Subgroups — #26106340

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 348 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×#15689178	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	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
9	#15689178	—	2	2	no	—
10	#15689178	—	2	2	no	—
11	#15689178	C₄	1	4	yes	complemented
12	#15689178	C₄	1	4	yes	complemented
13	C₂×Dic₄₇⋊C₈	—	2	1	no	—
14	C₂×Dic₄₇⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
16	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093935	C₂²	1	1	yes	characteristic
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	C₂²×C₄₇⋊C₈	—	2	1	no	—
33	C₂²×C₄₇⋊C₈	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
40	C₄²×D₄₇	C₂×C₄	1	2	yes	—
41	C₄²×D₄₇	C₂×C₄	1	2	yes	—
42	C₄²×D₄₇	D₄	1	2	yes	—
43	C₄²×D₄₇	Q₈	1	2	yes	—
44	Dic₄₇⋊C₈	—	2	4	no	—
45	Dic₄₇⋊C₈	—	2	4	no	—
46	(C₂×D₄₇)⋊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	C₄₇⋊#95	—	2	1	no	—
53	C₄₇⋊#95	—	2	1	no	—
54	C₄₇⋊#95	—	2	2	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₂²×C₈×C₄₇	—	2	1	no	—
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#97×C₄₇	—	2	1	no	—
61	#97×C₄₇	—	2	1	no	—
62	#97×C₄₇	C₂×C₄	1	2	yes	—
63	#97×C₄₇	C₂×C₄	1	2	yes	—
64	#95×C₄₇	—	2	1	no	—
65	#95×C₄₇	—	2	1	no	—
66	C₂³×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	—	2	2	no	—
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×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₄₇	—	4	1	no	—
76	C₂×C₄×D₄₇	—	4	1	no	—
77	C₂×C₄×D₄₇	—	4	2	no	—
78	C₂×C₄×D₄₇	—	4	2	no	—
79	C₂×C₄×D₄₇	C₄²	1	2	yes	—
80	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	4	2	no	—
88	C₂×C₈×C₄₇	—	4	2	no	—
89	C₄⋊C₄×C₄₇	—	4	2	no	—
90	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
94	(C₂×D₄₇)⋊C₄	—	4	4	no	—
95	C₄⋊Dic₄₇	—	4	2	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₄×Dic₄₇	C₄²	1	2	yes	—
98	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
99	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
100	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
101	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	4	2	no	—
106	C₂×C₄₇⋊C₈	—	4	2	no	—
107	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×D₄₇	—	2	1	no	—
109	C₂²×D₄₇	—	2	1	no	—
110	C₂²×D₄₇	—	2	2	no	—
111	C₂²×D₄₇	—	2	2	no	—
112	C₂²×D₄₇	#95	1	2	yes	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	1	no	—
127	C₂×Dic₄₇	—	2	1	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	#95	1	2	yes	—
134	C₂×Dic₄₇	#95	1	2	yes	—
135	C₂×Dic₄₇	#95	1	2	yes	—
136	C₄×D₄₇	—	4	2	no	—
137	C₄×D₄₇	—	4	2	no	—
138	C₄×D₄₇	—	4	2	no	—
139	C₄×D₄₇	—	4	2	no	—
140	C₄×D₄₇	—	4	2	no	—
141	C₄×D₄₇	—	4	2	no	—
142	C₈×C₄₇	—	4	2	no	—
143	C₈×C₄₇	—	4	2	no	—
144	C₄₇⋊C₈	—	4	2	no	—
145	C₄₇⋊C₈	—	4	2	no	—
146	#8032	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₄₇	#413	1	1	yes	characteristic
149	C₂²×C₄₇	#413	1	1	yes	characteristic
150	C₂²×C₄₇	#421	1	2	yes	—
151	C₂²×C₄₇	#421	1	2	yes	—
152	C₂×D₄₇	—	2	2	no	—
153	C₂×D₄₇	—	2	2	no	—
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	—	4	2	no	—
156	C₂×D₄₇	—	4	2	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	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	—	8	2	no	—
164	Dic₄₇	—	2	2	no	—
165	Dic₄₇	—	2	2	no	—
166	Dic₄₇	—	2	2	no	—
167	Dic₄₇	—	8	2	no	—
168	C₂×#432	—	47	1	no	—
169	C₂×#432	—	47	1	no	—
170	#2283	—	47	1	no	—
171	#1295	—	47	1	no	—
172	#1295	—	47	1	no	—
173	#1295	—	47	1	no	—
174	#1295	D₄₇	1	1	yes	characteristic, complemented
175	C₂×C₄₇	#1535	1	2	yes	—
176	C₂×C₄₇	#1535	1	2	yes	—
177	C₂×C₄₇	#1733	1	1	yes	characteristic
178	C₂×C₄₇	#1733	1	1	yes	characteristic
179	C₂×C₄₇	#1742	1	1	yes	characteristic
180	D₄₇	—	4	2	no	—
181	C₂²×C₄²	—	47	1	no	—
182	#432	—	47	4	no	—
183	#432	—	47	4	no	—
184	#432	—	94	2	no	—
