Subgroups — #26220701

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252190	C₂	1	1	yes	maximal, characteristic
2	#20252030	C₂	1	2	yes	maximal, complemented
3	#20244281	C₂	1	1	yes	maximal, characteristic
4	#20244276	C₂	1	1	yes	maximal, characteristic
5	#20244141	C₂	1	1	yes	maximal, characteristic, complemented
6	#20244136	C₂	1	1	yes	maximal, characteristic, complemented
7	#20244099	C₂	1	2	yes	maximal, complemented
8	#20244092	C₂	1	2	yes	maximal, complemented
9	#20243294	C₂	1	1	yes	maximal, characteristic, complemented
10	#20243199	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	C₄×#10625350	C₂²	1	2	yes	—
14	#15689236	C₄	1	2	yes	complemented
15	#15689231	C₄	1	2	yes	complemented
16	#15689225	C₄	1	4	yes	complemented
17	#15689182	—	2	4	no	—
18	#15689174	—	2	4	no	—
19	C₂×#13093960	C₂²	1	1	yes	characteristic
20	C₂×#13093960	C₂²	1	1	yes	characteristic
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689149	C₂²	1	2	yes	—
23	#15689149	C₂²	1	2	yes	—
24	C₂×#13093952	C₂²	1	2	yes	—
25	C₂×#13093952	C₂²	1	2	yes	—
26	#15689120	C₂²	1	2	yes	—
27	#15689089	C₂²	1	2	yes	—
28	#15689089	C₂²	1	2	yes	—
29	C₄×#10625329	C₂²	1	2	yes	—
30	#15688888	C₂²	1	2	yes	—
31	#15688883	C₂²	1	2	yes	—
32	#15688883	C₂²	1	2	yes	—
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#376	C₂²	1	1	yes	characteristic
40	C₂×#10625350	C₂³	1	2	yes	—
41	#13094046	—	2	2	no	—
42	#13094046	—	2	2	no	—
43	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
44	C₂×#10625325	C₂×C₄	1	2	yes	—
45	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625329	C₂³	1	2	yes	—
47	C₂×#10625329	C₂³	1	2	yes	—
48	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
49	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
50	C₄×C₂²⋊D₄₇	—	2	4	no	—
51	#13093983	C₂³	1	2	yes	—
52	#13093981	—	2	2	no	—
53	C₄²×D₄₇	—	2	2	no	—
54	#13093960	—	2	2	no	—
55	#13093960	—	2	2	no	—
56	#13093958	C₂×C₄	1	2	yes	—
57	#13093956	C₂×C₄	1	2	yes	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	Dic₄₇⋊C₈	—	2	4	no	—
60	#13093952	—	2	4	no	—
61	#13093952	—	2	4	no	—
62	#13093949	C₂×C₄	1	4	yes	—
63	#13093946	C₂×C₄	1	4	yes	—
64	(C₂×D₄₇)⋊C₈	—	2	4	no	—
65	(C₂×D₄₇)⋊C₈	—	2	4	no	—
66	#13093935	—	2	2	no	—
67	C₄×C₄₇⋊C₈	—	2	2	no	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	#97×C₄₇	—	2	2	no	—
76	#97×C₄₇	—	2	2	no	—
77	#97×C₄₇	C₂×C₄	1	2	yes	—
78	#97×C₄₇	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₄₇	—	2	2	no	—
80	C₄×C₈×C₄₇	—	2	2	no	—
81	C₄×C₈×C₄₇	D₄	1	4	yes	—
82	#95×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	2	1	no	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₇	—	2	2	no	—
88	#10625350	—	2	2	no	—
89	#10625350	—	2	2	no	—
90	#10625350	—	2	2	no	—
91	#10625350	—	2	2	no	—
92	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₄₇	—	2	4	no	—
94	C₂×C₄×D₄₇	—	2	2	no	—
95	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₇	—	2	1	no	—
97	C₂×C₈×C₄₇	—	2	1	no	—
98	C₂×C₈×C₄₇	—	2	1	no	—
99	C₂×C₈×C₄₇	—	2	1	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₂×C₈×C₄₇	#41	1	2	yes	—
109	C₄²×C₄₇	—	2	1	no	—
110	C₄²×C₄₇	—	2	1	no	—
111	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
112	C₄²×C₄₇	C₂×D₄	1	2	yes	—
113	C₄²×C₄₇	#41	1	2	yes	—
114	C₄²×C₄₇	#41	1	2	yes	—
