Subgroups — #26246000

Order 12032.

744 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094055	C₂	1	2	yes	maximal
2	#20252037	C₂	1	1	yes	maximal, characteristic
3	#20244899	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244834	C₂	1	1	yes	maximal, characteristic
5	#20244753	C₂	1	2	yes	maximal, complemented
6	#20244688	C₂	1	2	yes	maximal
7	#20244653	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244507	C₂	1	1	yes	maximal, characteristic
9	#20243294	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13093943	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094055	C₂²	1	2	yes	—
13	C₄×#10625350	C₂²	1	1	yes	characteristic
14	#15689228	—	2	2	no	—
15	#15689221	—	2	2	no	—
16	C₄×#10625337	—	2	2	no	—
17	C₄×#10625336	—	2	2	no	—
18	#15689185	—	2	2	no	—
19	#15689178	—	2	2	no	—
20	C₂×#13093960	C₂²	1	2	yes	—
21	#15689163	C₂²	1	2	yes	—
22	#15689149	C₂²	1	1	yes	characteristic
23	#15689149	C₂²	1	1	yes	characteristic
24	#15689132	C₂²	1	2	yes	—
25	#15689122	C₂²	1	2	yes	—
26	C₂×#13093943	C₂²	1	2	yes	—
27	C₂×#13093943	C₂²	1	2	yes	—
28	#15689089	C₂²	1	1	yes	characteristic
29	#15689089	C₂²	1	1	yes	characteristic
30	C₄×#10625329	C₂²	1	2	yes	—
31	C₄×#10625329	C₂²	1	1	yes	characteristic
32	#15688892	C₂²	1	2	yes	—
33	#15688892	C₂²	1	2	yes	—
34	#15688888	C₂²	1	1	yes	characteristic
35	#15688888	C₂²	1	1	yes	characteristic
36	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
38	C₄₇×#441	—	2	2	no	—
39	C₄₇×#441	—	2	2	no	—
40	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#103×C₄₇	C₂²	1	2	yes	—
42	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625350	C₂³	1	1	yes	characteristic
44	#13094055	—	2	4	no	—
45	#13094055	—	2	4	no	—
46	C₂×#10625337	—	2	2	no	—
47	C₂×#10625336	—	2	2	no	—
48	C₂×#10625329	C₂³	1	1	yes	characteristic
49	C₂×#10625329	C₂³	1	1	yes	characteristic
50	C₂×#10625329	C₂³	1	2	yes	—
51	C₂×#10625329	C₂³	1	2	yes	—
52	C₄×Dic₉₄	D₄	1	2	yes	—
53	C₄×D₁₈₈	D₄	1	2	yes	—
54	C₄×C₂²⋊D₄₇	—	2	2	no	—
55	#13093983	C₂³	1	1	yes	characteristic
56	#13093981	—	2	1	no	—
57	C₄²×D₄₇	—	2	1	no	—
58	#13093960	—	2	4	no	—
59	#13093957	—	2	2	no	—
60	#13093955	—	2	2	no	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	Dic₄₇⋊C₈	—	2	2	no	—
63	#13093948	—	2	2	no	—
64	#13093948	—	2	2	no	—
65	#13093947	—	2	2	no	—
66	#13093944	—	2	2	no	—
67	#13093943	—	2	4	no	—
68	#13093943	—	2	4	no	—
69	(C₂×D₄₇)⋊C₈	—	2	2	no	—
70	(C₂×D₄₇)⋊C₈	—	2	2	no	—
71	#13093935	—	2	2	no	—
72	#13093935	—	2	1	no	—
73	C₄×C₄₇⋊C₈	—	2	2	no	—
74	C₄×C₄₇⋊C₈	—	2	1	no	—
75	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
76	C₂×Q₁₆×C₄₇	—	2	4	no	—
77	C₂×Q₁₆×C₄₇	—	2	2	no	—
78	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄×Q₈×C₄₇	—	2	2	no	—
81	C₄×Q₈×C₄₇	—	2	2	no	—
82	C₄×Q₈×C₄₇	—	2	2	no	—
