Subgroups — #26373115

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255782	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255063	C₂	1	1	yes	maximal, characteristic
3	#20245611	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245572	C₂	1	2	yes	maximal, complemented
5	#20245536	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245481	C₂	1	2	yes	maximal, complemented
7	#20245315	C₂	1	2	yes	maximal, complemented
8	C₄×#13093967	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245207	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₄₇	C₂	1	2	yes	maximal, complemented
11	C₄₇×#2207	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₇	C₂²	1	2	yes	—
13	#15689833	—	2	1	no	—
14	#15689758	—	2	1	no	—
15	C₂×#13093967	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₄₇	C₂²	1	2	yes	complemented
17	#15689528	C₂²	1	1	yes	characteristic, complemented
18	#15689457	—	2	2	no	—
19	#15689418	—	2	1	no	—
20	#15689361	C₂²	1	2	yes	complemented
21	#15689356	C₂²	1	2	yes	—
22	#15689315	—	2	1	no	—
23	#15689255	—	2	2	no	—
24	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
25	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
26	#107×D₄₇	C₂²	1	2	yes	complemented
27	#15688951	C₂²	1	2	yes	complemented
28	#15688950	C₂²	1	1	yes	characteristic, complemented
29	#15688927	C₂²	1	1	yes	characteristic, complemented
30	#15688925	C₂²	1	2	yes	—
31	#15688924	C₂²	1	2	yes	complemented
32	#15688924	C₂²	1	2	yes	complemented
33	#15688923	C₂²	1	1	yes	characteristic
34	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	complemented
35	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	complemented
37	C₄₇×#493	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
40	C₄₇×#357	C₂²	1	2	yes	complemented
41	C₂×D₄×D₄₇	—	2	2	no	—
42	C₂×D₄×D₄₇	—	4	1	no	—
43	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
44	C₂×#10625334	—	2	2	no	—
45	D₈×D₄₇	—	2	4	no	—
46	D₈×D₄₇	—	4	2	no	—
47	C₂×C₈×D₄₇	C₂³	1	2	yes	—
48	#13094024	—	2	1	no	—
49	#13094016	—	4	1	no	—
50	#13094002	—	4	2	no	—
51	#13094000	—	2	2	no	—
52	#13093995	—	2	2	no	—
53	#13093995	—	4	1	no	—
54	#13093992	—	4	2	no	—
55	C₄×D₁₈₈	—	2	1	no	—
56	C₄×D₁₈₈	—	4	1	no	—
57	C₄×D₁₈₈	D₄	1	1	yes	characteristic
58	C₄⋊C₄×D₄₇	—	2	2	no	—
59	#13093967	—	2	2	no	—
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₁₆×D₄₇	—	4	2	no	—
63	#13093958	—	2	1	no	—
64	#13093958	C₂³	1	2	yes	—
65	#13093958	C₂³	1	1	yes	characteristic
66	#13093957	—	2	2	no	—
67	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
68	#13093948	—	2	2	no	—
69	#13093946	—	2	1	no	—
70	#13093945	—	2	2	no	—
71	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
72	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×D₈×C₄₇	—	2	1	no	—
74	C₂×D₈×C₄₇	C₂³	1	2	yes	—
75	#127×C₄₇	—	2	1	no	—
76	C₄×Q₈×C₄₇	—	2	1	no	—
77	SD₃₂×C₄₇	—	4	2	no	—
78	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#103×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂³×D₄₇	—	4	2	no	—
83	C₂×Q₈×C₄₇	—	2	1	no	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	4	1	no	—
86	C₂×D₄×C₄₇	—	4	1	no	—
87	C₂×C₂²⋊D₄₇	—	4	2	no	—
88	C₂×C₂²⋊D₄₇	—	4	2	no	—
89	D₄×D₄₇	—	4	4	no	—
90	D₄×D₄₇	—	8	2	no	—
91	D₄×D₄₇	—	8	4	no	—
92	C₂×D₁₈₈	—	2	1	no	—
93	C₂×D₁₈₈	D₈	1	1	yes	characteristic
94	C₂×C₄×D₄₇	—	4	1	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
97	C₂×Dic₉₄	—	4	1	no	—
98	Q₁₆×C₄₇	—	2	1	no	—
99	Q₁₆×C₄₇	—	2	1	no	—
100	D₈×C₄₇	—	2	2	no	—
101	D₈×C₄₇	—	2	2	no	—
102	D₈×C₄₇	—	8	1	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₄₇	—	2	1	no	—
106	C₄⋊C₄×C₄₇	—	4	1	no	—
107	C₄²×C₄₇	—	4	1	no	—
