Subgroups — #26220708

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#20252040	C₂	1	2	yes	maximal, complemented
3	#20244279	C₂	1	1	yes	maximal, characteristic
4	#20244274	C₂	1	1	yes	maximal, characteristic
5	#20244139	C₂	1	1	yes	maximal, characteristic, complemented
6	#20244134	C₂	1	1	yes	maximal, characteristic, complemented
7	#20244097	C₂	1	2	yes	maximal, complemented
8	#20244090	C₂	1	2	yes	maximal, complemented
9	C₄×#13093950	C₂	1	2	yes	maximal, complemented
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1778	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	#15689178	—	2	4	no	—
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689151	C₂²	1	2	yes	—
21	#15689151	C₂²	1	2	yes	—
22	C₂×#13093950	C₂²	1	2	yes	—
23	C₂×#13093950	C₂²	1	2	yes	—
24	#15689120	C₂²	1	2	yes	—
25	#15689091	C₂²	1	2	yes	—
26	#15689091	C₂²	1	2	yes	—
27	C₂×#13093935	C₂²	1	1	yes	characteristic
28	C₂×#13093934	C₂²	1	1	yes	characteristic
29	C₂×#13093934	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
33	#15688884	C₂²	1	2	yes	—
34	#15688884	C₂²	1	2	yes	—
35	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
38	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#341	C₂²	1	2	yes	—
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	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
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	C₂×C₄×Dic₄₇	—	2	2	no	—
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
56	C₄²×D₄₇	—	2	2	no	—
57	#13093958	C₂×C₄	1	2	yes	—
58	#13093956	C₂×C₄	1	2	yes	—
59	Dic₄₇⋊C₈	—	2	4	no	—
60	Dic₄₇⋊C₈	—	2	4	no	—
61	#13093950	—	2	4	no	—
62	#13093950	—	2	4	no	—
63	#13093949	C₂×C₄	1	4	yes	—
64	#13093946	C₂×C₄	1	4	yes	—
65	(C₂×D₄₇)⋊C₈	—	2	4	no	—
66	(C₂×D₄₇)⋊C₈	—	2	4	no	—
67	#13093935	D₄	1	4	yes	—
68	#13093934	—	2	2	no	—
69	#13093934	—	2	2	no	—
70	C₈×Dic₄₇	—	2	2	no	—
71	C₈×Dic₄₇	—	2	2	no	—
72	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
73	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	#108×C₄₇	—	2	2	no	—
78	#108×C₄₇	—	2	2	no	—
79	#97×C₄₇	—	2	2	no	—
80	#97×C₄₇	C₂×C₄	1	2	yes	—
81	#97×C₄₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	—	2	1	no	—
87	C₂²×Dic₄₇	—	2	1	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	2	no	—
94	C₂×C₄×D₄₇	—	2	4	no	—
95	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
96	M₁₆×C₄₇	—	2	2	no	—
97	M₁₆×C₄₇	—	2	2	no	—
98	M₁₆×C₄₇	—	2	2	no	—
99	M₁₆×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₂×C₈×C₄₇	#41	1	2	yes	—
109	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₄₇	C₂×D₄	1	2	yes	—
111	C₂²⋊Dic₄₇	—	2	2	no	—
112	(C₂×D₄₇)⋊C₄	—	2	4	no	—
113	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
114	Dic₄₇⋊C₄	—	2	4	no	—
115	C₄×Dic₄₇	—	2	4	no	—
116	C₄×Dic₄₇	—	2	2	no	—
117	C₄×Dic₄₇	—	2	1	no	—
118	C₄×Dic₄₇	—	2	1	no	—
119	C₄×Dic₄₇	—	4	4	no	—
120	C₄×Dic₄₇	#41	1	2	yes	—
121	C₄×Dic₄₇	#41	1	2	yes	—
122	C₂×C₄₇⋊C₈	—	2	1	no	—
123	C₂×C₄₇⋊C₈	—	2	1	no	—
124	C₂×C₄₇⋊C₈	—	2	1	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
128	Dic₁₈₈	—	2	2	no	—
129	D₃₇₆	—	2	2	no	—
130	#10625325	—	2	4	no	—
131	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₂²×D₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₄₇	—	4	2	no	—
