Subgroups — #26371125

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	#20253037	C₂	1	1	yes	maximal, characteristic
2	#20251985	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245582	C₂	1	2	yes	maximal, complemented
4	#20245581	C₂	1	2	yes	maximal, complemented
5	#20245575	C₂	1	2	yes	maximal, complemented
6	#20245570	C₂	1	2	yes	maximal, complemented
7	#20245177	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245103	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15688925	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245001	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
13	C₂×#13094030	C₂²	1	1	yes	characteristic
14	#15689692	—	2	1	no	—
15	#15689603	—	2	1	no	—
16	C₂×#13093964	C₂²	1	1	yes	characteristic
17	#15689377	—	2	2	no	—
18	#15689360	C₂²	1	2	yes	—
19	#15689360	C₂²	1	2	yes	—
20	#15689166	—	2	1	no	—
21	#15689162	—	2	2	no	—
22	#15689160	C₂²	1	2	yes	complemented
23	C₂×#13093958	C₂²	1	1	yes	characteristic
24	#107×D₄₇	C₂²	1	2	yes	complemented
25	#15689127	—	2	1	no	—
26	#15688950	C₂²	1	2	yes	complemented
27	#15688948	C₂²	1	2	yes	complemented
28	#15688925	C₂²	1	2	yes	—
29	#15688925	C₂²	1	2	yes	—
30	#15688923	C₂²	1	2	yes	—
31	#15688923	C₂²	1	2	yes	—
32	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	complemented
33	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#427	C₂²	1	1	yes	characteristic
36	C₄₇×#427	C₂²	1	1	yes	characteristic
37	C₄₇×#367	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#367	C₂²	1	1	yes	characteristic, complemented
39	C₄₇×#366	C₂²	1	1	yes	characteristic, complemented
40	C₄₇×#366	C₂²	1	1	yes	characteristic, complemented
41	C₂×#10625350	D₄	1	1	yes	characteristic
42	C₂²×D₁₈₈	—	2	1	no	—
43	C₂×D₃₇₆	—	2	2	no	—
44	C₂×D₃₇₆	—	2	2	no	—
45	C₂×D₃₇₆	—	2	1	no	—
46	C₂×D₃₇₆	C₂³	1	2	yes	—
47	#13094030	—	2	2	no	—
48	#13094030	—	2	2	no	—
49	C₂×#10625329	D₄	1	1	yes	characteristic
50	C₂×#10625324	D₄	1	2	yes	—
51	C₂×C₈×D₄₇	C₂³	1	2	yes	—
52	#13094021	—	4	1	no	—
53	#13094012	—	4	1	no	—
54	#13093999	—	2	2	no	—
55	#13093999	—	2	2	no	—
56	#13093994	—	4	2	no	—
57	C₄×Dic₉₄	—	4	1	no	—
58	C₄×D₁₈₈	—	4	1	no	—
59	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
60	#13093982	—	2	2	no	—
61	C₂×C₄⋊Dic₄₇	—	2	1	no	—
62	C₄⋊C₄×D₄₇	—	2	2	no	—
63	#13093964	—	2	2	no	—
64	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
65	#13093958	—	2	1	no	—
66	#13093958	C₂³	1	2	yes	—
67	#13093957	—	2	2	no	—
68	#13093955	—	2	2	no	—
69	#13093944	—	2	2	no	—
70	#13093944	—	2	2	no	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	#117×C₄₇	—	2	1	no	—
73	#117×C₄₇	—	2	1	no	—
74	C₂×C₁₆×C₄₇	—	2	2	no	—
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
77	#107×C₄₇	C₂³	1	1	yes	characteristic
78	#107×C₄₇	C₂³	1	1	yes	characteristic
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#107×C₄₇	C₂³	1	1	yes	characteristic
81	#106×C₄₇	—	2	1	no	—
82	#106×C₄₇	—	2	1	no	—
