Subgroups — #26371123

Order 12032.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253036	C₂	1	1	yes	maximal, characteristic
2	#20251988	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245580	C₂	1	2	yes	maximal, complemented
4	#20245580	C₂	1	2	yes	maximal, complemented
5	#20245574	C₂	1	2	yes	maximal, complemented
6	#20245569	C₂	1	2	yes	maximal, complemented
7	#20245176	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245105	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15688925	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2113	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
13	#15689687	—	2	1	no	—
14	#15689606	—	2	1	no	—
15	#15689376	—	2	2	no	—
16	#15689359	C₂²	1	2	yes	—
17	#15689359	C₂²	1	2	yes	—
18	#15689165	—	2	2	no	—
19	#15689165	—	2	1	no	—
20	#15689161	C₂²	1	1	yes	characteristic
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689161	C₂²	1	2	yes	complemented
23	C₂×#13093958	C₂²	1	1	yes	characteristic
24	C₂×#13093958	C₂²	1	2	yes	complemented
25	#15689129	—	2	1	no	—
26	#15688951	C₂²	1	1	yes	characteristic, complemented
27	#15688951	C₂²	1	2	yes	complemented
28	#15688951	C₂²	1	1	yes	characteristic, complemented
29	#15688949	C₂²	1	1	yes	characteristic, complemented
30	#15688949	C₂²	1	2	yes	complemented
31	#15688949	C₂²	1	1	yes	characteristic, complemented
32	#15688925	C₂²	1	2	yes	—
33	#15688925	C₂²	1	2	yes	—
34	#15688925	C₂²	1	2	yes	—
35	#15688925	C₂²	1	2	yes	—
36	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	C₂²×D₁₈₈	—	2	1	no	—
42	C₂×D₃₇₆	—	2	2	no	—
43	C₂×D₃₇₆	—	2	2	no	—
44	C₂×D₃₇₆	—	2	1	no	—
45	C₂×D₃₇₆	C₂³	1	2	yes	—
46	#13094020	—	4	1	no	—
47	#13094013	—	4	1	no	—
48	#13093998	—	2	2	no	—
49	#13093998	—	2	2	no	—
50	#13093992	—	4	2	no	—
51	Q₈×Dic₄₇	—	4	1	no	—
52	D₄×Dic₄₇	—	4	2	no	—
53	D₄×Dic₄₇	—	4	1	no	—
54	#13093983	—	2	1	no	—
55	#13093983	—	2	2	no	—
56	#13093983	—	2	1	no	—
57	C₂×C₄⋊Dic₄₇	—	2	1	no	—
58	C₂×C₄⋊Dic₄₇	—	2	2	no	—
59	#13093958	—	2	1	no	—
60	#13093958	—	2	2	no	—
61	#13093958	C₂³	1	1	yes	characteristic
62	#13093958	C₂³	1	1	yes	characteristic
63	#13093958	C₂³	1	2	yes	—
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093958	C₂³	1	1	yes	characteristic
66	#13093956	—	2	1	no	—
67	#13093956	—	2	1	no	—
68	#13093956	—	2	2	no	—
69	#13093946	—	2	2	no	—
70	#13093946	—	2	2	no	—
71	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
72	#131×C₄₇	—	2	2	no	—
73	#131×C₄₇	—	2	2	no	—
74	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
75	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	M₃₂×C₄₇	—	2	2	no	—
79	C₂×C₁₆×C₄₇	—	2	2	no	—
80	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
83	C₂³×D₄₇	—	4	1	no	—
84	#41×C₄₇	—	2	2	no	—
85	#41×C₄₇	—	2	2	no	—
86	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
87	C₂×D₄×C₄₇	D₈	1	2	yes	—
88	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₄₇	—	4	2	no	—
92	C₂×C₂²⋊D₄₇	—	4	2	no	—
93	C₂²×Dic₄₇	—	4	2	no	—
94	C₂²×Dic₄₇	—	4	1	no	—
95	C₂×D₁₈₈	—	2	2	no	—
96	C₂×D₁₈₈	—	4	2	no	—
97	C₂×D₁₈₈	—	4	2	no	—
98	C₂×D₁₈₈	—	4	1	no	—
99	M₁₆×C₄₇	—	2	2	no	—
100	M₁₆×C₄₇	—	2	4	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	#41	1	2	yes	—
106	C₂×C₈×C₄₇	#41	1	2	yes	—
107	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₄₇	—	4	2	no	—
109	C₂²⋊Dic₄₇	—	4	1	no	—
110	C₂²⋊Dic₄₇	—	4	1	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	4	2	no	—
119	C₄⋊Dic₄₇	—	4	1	no	—
120	C₄×Dic₄₇	—	4	2	no	—
121	C₄×Dic₄₇	—	4	1	no	—
122	C₄×Dic₄₇	—	4	1	no	—
123	D₃₇₆	—	2	2	no	—
124	D₃₇₆	—	2	2	no	—
125	D₃₇₆	—	4	2	no	—
126	C₁₆×C₄₇	—	2	1	no	—
127	C₁₆×C₄₇	—	2	1	no	—
128	C₁₆×C₄₇	—	2	2	no	—
129	C₂³×C₄₇	C₂×D₈	1	2	yes	—
130	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂²×D₄₇	—	4	2	no	—
132	C₂²×D₄₇	—	8	2	no	—
133	C₂²×D₄₇	—	16	1	no	—
134	C₂²×D₄₇	—	16	1	no	—