185	#432	—	94	2	no	—
186	C₂×C₄⋊C₈	—	94	1	no	—
187	C₂×C₄⋊C₈	—	94	1	no	—
188	C₂×C₂²⋊C₈	—	94	1	no	—
189	C₂×C₂²⋊C₈	—	94	1	no	—
190	C₂×#97	—	47	1	no	—
191	C₂×#97	—	47	1	no	—
192	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
194	#383	—	94	1	no	—
195	#382	—	47	1	no	—
196	#382	C₂×D₄₇	1	1	yes	characteristic
197	#381	—	94	1	no	—
198	#380	—	47	2	no	—
199	C₄×C₄⋊C₄	—	47	2	no	—
200	C₄×C₂²⋊C₄	—	94	1	no	—
201	#376	—	94	1	no	—
202	C₄₇	#8032	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	47	1	no	—
204	C₂³×C₄	—	47	1	no	—
205	C₂³×C₄	—	47	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	94	1	no	—
209	C₂²×C₈	—	94	1	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
212	C₂×C₂²⋊C₄	—	94	2	no	—
213	C₂×C₄²	—	47	1	no	—
214	C₂×C₄²	—	47	1	no	—
215	C₂×C₄²	—	47	1	no	—
216	C₂×C₄²	—	47	2	no	—
217	C₂×C₄²	—	47	2	no	—
218	C₂×C₄²	—	47	2	no	—
219	C₂×C₄²	—	47	2	no	—
220	C₂×C₄²	—	94	2	no	—
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₄⋊C₈	—	94	4	no	—
223	C₄⋊C₈	—	94	4	no	—
224	C₂²⋊C₈	—	94	4	no	—
225	C₂²⋊C₈	—	94	4	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	#97	—	47	2	no	—
229	#97	—	47	2	no	—
230	#97	—	94	1	no	—
231	#97	—	94	1	no	—
232	#97	C₄×D₄₇	1	2	yes	—
233	#97	C₄×D₄₇	1	2	yes	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	94	2	no	—
237	#95	—	94	2	no	—
238	#95	—	94	1	no	—
239	#95	—	94	1	no	—
240	C₂⁴	—	47	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	47	1	no	—
243	C₂²×C₄	—	47	1	no	—
244	C₂²×C₄	—	47	1	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	2	no	—
247	C₂²×C₄	—	47	2	no	—
248	C₂²×C₄	—	47	2	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	—	47	2	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	—	188	1	no	—
253	C₂²×C₄	—	188	1	no	—
254	C₂²×C₄	—	188	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	188	2	no	—
266	C₂×C₈	—	188	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	188	2	no	—
269	C₂²⋊C₄	—	188	4	no	—
270	C₄²	—	47	2	no	—
271	C₄²	—	47	2	no	—
272	C₄²	—	47	2	no	—
273	C₄²	—	47	2	no	—
274	C₄²	—	47	2	no	—
275	C₄²	—	47	2	no	—
276	C₄²	—	188	4	no	—
277	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
278	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
279	C₄²	D₄×D₄₇	1	1	yes	characteristic
280	C₄²	Q₈×D₄₇	1	1	yes	characteristic
281	C₂³	—	47	2	no	—
282	C₂³	—	94	1	no	—
283	C₂³	—	94	1	no	—
284	C₂³	—	94	2	no	—
285	C₂³	—	94	2	no	—
286	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	47	2	no	—
295	C₂×C₄	—	47	2	no	—
296	C₂×C₄	—	47	2	no	—
297	C₂×C₄	—	94	1	no	—
298	C₂×C₄	—	94	1	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	94	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	—	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	2	no	—
310	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
313	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₈	—	188	2	no	—
319	C₈	—	188	2	no	—
320	C₂²	—	94	2	no	—
321	C₂²	—	94	2	no	—
322	C₂²	—	94	2	no	—
323	C₂²	—	188	2	no	—
324	C₂²	—	188	2	no	—
325	C₂²	#95×D₄₇	1	1	yes	characteristic
326	C₂²	#15689104	1	1	yes	characteristic
327	C₂²	#15689106	1	1	yes	characteristic
328	C₂²	#15689138	1	2	yes	—
329	C₂²	#15689138	1	2	yes	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	94	2	no	—
338	C₄	—	94	2	no	—
339	C₄	—	94	2	no	—
340	C₄	—	376	2	no	—
341	C₂	—	188	2	no	—
342	C₂	#20242016	1	2	yes	—
343	C₂	#20242018	1	2	yes	—
344	C₂	#20243096	1	1	yes	characteristic
345	C₂	#20243096	1	1	yes	characteristic
346	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	#8032
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
12. #15689178	C₄	semidirect
174. #1295	D₄₇	semidirect
202. C₄₇	#8032	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
#20245781	2
#20241270	2
#20241269	2
C₄₇×#1295	2
#8032	47