Subgroups — #26216651

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	#20251813	C₂	1	1	yes	maximal, characteristic
2	#20243534	C₂	1	2	yes	maximal, complemented
3	#20243513	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243511	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243491	C₂	1	2	yes	maximal
6	#20243435	C₂	1	1	yes	maximal, characteristic
7	#20243431	C₂	1	1	yes	maximal, characteristic
8	#20243272	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243189	C₂	1	2	yes	maximal
10	#20243143	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094030	C₂²	1	1	yes	characteristic
13	C₂×#13094030	C₂²	1	1	yes	characteristic
14	C₄×#10625350	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	#15689080	C₄	1	2	yes	complemented
20	#15689080	C₄	1	2	yes	complemented
21	#15689078	C₂²	1	2	yes	—
22	#15689078	C₂²	1	1	yes	characteristic
23	#15689078	C₂²	1	1	yes	characteristic
24	C₄×#10625329	C₂²	1	1	yes	characteristic
25	#15689075	C₄	1	2	yes	complemented
26	#97×D₄₇	C₄	1	2	yes	complemented
27	#15688895	C₂²	1	2	yes	—
28	#15688893	C₂²	1	2	yes	—
29	#15688883	C₂²	1	2	yes	—
30	C₄₇⋊#338	C₂²	1	2	yes	—
31	#15688880	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	2	yes	—
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#339	C₂²	1	2	yes	—
39	C₂×#10625350	C₂³	1	1	yes	characteristic
40	#13094030	—	2	1	no	—
41	#13094030	—	2	1	no	—
42	#13094030	—	2	1	no	—
43	#13094030	—	2	1	no	—
44	#13094030	—	2	2	no	—
45	#13094030	—	2	2	no	—
46	C₂×#10625329	C₂³	1	2	yes	—
47	C₂×#10625329	C₂³	1	1	yes	characteristic
48	C₂×#10625329	C₂³	1	1	yes	characteristic
49	C₂×#10625324	C₂×C₄	1	2	yes	—
50	C₂×#10625324	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₉₄	—	2	1	no	—
54	C₄×D₁₈₈	—	2	1	no	—
55	C₄×C₂²⋊D₄₇	—	2	2	no	—
56	#13093983	C₂³	1	2	yes	—
57	#13093983	C₂³	1	1	yes	characteristic
58	#13093981	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	2	yes	—
68	#13093934	—	2	2	no	—
69	#13093934	C₂×C₄	1	2	yes	—
70	#13093934	C₂×C₄	1	2	yes	—
71	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
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	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	#117×C₄₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	#108×C₄₇	—	2	1	no	—
81	#108×C₄₇	—	2	1	no	—
82	#107×C₄₇	—	2	1	no	—
83	#106×C₄₇	—	2	1	no	—
84	#97×C₄₇	—	2	2	no	—
85	#97×C₄₇	C₂×C₄	1	1	yes	characteristic
86	#97×C₄₇	C₂×C₄	1	1	yes	characteristic
87	#97×C₄₇	C₂×C₄	1	1	yes	characteristic
88	#97×C₄₇	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₄₇	—	2	2	no	—
90	C₂²×C₄×C₄₇	—	2	1	no	—
91	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	#41	1	2	yes	—
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	2	no	—
102	M₁₆×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	#41	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	2	yes	—
113	C₂×C₈×C₄₇	#41	1	2	yes	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₂²⋊C₄×C₄₇	—	2	2	no	—
116	C₄²×C₄₇	—	2	1	no	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊Dic₄₇	—	2	1	no	—
121	C₂²⋊Dic₄₇	—	2	2	no	—
122	(C₂×D₄₇)⋊C₄	—	2	2	no	—
123	C₄⋊Dic₄₇	—	2	1	no	—
124	C₄⋊Dic₄₇	—	2	2	no	—
125	Dic₄₇⋊C₄	—	2	2	no	—
126	C₄×Dic₄₇	—	2	2	no	—
127	C₄×Dic₄₇	—	2	2	no	—
128	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
129	#10625329	—	2	1	no	—
