Subgroups — #26216648

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251815	C₂	1	1	yes	maximal, characteristic
2	#20243536	C₂	1	2	yes	maximal, complemented
3	#20243512	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243490	C₂	1	2	yes	maximal
5	#20243435	C₂	1	1	yes	maximal, characteristic
6	#20243431	C₂	1	1	yes	maximal, characteristic
7	#20243275	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243274	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243187	C₂	1	2	yes	maximal
10	#20243142	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094032	C₂²	1	1	yes	characteristic
13	C₂×#13094030	C₂²	1	1	yes	characteristic
14	#15689591	C₂²	1	1	yes	characteristic
15	C₂×#13093960	C₂²	1	1	yes	characteristic
16	#15689161	C₂²	1	1	yes	characteristic
17	#15689105	C₂²	1	2	yes	—
18	#15689096	C₂²	1	2	yes	—
19	#15689081	C₂²	1	2	yes	—
20	#15689080	C₄	1	2	yes	complemented
21	#15689079	C₂²	1	2	yes	—
22	#15689079	C₂²	1	1	yes	characteristic
23	#15689077	C₄	1	2	yes	complemented
24	M₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₄×#10625324	C₄	1	2	yes	complemented
26	C₂×#13093934	C₂²	1	1	yes	characteristic
27	#97×D₄₇	C₄	1	2	yes	complemented
28	#15688894	C₂²	1	2	yes	—
29	#15688884	C₂²	1	2	yes	—
30	#15688882	C₂²	1	2	yes	—
31	C₄₇⋊#338	C₂²	1	2	yes	—
32	#15688879	C₂²	1	2	yes	—
33	#15688879	C₂²	1	2	yes	—
34	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#427	C₂²	1	1	yes	characteristic
36	C₄₇×#405	C₂²	1	1	yes	characteristic
37	C₄₇×#344	C₂²	1	2	yes	—
38	C₄₇×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#10625350	C₂³	1	1	yes	characteristic
40	#13094032	—	2	2	no	—
41	#13094032	—	2	2	no	—
42	#13094030	—	2	1	no	—
43	#13094030	—	2	1	no	—
44	#13094030	—	2	1	no	—
45	#13094030	—	2	1	no	—
46	C₂×#10625329	C₂³	1	2	yes	—
47	C₂×#10625329	C₂³	1	1	yes	characteristic
48	C₂×#10625324	C₂×C₄	1	2	yes	—
49	C₂×#10625324	C₂×C₄	1	2	yes	—
50	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
53	#13093989	—	2	1	no	—
54	#13093986	—	2	1	no	—
55	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
56	#13093983	C₂³	1	2	yes	—
57	#13093981	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
59	C₄²×D₄₇	C₂×C₄	1	2	yes	—
60	#13093960	—	2	1	no	—
61	#13093960	—	2	1	no	—
62	#13093958	—	2	1	no	—
63	#13093956	—	2	1	no	—
64	#13093942	—	2	2	no	—
65	#13093940	—	2	2	no	—
66	#13093938	—	2	4	no	—
67	#13093935	—	2	2	no	—
68	#13093935	C₂×C₄	1	2	yes	—
69	#13093934	—	2	2	no	—
70	#13093934	C₂×C₄	1	1	yes	characteristic
71	#13093934	C₂×C₄	1	2	yes	—
72	#13093934	C₂×C₄	1	1	yes	characteristic
73	#13093934	C₂×C₄	1	1	yes	characteristic
74	#13093934	C₂×C₄	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	—	2	2	no	—
76	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
77	C₈×Dic₄₇	—	2	2	no	—
78	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
80	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	#117×C₄₇	C₂³	1	2	yes	—
83	#117×C₄₇	C₂³	1	1	yes	characteristic