135	Q₈×C₄₇	—	2	2	no	—
136	D₄×C₄₇	—	2	4	no	—
137	D₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	#123	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	#137	1	2	yes	—
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₇	—	16	2	no	—
146	C₂²⋊D₄₇	—	16	2	no	—
147	C₂×Dic₄₇	—	4	1	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	1	no	—
150	C₂×Dic₄₇	—	4	1	no	—
151	C₂×Dic₄₇	—	4	1	no	—
152	C₂×Dic₄₇	—	16	1	no	—
153	C₂×Dic₄₇	—	16	2	no	—
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	2	no	—
156	D₁₈₈	—	8	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	#123	1	2	yes	—
159	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
160	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
161	#33650	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	—	2	2	no	—
166	C₂²×C₄₇	—	2	2	no	—
167	C₂²×C₄₇	#480	1	2	yes	—
168	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₄₇	—	8	2	no	—
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	8	2	no	—
172	C₂×D₄₇	—	16	4	no	—
173	C₄×C₄₇	—	2	2	no	—
174	C₄×C₄₇	#480	1	2	yes	—
175	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
177	Dic₄₇	—	8	2	no	—
178	Dic₄₇	—	8	1	no	—
179	Dic₄₇	—	8	1	no	—
180	Dic₄₇	—	8	1	no	—
181	Dic₄₇	—	8	1	no	—
182	#3058	—	47	1	no	—
183	#2913	—	47	1	no	—
184	#2234	—	47	2	no	—
185	#2234	—	47	2	no	—
186	#2233	—	47	2	no	—
187	#2232	—	47	2	no	—
188	#2159	—	47	1	no	—
189	#2158	—	47	1	no	—
190	#2140	—	47	1	no	—
191	C₂×#357	—	47	1	no	—
192	#2113	D₄₇	1	1	yes	characteristic, complemented
193	C₂×C₄₇	—	2	2	no	—
194	C₂×C₄₇	—	4	2	no	—
195	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
196	C₂×C₄₇	#3415	1	1	yes	characteristic
197	C₂×C₄₇	#3416	1	1	yes	characteristic
198	D₄₇	—	16	2	no	—
199	C₂²×D₈	—	47	1	no	—
200	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
201	#535	—	94	1	no	—
202	#520	—	94	1	no	—
203	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
204	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
205	#469	—	94	2	no	—
206	#466	—	47	2	no	—
207	#466	—	47	2	no	—
208	#428	—	94	1	no	—
209	#428	—	94	2	no	—
210	#427	—	47	1	no	—
211	#427	—	47	2	no	—
212	#427	—	47	1	no	—
213	C₂×#107	—	47	1	no	—
214	C₂×#107	—	47	2	no	—
215	#418	—	94	1	no	—
216	#367	—	47	1	no	—
217	#367	—	47	1	no	—
218	#367	—	47	2	no	—
219	#366	—	47	1	no	—
220	#366	—	47	1	no	—
221	#366	—	47	2	no	—
222	#357	—	47	2	no	—
223	#357	—	47	2	no	—
224	#357	—	47	2	no	—
225	#357	—	47	2	no	—
226	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	complemented
227	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	complemented
228	C₄₇	#33650	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	D₁₈₈	1	1	yes	characteristic
230	C₂²×D₄	—	94	1	no	—
231	C₂×D₈	—	47	2	no	—
232	C₂×D₈	—	94	1	no	—
233	C₂×D₈	—	94	2	no	—
234	C₂×D₈	—	94	2	no	—
235	#131	—	2	2	no	—
236	#131	—	2	2	no	—
237	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
238	C₂×M₁₆	D₁₈₈	1	2	yes	—
239	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
240	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	#127	—	188	1	no	—
242	#124	—	188	1	no	—
243	#121	—	94	2	no	—
244	#121	—	94	2	no	—
245	#120	—	188	2	no	—
246	C₄×Q₈	—	188	1	no	—
247	C₄×D₄	—	188	2	no	—
248	C₄×D₄	—	188	1	no	—
249	#117	—	94	2	no	—
250	#117	—	94	1	no	—
251	#117	—	94	1	no	—
252	C₂×C₄⋊C₄	—	94	2	no	—
253	C₂×C₄⋊C₄	—	94	1	no	—
254	M₃₂	—	2	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
257	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
258	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
259	#107	—	47	2	no	—
260	#107	—	47	1	no	—
261	#107	—	47	1	no	—
262	#107	—	47	1	no	—
263	#107	—	47	1	no	—
264	#107	—	94	2	no	—
265	#107	—	94	1	no	—
266	#106	—	94	2	no	—
267	#106	—	94	1	no	—
268	#106	—	94	1	no	—
269	#102	—	94	2	no	—
270	#102	—	94	2	no	—