108	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂²⋊Dic₄₇	—	4	2	no	—
110	#10625334	—	4	4	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	(C₂×D₄₇)⋊C₄	—	4	1	no	—
113	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	4	1	no	—
117	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
118	Dic₄₇⋊C₄	—	4	2	no	—
119	Dic₄₇⋊C₄	—	4	2	no	—
120	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
121	D₃₇₆	—	2	1	no	—
122	D₃₇₆	#41	1	2	yes	—
123	C₈×D₄₇	—	2	2	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₁₆×C₄₇	—	2	2	no	—
126	C₄₇⋊C₁₆	—	2	1	no	—
127	C₄₇⋊C₁₆	—	2	1	no	—
128	C₂³×C₄₇	—	4	2	no	—
129	C₂²×D₄₇	—	4	1	no	—
130	C₂²×D₄₇	—	8	4	no	—
131	C₂²×D₄₇	—	8	4	no	—
132	C₂²×D₄₇	—	16	2	no	—
133	C₂²×D₄₇	C₂×D₈	1	2	yes	—
134	Q₈×C₄₇	—	4	1	no	—
135	Q₈×C₄₇	—	4	1	no	—
136	D₄×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	8	2	no	—
139	D₄×C₄₇	—	16	1	no	—
140	C₂×C₄×C₄₇	—	4	1	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₇	—	8	4	no	—
146	C₂²⋊D₄₇	—	8	4	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	1	no	—
149	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
150	D₁₈₈	—	2	2	no	—
151	D₁₈₈	—	4	1	no	—
152	D₁₈₈	SD₃₂	1	4	yes	complemented
153	C₄×D₄₇	—	2	2	no	—
154	C₄×D₄₇	—	2	2	no	—
155	C₄×D₄₇	—	8	2	no	—
156	C₄×D₄₇	—	16	2	no	—
157	Dic₉₄	—	16	1	no	—
158	C₈×C₄₇	—	2	1	no	—
159	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
160	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₄₇⋊C₈	—	2	2	no	—
162	#33862	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	—	8	2	no	—
164	C₂²×C₄₇	—	8	2	no	—
165	C₂²×C₄₇	—	8	2	no	—
166	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₇	—	2	2	no	—
168	C₂×D₄₇	—	2	2	no	—
169	C₂×D₄₇	—	4	2	no	—
170	C₂×D₄₇	—	8	1	no	—
171	C₂×D₄₇	—	8	1	no	—
172	C₂×D₄₇	—	16	8	no	—
173	C₂×D₄₇	C₂×SD₃₂	1	4	yes	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	8	1	no	—
176	C₄×C₄₇	—	8	1	no	—
177	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
178	C₄×C₄₇	#546	1	1	yes	characteristic
179	C₄×C₄₇	#577	1	1	yes	characteristic
180	Dic₄₇	—	2	2	no	—
181	Dic₄₇	—	8	1	no	—
182	Dic₄₇	—	8	2	no	—
183	#3338	—	47	1	no	—
184	D₄×D₈	—	47	1	no	—
185	#2240	—	47	1	no	—
186	#2233	—	47	2	no	—
187	#2226	—	47	1	no	—
188	#2217	—	47	2	no	—
189	#2207	D₄₇	1	1	yes	characteristic, complemented
190	#2187	—	47	2	no	—
191	C₄×SD₃₂	—	47	1	no	—
192	D₄×C₁₆	—	47	1	no	—
193	C₂×#357	—	47	2	no	—
194	C₂×C₄₇	—	16	2	no	—
195	C₂×C₄₇	#3294	1	1	yes	characteristic
196	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
197	C₂×C₄₇	#3415	1	1	yes	characteristic
198	D₄₇	—	2	4	no	—
199	D₄₇	—	8	2	no	—
200	C₂²×D₈	—	47	2	no	—
201	#554	—	94	1	no	—
202	D₄²	—	94	1	no	—
203	C₂×SD₃₂	—	47	1	no	—
204	C₂×SD₃₂	C₂×D₄₇	1	2	yes	complemented
205	C₂²×C₁₆	—	47	2	no	—
206	#500	—	47	1	no	—
207	#493	C₂×D₄₇	1	1	yes	characteristic
208	#484	—	94	2	no	—
209	#477	—	94	1	no	—
210	#467	—	47	2	no	—
211	#466	—	47	2	no	—
212	#459	—	94	1	no	—
213	#447	—	94	2	no	—
214	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
215	C₄×D₈	—	47	1	no	—
216	C₈×D₄	—	47	1	no	—
217	C₂×#107	—	47	2	no	—
218	#367	—	47	2	no	—
219	#367	—	47	1	no	—
220	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
221	#358	—	47	1	no	—
222	#357	—	47	2	no	—
223	#357	—	47	2	no	—
224	#357	—	47	2	no	—
225	#357	—	47	1	no	—
226	#357	C₂×D₄₇	1	2	yes	complemented
227	C₂²⋊C₁₆	—	47	2	no	—
228	C₄×C₁₆	—	47	1	no	—
229	C₄₇	#33862	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	94	2	no	—
231	C₂²×D₄	—	188	1	no	—
232	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×D₈	—	2	1	no	—
234	C₂×D₈	—	47	1	no	—
