Subgroups — #26371326

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253968	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245570	C₂	1	2	yes	maximal, complemented
4	#20245537	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245429	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245428	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245306	C₂	1	2	yes	maximal, complemented
8	#20245255	C₂	1	2	yes	maximal, complemented
9	#20245251	C₂	1	2	yes	maximal, complemented
10	#20245190	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	C₂²	1	1	yes	characteristic
13	#15689674	—	2	1	no	—
14	C₄×Q₈×D₄₇	—	2	1	no	—
15	C₂×#13093969	C₂²	1	2	yes	—
16	C₂×#13093967	C₂²	1	2	yes	—
17	#15689500	C₂²	1	1	yes	characteristic
18	#15689471	—	2	1	no	—
19	#15689360	—	2	1	no	—
20	#15689360	C₂²	1	2	yes	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	Q₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15689217	C₂²	1	1	yes	characteristic
24	C₄×#10625337	—	2	1	no	—
25	#15689215	—	2	1	no	—
26	#15689205	—	2	1	no	—
27	#15689205	C₂²	1	2	yes	—
28	#107×D₄₇	C₂²	1	1	yes	characteristic
29	#103×D₄₇	—	2	1	no	—
30	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
31	#15688948	C₂²	1	2	yes	—
32	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
33	#15688926	C₂²	1	2	yes	—
34	#15688925	C₂²	1	2	yes	—
35	#15688923	C₂²	1	2	yes	—
36	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
37	#15688898	C₂²	1	2	yes	—
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#366	C₂²	1	2	yes	—
41	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₇×#358	C₂²	1	2	yes	—
43	C₂×#10625354	—	4	1	no	—
44	C₂×Q₈×D₄₇	—	2	1	no	—
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625337	—	2	1	no	—
47	Q₁₆×D₄₇	—	2	2	no	—
48	Q₁₆×D₄₇	—	2	2	no	—
49	C₂×D₃₇₆	—	2	1	no	—
50	C₂×D₃₇₆	C₂³	1	2	yes	—
51	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
52	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
53	#13094021	—	2	1	no	—
54	#13094013	—	4	1	no	—
55	#13093999	—	2	2	no	—
56	#13093999	—	4	1	no	—
57	Q₈×Dic₄₇	—	2	1	no	—
58	C₄×Dic₉₄	—	2	1	no	—
59	C₄×Dic₉₄	—	4	1	no	—
60	#13093989	—	2	1	no	—
61	#13093989	—	4	1	no	—
62	#13093986	—	2	2	no	—
63	#13093986	—	4	1	no	—
64	C₄⋊C₄×D₄₇	—	2	1	no	—
65	C₄⋊C₄×D₄₇	—	4	1	no	—
66	C₄²×D₄₇	—	4	1	no	—
67	C₄²×D₄₇	D₄	1	1	yes	characteristic
68	#13093969	—	2	4	no	—
69	#13093967	—	2	4	no	—
70	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
71	#13093958	C₂³	1	1	yes	characteristic
72	#13093957	—	2	1	no	—
73	#13093949	—	2	1	no	—
74	#13093948	—	2	1	no	—
75	#13093947	—	2	1	no	—
76	#13093944	—	2	2	no	—
77	#13093944	—	2	2	no	—
78	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
79	C₈×Dic₄₇	D₄	1	1	yes	characteristic
80	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₄₇	—	2	1	no	—
83	C₄×Q₈×C₄₇	—	2	1	no	—
84	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
85	#107×C₄₇	—	2	1	no	—
86	#107×C₄₇	C₂³	1	1	yes	characteristic
87	#107×C₄₇	C₂³	1	2	yes	—
88	#103×C₄₇	—	2	1	no	—