130	#10625329	—	2	1	no	—
131	#10625329	—	2	1	no	—
132	#10625329	—	2	1	no	—
133	#10625329	—	2	2	no	—
134	#10625329	—	2	2	no	—
135	C₂×C₄₇⋊C₈	—	2	2	no	—
136	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₄₇⋊C₈	C₂²×C₄	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	1	no	—
147	C₂×C₄×C₄₇	—	2	1	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	8	1	no	—
150	C₂×C₄×C₄₇	C₂³×C₄	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	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₂²⋊D₄₇	—	2	2	no	—
157	C₂²⋊D₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	4	2	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	—	4	2	no	—
169	C₈×C₄₇	#117	1	1	yes	characteristic
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	2	yes	—
174	C₈×C₄₇	#117	1	2	yes	—
175	C₄₇⋊C₈	—	2	2	no	—
176	C₄₇⋊C₈	—	2	2	no	—
177	C₄₇⋊C₈	—	4	2	no	—
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	1	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	—	8	2	no	—
188	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
189	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
190	C₄×C₄₇	#529	1	1	yes	characteristic
191	C₄×C₄₇	#529	1	1	yes	characteristic
192	Dic₄₇	—	2	2	no	—
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	8	2	no	—
195	#2875	—	47	1	no	—
196	#1841	—	47	2	no	—
197	#1833	—	47	1	no	—
198	#1833	—	47	1	no	—
199	#1827	—	47	2	no	—
200	#1813	—	47	1	no	—
201	#1812	—	47	1	no	—
202	#1773	—	47	1	no	—
203	#1773	D₄₇	1	1	yes	characteristic, complemented
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	1	no	—
222	#405	—	47	2	no	—
223	#405	—	47	2	no	—
224	#405	—	47	2	no	—
225	#405	C₂×D₄₇	1	2	yes	—
226	C₄×M₁₆	—	47	1	no	—
227	C₄×M₁₆	—	47	2	no	—
228	C₂×#97	—	47	2	no	—
229	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
230	#344	—	47	2	no	—
231	#344	—	47	2	no	—
232	#339	—	47	2	no	—
233	#339	C₂×D₄₇	1	2	yes	—
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)	—	47	2	no	—
238	C₄₇	#20788	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	47	1	no	—
240	#131	—	94	1	no	—
241	#131	—	94	1	no	—
242	#131	—	94	1	no	—
243	#131	—	94	1	no	—
244	#131	—	94	2	no	—
245	#131	—	94	2	no	—
246	C₂×M₁₆	—	47	1	no	—
247	C₂×M₁₆	—	47	1	no	—
248	C₂×M₁₆	—	47	2	no	—
249	C₂×M₁₆	—	47	2	no	—
250	C₂×M₁₆	—	47	2	no	—
251	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
252	C₂²×C₈	—	47	2	no	—
253	C₂²×C₈	—	47	2	no	—
254	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
255	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
256	C₄×Q₈	—	94	1	no	—
257	C₄×D₄	—	94	1	no	—
258	C₄×D₄	—	94	2	no	—
259	#117	—	47	1	no	—
260	#117	—	47	2	no	—
261	#117	—	47	2	no	—
262	#117	C₂²×D₄₇	1	2	yes	—
263	C₂×C₄²	—	47	2	no	—
264	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
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	—	2	2	no	—
277	#97	—	47	2	no	—
278	#97	—	47	2	no	—
279	#97	—	47	2	no	—
280	#97	—	94	2	no	—
281	#97	C₄×D₄₇	1	1	yes	characteristic
282	#97	C₄×D₄₇	1	1	yes	characteristic
283	#97	C₄×D₄₇	1	1	yes	characteristic
284	#97	C₄×D₄₇	1	1	yes	characteristic
285	C₄×C₈	—	2	2	no	—
286	C₄×C₈	—	47	2	no	—
287	C₄×C₈	—	47	2	no	—
288	C₄×C₈	—	47	2	no	—
289	C₄×C₈	—	94	2	no	—
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	—	2	1	no	—
298	C₂²×C₄	—	47	2	no	—
299	C₂²×C₄	—	94	2	no	—
