Subgroups — #26235284

Order 12032.

622 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	#20252339	C₂	1	2	yes	maximal
2	C₂×#15689251	C₂	1	1	yes	maximal, characteristic
3	#20244162	C₂	1	2	yes	maximal
4	#20244162	C₂	1	2	yes	maximal
5	#20244161	C₂	1	2	yes	maximal
6	#20243527	C₂	1	1	yes	maximal, characteristic
7	#20243524	C₂	1	1	yes	maximal, characteristic
8	#20243294	C₂	1	1	yes	maximal, characteristic
9	#20243293	C₂	1	1	yes	maximal, characteristic
10	#20243289	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1838	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	#15689252	—	2	2	no	—
14	#15689252	—	2	2	no	—
15	#15689251	C₄	1	4	yes	—
16	#15689251	C₄	1	4	yes	—
17	#15689250	—	2	2	no	—
18	#15689247	—	2	2	no	—
19	C₂×#13093960	C₂²	1	2	yes	—
20	C₂×#13093959	C₂²	1	2	yes	—
21	C₂×#13093959	C₂²	1	2	yes	—
22	#15689146	C₂²	1	2	yes	—
23	#15689146	C₂²	1	2	yes	—
24	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×#10625329	C₂²	1	2	yes	—
28	M₁₆×Dic₄₇	C₂²	1	2	yes	—
29	M₁₆×Dic₄₇	C₂²	1	2	yes	—
30	C₂×#13093934	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
32	#15688888	C₂²	1	1	yes	characteristic
33	#15688888	C₂²	1	1	yes	characteristic
34	#15688887	C₂²	1	2	yes	—
35	#15688887	C₂²	1	2	yes	—
36	C₄₇×#423	C₂²	1	2	yes	—
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#341	C₂²	1	1	yes	characteristic
39	C₄₇×#341	C₂²	1	1	yes	characteristic
40	C₄₇×#341	C₂²	1	2	yes	—
41	C₂²×Dic₉₄	Q₈	1	1	yes	characteristic
42	C₂×#10625329	C₂³	1	2	yes	—
43	C₂×#10625329	C₂³	1	2	yes	—
44	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
47	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
48	C₄×Dic₉₄	D₄	1	2	yes	—
49	C₄×Dic₉₄	D₄	1	2	yes	—
50	C₂×C₄⋊Dic₄₇	Q₈	1	1	yes	characteristic
51	C₂×Dic₄₇⋊C₄	Q₈	1	1	yes	characteristic
52	C₂×Dic₄₇⋊C₄	Q₈	1	1	yes	characteristic
53	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	#13093960	—	2	4	no	—
56	#13093959	—	2	4	no	—
57	#13093959	—	2	4	no	—
58	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
59	Dic₄₇⋊C₈	—	2	2	no	—
60	Dic₄₇⋊C₈	—	2	2	no	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	Dic₄₇⋊C₈	—	2	2	no	—
63	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
64	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
66	#13093935	—	2	2	no	—
67	#13093934	—	2	2	no	—
68	#13093934	—	2	2	no	—
69	#13093934	C₂×C₄	1	4	yes	—
70	C₄×C₄₇⋊C₈	—	2	2	no	—
71	C₈×Dic₄₇	—	2	2	no	—
72	C₈×Dic₄₇	—	2	2	no	—
73	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
74	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
75	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
76	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₇	—	2	2	no	—
81	C₄⋊C₈×C₄₇	—	2	2	no	—
82	#97×C₄₇	C₂×C₄	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
87	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
88	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
89	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
90	C₂×Dic₉₄	—	4	2	no	—
91	C₂×Dic₉₄	C₄⋊C₄	1	2	yes	—
92	C₂×Dic₉₄	C₄⋊C₄	1	2	yes	—
93	M₁₆×C₄₇	—	2	2	no	—
94	M₁₆×C₄₇	—	2	2	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
106	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
107	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
