₄₇	#104	1	4	yes	—
138	C₄×D₄₇	—	2	4	no	—
139	C₄×D₄₇	—	2	4	no	—
140	C₄×D₄₇	—	4	2	no	—
141	C₄×D₄₇	—	4	2	no	—
142	C₄×D₄₇	—	4	2	no	—
143	C₄×D₄₇	—	4	2	no	—
144	C₈×C₄₇	—	2	4	no	—
145	C₈×C₄₇	—	4	2	no	—
146	C₄₇⋊C₈	—	2	4	no	—
147	C₄₇⋊C₈	—	4	2	no	—
148	#8035	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
150	C₂²×C₄₇	#413	1	1	yes	characteristic
151	C₂²×C₄₇	#413	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#104	1	2	yes	—
153	C₂²×C₄₇	#421	1	2	yes	—
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	—	2	2	no	—
156	C₂×D₄₇	—	2	2	no	—
157	C₂×D₄₇	—	4	2	no	—
158	C₂×D₄₇	—	4	2	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	8	2	no	—
166	Dic₄₇	—	2	2	no	—
167	Dic₄₇	—	2	2	no	—
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	#339	1	4	yes	—
170	C₂×#432	—	47	1	no	—
171	C₂×C₄×M₁₆	—	47	1	no	—
172	#2283	—	47	1	no	—
173	#1295	—	47	1	no	—
174	#1295	—	47	1	no	—
175	#1295	—	47	1	no	—
176	#1295	D₄₇	1	1	yes	characteristic, complemented
177	C₂×C₄₇	#1529	1	2	yes	—
178	C₂×C₄₇	#1529	1	2	yes	—
179	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
180	C₂×C₄₇	#1733	1	1	yes	characteristic
181	C₂×C₄₇	#1742	1	1	yes	characteristic
182	D₄₇	—	4	2	no	—
183	C₂²×M₁₆	—	94	1	no	—
184	C₂²×C₄²	—	47	1	no	—
185	#432	—	47	4	no	—
186	#432	—	94	2	no	—
187	C₂×C₄⋊C₈	—	94	1	no	—
188	C₂×C₂²⋊C₈	—	94	1	no	—
189	C₄×M₁₆	—	47	4	no	—
190	C₄×M₁₆	—	94	2	no	—
191	C₂×#97	—	47	1	no	—
192	C₂×#97	—	47	1	no	—
193	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
194	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₄×C₈	—	94	1	no	—
196	#383	—	94	1	no	—
197	#382	—	47	1	no	—
198	#382	C₂×D₄₇	1	1	yes	characteristic
199	#381	—	94	1	no	—
200	#380	—	47	2	no	—
201	C₄×C₄⋊C₄	—	47	2	no	—
202	C₄×C₂²⋊C₄	—	94	1	no	—
203	#376	—	94	1	no	—
204	C₄₇	#8035	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₄	—	47	1	no	—
206	C₂³×C₄	—	47	1	no	—
207	C₂³×C₄	—	47	1	no	—
208	C₂×M₁₆	—	94	4	no	—
209	C₂×M₁₆	—	94	8	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	94	1	no	—
213	C₂²×C₈	—	94	1	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
216	C₂×C₂²⋊C₄	—	94	2	no	—
217	C₂×C₄²	—	47	1	no	—
218	C₂×C₄²	—	47	1	no	—
219	C₂×C₄²	—	47	1	no	—
220	C₂×C₄²	—	47	2	no	—
221	C₂×C₄²	—	47	2	no	—
222	C₂×C₄²	—	47	2	no	—
223	C₂×C₄²	—	47	2	no	—
224	C₂×C₄²	—	94	2	no	—
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	C₄⋊C₈	—	94	4	no	—
227	C₂²⋊C₈	—	94	4	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	—	47	2	no	—
231	#97	—	47	2	no	—
232	#97	—	94	1	no	—
233	#97	—	94	1	no	—
234	#97	C₄×D₄₇	1	2	yes	—
235	#97	C₄×D₄₇	1	2	yes	—
236	C₄×C₈	—	94	4	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	#95	—	94	2	no	—
240	#95	—	94	2	no	—
241	#95	—	94	1	no	—
242	#95	—	94	1	no	—
243	C₂⁴	—	47	1	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	47	1	no	—
246	C₂²×C₄	—	47	1	no	—
247	C₂²×C₄	—	47	1	no	—
248	C₂²×C₄	—	47	2	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	—	47	2	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	2	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	94	4	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	—	188	1	no	—
257	C₂²×C₄	—	188	1	no	—
258	C₂²×C₄	—	188	2	no	—
259	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
261	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
262	M₁₆	—	94	16	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	94	4	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	—	188	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	188	2	no	—
273	C₂²⋊C₄	—	188	4	no	—
274	C₄²	—	47	2	no	—
275	C₄²	—	47	2	no	—
276	C₄²	—	47	2	no	—
277	C₄²	—	47	2	no	—
278	C₄²	—	47	2	no	—
279	C₄²	—	47	2	no	—
280	C₄²	—	188	4	no	—
281	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
282	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
283	C₄²	D₄×D₄₇	1	1	yes	characteristic
284	C₄²	Q₈×D₄₇	1	1	yes	characteristic
285	C₂³	—	47	2	no	—
286	C₂³	—	94	1	no	—
287	C₂³	—	94	1	no	—
288	C₂³	—	94	2	no	—
289	C₂³	—	94	2	no	—
290	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	47	2	no	—
299	C₂×C₄	—	47	4	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	47	2	no	—
302	C₂×C₄	—	94	4	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	—	94	1	no	—
305	C₂×C₄	—	94	1	no	—
306	C₂×C₄	—	94	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
318	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
319	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	94	4	no	—
323	C₈	—	188	2	no	—
324	C₂²	—	94	2	no	—
325	C₂²	—	94	2	no	—
326	C₂²	—	94	2	no	—
327	C₂²	—	188	2	no	—
328	C₂²	—	188	2	no	—
329	C₂²	#95×D₄₇	1	1	yes	characteristic
330	C₂²	#15689104	1	1	yes	characteristic
331	C₂²	#15689106	1	1	yes	characteristic
332	C₂²	#104×D₄₇	1	2	yes	—
333	C₂²	#15689138	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	47	4	no	—
342	C₄	—	94	2	no	—
343	C₄	—	94	2	no	—
344	C₄	—	376	2	no	—
345	C₂	—	188	2	no	—
346	C₂	#20241992	1	2	yes	—
347	C₂	#20241994	1	2	yes	—
348	C₂	#339×D₄₇	1	1	yes	characteristic
349	C₂	#20243096	1	1	yes	characteristic
350	C₂	#20243125	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1295	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂³×D₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1295	D₄₇
hypocenter	C₄₇	#8035
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

Complements

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

N	H ≅ G/N	type
4. #20241270	C₂	semidirect
5. #20241270	C₂	semidirect
6. #20241269	C₂	semidirect
7. C₄₇×#1295	C₂	semidirect
11. #15689178	C₄	semidirect
15. #15689075	C₄	semidirect
176. #1295	D₄₇	semidirect
204. C₄₇	#8035	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689178	2
C₂×#15689075	2
#20245781	2
#20241270	2
#20241269	2
C₄₇×#1295	2
#8035	47