142	C₂²⋊D₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	D₁₈₈	—	2	2	no	—
150	D₁₈₈	—	2	2	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₄×D₄₇	—	4	2	no	—
153	C₄×D₄₇	—	8	4	no	—
154	Dic₉₄	—	2	2	no	—
155	Dic₉₄	—	2	2	no	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₈×C₄₇	—	4	2	no	—
159	C₈×C₄₇	#117	1	2	yes	—
160	C₈×C₄₇	#117	1	2	yes	—
161	C₄₇⋊C₈	—	2	2	no	—
162	C₄₇⋊C₈	—	2	2	no	—
163	#21297	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₄₇	#529	1	1	yes	characteristic
167	C₂²×C₄₇	#529	1	1	yes	characteristic
168	C₂×D₄₇	—	4	4	no	—
169	C₂×D₄₇	—	4	2	no	—
170	C₄×C₄₇	—	2	4	no	—
171	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
172	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
173	C₄×C₄₇	#532	1	1	yes	characteristic
174	C₄×C₄₇	#532	1	1	yes	characteristic
175	Dic₄₇	—	2	4	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	—	4	4	no	—
178	Dic₄₇	—	4	2	no	—
179	#2943	—	47	1	no	—
180	#2923	—	47	2	no	—
181	#1975	—	47	1	no	—
182	#1974	—	47	1	no	—
183	#1946	—	47	1	no	—
184	#1945	—	47	1	no	—
185	#1938	—	47	2	no	—
186	#1937	—	47	2	no	—
187	#1778	D₄₇	1	1	yes	characteristic, complemented
188	C₄×#104	—	47	2	no	—
189	C₄×#97	—	47	1	no	—
190	C₂×C₄₇	—	2	1	no	—
191	C₂×C₄₇	—	2	1	no	—
192	C₂×C₄₇	#2293	1	1	yes	characteristic
193	C₂×C₄₇	#2956	1	2	yes	—
194	D₄₇	—	8	2	no	—
195	C₂×#135	—	47	1	no	—
196	C₄×#41	—	47	2	no	—
197	#442	—	47	2	no	—
198	#441	—	47	2	no	—
199	#440	—	47	4	no	—
200	#433	—	94	4	no	—
201	#432	—	94	4	no	—
202	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
203	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
204	#427	—	47	1	no	—
205	#424	—	47	2	no	—
206	#424	—	47	2	no	—
207	C₂×#104	—	47	2	no	—
208	C₂×#104	—	47	2	no	—
209	#416	—	47	2	no	—
210	#408	—	47	2	no	—
211	#408	—	47	2	no	—
212	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
213	C₂×#97	—	47	1	no	—
214	C₂×#97	—	47	1	no	—
215	C₂×#97	—	47	1	no	—
216	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
217	C₂×C₄×C₈	—	47	1	no	—
218	C₂×C₄×C₈	—	47	1	no	—
219	C₄³	—	47	1	no	—
220	#341	C₂×D₄₇	1	2	yes	—
221	#339	—	47	2	no	—
222	#339	—	47	2	no	—
223	C₄₇	#21297	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	47	2	no	—
225	#135	—	94	2	no	—
226	#135	—	94	2	no	—
227	C₂×Q₁₆	—	47	1	no	—
228	C₂×SD₁₆	—	47	2	no	—
229	C₂×D₈	—	47	1	no	—
230	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
231	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
232	C₂²×C₈	—	47	1	no	—
233	C₂²×C₈	—	47	1	no	—
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄×Q₈	—	47	4	no	—
237	C₄×D₄	—	47	4	no	—
238	C₄×D₄	—	94	4	no	—
239	#117	—	47	2	no	—
240	#117	—	94	2	no	—
241	C₂×C₄²	—	47	1	no	—
242	C₂×C₄²	—	47	1	no	—
243	C₂×C₄²	—	94	2	no	—
244	C₂×C₄²	—	94	2	no	—
245	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
246	#108	—	2	2	no	—
247	#108	—	2	2	no	—
248	#107	—	47	2	no	—
249	#106	—	47	2	no	—
250	C₄⋊C₈	—	94	4	no	—
251	C₄⋊C₈	—	94	4	no	—
252	#104	—	94	4	no	—
253	#104	—	94	4	no	—
254	#103	—	47	4	no	—
255	#102	—	47	4	no	—
256	C₂²⋊C₈	—	94	4	no	—
257	C₂²⋊C₈	—	94	4	no	—
258	#97	—	2	2	no	—
259	#97	—	47	4	no	—
260	#97	—	94	2	no	—
261	#97	—	94	2	no	—
262	#97	C₄×D₄₇	1	2	yes	—