235	C₂×D₈	—	94	2	no	—
236	C₂×D₈	—	94	4	no	—
237	C₂×D₈	—	188	2	no	—
238	C₂×D₈	C₂²×D₄₇	1	2	yes	—
239	C₂²×C₈	—	47	2	no	—
240	C₄⋊Q₈	—	94	1	no	—
241	#127	—	2	1	no	—
242	#125	—	188	1	no	—
243	C₂²⋊Q₈	—	94	2	no	—
244	C₂²⋊Q₈	—	188	2	no	—
245	#121	—	94	2	no	—
246	#121	—	188	1	no	—
247	#120	—	188	2	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×D₄	—	47	1	no	—
250	C₄×D₄	—	94	1	no	—
251	C₄×D₄	—	188	1	no	—
252	C₂×C₄⋊C₄	—	94	2	no	—
253	SD₃₂	—	4	2	no	—
254	SD₃₂	—	94	2	no	—
255	C₂×C₁₆	—	47	1	no	—
256	C₂×C₁₆	—	47	1	no	—
257	C₂×C₁₆	—	188	2	no	—
258	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
259	#107	—	47	2	no	—
260	#107	—	47	1	no	—
261	#107	—	94	2	no	—
262	#107	—	94	1	no	—
263	#107	C₂²×D₄₇	1	1	yes	characteristic
264	C₄⋊C₈	—	47	1	no	—
265	#103	—	2	1	no	—
266	#103	—	94	2	no	—
267	#102	—	94	2	no	—
268	#102	—	94	1	no	—
269	C₂²⋊C₈	—	47	2	no	—
270	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
271	C₂⁴	—	188	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	188	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	47	1	no	—
278	C₂×D₄	—	94	1	no	—
279	C₂×D₄	—	376	2	no	—
280	C₂×D₄	—	376	4	no	—
281	C₂×D₄	—	188	2	no	—
282	C₂×D₄	—	188	4	no	—
283	C₂×D₄	—	188	2	no	—
284	C₂²×C₄	—	47	2	no	—
285	C₂²×C₄	—	188	2	no	—
286	C₂²×C₄	—	188	1	no	—
287	Q₁₆	—	2	1	no	—
288	Q₁₆	—	2	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	8	1	no	—
292	D₈	—	47	2	no	—
293	D₈	—	94	1	no	—
294	D₈	—	188	4	no	—
295	C₂×C₈	—	47	2	no	—
296	C₂×C₈	—	94	2	no	—
297	C₂×C₈	—	94	2	no	—
298	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	47	1	no	—
303	C₄⋊C₄	—	94	1	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	188	2	no	—
306	C₄⋊C₄	—	188	2	no	—
307	C₄⋊C₄	—	188	1	no	—
308	C₂²⋊C₄	—	47	2	no	—
309	C₂²⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	188	2	no	—
311	C₂²⋊C₄	—	188	1	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₄×D₄₇	1	1	yes	characteristic
314	C₁₆	—	2	2	no	—
315	C₁₆	—	94	1	no	—
316	C₁₆	—	94	1	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	47	2	no	—
319	C₂³	—	376	4	no	—
320	C₂³	—	376	4	no	—
321	C₂³	—	188	1	no	—
322	C₂³	—	752	2	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	752	1	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	16	1	no	—
330	D₄	—	47	4	no	—
331	D₄	—	94	2	no	—
332	D₄	—	376	4	no	—
333	D₄	—	376	4	no	—
334	D₄	—	188	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	47	2	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	94	2	no	—
340	C₂×C₄	—	376	2	no	—
341	C₂×C₄	—	188	1	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	—	752	2	no	—
344	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
345	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	94	2	no	—
349	C₈	#41×D₄₇	1	1	yes	characteristic
350	C₈	#13094069	1	1	yes	characteristic
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	47	4	no	—
355	C₂²	—	94	2	no	—
356	C₂²	—	94	2	no	—
357	C₂²	—	376	1	no	—
358	C₂²	—	376	1	no	—
359	C₂²	—	188	2	no	—
360	C₂²	—	752	8	no	—
361	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	94	2	no	—
366	C₄	—	376	2	no	—
367	C₄	—	376	1	no	—
368	C₄	SD₃₂×D₄₇	1	2	yes	—
369	C₄	#15689770	1	1	yes	characteristic
370	C₄	#15689933	1	1	yes	characteristic
371	C₂	—	16	2	no	—
372	C₂	—	94	4	no	—
373	C₂	—	376	2	no	—
374	C₂	#20255204	1	1	yes	characteristic
375	C₂	C₂×SD₃₂×D₄₇	1	1	yes	characteristic
376	C₂	#20256528	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×#2207	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255063	C₂
hypercenter	#2207	D₄₇
hypocenter	C₄₇	#33862
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2207

