Subgroups — #26116089

Order 12032.

694 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689499	C₂	1	1	yes	maximal, characteristic
2	#20244195	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244194	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244193	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243515	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625334	—	2	2	no	—
9	C₂×#13094021	C₂²	1	1	yes	characteristic
10	#15689499	—	2	2	no	—
11	#15689499	C₄	1	4	yes	complemented
12	C₂×#13093946	—	2	2	no	—
13	C₂×#13093945	—	2	2	no	—
14	C₂×#13093944	—	2	2	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15688993	C₂²	1	1	yes	characteristic
18	#15688885	—	2	2	no	—
19	#15688845	—	2	2	no	—
20	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₄₇	—	2	2	no	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	1	yes	characteristic
24	C₂²×D₁₈₈	—	2	1	no	—
25	C₂²×D₁₈₈	—	2	1	no	—
26	C₂²×D₁₈₈	—	2	1	no	—
27	C₂×#10625334	—	2	8	no	—
28	C₂×#10625334	—	4	4	no	—
29	C₂×#10625334	—	4	4	no	—
30	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
32	#13094021	—	2	1	no	—
33	#13094021	—	2	1	no	—
34	#13094021	—	2	1	no	—
35	#13094021	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄⋊Dic₄₇	—	2	1	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	#13093946	—	4	4	no	—
40	#13093945	—	4	4	no	—
41	#13093944	—	4	4	no	—
42	C₄×C₄₇⋊C₈	—	2	1	no	—
43	C₄×C₄₇⋊C₈	—	2	1	no	—
44	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
46	C₂²×D₄×C₄₇	—	2	1	no	—
47	C₂²×D₄×C₄₇	—	2	1	no	—
48	C₂²×D₄×C₄₇	—	2	1	no	—
49	C₂²×C₈×C₄₇	D₄	1	2	yes	—
50	#127×C₄₇	—	2	1	no	—
51	#127×C₄₇	—	2	1	no	—
52	#127×C₄₇	—	2	1	no	—
53	#127×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#102×C₄₇	—	4	4	no	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₄×C₈×C₄₇	—	2	1	no	—
61	C₂⁴×C₄₇	—	4	1	no	—
62	C₂³×D₄₇	—	4	1	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	2	2	no	—
66	C₂×D₄×C₄₇	—	4	1	no	—
67	C₂×D₄×C₄₇	—	4	1	no	—
68	C₂×D₄×C₄₇	—	4	1	no	—
69	C₂×D₄×C₄₇	—	4	2	no	—
70	C₂×D₄×C₄₇	—	4	2	no	—
71	C₂×D₄×C₄₇	—	4	2	no	—
72	C₂×D₄×C₄₇	—	4	2	no	—
73	C₂×D₄×C₄₇	—	4	2	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	4	1	no	—
80	C₂×D₁₈₈	—	2	2	no	—
81	C₂×D₁₈₈	—	2	2	no	—
82	C₂×D₁₈₈	—	2	2	no	—
83	C₂×D₁₈₈	—	4	2	no	—
84	C₂×D₁₈₈	—	4	2	no	—
85	C₂×D₁₈₈	—	4	2	no	—
86	C₂×D₁₈₈	—	4	2	no	—
87	C₂×D₁₈₈	—	4	2	no	—
88	C₂×D₁₈₈	—	4	2	no	—
89	C₂×D₁₈₈	—	4	1	no	—
90	C₂×D₁₈₈	—	4	1	no	—
91	C₂×D₁₈₈	—	4	1	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₄⋊C₄×C₄₇	—	4	2	no	—
96	C₄⋊C₄×C₄₇	—	4	2	no	—
97	C₄⋊C₄×C₄₇	—	4	2	no	—
98	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₇	#41	1	1	yes	characteristic
101	C₄²×C₄₇	#41	1	1	yes	characteristic