83	C₄×Q₈×C₄₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₄₇	—	2	2	no	—
87	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
88	#106×C₄₇	—	2	2	no	—
89	#103×C₄₇	—	2	4	no	—
90	#97×C₄₇	D₄	1	2	yes	—
91	#97×C₄₇	D₄	1	2	yes	—
92	C₂×Q₈×C₄₇	—	2	2	no	—
93	C₂×Q₈×C₄₇	—	2	2	no	—
94	C₂×Q₈×C₄₇	—	2	2	no	—
95	C₂×Q₈×C₄₇	—	2	2	no	—
96	C₂×Q₈×C₄₇	—	2	2	no	—
97	C₂×Q₈×C₄₇	#41	1	4	yes	—
98	C₂²×C₄×C₄₇	—	2	2	no	—
99	C₂²×C₄×C₄₇	—	2	2	no	—
100	C₂²×C₄×C₄₇	#41	1	2	yes	—
101	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₂²⋊D₄₇	—	2	1	no	—
103	#10625350	—	2	2	no	—
104	#10625350	—	2	2	no	—
105	C₂×D₁₈₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₄₇	—	2	1	no	—
107	C₂×C₄×D₄₇	—	2	2	no	—
108	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
109	Q₁₆×C₄₇	—	2	8	no	—
110	C₂×C₈×C₄₇	—	2	2	no	—
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	4	2	no	—
117	C₄⋊C₄×C₄₇	—	4	2	no	—
118	C₄²×C₄₇	—	4	2	no	—
119	C₄²×C₄₇	—	4	2	no	—
120	C₄²×C₄₇	C₂×D₄	1	2	yes	—
121	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
123	C₂²⋊Dic₄₇	—	2	1	no	—
124	#10625337	—	2	8	no	—
125	#10625336	—	2	8	no	—
126	(C₂×D₄₇)⋊C₄	—	2	2	no	—
127	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
128	Dic₄₇⋊C₄	—	2	2	no	—
129	C₄×Dic₄₇	—	2	2	no	—
130	C₄×Dic₄₇	—	2	1	no	—
131	#10625329	—	2	2	no	—
132	#10625329	—	2	2	no	—
133	#10625329	—	2	2	no	—
134	#10625329	—	2	2	no	—
135	#10625329	—	2	2	no	—
136	#10625329	#41	1	4	yes	—
137	C₂×C₄₇⋊C₈	—	2	2	no	—
138	C₂×C₄₇⋊C₈	—	2	1	no	—
139	C₂×C₄₇⋊C₈	—	2	1	no	—
140	C₂×C₄₇⋊C₈	—	2	2	no	—
141	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂²×D₄₇	—	2	1	no	—
143	Q₈×C₄₇	—	2	4	no	—
144	Q₈×C₄₇	—	2	4	no	—
145	Q₈×C₄₇	—	4	2	no	—
146	Q₈×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	—	2	4	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	4	2	no	—
150	C₂×C₄×C₄₇	—	4	2	no	—
151	C₂×C₄×C₄₇	—	4	2	no	—
152	C₂×C₄×C₄₇	—	8	2	no	—
153	C₂×C₄×C₄₇	#121	1	2	yes	—
154	C₂×C₄×C₄₇	#121	1	2	yes	—
155	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
158	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
159	C₂²⋊D₄₇	—	4	2	no	—
160	C₂×Dic₄₇	—	2	1	no	—
161	C₂×Dic₄₇	—	2	2	no	—
162	D₁₈₈	—	2	2	no	—
163	C₄×D₄₇	—	4	1	no	—
164	C₄×D₄₇	—	4	1	no	—
165	C₄×D₄₇	—	8	2	no	—
166	Dic₉₄	—	2	2	no	—
167	C₈×C₄₇	—	2	2	no	—
168	C₈×C₄₇	—	2	2	no	—
169	C₄₇⋊C₈	—	2	4	no	—
170	C₄₇⋊C₈	—	4	2	no	—
171	C₄₇⋊C₈	—	4	1	no	—
172	C₄₇⋊C₈	—	4	1	no	—
173	#23720	—	47	1	no	maximal, Sylow
174	C₂²×C₄₇	—	2	2	no	—
175	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
176	C₂²×C₄₇	#529	1	2	yes	—
177	C₂×D₄₇	—	4	1	no	—
178	C₂×D₄₇	—	4	2	no	—
179	C₄×C₄₇	—	2	2	no	—
180	C₄×C₄₇	—	2	2	no	—
181	C₄×C₄₇	—	4	4	no	—
182	C₄×C₄₇	—	8	2	no	—
183	C₄×C₄₇	#525	1	2	yes	—
184	C₄×C₄₇	#525	1	1	yes	characteristic