83	C₂³×D₄₇	—	4	1	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₇	D₈	1	2	yes	—
86	C₂²×Dic₄₇	—	4	1	no	—
87	#10625350	—	2	2	no	—
88	#10625350	—	2	2	no	—
89	C₂×D₁₈₈	—	2	2	no	—
90	C₂×D₁₈₈	—	4	2	no	—
91	C₂×D₁₈₈	—	4	2	no	—
92	C₂×D₁₈₈	—	4	1	no	—
93	C₂×D₁₈₈	—	4	2	no	—
94	C₂×D₁₈₈	—	4	2	no	—
95	C₂×D₁₈₈	D₈	1	1	yes	characteristic
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
98	C₂×Dic₉₄	D₈	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	#41	1	2	yes	—
101	C₂×C₈×C₄₇	#41	1	2	yes	—
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	1	no	—
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₄⋊C₄×C₄₇	—	2	1	no	—
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₂²⋊C₄×C₄₇	—	4	1	no	—
108	C₂²⋊C₄×C₄₇	—	4	1	no	—
109	C₄²×C₄₇	—	4	1	no	—
110	C₄²×C₄₇	—	4	1	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	(C₂×D₄₇)⋊C₄	—	4	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	C₄×Dic₄₇	—	4	2	no	—
118	#10625329	—	2	2	no	—
119	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
120	D₃₇₆	—	2	2	no	—
121	D₃₇₆	—	2	2	no	—
122	D₃₇₆	—	4	2	no	—
123	#10625324	—	2	4	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₈×D₄₇	—	2	2	no	—
126	C₁₆×C₄₇	—	2	1	no	—
127	C₁₆×C₄₇	—	2	1	no	—
128	C₄₇⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₄₇	—	4	2	no	—
131	C₂²×D₄₇	—	8	2	no	—
132	C₂²×D₄₇	—	16	1	no	—
133	C₂²×D₄₇	—	16	1	no	—
134	C₂²×D₄₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	1	no	—
139	C₂×C₄×C₄₇	#123	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#137	1	2	yes	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	16	1	no	—
146	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
147	D₁₈₈	—	2	2	no	—
148	D₁₈₈	—	4	2	no	—
149	D₁₈₈	—	4	2	no	—
150	D₁₈₈	—	8	2	no	—
151	D₁₈₈	—	16	2	no	—
152	D₁₈₈	—	16	2	no	—
153	C₄×D₄₇	—	2	2	no	—
154	C₄×D₄₇	—	2	2	no	—
155	C₄×D₄₇	—	16	2	no	—
156	Dic₉₄	—	2	2	no	—
157	C₈×C₄₇	#123	1	2	yes	—
158	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
159	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
160	C₄₇⋊C₈	—	2	2	no	—
161	#33650	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	#480	1	2	yes	—
165	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₄₇	—	2	2	no	—
167	C₂×D₄₇	—	2	2	no	—
168	C₂×D₄₇	—	2	2	no	—
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	—	8	1	no	—
174	C₄×C₄₇	—	8	1	no	—
175	C₄×C₄₇	—	8	1	no	—
176	C₄×C₄₇	—	8	1	no	—
177	C₄×C₄₇	#480	1	2	yes	—
178	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
179	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
180	Dic₄₇	—	2	2	no	—
181	Dic₄₇	—	8	2	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	D₄₇	1	1	yes	characteristic, complemented
190	#2140	—	47	1	no	—
191	C₂×#357	—	47	1	no	—
192	#2113	—	47	1	no	—
193	C₂×C₄₇	—	2	2	no	—
194	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
195	C₂×C₄₇	#3415	1	1	yes	characteristic
196	C₂×C₄₇	#3416	1	1	yes	characteristic
197	D₄₇	—	4	2	no	—
198	D₄₇	—	16	2	no	—
199	C₂²×D₈	—	47	1	no	—