102	#10625334	—	4	16	no	—
103	C₄⋊Dic₄₇	—	4	2	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	—	4	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
111	C₂×C₄₇⋊C₈	D₈	1	4	yes	complemented
112	C₂³×C₄₇	—	4	2	no	—
113	C₂³×C₄₇	—	8	1	no	—
114	C₂³×C₄₇	—	8	1	no	—
115	C₂³×C₄₇	—	8	1	no	—
116	C₂³×C₄₇	—	8	1	no	—
117	C₂³×C₄₇	—	8	1	no	—
118	C₂³×C₄₇	—	8	1	no	—
119	C₂³×C₄₇	#127	1	1	yes	characteristic
120	C₂²×D₄₇	—	4	2	no	—
121	C₂²×D₄₇	—	8	1	no	—
122	C₂²×D₄₇	—	8	1	no	—
123	C₂²×D₄₇	—	8	1	no	—
124	C₂²×D₄₇	—	8	1	no	—
125	C₂²×D₄₇	—	8	1	no	—
126	C₂²×D₄₇	—	8	1	no	—
127	D₄×C₄₇	—	4	4	no	—
128	D₄×C₄₇	—	4	4	no	—
129	D₄×C₄₇	—	4	4	no	—
130	D₄×C₄₇	—	8	2	no	—
131	D₄×C₄₇	—	8	2	no	—
132	D₄×C₄₇	—	8	2	no	—
133	C₂×C₄×C₄₇	—	8	1	no	—
134	C₂×C₄×C₄₇	—	8	1	no	—
135	C₂×C₄×C₄₇	—	8	1	no	—
136	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₇	#135	1	2	yes	—
144	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
146	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×Dic₄₇	—	8	1	no	—
149	C₂×Dic₄₇	—	8	1	no	—
150	C₂×Dic₄₇	—	8	1	no	—
151	D₁₈₈	—	4	4	no	—
152	D₁₈₈	—	4	4	no	—
153	D₁₈₈	—	4	4	no	—
154	D₁₈₈	—	8	2	no	—
155	D₁₈₈	—	8	2	no	—
156	D₁₈₈	—	8	2	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₄₇⋊C₈	—	2	4	no	—
159	C₄₇⋊C₈	—	2	4	no	—
160	#9832	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	—	8	2	no	—
163	C₂²×C₄₇	—	8	2	no	—
164	C₂²×C₄₇	—	16	1	no	—
165	C₂²×C₄₇	—	16	1	no	—
166	C₂²×C₄₇	—	16	1	no	—
167	C₂²×C₄₇	—	16	1	no	—
168	C₂²×C₄₇	#390	1	1	yes	characteristic
169	C₂²×C₄₇	#493	1	1	yes	characteristic
170	C₂²×C₄₇	#495	1	1	yes	characteristic
171	C₂²×C₄₇	#496	1	1	yes	characteristic
172	C₂²×C₄₇	#496	1	1	yes	characteristic
173	C₂²×C₄₇	#496	1	1	yes	characteristic
174	C₂²×C₄₇	#496	1	1	yes	characteristic
175	C₂×D₄₇	—	8	2	no	—
176	C₂×D₄₇	—	8	2	no	—
177	C₂×D₄₇	—	8	2	no	—
178	C₂×D₄₇	—	16	1	no	—
179	C₂×D₄₇	—	16	1	no	—
180	C₂×D₄₇	—	16	1	no	—
181	C₂×D₄₇	—	16	1	no	—
182	C₄×C₄₇	—	16	1	no	—
183	C₄×C₄₇	#418	1	2	yes	—
184	C₄×C₄₇	C₄×D₈	1	2	yes	—
185	C₄×C₄₇	#442	1	2	yes	—
186	C₄×C₄₇	#466	1	2	yes	—
187	C₄×C₄₇	#469	1	2	yes	—
188	C₄×C₄₇	#490	1	2	yes	—
189	Dic₄₇	—	16	1	no	—
190	C₂×#493	—	47	1	no	—
191	#1957	—	47	1	no	—
192	#1957	—	47	1	no	—
193	#1957	—	47	1	no	—
194	#1957	D₄₇	1	1	yes	characteristic, complemented
195	#1834	—	47	1	no	—
196	C₂×#322	—	47	1	no	—
197	C₂×C₄₇	—	16	2	no	—
198	C₂×C₄₇	#1717	1	1	yes	characteristic
199	C₂×C₄₇	#1717	1	1	yes	characteristic
200	C₂×C₄₇	#1721	1	1	yes	characteristic
201	C₂×C₄₇	#1721	1	1	yes	characteristic
202	C₂×C₄₇	#1961	1	1	yes	characteristic
203	C₂×C₄₇	#1970	1	1	yes	characteristic
204	C₂×C₄₇	#1972	1	1	yes	characteristic
205	D₄₇	—	16	2	no	—
206	C₂²×D₈	—	94	2	no	—