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	C₂×Q₈×C₄₇	—	4	1	no	—
92	#10625354	—	8	4	no	—
93	Q₈×D₄₇	—	2	4	no	—
94	Q₈×D₄₇	—	4	2	no	—
95	C₂×D₁₈₈	—	2	2	no	—
96	C₂×D₁₈₈	—	4	2	no	—
97	C₂×D₁₈₈	—	4	1	no	—
98	C₂×D₁₈₈	—	4	1	no	—
99	C₂×C₄×D₄₇	—	4	1	no	—
100	C₂×C₄×D₄₇	—	4	1	no	—
101	C₂×C₄×D₄₇	—	4	2	no	—
102	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
103	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
104	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₉₄	—	2	1	no	—
106	C₂×Dic₉₄	—	4	1	no	—
107	Q₁₆×C₄₇	—	2	1	no	—
108	Q₁₆×C₄₇	#41	1	2	yes	complemented
109	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₄₇	—	2	1	no	—
112	C₄⋊C₄×C₄₇	—	2	2	no	—
113	C₄⋊C₄×C₄₇	—	4	1	no	—
114	C₄⋊C₄×C₄₇	—	4	1	no	—
115	C₄²×C₄₇	—	4	1	no	—
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	#10625337	—	2	4	no	—
118	(C₂×D₄₇)⋊C₄	—	4	2	no	—
119	(C₂×D₄₇)⋊C₄	—	4	2	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	—	4	1	no	—
122	Dic₄₇⋊C₄	—	4	1	no	—
123	Dic₄₇⋊C₄	—	4	1	no	—
124	Dic₄₇⋊C₄	—	4	1	no	—
125	C₄×Dic₄₇	—	4	1	no	—
126	C₄×Dic₄₇	—	4	1	no	—
127	C₄×Dic₄₇	—	4	1	no	—
128	C₄×Dic₄₇	D₈	1	1	yes	characteristic
129	C₄×Dic₄₇	D₈	1	1	yes	characteristic
130	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
133	Dic₁₈₈	—	2	1	no	—
134	Dic₁₈₈	#41	1	2	yes	complemented
135	D₃₇₆	—	2	2	no	—
136	D₃₇₆	—	2	2	no	—
137	D₃₇₆	—	4	2	no	—
138	C₈×D₄₇	—	2	1	no	—
139	C₈×D₄₇	—	2	1	no	—
140	C₈×D₄₇	—	4	1	no	—
141	C₁₆×C₄₇	—	2	2	no	—
142	C₄₇⋊C₁₆	—	2	2	no	—
143	C₂²×D₄₇	—	4	2	no	—
144	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
145	Q₈×C₄₇	—	2	2	no	—
146	Q₈×C₄₇	—	4	1	no	—
147	Q₈×C₄₇	—	8	2	no	—
148	C₂×C₄×C₄₇	—	4	1	no	—
149	C₂×C₄×C₄₇	—	4	1	no	—
150	C₂×C₄×C₄₇	—	4	2	no	—
151	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₄₇	—	4	1	no	—
155	C₂×Dic₄₇	—	4	1	no	—
156	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
158	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
159	D₁₈₈	—	4	2	no	—
160	D₁₈₈	—	4	2	no	—
161	D₁₈₈	—	8	2	no	—
162	D₁₈₈	—	8	4	no	—
163	D₁₈₈	—	8	2	no	—
164	C₄×D₄₇	—	2	2	no	—
165	C₄×D₄₇	—	2	1	no	—
166	C₄×D₄₇	—	2	1	no	—
167	C₄×D₄₇	—	2	2	no	—
168	C₄×D₄₇	—	4	4	no	—
169	C₄×D₄₇	—	8	4	no	—
170	C₄×D₄₇	—	8	2	no	—
171	Dic₉₄	—	2	2	no	—
172	Dic₉₄	—	4	4	no	—
173	Dic₉₄	—	4	1	no	—
174	C₈×C₄₇	—	2	1	no	—
175	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
176	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
177	C₄₇⋊C₈	—	2	1	no	—
178	C₄₇⋊C₈	—	2	1	no	—
179	#33672	—	47	1	no	maximal, Sylow
180	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₄₇	—	2	1	no	—
182	C₂×D₄₇	—	2	1	no	—
183	C₂×D₄₇	—	8	2	no	—
184	C₂×D₄₇	—	8	2	no	—
185	C₂×D₄₇	—	8	2	no	—
186	C₂×D₄₇	#509	1	2	yes	—
187	C₄×C₄₇	—	2	1	no	—
188	C₄×C₄₇	—	4	2	no	—
189	C₄×C₄₇	—	8	2	no	—
190	C₄×C₄₇	—	8	1	no	—
191	C₄×C₄₇	#510	1	2	yes	—
192	C₄×C₄₇	#525	1	1	yes	characteristic
193	C₄×C₄₇	#575	1	1	yes	characteristic
194	Dic₄₇	—	2	1	no	—
195	Dic₄₇	—	2	1	no	—
196	Dic₄₇	—	2	1	no	—
197	Dic₄₇	—	4	2	no	—
198	Dic₄₇	—	8	1	no	—
199	#3169	—	47	1	no	—
200	C₂×C₄×Q₁₆	—	47	1	no	—
