Subgroups — #26220709

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	#20252189	C₂	1	1	yes	maximal, characteristic
2	#20252035	C₂	1	2	yes	maximal, complemented
3	#20244277	C₂	1	1	yes	maximal, characteristic
4	#20244141	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244134	C₂	1	1	yes	maximal, characteristic, complemented
6	#20244096	C₂	1	2	yes	maximal, complemented
7	#20244095	C₂	1	2	yes	maximal, complemented
8	#20243296	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×Dic₄₇	C₂	1	2	yes	maximal, complemented
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1975	C₂	1	1	yes	maximal, characteristic
12	#41×Dic₄₇	C₂²	1	2	yes	—
13	#15689235	C₄	1	2	yes	complemented
14	#15689230	C₄	1	2	yes	complemented
15	#15689224	C₄	1	4	yes	complemented
16	#15689181	—	2	4	no	—
17	#15689177	—	2	4	no	—
18	C₂×#13093960	C₂²	1	1	yes	characteristic
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689151	C₂²	1	2	yes	—
21	#15689149	C₂²	1	2	yes	—
22	C₂×#13093951	C₂²	1	2	yes	—
23	#15689119	C₂²	1	2	yes	—
24	#15689090	C₂²	1	2	yes	—
25	M₁₆×Dic₄₇	C₂²	1	2	yes	—
26	C₂×#13093935	C₂²	1	1	yes	characteristic
27	C₂×#13093934	C₂²	1	1	yes	characteristic
28	C₂×#13093934	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
32	#15688887	C₂²	1	2	yes	—
33	#15688884	C₂²	1	2	yes	—
34	#15688883	C₂²	1	2	yes	—
35	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#104×C₄₇	C₂²	1	2	yes	—
38	C₄₇×#408	C₂²	1	2	yes	—
39	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×#10625329	C₂³	1	2	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	Q₈×Dic₄₇	C₂×C₄	1	4	yes	—
44	D₄×Dic₄₇	—	2	4	no	—
45	D₄×Dic₄₇	C₂×C₄	1	4	yes	—
46	#13093983	—	2	2	no	—
47	#13093983	C₂³	1	2	yes	—
48	C₂×C₄×Dic₄₇	—	2	2	no	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	#13093960	—	2	2	no	—
53	#13093958	C₂×C₄	1	2	yes	—
54	#13093956	C₂×C₄	1	2	yes	—
55	Dic₄₇⋊C₈	—	2	4	no	—
56	Dic₄₇⋊C₈	—	2	4	no	—
57	#13093951	—	2	4	no	—
58	#13093948	C₂×C₄	1	4	yes	—
59	#13093945	C₂×C₄	1	4	yes	—
60	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
61	#13093935	—	2	2	no	—
62	#13093934	—	2	2	no	—
63	#13093934	C₂×C₄	1	2	yes	—
64	#13093934	C₂×C₄	1	2	yes	—
65	#13093934	D₄	1	4	yes	—
66	C₈×Dic₄₇	—	2	2	no	—
67	C₈×Dic₄₇	—	2	2	no	—
68	C₈×Dic₄₇	—	2	2	no	—
69	C₂×#41×C₄₇	C₂³	1	2	yes	—
70	#135×C₄₇	—	2	2	no	—
71	#135×C₄₇	—	2	2	no	—
72	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	—
74	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	#108×C₄₇	—	2	2	no	—
80	#104×C₄₇	—	2	4	no	—
81	C₂²⋊C₈×C₄₇	—	2	4	no	—
82	#97×C₄₇	—	2	2	no	—
83	#41×C₄₇	—	2	2	no	—
84	#41×C₄₇	—	2	2	no	—
85	#41×C₄₇	—	2	2	no	—
86	#41×C₄₇	—	2	2	no	—
87	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₄₇	—	2	1	no	—
91	C₂²×C₄×C₄₇	—	2	2	no	—
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	1	no	—
94	C₂²×Dic₄₇	—	2	4	no	—
95	C₂²×Dic₄₇	#41	1	2	yes	—
96	Q₁₆×C₄₇	—	2	2	no	—
97	SD₁₆×C₄₇	—	2	4	no	—
98	D₈×C₄₇	—	2	2	no	—
99	M₁₆×C₄₇	—	2	2	no	—
100	M₁₆×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	2	1	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	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	#41	1	2	yes	—
111	C₂²⋊Dic₄₇	—	2	2	no	—
112	C₂²⋊Dic₄₇	—	2	4	no	—
113	C₄⋊Dic₄₇	—	2	4	no	—
114	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
115	C₄×Dic₄₇	—	2	2	no	—
116	C₄×Dic₄₇	—	2	1	no	—
117	C₄×Dic₄₇	—	2	4	no	—
118	C₄×Dic₄₇	—	4	4	no	—
119	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
120	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
121	C₄×Dic₄₇	#41	1	2	yes	—
122	#10625329	—	2	2	no	—
123	#10625329	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	2	1	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	—	2	1	no	—
128	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
131	C₂³×C₄₇	—	2	2	no	—
132	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
133	Q₈×C₄₇	—	2	2	no	—
134	Q₈×C₄₇	—	2	2	no	—
135	D₄×C₄₇	—	2	2	no	—
136	D₄×C₄₇	—	2	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₄₇	—	2	4	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	4	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	8	4	no	—
155	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₈×C₄₇	#117	1	2	yes	—
159	C₈×C₄₇	#117	1	2	yes	—
160	C₄₇⋊C₈	—	2	2	no	—
161	C₄₇⋊C₈	—	2	2	no	—
162	C₄₇⋊C₈	—	4	2	no	—
163	#21297	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	—	4	4	no	—
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₄₇	#529	1	1	yes	characteristic