108	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
109	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
110	Dic₄₇⋊C₄	C₄⋊C₄	1	2	yes	—
111	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
114	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
116	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
117	#10625329	—	2	2	no	—
118	#10625329	—	2	2	no	—
119	#10625329	—	2	2	no	—
120	#10625329	#41	1	4	yes	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
127	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	4	1	no	—
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×Dic₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×Dic₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×Dic₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	Dic₉₄	—	2	4	no	—
147	Dic₉₄	—	4	2	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	—	2	2	no	—
150	C₈×C₄₇	—	4	2	no	—
151	C₈×C₄₇	—	4	2	no	—
152	C₄₇⋊C₈	—	2	2	no	—
153	C₄₇⋊C₈	—	2	2	no	—
154	C₄₇⋊C₈	—	2	4	no	—
155	C₄₇⋊C₈	—	4	2	no	—
156	#22747	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	—	2	1	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄₇	#557	1	2	yes	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
164	C₄×C₄₇	#519	1	1	yes	characteristic
165	C₄×C₄₇	#548	1	2	yes	—
166	Dic₄₇	—	2	2	no	—
167	Dic₄₇	—	2	1	no	—
168	Dic₄₇	—	2	1	no	—
169	Dic₄₇	—	2	1	no	—
170	Dic₄₇	—	2	2	no	—
171	Dic₄₇	#430	1	2	yes	—
172	Q₈×M₁₆	—	47	2	no	—
173	C₂×#446	—	47	1	no	—
174	#1950	—	47	2	no	—
175	#1950	—	47	2	no	—
176	#1950	—	47	2	no	—
177	#1838	—	47	1	no	—
178	#1838	D₄₇	1	1	yes	characteristic
179	#1837	—	47	1	no	—
180	#1778	—	47	1	no	—
181	#1778	—	47	1	no	—
182	C₄×#108	—	47	1	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	#2351	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#430	1	1	yes	characteristic
186	C₂×C₄₇	#2916	1	1	yes	characteristic
187	C₂×C₄×Q₈	—	47	1	no	—
188	#446	—	47	4	no	—
189	#446	—	47	4	no	—
190	#446	—	94	2	no	—
191	#446	—	94	2	no	—
192	#446	—	94	2	no	—
193	C₈×Q₈	—	94	2	no	—
194	C₂×#108	—	47	2	no	—
195	C₂×#108	—	47	2	no	—
196	C₂×#108	—	47	2	no	—
197	#423	—	47	2	no	—
198	#423	—	47	2	no	—
199	#423	C₂×D₄₇	1	2	yes	—
200	C₂×C₄⋊C₈	—	47	1	no	—
201	C₂×C₄⋊C₈	—	47	1	no	—
202	C₂×C₄⋊C₈	—	47	1	no	—
203	C₄×M₁₆	—	47	2	no	—
204	C₄×M₁₆	—	47	2	no	—
205	C₄×M₁₆	—	47	2	no	—
206	C₂×#97	—	47	1	no	—
207	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
208	C₂×C₄×C₈	—	47	1	no	—
209	#341	—	47	2	no	—
210	#341	—	47	2	no	—
211	#341	—	47	1	no	—
212	#341	—	47	1	no	—
213	#341	C₂×D₄₇	1	1	yes	characteristic
214	#341	C₂×D₄₇	1	1	yes	characteristic
215	#341	C₂×D₄₇	1	2	yes	—
216	C₄₇	#22747	1	1	yes	characteristic, complemented, Sylow
217	C₂²×Q₈	—	47	1	no	—
218	C₂×M₁₆	—	47	2	no	—
219	C₂×M₁₆	—	47	2	no	—
220	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
221	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
222	C₂²×C₈	—	47	1	no	—
223	C₂²×C₈	—	47	1	no	—
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	C₄×Q₈	—	47	2	no	—
227	C₄×Q₈	—	47	2	no	—
228	C₄×Q₈	—	47	2	no	—
229	C₄×Q₈	—	47	2	no	—
230	C₂×C₄⋊C₄	—	47	1	no	—
231	C₂×C₄⋊C₄	—	47	1	no	—
232	C₂×C₄⋊C₄	—	47	1	no	—
233	C₂×C₄²	—	47	1	no	—
234	C₂×C₄²	—	47	1	no	—