201	#2232	—	47	2	no	—
202	#2227	—	47	1	no	—
203	#2227	D₄₇	1	1	yes	characteristic, complemented
204	#2207	—	47	1	no	—
205	#2207	—	47	1	no	—
206	#2185	—	47	2	no	—
207	#2175	—	47	2	no	—
208	#2175	—	47	2	no	—
209	#2163	—	47	1	no	—
210	C₂×C₄₇	#3100	1	1	yes	characteristic
211	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
212	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
213	D₄₇	—	2	2	no	—
214	D₄₇	—	16	2	no	—
215	C₂²×Q₁₆	—	47	1	no	—
216	#532	—	94	1	no	—
217	C₂×C₄×Q₈	—	94	1	no	—
218	C₂×SD₃₂	—	47	2	no	—
219	C₂×SD₃₂	—	47	2	no	—
220	#500	C₂×D₄₇	1	1	yes	characteristic
221	#493	—	47	1	no	—
222	#486	—	94	1	no	—
223	#466	—	47	2	no	—
224	#466	—	94	1	no	—
225	C₄×Q₁₆	—	47	1	no	—
226	C₄×Q₁₆	—	47	1	no	—
227	C₄×Q₁₆	—	47	1	no	—
228	C₄×Q₁₆	—	94	1	no	—
229	C₄×Q₁₆	—	94	1	no	—
230	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
231	C₄×D₈	—	47	2	no	—
232	C₄×D₈	—	94	1	no	—
233	C₂×#107	—	47	1	no	—
234	C₂×#103	—	94	1	no	—
235	C₂×C₄×C₈	—	47	1	no	—
236	#366	—	47	2	no	—
237	#366	C₂×D₄₇	1	2	yes	—
238	C₄⋊C₁₆	—	47	1	no	—
239	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
240	#358	—	47	2	no	—
241	#358	C₂×D₄₇	1	2	yes	—
242	#357	—	47	2	no	—
243	#357	—	47	2	no	—
244	C₂²⋊C₁₆	—	47	2	no	—
245	#346	—	47	2	no	—
246	C₄₇	#33672	1	1	yes	characteristic, complemented, Sylow
247	C₂×#41	—	188	1	no	—
248	C₂²×Q₈	—	94	1	no	—
249	C₂×Q₁₆	—	47	1	no	—
250	C₂×Q₁₆	—	94	1	no	—
251	C₂×Q₁₆	—	94	2	no	—
252	C₂×Q₁₆	—	94	2	no	—
253	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
254	C₂×D₈	—	47	2	no	—
255	C₂×D₈	—	94	1	no	—
256	C₂²×C₈	—	47	1	no	—
257	C₂²×C₈	—	47	1	no	—
258	C₄⋊Q₈	—	2	1	no	—
259	#127	—	94	1	no	—
260	#124	—	188	1	no	—
261	#121	—	94	2	no	—
262	#121	—	188	1	no	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	—	94	1	no	—
265	C₄×Q₈	—	94	1	no	—
266	C₄×Q₈	—	94	1	no	—
267	C₄×Q₈	—	188	1	no	—
268	C₄×Q₈	—	188	1	no	—
269	C₄×D₄	—	94	2	no	—
270	C₄×D₄	—	188	1	no	—
271	C₂×C₄⋊C₄	—	94	1	no	—
272	C₂×C₄⋊C₄	—	188	1	no	—
273	C₂×C₄²	—	47	1	no	—
274	C₂×C₄²	—	188	1	no	—
275	SD₃₂	—	94	4	no	—
276	SD₃₂	—	94	4	no	—
277	C₂×C₁₆	—	47	2	no	—
278	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
279	#107	—	2	1	no	—
280	#107	—	47	1	no	—
281	#107	—	94	1	no	—
282	#107	C₂²×D₄₇	1	1	yes	characteristic
283	#107	C₂²×D₄₇	1	2	yes	—
284	#103	—	2	1	no	—
285	#103	—	94	1	no	—
286	#103	—	94	1	no	—
287	#103	—	94	1	no	—
288	#102	—	94	2	no	—
289	#102	—	94	2	no	—
290	C₄×C₈	—	47	1	no	—
291	C₄×C₈	—	47	1	no	—
292	C₄×C₈	—	47	1	no	—
293	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
294	#41	—	376	4	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	94	4	no	—
298	C₂×Q₈	—	94	1	no	—
299	C₂×Q₈	—	188	1	no	—
300	C₂×Q₈	—	188	2	no	—
301	C₂×D₄	—	94	2	no	—
302	C₂×D₄	—	188	2	no	—
303	C₂×D₄	—	188	1	no	—
304	C₂×D₄	—	188	1	no	—
305	C₂²×C₄	—	47	1	no	—
306	C₂²×C₄	—	47	1	no	—
307	C₂²×C₄	—	47	1	no	—
308	C₂²×C₄	—	188	2	no	—
309	C₂²×C₄	—	188	1	no	—
310	C₂²×C₄	—	188	1	no	—
311	Q₁₆	—	2	1	no	—
312	Q₁₆	—	47	2	no	—
313	Q₁₆	—	94	4	no	—