207	C₂×#127	—	47	1	no	—
208	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
209	#493	—	47	4	no	—
210	#493	—	94	2	no	—
211	C₂×#102	—	2	2	no	—
212	C₂×#102	—	94	2	no	—
213	C₂×#102	—	94	2	no	—
214	C₂×#102	—	94	2	no	—
215	C₂×C₄×C₈	—	47	1	no	—
216	C₂×C₄×C₈	—	47	1	no	—
217	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
218	#384	—	47	1	no	—
219	#384	C₂×D₄₇	1	1	yes	characteristic
220	#340	—	94	2	no	—
221	#322	—	94	2	no	—
222	C₄₇	#9832	1	1	yes	characteristic, complemented, Sylow
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	2	1	no	—
226	C₂²×D₄	—	94	1	no	—
227	C₂²×D₄	—	94	1	no	—
228	C₂²×D₄	—	94	1	no	—
229	C₂×D₈	—	94	8	no	—
230	C₂×D₈	—	188	4	no	—
231	C₂×D₈	—	188	4	no	—
232	C₂²×C₈	—	47	2	no	—
233	C₂²×C₈	—	47	2	no	—
234	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
235	#127	—	2	1	no	—
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	—	47	2	no	—
239	#127	—	94	1	no	—
240	#127	—	94	1	no	—
241	#127	—	94	1	no	—
242	#127	C₄×D₄₇	1	2	yes	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	94	1	no	—
247	C₂×C₄⋊C₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
250	#102	—	4	4	no	—
251	#102	—	188	4	no	—
252	#102	—	188	4	no	—
253	#102	—	188	4	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	47	2	no	—
257	C₄×C₈	—	47	2	no	—
258	C₄×C₈	—	94	1	no	—
259	C₄×C₈	—	94	1	no	—
260	C₂⁴	—	4	1	no	—
261	C₂⁴	—	188	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	94	2	no	—
275	C₂×D₄	—	94	2	no	—
276	C₂×D₄	—	94	2	no	—
277	C₂×D₄	—	188	1	no	—
278	C₂×D₄	—	188	1	no	—
279	C₂×D₄	—	188	1	no	—
280	C₂×D₄	—	188	2	no	—
281	C₂×D₄	—	188	2	no	—
282	C₂×D₄	—	188	2	no	—
283	C₂×D₄	—	188	2	no	—
284	C₂×D₄	—	188	2	no	—
285	C₂×D₄	—	188	2	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	188	1	no	—
288	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
291	D₈	—	188	16	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	47	4	no	—
296	C₂×C₈	—	47	4	no	—
297	C₂×C₈	—	94	2	no	—
298	C₂×C₈	—	94	2	no	—
299	C₂×C₈	—	94	2	no	—
300	C₂×C₈	—	94	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	188	2	no	—
305	C₄⋊C₄	—	188	2	no	—
306	C₄⋊C₄	—	188	2	no	—
307	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
308	C₄²	#10625350	1	1	yes	characteristic
309	C₄²	D₄×D₄₇	1	1	yes	characteristic
310	C₄²	#10625352	1	1	yes	characteristic
311	C₂³	—	4	2	no	—
312	C₂³	—	8	1	no	—
313	C₂³	—	8	1	no	—
314	C₂³	—	8	1	no	—
315	C₂³	—	8	1	no	—
316	C₂³	—	8	1	no	—
317	C₂³	—	8	1	no	—
318	C₂³	—	376	1	no	—
319	C₂³	—	376	1	no	—
320	C₂³	—	376	1	no	—
321	C₂³	—	376	1	no	—
322	C₂³	—	376	1	no	—
323	C₂³	—	376	1	no	—
324	C₂³	—	188	2	no	—
325	C₂³	#13094020	1	1	yes	characteristic