115	C₂²⋊Dic₄₇	—	2	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	4	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
118	Dic₄₇⋊C₄	—	2	4	no	—
119	C₄×Dic₄₇	—	2	4	no	—
120	C₄×Dic₄₇	—	2	2	no	—
121	#10625329	—	2	2	no	—
122	#10625329	—	2	2	no	—
123	#10625329	—	2	2	no	—
124	#10625329	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	Dic₁₈₈	—	2	2	no	—
128	D₃₇₆	—	2	2	no	—
129	#10625325	—	2	4	no	—
130	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₂²×D₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
138	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₄₇	—	4	2	no	—
145	C₂²⋊D₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	4	no	—
148	D₁₈₈	—	2	2	no	—
149	D₁₈₈	—	2	2	no	—
150	C₄×D₄₇	—	4	8	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₄×D₄₇	—	4	2	no	—
153	Dic₉₄	—	2	2	no	—
154	Dic₉₄	—	2	2	no	—
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	#117	1	2	yes	—
159	C₈×C₄₇	#117	1	2	yes	—
160	C₄₇⋊C₈	—	4	2	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	#21296	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₄₇	#529	1	1	yes	characteristic
166	C₂²×C₄₇	#529	1	1	yes	characteristic
167	C₂×D₄₇	—	4	4	no	—
168	C₂×D₄₇	—	4	2	no	—
169	C₄×C₄₇	—	2	4	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
173	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
174	C₄×C₄₇	#532	1	1	yes	characteristic
175	C₄×C₄₇	#532	1	1	yes	characteristic
176	Dic₄₇	—	4	4	no	—
177	Dic₄₇	—	4	2	no	—
178	#2943	—	47	1	no	—
179	#2922	—	47	2	no	—
180	#1975	—	47	1	no	—
181	#1974	—	47	1	no	—
182	#1946	—	47	1	no	—
183	#1945	—	47	1	no	—
184	#1938	—	47	2	no	—
185	#1937	—	47	2	no	—
186	#1778	—	47	1	no	—
187	#1759	—	47	2	no	—
188	#1726	D₄₇	1	1	yes	characteristic, complemented
189	C₂×C₄₇	—	2	1	no	—
190	C₂×C₄₇	—	2	1	no	—
191	C₂×C₄₇	#2293	1	1	yes	characteristic
192	C₂×C₄₇	#2955	1	2	yes	—
193	D₄₇	—	8	2	no	—
194	C₂×#135	—	47	1	no	—
195	C₄×#41	—	47	2	no	—
196	#442	—	47	2	no	—
197	#441	—	47	2	no	—
198	#440	—	47	4	no	—
199	#433	—	94	4	no	—
200	#431	—	94	4	no	—
201	C₂×#108	—	47	1	no	—
202	C₂×#108	—	47	1	no	—
203	#427	—	47	1	no	—
204	#424	—	47	2	no	—
205	#424	—	47	2	no	—
206	C₂×#104	—	47	2	no	—
207	C₂×#104	—	47	2	no	—
208	#416	—	47	2	no	—
209	#408	—	47	2	no	—
210	#408	—	47	2	no	—
211	C₄×M₁₆	—	47	2	no	—
212	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
214	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
217	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
218	#376	C₂×D₄₇	1	1	yes	characteristic
219	#341	—	47	2	no	—
220	#339	—	47	2	no	—
221	#339	—	47	2	no	—
222	C₄₇	#21296	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	47	2	no	—
224	#135	—	94	2	no	—
225	#135	—	94	2	no	—
226	C₂×Q₁₆	—	47	1	no	—
227	C₂×SD₁₆	—	47	2	no	—
228	C₂×D₈	—	47	1	no	—
229	C₂×M₁₆	—	47	2	no	—
230	C₂×M₁₆	—	47	2	no	—
231	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₄×Q₈	—	47	4	no	—
236	C₄×D₄	—	47	4	no	—
237	C₄×D₄	—	94	4	no	—
238	#117	—	47	2	no	—
239	#117	—	94	2	no	—
240	C₂×C₄²	—	94	2	no	—
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
244	#108	—	94	2	no	—
245	#108	—	94	2	no	—
246	#107	—	47	2	no	—
247	#106	—	47	2	no	—
248	C₄⋊C₈	—	94	4	no	—
249	C₄⋊C₈	—	94	4	no	—
250	#104	—	94	4	no	—
251	#104	—	94	4	no	—