185	C₄×C₄₇	#532	1	1	yes	characteristic
186	Dic₄₇	—	4	1	no	—
187	Dic₄₇	—	4	2	no	—
188	C₄×#137	—	47	2	no	—
189	C₂×#441	D₄₇	1	1	yes	characteristic, complemented
190	#2923	—	47	1	no	—
191	#2090	—	47	1	no	—
192	#2079	—	47	1	no	—
193	#2064	—	47	2	no	—
194	#2052	—	47	2	no	—
195	#2047	—	47	1	no	—
196	#2020	—	47	1	no	—
197	#1778	—	47	1	no	—
198	C₄×#101	—	47	2	no	—
199	C₂×C₄₇	—	2	2	no	—
200	C₂×C₄₇	#2392	1	1	yes	characteristic
201	C₂×C₄₇	#3069	1	2	yes	—
202	D₄₇	—	8	1	no	—
203	C₂×#137	—	47	2	no	—
204	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
205	C₄×#41	—	47	1	no	—
206	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
207	#441	—	2	2	no	—
208	#441	—	2	2	no	—
209	#441	—	94	2	no	—
210	#440	—	94	2	no	—
211	C₄×Q₁₆	—	94	2	no	—
212	C₄×SD₁₆	—	94	2	no	—
213	#433	—	94	2	no	—
214	#432	—	94	2	no	—
215	C₂×#108	—	47	2	no	—
216	#428	—	47	2	no	—
217	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
218	#424	—	47	1	no	—
219	#424	—	47	1	no	—
220	#419	—	47	2	no	—
221	#417	—	47	2	no	—
222	C₂×#103	C₂×D₄₇	1	2	yes	—
223	C₂×#101	—	47	2	no	—
224	C₂×#101	—	47	2	no	—
225	#408	—	47	1	no	—
226	#408	—	47	1	no	—
227	C₄×M₁₆	—	47	1	no	—
228	C₄×M₁₆	—	47	2	no	—
229	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
230	#343	—	47	2	no	—
231	#343	—	47	2	no	—
232	#341	—	47	1	no	—
233	#341	—	47	1	no	—
234	C₄₇	#23720	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	—	47	1	no	—
236	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
237	#137	—	94	4	no	—
238	#137	—	94	4	no	—
239	C₂×Q₁₆	—	2	4	no	—
240	C₂×Q₁₆	—	2	2	no	—
241	C₂×Q₁₆	—	94	2	no	—
242	C₂×SD₁₆	—	94	2	no	—
243	C₂×M₁₆	—	47	2	no	—
244	C₂×M₁₆	—	47	2	no	—
245	C₂×M₁₆	—	47	1	no	—
246	C₂×M₁₆	—	47	1	no	—
247	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
249	C₄×Q₈	—	2	2	no	—
250	C₄×Q₈	—	2	2	no	—
251	C₄×Q₈	—	2	2	no	—
252	C₄×Q₈	—	2	2	no	—
253	C₄×Q₈	—	47	2	no	—
254	C₄×D₄	—	47	2	no	—
255	C₄×D₄	—	94	2	no	—
256	#117	—	47	1	no	—
257	#117	—	94	1	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	94	1	no	—
262	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
263	#108	—	94	4	no	—
264	#107	—	94	2	no	—
265	#106	—	2	2	no	—
266	#106	—	94	2	no	—
267	C₄⋊C₈	—	94	2	no	—
268	C₄⋊C₈	—	94	2	no	—
269	#103	—	2	4	no	—
270	#103	—	94	2	no	—
271	#103	—	94	2	no	—
272	#103	—	94	2	no	—
273	#102	—	94	2	no	—
274	#101	—	94	4	no	—
275	#101	—	94	4	no	—
276	C₂²⋊C₈	—	94	2	no	—
277	C₂²⋊C₈	—	94	2	no	—
278	#97	—	94	2	no	—
279	#97	—	94	1	no	—
280	#97	C₂²⋊D₄₇	1	2	yes	—
281	#97	C₂²⋊D₄₇	1	2	yes	—
282	C₄×C₈	—	94	2	no	—
283	C₄×C₈	—	94	1	no	—
284	#41	—	94	2	no	—
285	#41	—	94	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	47	1	no	—
292	C₂×Q₈	#10625350	1	4	yes	—
293	C₂×D₄	—	47	1	no	—
294	C₂×D₄	—	94	1	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	94	2	no	—