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	#108	—	94	4	no	—
237	#108	—	94	4	no	—
238	#108	—	94	4	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	47	2	no	—
242	C₄⋊C₈	—	47	2	no	—
243	C₄⋊C₈	—	47	4	no	—
244	C₄⋊C₈	—	47	4	no	—
245	C₄⋊C₈	—	94	2	no	—
246	C₄⋊C₈	—	94	2	no	—
247	C₄⋊C₈	—	94	2	no	—
248	C₄⋊C₈	—	94	2	no	—
249	#97	—	47	4	no	—
250	#97	—	94	2	no	—
251	#97	—	94	2	no	—
252	#97	—	94	2	no	—
253	#97	C₄×D₄₇	1	4	yes	—
254	C₄×C₈	—	47	2	no	—
255	C₄×C₈	—	47	2	no	—
256	C₄×C₈	—	94	2	no	—
257	C₄×C₈	—	94	2	no	—
258	C₄×C₈	—	94	2	no	—
259	C₂×Q₈	—	47	2	no	—
260	C₂×Q₈	—	47	2	no	—
261	C₂×Q₈	—	188	2	no	—
262	C₂²×C₄	—	47	1	no	—
263	C₂²×C₄	—	47	1	no	—
264	C₂²×C₄	—	47	1	no	—
265	C₂²×C₄	—	47	1	no	—
266	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
267	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	47	4	no	—
274	M₁₆	—	94	2	no	—
275	M₁₆	—	94	2	no	—
276	M₁₆	—	94	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	47	2	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	94	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
289	C₄⋊C₄	—	47	2	no	—
290	C₄⋊C₄	—	47	2	no	—
291	C₄⋊C₄	—	47	2	no	—
292	C₄⋊C₄	—	47	2	no	—
293	C₄⋊C₄	—	47	2	no	—
294	C₄⋊C₄	—	47	2	no	—
295	C₄²	—	47	1	no	—
296	C₄²	—	47	1	no	—
297	C₄²	—	47	2	no	—
298	C₄²	—	47	1	no	—
299	C₄²	—	47	2	no	—
300	C₄²	—	47	1	no	—
301	C₄²	C₂×C₄×D₄₇	1	2	yes	—
302	C₄²	D₄×D₄₇	1	2	yes	—
303	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
304	Q₈	—	94	4	no	—
305	Q₈	—	188	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	47	1	no	—
309	C₂×C₄	—	47	1	no	—
310	C₂×C₄	—	47	1	no	—
311	C₂×C₄	—	47	2	no	—
312	C₂×C₄	—	47	1	no	—
313	C₂×C₄	—	47	2	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	188	1	no	—
316	C₂×C₄	—	188	1	no	—
317	C₂×C₄	—	188	1	no	—
318	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
319	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
320	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
322	C₂×C₄	#41×D₄₇	1	2	yes	—
323	C₂×C₄	#13094071	1	2	yes	—
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	94	4	no	—
329	C₈	—	94	2	no	—
330	C₈	—	94	2	no	—
331	C₈	—	188	2	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
335	C₂²	#15689850	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	47	2	no	—
339	C₄	—	94	1	no	—
340	C₄	—	94	2	no	—
341	C₄	—	94	1	no	—
342	C₄	—	94	1	no	—
343	C₄	—	94	2	no	—
344	C₄	#15689591	1	1	yes	characteristic
345	C₄	#15689600	1	1	yes	characteristic
346	C₄	#15689781	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	#20246305	1	1	yes	characteristic
349	C₂	#430×D₄₇	1	1	yes	characteristic
350	C₂	#20251998	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1838	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₄₇	C₂²×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1838	D₄₇
hypocenter	C₄₇	#22747
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

Complements

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

N	H ≅ G/N	type
216. C₄₇	#22747	semidirect

Maximal subgroups

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

subgroup	index
#20252339	2
C₂×#15689251	2
#20244162	2
#20244161	2
#20243527	2
#20243524	2
#20243294	2
#20243293	2
#20243289	2
C₄₇×#1838	2
#22747	47