84	#108×C₄₇	—	2	1	no	—
85	#108×C₄₇	—	2	1	no	—
86	#107×C₄₇	—	2	1	no	—
87	#106×C₄₇	—	2	1	no	—
88	#97×C₄₇	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₂²⋊D₄₇	#41	1	2	yes	—
92	C₂²×Dic₄₇	—	2	1	no	—
93	#10625350	—	2	1	no	—
94	#10625350	—	2	1	no	—
95	#10625350	—	2	1	no	—
96	#10625350	—	2	1	no	—
97	C₂×D₁₈₈	#41	1	1	yes	characteristic
98	C₂×C₄×D₄₇	—	2	2	no	—
99	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
100	C₂×Dic₉₄	#41	1	1	yes	characteristic
101	M₁₆×C₄₇	—	2	1	no	—
102	M₁₆×C₄₇	—	2	1	no	—
103	M₁₆×C₄₇	—	2	2	no	—
104	M₁₆×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
112	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
113	C₄⋊C₄×C₄₇	—	2	1	no	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₂²⋊C₄×C₄₇	—	2	1	no	—
116	C₂²⋊C₄×C₄₇	—	2	2	no	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
119	C₂²⋊Dic₄₇	—	2	2	no	—
120	(C₂×D₄₇)⋊C₄	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	Dic₄₇⋊C₄	—	2	2	no	—
123	C₄×Dic₄₇	—	2	1	no	—
124	C₄×Dic₄₇	—	2	2	no	—
125	C₄×Dic₄₇	—	2	2	no	—
126	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
128	#10625329	—	2	1	no	—
129	#10625329	—	2	1	no	—
130	#10625329	—	2	2	no	—
131	#10625329	—	2	2	no	—
132	C₂×C₄₇⋊C₈	—	2	2	no	—
133	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
134	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	#41	1	2	yes	—
137	C₂×C₄₇⋊C₈	#41	1	2	yes	—
138	#10625324	—	2	2	no	—
139	#10625324	—	2	2	no	—
140	#10625324	—	2	4	no	—
141	C₈×D₄₇	—	2	2	no	—
142	C₈×D₄₇	—	2	2	no	—
143	C₈×D₄₇	—	2	4	no	—
144	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂²×D₄₇	#117	1	2	yes	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	4	2	no	—
148	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₄₇	—	2	2	no	—
155	C₂²⋊D₄₇	—	2	2	no	—
156	C₂×Dic₄₇	—	2	1	no	—
157	C₂×Dic₄₇	—	2	1	no	—
158	C₂×Dic₄₇	—	4	2	no	—
159	C₂×Dic₄₇	—	8	1	no	—
160	C₂×Dic₄₇	#117	1	2	yes	—
161	D₁₈₈	—	2	1	no	—
162	D₁₈₈	—	2	1	no	—
163	C₄×D₄₇	—	2	2	no	—
164	C₄×D₄₇	—	2	2	no	—
165	C₄×D₄₇	—	8	2	no	—
166	Dic₉₄	—	2	1	no	—
167	Dic₉₄	—	2	1	no	—
168	C₈×C₄₇	—	2	2	no	—
169	C₈×C₄₇	—	4	2	no	—
170	C₈×C₄₇	#117	1	1	yes	characteristic
171	C₈×C₄₇	#117	1	1	yes	characteristic
172	C₈×C₄₇	#117	1	1	yes	characteristic
173	C₈×C₄₇	#117	1	1	yes	characteristic
174	C₄₇⋊C₈	—	2	2	no	—
175	C₄₇⋊C₈	—	4	2	no	—
176	C₄₇⋊C₈	#117	1	2	yes	—
177	C₄₇⋊C₈	#117	1	2	yes	—
178	#20788	—	47	1	no	maximal, Sylow
179	C₂²×C₄₇	—	4	1	no	—
180	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
181	C₂²×C₄₇	#529	1	1	yes	characteristic
182	C₂²×C₄₇	#529	1	1	yes	characteristic
183	C₂×D₄₇	—	2	2	no	—
184	C₂×D₄₇	—	4	2	no	—
185	C₄×C₄₇	—	4	2	no	—
186	C₄×C₄₇	—	8	2	no	—
187	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
188	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
189	C₄×C₄₇	#529	1	1	yes	characteristic
190	C₄×C₄₇	#529	1	1	yes	characteristic
191	Dic₄₇	—	2	2	no	—
192	Dic₄₇	—	4	1	no	—
193	Dic₄₇	—	4	1	no	—