300	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	94	1	no	—
304	M₁₆	—	94	1	no	—
305	M₁₆	—	94	1	no	—
306	M₁₆	—	94	1	no	—
307	M₁₆	—	94	2	no	—
308	M₁₆	—	94	2	no	—
309	M₁₆	—	94	4	no	—
310	M₁₆	—	94	2	no	—
311	M₁₆	—	94	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	47	2	no	—
314	C₂×C₈	—	47	2	no	—
315	C₂×C₈	—	94	2	no	—
316	C₂×C₈	—	94	2	no	—
317	C₂×C₈	—	94	4	no	—
318	C₂×C₈	—	94	2	no	—
319	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
320	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
321	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
323	C₂×C₈	#10625350	1	1	yes	characteristic
324	C₂×C₈	#10625350	1	1	yes	characteristic
325	C₂×C₈	#10625350	1	1	yes	characteristic
326	C₂×C₈	#10625350	1	1	yes	characteristic
327	C₂×C₈	#10625352	1	2	yes	—
328	C₂×C₈	#10625354	1	2	yes	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	94	2	no	—
331	C₄⋊C₄	—	94	1	no	—
332	C₄⋊C₄	—	94	2	no	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	94	2	no	—
335	C₂²⋊C₄	—	94	1	no	—
336	C₂²⋊C₄	—	94	2	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	47	2	no	—
340	C₄²	—	94	2	no	—
341	C₄²	—	94	2	no	—
342	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
343	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
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	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	1	no	—
356	C₂×C₄	—	47	2	no	—
357	C₂×C₄	—	94	2	no	—
358	C₂×C₄	—	94	2	no	—
359	C₂×C₄	—	94	2	no	—
360	C₂×C₄	—	376	2	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	C₂×#10625352	1	1	yes	characteristic
365	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
366	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
367	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
368	C₈	—	4	2	no	—
369	C₈	—	94	2	no	—
370	C₈	—	94	2	no	—
371	C₈	—	188	2	no	—
372	C₈	#13093981	1	1	yes	characteristic
373	C₈	#13093981	1	1	yes	characteristic
374	C₈	#13093981	1	1	yes	characteristic
375	C₈	#13093981	1	1	yes	characteristic
376	C₈	#13093982	1	2	yes	—
377	C₈	#13093982	1	2	yes	—
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	#15689655	1	1	yes	characteristic
383	C₂²	#15689660	1	1	yes	characteristic
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	94	2	no	—
388	C₄	—	376	2	no	—
389	C₄	—	188	2	no	—
390	C₄	C₂×#13093981	1	1	yes	characteristic
391	C₄	C₂×#13093982	1	1	yes	characteristic
392	C₄	#15689655	1	1	yes	characteristic
393	C₄	#15689660	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	#20245784	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13093982
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1773	C₂
socle	C₂×C₄₇	#2283
prime radical	C₂×#10625350	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1773	D₄₇
hypocenter	C₄₇	#20788
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
2. #20243534	C₂	semidirect
3. #20243513	C₂	semidirect
4. #20243511	C₂	semidirect
8. #20243272	C₂	semidirect
10. #20243143	C₂	semidirect
11. C₄₇×#1773	C₂	semidirect
19. #15689080	C₄	semidirect
20. #15689080	C₄	semidirect
25. #15689075	C₄	semidirect
26. #97×D₄₇	C₄	semidirect
203. #1773	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
#20251813	2
#20243534	2
#20243513	2
#20243511	2
#20243491	2
#20243435	2
#20243431	2
#20243272	2
#20243189	2
#20243143	2
C₄₇×#1773	2
#20788	47