271	C₂⁴	—	188	1	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
275	C₂×D₄	—	94	2	no	—
276	C₂×D₄	—	188	2	no	—
277	C₂×D₄	—	188	2	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂×D₄	—	188	1	no	—
280	C₂×D₄	—	188	2	no	—
281	C₂×D₄	D₃₇₆	1	2	yes	—
282	C₂×D₄	D₃₇₆	1	1	yes	characteristic
283	C₂²×C₄	—	188	2	no	—
284	C₂²×C₄	—	188	1	no	—
285	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
286	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
287	D₈	—	94	2	no	—
288	D₈	—	94	2	no	—
289	D₈	—	188	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
295	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
296	C₂×C₈	#10625352	1	2	yes	—
297	C₂×C₈	#10625352	1	2	yes	—
298	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
299	C₄⋊C₄	—	94	1	no	—
300	C₄⋊C₄	—	94	1	no	—
301	C₄⋊C₄	—	94	1	no	—
302	C₄⋊C₄	—	94	1	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	188	2	no	—
305	C₄⋊C₄	—	188	1	no	—
306	C₂²⋊C₄	—	188	2	no	—
307	C₂²⋊C₄	—	188	1	no	—
308	C₂²⋊C₄	—	188	1	no	—
309	C₂²⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	188	2	no	—
311	C₄²	—	188	2	no	—
312	C₄²	—	188	1	no	—
313	C₄²	—	188	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	2	2	no	—
317	C₂³	—	376	2	no	—
318	C₂³	—	188	2	no	—
319	C₂³	—	752	1	no	—
320	C₂³	—	752	1	no	—
321	C₂³	C₂×D₃₇₆	1	2	yes	—
322	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	D₄	—	2	4	no	—
325	D₄	—	2	2	no	—
326	D₄	—	376	2	no	—
327	D₄	—	188	2	no	—
328	D₄	—	188	2	no	—
329	D₄	—	752	2	no	—
330	D₄	—	752	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	188	1	no	—
334	C₂×C₄	—	188	1	no	—
335	C₂×C₄	—	188	1	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	1	no	—
338	C₂×C₄	—	752	1	no	—
339	C₂×C₄	—	752	2	no	—
340	C₂×C₄	#13094008	1	2	yes	—
341	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
342	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
343	C₂×C₄	#13094057	1	2	yes	—
344	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	#13094008	1	2	yes	—
347	C₈	C₂×#10625352	1	1	yes	characteristic
348	C₈	C₂×#10625352	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	376	2	no	—
355	C₂²	—	376	2	no	—
356	C₂²	—	376	2	no	—
357	C₂²	—	752	4	no	—
358	C₂²	#15689437	1	2	yes	—
359	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	376	2	no	—
362	C₄	—	376	1	no	—
363	C₄	—	376	1	no	—
364	C₄	—	376	1	no	—
365	C₄	—	376	1	no	—
366	C₄	#15689437	1	2	yes	—
367	C₄	C₂×#13094008	1	1	yes	characteristic
368	C₄	C₂×#13094057	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	4	2	no	—
371	C₂	—	752	2	no	—
372	C₂	C₂×#15689437	1	1	yes	characteristic
373	C₂	#20256529	1	1	yes	characteristic
374	C₂	#20256536	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2113	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20253036	C₂
hypercenter	#2113	D₄₇
hypocenter	C₄₇	#33650
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #20251988	C₂	semidirect
3. #20245580	C₂	semidirect
4. #20245580	C₂	semidirect
5. #20245574	C₂	semidirect
6. #20245569	C₂	semidirect
7. #20245176	C₂	semidirect
8. #20245172	C₂	semidirect
9. #20245105	C₂	semidirect
10. C₂×#15688925	C₂	semidirect
11. C₄₇×#2113	C₂	semidirect
22. #15689161	C₂²	semidirect
24. C₂×#13093958	C₂²	semidirect
26. #15688951	C₂²	semidirect
27. #15688951	C₂²	semidirect
28. #15688951	C₂²	semidirect
29. #15688949	C₂²	semidirect
30. #15688949	C₂²	semidirect
31. #15688949	C₂²	semidirect
39. C₄₇×C₂²⋊C₁₆	C₂²	semidirect
40. C₄₇×C₂²⋊C₁₆	C₂²	semidirect
192. #2113	D₄₇	semidirect
226. C₂²⋊C₁₆	C₂×D₄₇	semidirect
227. C₂²⋊C₁₆	C₂×D₄₇	semidirect
228. C₄₇	#33650	semidirect

Maximal subgroups

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

subgroup	index
#20253036	2
#20251988	2
#20245580	2
#20245574	2
#20245569	2
#20245176	2
#20245172	2
#20245105	2
C₂×#15688925	2
C₄₇×#2113	2
#33650	47