194	Dic₄₇	—	8	2	no	—
195	#2875	—	47	1	no	—
196	#1841	—	47	2	no	—
197	#1833	—	47	1	no	—
198	#1833	D₄₇	1	1	yes	characteristic, complemented
199	#1827	—	47	2	no	—
200	#1813	—	47	1	no	—
201	#1812	—	47	1	no	—
202	#1773	—	47	1	no	—
203	#1773	—	47	1	no	—
204	#1757	—	47	2	no	—
205	#1747	—	47	2	no	—
206	C₂×C₄₇	—	2	1	no	—
207	C₂×C₄₇	—	2	1	no	—
208	C₂×C₄₇	—	2	1	no	—
209	C₂×C₄₇	#2283	1	1	yes	characteristic
210	D₄₇	—	4	2	no	—
211	C₂×#131	—	47	1	no	—
212	C₂×#131	—	47	1	no	—
213	C₄×#41	—	47	1	no	—
214	C₂×#108	—	47	1	no	—
215	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
216	#427	—	47	1	no	—
217	#427	C₂×D₄₇	1	1	yes	characteristic
218	#413	—	47	2	no	—
219	#410	—	47	2	no	—
220	#405	—	47	1	no	—
221	#405	—	47	2	no	—
222	#405	—	47	2	no	—
223	#405	—	47	2	no	—
224	#405	—	47	2	no	—
225	#405	C₂×D₄₇	1	1	yes	characteristic
226	C₄×M₁₆	—	47	1	no	—
227	C₄×M₁₆	—	47	2	no	—
228	C₂×#97	—	47	1	no	—
229	C₂×#97	—	47	2	no	—
230	#344	—	47	2	no	—
231	#344	C₂×D₄₇	1	2	yes	—
232	#339	—	47	2	no	—
233	#339	—	47	2	no	—
234	#338	—	47	2	no	—
235	H(3,ℤ/4)	—	47	2	no	—
236	H(3,ℤ/4)	—	47	2	no	—
237	H(3,ℤ/4)	C₂×D₄₇	1	2	yes	—
238	C₄₇	#20788	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	47	1	no	—
240	#131	—	94	2	no	—
241	#131	—	94	2	no	—
242	#131	—	94	1	no	—
243	#131	—	94	1	no	—
244	#131	—	94	1	no	—
245	#131	—	94	1	no	—
246	C₂×M₁₆	—	47	1	no	—
247	C₂×M₁₆	—	47	2	no	—
248	C₂×M₁₆	—	47	2	no	—
249	C₂×M₁₆	—	47	2	no	—
250	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
251	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
252	C₂²×C₈	—	47	1	no	—
253	C₂²×C₈	—	47	2	no	—
254	C₂²×C₈	—	47	2	no	—
255	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
256	C₄×Q₈	—	94	1	no	—
257	C₄×D₄	—	94	2	no	—
258	C₄×D₄	—	94	1	no	—
259	#117	—	47	2	no	—
260	#117	—	47	2	no	—
261	#117	C₂²×D₄₇	1	2	yes	—
262	#117	C₂²×D₄₇	1	1	yes	characteristic
263	C₂×C₄²	—	47	1	no	—
264	C₂×C₄²	—	47	2	no	—
265	#108	—	2	1	no	—
266	#108	—	2	1	no	—
267	#108	—	94	1	no	—
268	#108	—	94	1	no	—
269	#107	—	2	1	no	—
270	#107	—	94	1	no	—
271	#106	—	2	1	no	—
272	#106	—	94	1	no	—
273	#101	—	94	2	no	—
274	#100	—	94	2	no	—
275	#99	—	94	4	no	—
276	#97	—	47	1	no	—
277	#97	—	47	1	no	—
278	#97	—	47	2	no	—
279	#97	—	47	1	no	—
280	#97	—	47	1	no	—
281	#97	—	47	2	no	—
282	#97	—	94	2	no	—
283	#97	—	94	2	no	—
284	#97	C₄×D₄₇	1	2	yes	—
285	C₄×C₈	—	47	2	no	—
286	C₄×C₈	—	47	2	no	—
287	C₄×C₈	—	94	2	no	—
288	C₄×C₈	—	94	2	no	—
289	C₄×C₈	C₄×D₄₇	1	2	yes	—
290	#41	—	94	1	no	—
291	#41	—	94	1	no	—
292	#41	—	94	1	no	—
293	#41	—	94	1	no	—
294	C₂×Q₈	—	47	1	no	—
295	C₂×D₄	—	47	2	no	—
296	C₂×D₄	—	47	1	no	—
297	C₂²×C₄	—	47	2	no	—
298	C₂²×C₄	—	94	2	no	—
299	C₂²×C₄	—	94	1	no	—
300	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	94	4	no	—
306	M₁₆	—	94	1	no	—
307	M₁₆	—	94	1	no	—
308	M₁₆	—	94	2	no	—
309	M₁₆	—	94	2	no	—
310	M₁₆	—	94	2	no	—
311	M₁₆	—	94	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	47	1	no	—
314	C₂×C₈	—	47	1	no	—