298	C₂²×C₄	—	94	1	no	—
299	C₂²×C₄	#10625352	1	2	yes	—
300	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
301	Q₁₆	—	2	8	no	—
302	Q₁₆	—	94	8	no	—
303	SD₁₆	—	94	8	no	—
304	M₁₆	—	47	4	no	—
305	M₁₆	—	94	2	no	—
306	M₁₆	—	94	2	no	—
307	M₁₆	—	94	2	no	—
308	M₁₆	—	94	2	no	—
309	M₁₆	—	94	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	94	2	no	—
313	C₂×C₈	—	94	2	no	—
314	C₂×C₈	—	94	1	no	—
315	C₂×C₈	—	94	1	no	—
316	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
317	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₄⋊C₄	—	47	2	no	—
323	C₄⋊C₄	—	94	2	no	—
324	C₂²⋊C₄	—	94	1	no	—
325	C₂²⋊C₄	—	94	2	no	—
326	C₄²	—	4	2	no	—
327	C₄²	—	4	2	no	—
328	C₄²	—	94	2	no	—
329	C₄²	—	94	1	no	—
330	C₄²	D₄×D₄₇	1	2	yes	—
331	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
332	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂³	—	94	1	no	—
334	C₂³	C₂×#10625352	1	1	yes	characteristic
335	Q₈	—	2	4	no	—
336	Q₈	—	2	4	no	—
337	Q₈	—	4	2	no	—
338	Q₈	—	4	2	no	—
339	Q₈	—	94	2	no	—
340	D₄	—	94	2	no	—
341	D₄	—	188	2	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	94	2	no	—
349	C₂×C₄	—	94	1	no	—
350	C₂×C₄	—	376	2	no	—
351	C₂×C₄	—	188	1	no	—
352	C₂×C₄	—	188	1	no	—
353	C₂×C₄	#13093995	1	2	yes	—
354	C₂×C₄	#13093995	1	2	yes	—
355	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
357	C₂×C₄	#41×D₄₇	1	2	yes	—
358	C₂×C₄	C₂×#10625350	1	2	yes	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	94	4	no	—
362	C₈	—	188	2	no	—
363	C₈	—	188	1	no	—
364	C₈	—	188	1	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	188	1	no	—
367	C₂²	—	188	2	no	—
368	C₂²	C₂×#13093995	1	1	yes	characteristic
369	C₂²	#15689653	1	2	yes	—
370	C₄	—	2	2	no	—
371	C₄	—	2	2	no	—
372	C₄	—	4	4	no	—
373	C₄	—	8	2	no	—
374	C₄	—	188	1	no	—
375	C₄	—	188	2	no	—
376	C₄	#15689632	1	2	yes	—
377	C₄	#15689634	1	1	yes	characteristic
378	C₄	#15689669	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	376	1	no	—
381	C₂	#20246626	1	1	yes	characteristic
382	C₂	#20253121	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#441	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂×#10625350	C₂³
hypercenter	C₂×#441	D₄₇
hypocenter	C₄₇	#23720
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#441

Complements

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

N	H ≅ G/N	type
3. #20244899	C₂	semidirect
5. #20244753	C₂	semidirect
7. #20244653	C₂	semidirect
9. #20243294	C₂	semidirect
10. C₄×#13093943	C₂	semidirect
11. C₂×C₄₇×#441	C₂	semidirect
189. C₂×#441	D₄₇	semidirect
234. C₄₇	#23720	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094055	2
#20252037	2
#20244899	2
#20244834	2
#20244753	2
#20244688	2
#20244653	2
#20244507	2
#20243294	2
C₄×#13093943	2
C₂×C₄₇×#441	2
#23720	47