263	#97	C₄×D₄₇	1	2	yes	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	94	2	no	—
266	C₄×C₈	—	94	2	no	—
267	#41	—	94	2	no	—
268	#41	—	94	2	no	—
269	#41	—	94	2	no	—
270	#41	—	94	2	no	—
271	C₂×Q₈	—	47	2	no	—
272	C₂×D₄	—	47	2	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂²×C₄	—	94	1	no	—
275	C₂²×C₄	—	94	1	no	—
276	C₂²×C₄	—	94	4	no	—
277	C₂²×C₄	—	94	2	no	—
278	C₂²×C₄	#10625354	1	2	yes	—
279	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	Q₁₆	—	94	2	no	—
281	SD₁₆	—	94	4	no	—
282	D₈	—	94	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	47	2	no	—
293	C₂×C₈	—	94	1	no	—
294	C₂×C₈	—	94	1	no	—
295	C₂×C₈	—	94	1	no	—
296	C₂×C₈	—	94	1	no	—
297	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₈	#10625350	1	2	yes	—
301	C₂×C₈	#10625350	1	2	yes	—
302	C₄⋊C₄	—	47	4	no	—
303	C₄⋊C₄	—	94	4	no	—
304	C₂²⋊C₄	—	94	2	no	—
305	C₂²⋊C₄	—	94	4	no	—
306	C₄²	—	47	2	no	—
307	C₄²	—	47	2	no	—
308	C₄²	—	94	4	no	—
309	C₄²	—	94	2	no	—
310	C₄²	—	94	1	no	—
311	C₄²	—	94	1	no	—
312	C₄²	—	188	4	no	—
313	C₄²	C₂×C₄×D₄₇	1	2	yes	—
314	C₄²	D₄×D₄₇	1	2	yes	—
315	C₂³	—	94	2	no	—
316	C₂³	C₂×#10625354	1	1	yes	characteristic
317	Q₈	—	94	2	no	—
318	Q₈	—	94	2	no	—
319	D₄	—	94	2	no	—
320	D₄	—	94	2	no	—
321	D₄	—	188	2	no	—
322	D₄	—	188	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	94	4	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	4	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	376	4	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	#13093986	1	4	yes	—
334	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
336	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
337	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
339	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	94	2	no	—
344	C₈	—	94	2	no	—
345	C₈	#13093981	1	2	yes	—
346	C₈	#13093981	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	188	2	no	—
349	C₂²	—	188	4	no	—
350	C₂²	C₂×#13093986	1	1	yes	characteristic
351	C₂²	#15689660	1	1	yes	characteristic
352	C₂²	#15689660	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	94	4	no	—
355	C₄	—	188	2	no	—
356	C₄	—	188	2	no	—
357	C₄	—	188	4	no	—
358	C₄	C₂×#13093981	1	1	yes	characteristic
359	C₄	#15689591	1	1	yes	characteristic
360	C₄	#15689670	1	1	yes	characteristic
361	C₄	#15689670	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	376	2	no	—
365	C₂	#20245849	1	1	yes	characteristic
366	C₂	#20252297	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1778	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1778	D₄₇
hypocenter	C₄₇	#21297
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
2. #20252040	C₂	semidirect
5. #20244139	C₂	semidirect
6. #20244134	C₂	semidirect
7. #20244097	C₂	semidirect
8. #20244090	C₂	semidirect
9. C₄×#13093950	C₂	semidirect
10. #97×Dic₄₇	C₂	semidirect
11. C₄₇×#1778	C₂	semidirect
14. #15689236	C₄	semidirect
15. #15689231	C₄	semidirect
16. #15689225	C₄	semidirect
187. #1778	D₄₇	semidirect
223. C₄₇	#21297	semidirect

Maximal subgroups

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

subgroup	index
#20252190	2
#20252040	2
#20244279	2
#20244274	2
#20244139	2
#20244134	2
#20244097	2
#20244090	2
C₄×#13093950	2
#97×Dic₄₇	2
C₄₇×#1778	2
#21297	47