252	#103	—	47	4	no	—
253	#102	—	47	4	no	—
254	C₂²⋊C₈	—	94	4	no	—
255	C₂²⋊C₈	—	94	4	no	—
256	#97	—	2	2	no	—
257	#97	—	2	2	no	—
258	#97	—	94	2	no	—
259	#97	C₄×D₄₇	1	2	yes	—
260	#97	C₄×D₄₇	1	2	yes	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	94	2	no	—
264	C₄×C₈	D₁₈₈	1	4	yes	—
265	#95	—	2	2	no	—
266	#41	—	94	2	no	—
267	#41	—	94	2	no	—
268	#41	—	94	2	no	—
269	#41	—	94	2	no	—
270	C₂×Q₈	—	47	2	no	—
271	C₂×D₄	—	47	2	no	—
272	C₂×D₄	—	94	2	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	94	4	no	—
276	C₂²×C₄	—	94	2	no	—
277	C₂²×C₄	#10625350	1	2	yes	—
278	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
279	Q₁₆	—	94	2	no	—
280	SD₁₆	—	94	4	no	—
281	D₈	—	94	2	no	—
282	M₁₆	—	94	2	no	—
283	M₁₆	—	94	2	no	—
284	M₁₆	—	94	2	no	—
285	M₁₆	—	94	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
298	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
299	C₂×C₈	#10625350	1	2	yes	—
300	C₂×C₈	#10625350	1	2	yes	—
301	C₄⋊C₄	—	47	4	no	—
302	C₄⋊C₄	—	94	4	no	—
303	C₂²⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	94	4	no	—
308	C₄²	—	94	2	no	—
309	C₄²	C₂×C₄×D₄₇	1	2	yes	—
310	C₄²	C₂×D₁₈₈	1	2	yes	—
311	C₄²	#10625350	1	2	yes	—
312	C₄²	#10625350	1	2	yes	—
313	C₂³	—	94	2	no	—
314	C₂³	C₂×#10625350	1	1	yes	characteristic
315	Q₈	—	94	2	no	—
316	Q₈	—	94	2	no	—
317	D₄	—	94	2	no	—
318	D₄	—	94	2	no	—
319	D₄	—	188	2	no	—
320	D₄	—	188	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	94	4	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	188	8	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
332	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
334	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
335	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
336	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
337	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	188	2	no	—
342	C₈	—	188	2	no	—
343	C₈	#13093981	1	2	yes	—
344	C₈	#13093981	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	188	2	no	—
347	C₂²	—	188	4	no	—
348	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
349	C₂²	#15689657	1	1	yes	characteristic
350	C₂²	#15689657	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	188	2	no	—
355	C₄	—	188	4	no	—
356	C₄	C₂×#13093981	1	1	yes	characteristic
357	C₄	C₄×#10625350	1	1	yes	characteristic
358	C₄	#15689673	1	1	yes	characteristic
359	C₄	#15689673	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	376	2	no	—
363	C₂	#20245852	1	1	yes	characteristic
364	C₂	#20252292	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21296
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #20252030	C₂	semidirect
5. #20244141	C₂	semidirect
6. #20244136	C₂	semidirect
7. #20244099	C₂	semidirect
8. #20244092	C₂	semidirect
9. #20243294	C₂	semidirect
10. #20243199	C₂	semidirect
11. C₄₇×#1726	C₂	semidirect
14. #15689236	C₄	semidirect
15. #15689231	C₄	semidirect
16. #15689225	C₄	semidirect
188. #1726	D₄₇	semidirect
222. C₄₇	#21296	semidirect

Maximal subgroups

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

subgroup	index
#20252190	2
#20252030	2
#20244281	2
#20244276	2
#20244141	2
#20244136	2
#20244099	2
#20244092	2
#20243294	2
#20243199	2
C₄₇×#1726	2
#21296	47