Complements

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

N	H ≅ G/N	type
1. #20255782	C₂	semidirect
3. #20245611	C₂	semidirect
4. #20245572	C₂	semidirect
5. #20245536	C₂	semidirect
6. #20245481	C₂	semidirect
7. #20245315	C₂	semidirect
8. C₄×#13093967	C₂	semidirect
9. #20245207	C₂	semidirect
10. #357×D₄₇	C₂	semidirect
11. C₄₇×#2207	C₂	semidirect
16. C₂×C₁₆×D₄₇	C₂²	semidirect
17. #15689528	C₂²	semidirect
20. #15689361	C₂²	semidirect
26. #107×D₄₇	C₂²	semidirect
27. #15688951	C₂²	semidirect
28. #15688950	C₂²	semidirect
29. #15688927	C₂²	semidirect
31. #15688924	C₂²	semidirect
32. #15688924	C₂²	semidirect
34. (C₂×D₄₇)⋊C₁₆	C₂²	semidirect
35. C₄×C₄₇⋊C₁₆	C₂²	semidirect
36. C₂×SD₃₂×C₄₇	C₂²	semidirect
38. C₄×Q₁₆×C₄₇	C₂²	semidirect
39. C₄₇×C₄⋊C₁₆	C₂²	semidirect
40. C₄₇×#357	C₂²	semidirect
152. D₁₈₈	SD₃₂	semidirect
189. #2207	D₄₇	semidirect
204. C₂×SD₃₂	C₂×D₄₇	semidirect
214. C₄×Q₁₆	C₂×D₄₇	semidirect
220. C₄⋊C₁₆	C₂×D₄₇	semidirect
226. #357	C₂×D₄₇	semidirect
229. C₄₇	#33862	semidirect

Maximal subgroups

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

subgroup	index
#20255782	2
#20255063	2
#20245611	2
#20245572	2
#20245536	2
#20245481	2
#20245315	2
C₄×#13093967	2
#20245207	2
#357×D₄₇	2
C₄₇×#2207	2
#33862	47