314	Q₁₆	—	94	1	no	—
315	Q₁₆	#10625354	1	2	yes	complemented
316	D₈	—	94	2	no	—
317	D₈	—	94	2	no	—
318	D₈	—	188	2	no	—
319	C₂×C₈	—	47	1	no	—
320	C₂×C₈	—	47	1	no	—
321	C₂×C₈	—	94	1	no	—
322	C₂×C₈	—	94	1	no	—
323	C₂×C₈	—	188	1	no	—
324	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
325	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	94	1	no	—
331	C₄⋊C₄	—	188	1	no	—
332	C₄⋊C₄	—	188	1	no	—
333	C₄⋊C₄	—	188	1	no	—
334	C₄⋊C₄	—	188	1	no	—
335	C₂²⋊C₄	—	188	2	no	—
336	C₂²⋊C₄	—	188	2	no	—
337	C₄²	—	4	1	no	—
338	C₄²	—	47	1	no	—
339	C₄²	—	47	1	no	—
340	C₄²	—	47	1	no	—
341	C₄²	—	188	1	no	—
342	C₄²	—	188	1	no	—
343	C₄²	—	188	1	no	—
344	C₄²	D₄×D₄₇	1	1	yes	characteristic
345	C₁₆	—	2	2	no	—
346	C₁₆	—	94	2	no	—
347	C₂³	—	47	1	no	—
348	C₂³	—	188	2	no	—
349	Q₈	—	2	2	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	8	2	no	—
352	Q₈	—	94	2	no	—
353	Q₈	—	188	4	no	—
354	Q₈	—	188	1	no	—
355	D₄	—	376	2	no	—
356	D₄	—	376	4	no	—
357	D₄	—	376	2	no	—
358	D₄	—	188	2	no	—
359	D₄	—	188	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	47	1	no	—
364	C₂×C₄	—	47	1	no	—
365	C₂×C₄	—	47	1	no	—
366	C₂×C₄	—	94	2	no	—
367	C₂×C₄	—	94	2	no	—
368	C₂×C₄	—	94	1	no	—
369	C₂×C₄	—	94	1	no	—
370	C₂×C₄	—	376	4	no	—
371	C₂×C₄	—	376	2	no	—
372	C₂×C₄	—	188	4	no	—
373	C₂×C₄	—	188	1	no	—
374	C₂×C₄	—	188	1	no	—
375	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
376	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	94	1	no	—
380	C₈	—	94	1	no	—
381	C₈	#41×D₄₇	1	1	yes	characteristic
382	C₈	C₂×#10625354	1	1	yes	characteristic
383	C₂²	—	47	2	no	—
384	C₂²	—	94	1	no	—
385	C₂²	—	94	1	no	—
386	C₂²	—	376	2	no	—
387	C₂²	—	376	2	no	—
388	C₂²	—	376	2	no	—
389	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	8	2	no	—
393	C₄	—	8	1	no	—
394	C₄	—	94	1	no	—
395	C₄	—	94	1	no	—
396	C₄	—	94	1	no	—
397	C₄	—	376	1	no	—
398	C₄	—	188	2	no	—
399	C₄	#15689565	1	2	yes	—
400	C₄	#15689636	1	1	yes	characteristic
401	C₄	#15689922	1	1	yes	characteristic
402	C₂	—	94	2	no	—
403	C₂	—	752	2	no	—
404	C₂	#20253390	1	1	yes	characteristic
405	C₂	#509×D₄₇	1	1	yes	characteristic
406	C₂	C₂×#15689565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2227	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20253968	C₂
hypercenter	#2227	D₄₇
hypocenter	C₄₇	#33672
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
2. C₄×Q₁₆×D₄₇	C₂	semidirect
3. #20245570	C₂	semidirect
4. #20245537	C₂	semidirect
5. #20245429	C₂	semidirect
6. #20245428	C₂	semidirect
7. #20245306	C₂	semidirect
8. #20245255	C₂	semidirect
9. #20245251	C₂	semidirect
10. #20245190	C₂	semidirect
11. C₄₇×#2227	C₂	semidirect
108. Q₁₆×C₄₇	#41	semidirect
134. Dic₁₈₈	#41	semidirect
203. #2227	D₄₇	semidirect
246. C₄₇	#33672	semidirect
315. Q₁₆	#10625354	semidirect

Maximal subgroups

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

subgroup	index
#20253968	2
C₄×Q₁₆×D₄₇	2
#20245570	2
#20245537	2
#20245429	2
#20245428	2
#20245306	2
#20245255	2
#20245251	2
#20245190	2
C₄₇×#2227	2
#33672	47