169	C₂²×C₄₇	#529	1	1	yes	characteristic
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	—	2	4	no	—
177	Dic₄₇	—	2	4	no	—
178	Dic₄₇	—	4	4	no	—
179	#2943	—	47	1	no	—
180	#2923	—	47	2	no	—
181	#1975	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
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	—	8	2	no	—
193	C₂×C₄₇	#2293	1	1	yes	characteristic
194	C₂×C₄₇	#2956	1	2	yes	—
195	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
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	C₂×D₄₇	1	2	yes	—
209	#416	—	47	2	no	—
210	#408	—	47	2	no	—
211	#408	C₂×D₄₇	1	2	yes	—
212	C₄×M₁₆	—	47	2	no	—
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	—	47	2	no	—
221	#339	—	47	2	no	—
222	#339	—	47	2	no	—
223	C₄₇	#21297	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	C₂²×D₄₇	1	2	yes	—
225	#135	—	2	2	no	—
226	#135	—	2	2	no	—
227	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
228	C₂×SD₁₆	C₄×D₄₇	1	2	yes	—
229	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
230	C₂×M₁₆	—	47	2	no	—
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	—	94	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	—	2	4	no	—
253	#104	—	94	4	no	—
254	#103	—	47	4	no	—
255	#102	—	47	4	no	—
256	C₂²⋊C₈	—	2	4	no	—
257	C₂²⋊C₈	—	94	4	no	—
258	#97	—	2	2	no	—
259	#97	—	47	2	no	—
260	#97	—	47	2	no	—
261	#97	—	47	4	no	—
262	#97	—	94	2	no	—
263	#97	—	94	2	no	—
264	C₄×C₈	—	94	2	no	—
265	C₄×C₈	—	94	2	no	—
266	C₄×C₈	—	94	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	47	2	no	—
277	C₂²×C₄	—	94	4	no	—
278	C₂²×C₄	—	94	1	no	—
279	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	Q₁₆	—	2	2	no	—
281	SD₁₆	—	2	4	no	—
282	D₈	—	2	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	94	2	no	—
286	M₁₆	—	94	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	47	2	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	94	1	no	—
293	C₂×C₈	—	94	1	no	—
294	C₂×C₈	—	94	1	no	—
295	C₂×C₈	—	94	1	no	—
296	C₂×C₈	—	94	2	no	—
297	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
299	C₂×C₈	D₄×D₄₇	1	2	yes	—
300	C₂×C₈	#10625352	1	2	yes	—
301	C₂×C₈	#10625352	1	2	yes	—
302	C₄⋊C₄	—	47	4	no	—
303	C₄⋊C₄	—	94	4	no	—
304	C₂²⋊C₄	—	94	4	no	—
305	C₂²⋊C₄	—	94	2	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	47	2	no	—
308	C₄²	—	47	2	no	—
309	C₄²	—	47	2	no	—
310	C₄²	—	94	4	no	—
311	C₄²	—	94	2	no	—
312	C₄²	—	94	1	no	—
313	C₄²	—	188	4	no	—
314	C₄²	#10625352	1	2	yes	—
315	C₂³	—	2	2	no	—
316	C₂³	#41×D₄₇	1	1	yes	characteristic
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	47	4	no	—
329	C₂×C₄	—	94	4	no	—
330	C₂×C₄	—	94	2	no	—
331	C₂×C₄	—	94	4	no	—
332	C₂×C₄	—	376	4	no	—
333	C₂×C₄	—	188	2	no	—
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	C₂×#10625352	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
339	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	94	2	no	—
343	C₈	—	94	2	no	—
344	C₈	—	188	2	no	—
345	C₈	#13093980	1	2	yes	—
346	C₈	#13093980	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	4	no	—
350	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
351	C₂²	#15689652	1	1	yes	characteristic
352	C₂²	#15689654	1	1	yes	characteristic
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	94	4	no	—
356	C₄	—	94	4	no	—
357	C₄	—	188	4	no	—
358	C₄	C₂×#13093980	1	1	yes	characteristic
359	C₄	C₄×#10625352	1	1	yes	characteristic
360	C₄	#15689668	1	1	yes	characteristic
361	C₄	#15689671	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	8	2	no	—
365	C₂	#20245850	1	1	yes	characteristic
366	C₂	#20252296	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×#1975	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#135×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1975	D₄₇
hypocenter	C₄₇	#21297
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1975

Complements

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

N	H ≅ G/N	type
2. #20252035	C₂	semidirect
4. #20244141	C₂	semidirect
5. #20244134	C₂	semidirect
6. #20244096	C₂	semidirect
7. #20244095	C₂	semidirect
8. #20243296	C₂	semidirect
9. #104×Dic₄₇	C₂	semidirect
10. #97×Dic₄₇	C₂	semidirect
13. #15689235	C₄	semidirect
14. #15689230	C₄	semidirect
15. #15689224	C₄	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
#20252189	2
#20252035	2
#20244277	2
#20244141	2
#20244134	2
#20244096	2
#20244095	2
#20243296	2
#104×Dic₄₇	2
#97×Dic₄₇	2
C₄₇×#1975	2
#21297	47