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	4	4	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	2	no	—
331	D₄	—	8	2	no	—
332	D₄	—	376	2	no	—
333	D₄	—	376	2	no	—
334	D₄	—	376	2	no	—
335	D₄	—	188	4	no	—
336	D₄	—	188	4	no	—
337	D₄	—	188	4	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	376	1	no	—
342	C₂×C₄	—	376	1	no	—
343	C₂×C₄	—	376	1	no	—
344	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
345	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
347	C₂×C₄	#13093996	1	1	yes	characteristic
348	C₂×C₄	#13093997	1	1	yes	characteristic
349	C₂×C₄	#13094010	1	1	yes	characteristic
350	C₂×C₄	D₈×D₄₇	1	2	yes	—
351	C₂×C₄	#13094045	1	2	yes	—
352	C₂×C₄	#13094048	1	2	yes	—
353	C₂×C₄	#13094049	1	2	yes	—
354	C₂×C₄	#13094050	1	2	yes	—
355	C₂×C₄	#13094051	1	2	yes	—
356	C₈	—	4	2	no	—
357	C₈	—	94	4	no	—
358	C₈	—	94	4	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	16	1	no	—
363	C₂²	—	16	1	no	—
364	C₂²	—	16	1	no	—
365	C₂²	—	16	1	no	—
366	C₂²	—	376	2	no	—
367	C₂²	—	376	2	no	—
368	C₂²	—	376	2	no	—
369	C₂²	—	752	1	no	—
370	C₂²	—	752	1	no	—
371	C₂²	—	752	1	no	—
372	C₂²	—	752	1	no	—
373	C₂²	#15689023	1	1	yes	characteristic
374	C₂²	#15689498	1	1	yes	characteristic
375	C₂²	#15689504	1	1	yes	characteristic
376	C₂²	#15689509	1	1	yes	characteristic
377	C₂²	#15689510	1	1	yes	characteristic
378	C₂²	#15689511	1	1	yes	characteristic
379	C₂²	#15689513	1	1	yes	characteristic
380	C₄	—	16	1	no	—
381	C₄	—	752	1	no	—
382	C₄	#15689126	1	2	yes	—
383	C₄	#15689203	1	2	yes	—
384	C₄	#15689233	1	2	yes	—
385	C₄	#15689358	1	2	yes	—
386	C₄	#15689374	1	2	yes	—
387	C₄	#15689486	1	2	yes	—
388	C₂	—	16	2	no	—
389	C₂	—	752	2	no	—
390	C₂	#20243025	1	1	yes	characteristic
391	C₂	#20243029	1	1	yes	characteristic
392	C₂	#20243053	1	1	yes	characteristic
393	C₂	#20243056	1	1	yes	characteristic
394	C₂	#20244210	1	1	yes	characteristic
395	C₂	#20244251	1	1	yes	characteristic
396	C₂	#20244263	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1957	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#15689499	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1957	D₄₇
hypocenter	C₄₇	#9832
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

Complements

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

N	H ≅ G/N	type
2. #20244195	C₂	semidirect
3. #20244194	C₂	semidirect
4. #20244193	C₂	semidirect
5. #20243515	C₂	semidirect
6. C₂×#15688845	C₂	semidirect
7. C₄₇×#1957	C₂	semidirect
11. #15689499	C₄	semidirect
111. C₂×C₄₇⋊C₈	D₈	semidirect
194. #1957	D₄₇	semidirect
222. C₄₇	#9832	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689499	2
#20244195	2
#20244194	2
#20244193	2
#20243515	2
C₂×#15688845	2
C₄₇×#1957	2
#9832	47