315	C₂×C₈	—	47	2	no	—
316	C₂×C₈	—	47	2	no	—
317	C₂×C₈	—	47	2	no	—
318	C₂×C₈	—	94	4	no	—
319	C₂×C₈	—	94	2	no	—
320	C₂×C₈	—	94	2	no	—
321	C₂×C₈	—	94	2	no	—
322	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
325	C₂×C₈	#10625352	1	1	yes	characteristic
326	C₂×C₈	#10625352	1	1	yes	characteristic
327	C₂×C₈	#10625354	1	1	yes	characteristic
328	C₂×C₈	#10625354	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	94	2	no	—
332	C₄⋊C₄	—	94	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	94	2	no	—
336	C₂²⋊C₄	—	94	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	47	1	no	—
339	C₄²	—	47	1	no	—
340	C₄²	—	94	2	no	—
341	C₄²	—	94	1	no	—
342	C₄²	—	94	2	no	—
343	C₄²	C₂×C₄×D₄₇	1	2	yes	—
344	C₂³	—	47	2	no	—
345	C₂³	#41×D₄₇	1	1	yes	characteristic
346	Q₈	—	94	1	no	—
347	Q₈	—	94	1	no	—
348	D₄	—	94	2	no	—
349	D₄	—	94	2	no	—
350	D₄	—	94	1	no	—
351	D₄	—	94	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	47	2	no	—
355	C₂×C₄	—	94	2	no	—
356	C₂×C₄	—	94	2	no	—
357	C₂×C₄	—	94	1	no	—
358	C₂×C₄	—	94	1	no	—
359	C₂×C₄	—	376	2	no	—
360	C₂×C₄	—	376	1	no	—
361	C₂×C₄	—	188	2	no	—
362	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
363	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
364	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
365	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
367	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	47	2	no	—
371	C₈	—	47	2	no	—
372	C₈	—	94	2	no	—
373	C₈	—	188	2	no	—
374	C₈	#13093982	1	1	yes	characteristic
375	C₈	#13093982	1	1	yes	characteristic
376	C₈	#13093982	1	1	yes	characteristic
377	C₈	#13093982	1	1	yes	characteristic
378	C₂²	—	4	1	no	—
379	C₂²	—	94	2	no	—
380	C₂²	—	188	2	no	—
381	C₂²	#117×D₄₇	1	1	yes	characteristic
382	C₂²	#15689652	1	1	yes	characteristic
383	C₂²	#15689658	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	94	2	no	—
387	C₄	—	376	2	no	—
388	C₄	—	188	1	no	—
389	C₄	—	188	1	no	—
390	C₄	#117×D₄₇	1	1	yes	characteristic
391	C₄	C₂×#13093982	1	1	yes	characteristic
392	C₄	#15689652	1	1	yes	characteristic
393	C₄	#15689658	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	188	2	no	—
398	C₂	#20245781	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1833	C₂
socle	C₂×C₄₇	#2283
prime radical	C₂×#10625350	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1833	D₄₇
hypocenter	C₄₇	#20788
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #20243536	C₂	semidirect
3. #20243512	C₂	semidirect
7. #20243275	C₂	semidirect
8. #20243274	C₂	semidirect
10. #20243142	C₂	semidirect
11. C₄₇×#1833	C₂	semidirect
20. #15689080	C₄	semidirect
23. #15689077	C₄	semidirect
25. C₄×#10625324	C₄	semidirect
27. #97×D₄₇	C₄	semidirect
198. #1833	D₄₇	semidirect
238. C₄₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#20251815	2
#20243536	2
#20243512	2
#20243490	2
#20243435	2
#20243431	2
#20243275	2
#20243274	2
#20243187	2
#20243142	2
C₄₇×#1833	2
#20788	47