200	C₂×#131	—	47	1	no	—
201	#535	—	94	1	no	—
202	#520	—	94	1	no	—
203	C₂×M₃₂	—	47	1	no	—
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	2	no	—
209	#428	—	94	1	no	—
210	#427	—	47	2	no	—
211	#427	C₂×D₄₇	1	1	yes	characteristic
212	#427	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#107	—	47	1	no	—
214	C₂×#107	—	47	2	no	—
215	#418	—	94	1	no	—
216	#367	—	47	2	no	—
217	#367	C₂×D₄₇	1	1	yes	characteristic, complemented
218	#367	C₂×D₄₇	1	1	yes	characteristic, complemented
219	#366	—	47	2	no	—
220	#366	C₂×D₄₇	1	1	yes	characteristic, complemented
221	#366	C₂×D₄₇	1	1	yes	characteristic, complemented
222	#357	—	47	2	no	—
223	#357	—	47	2	no	—
224	#357	—	47	2	no	—
225	#357	—	47	2	no	—
226	C₂²⋊C₁₆	—	47	2	no	—
227	C₂²⋊C₁₆	—	47	2	no	—
228	C₄₇	#33650	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	47	1	no	—
230	C₂²×D₄	—	94	1	no	—
231	C₂×D₈	—	47	2	no	—
232	C₂×D₈	—	94	2	no	—
233	C₂×D₈	—	94	1	no	—
234	C₂×D₈	—	94	2	no	—
235	#131	—	94	2	no	—
236	#131	—	94	2	no	—
237	C₂×M₁₆	—	47	1	no	—
238	C₂×M₁₆	—	47	2	no	—
239	C₂²×C₈	—	47	2	no	—
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	1	no	—
248	C₄×D₄	—	188	2	no	—
249	#117	—	2	1	no	—
250	#117	—	2	1	no	—
251	#117	—	94	2	no	—
252	C₂×C₄⋊C₄	—	94	2	no	—
253	C₂×C₄⋊C₄	—	94	1	no	—
254	M₃₂	—	94	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	—	47	2	no	—
257	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
259	#107	—	47	2	no	—
260	#107	—	94	2	no	—
261	#107	—	94	1	no	—
262	#107	C₂²×D₄₇	1	1	yes	characteristic
263	#107	C₂²×D₄₇	1	1	yes	characteristic
264	#107	C₂²×D₄₇	1	1	yes	characteristic
265	#107	C₂²×D₄₇	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	2	1	no	—
268	#106	—	94	2	no	—
269	#102	—	94	2	no	—
270	#102	—	94	2	no	—
271	C₂⁴	—	188	1	no	—
272	#41	—	94	2	no	—
273	#41	—	94	2	no	—
274	C₂×Q₈	—	47	1	no	—
275	C₂×D₄	—	47	2	no	—
276	C₂×D₄	—	47	1	no	—
277	C₂×D₄	—	94	2	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂×D₄	—	188	2	no	—
280	C₂×D₄	—	188	2	no	—
281	C₂×D₄	—	188	1	no	—
282	C₂×D₄	—	188	2	no	—
283	C₂²×C₄	—	47	2	no	—
284	C₂²×C₄	—	188	2	no	—
285	C₂²×C₄	—	188	1	no	—
286	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
287	D₈	—	94	2	no	—
288	D₈	—	94	2	no	—
289	D₈	—	188	2	no	—
290	M₁₆	—	94	2	no	—
291	M₁₆	—	94	4	no	—
292	C₂×C₈	—	47	2	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	—	94	2	no	—
295	C₂×C₈	#10625350	1	2	yes	—
296	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	#10625354	1	2	yes	—
298	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	188	2	no	—
305	C₄⋊C₄	—	188	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	188	2	no	—
309	C₂²⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	188	2	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	4	1	no	—
313	C₄²	—	188	2	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	94	2	no	—
317	C₂³	—	47	2	no	—
318	C₂³	—	376	2	no	—
319	C₂³	—	188	2	no	—
320	C₂³	—	752	1	no	—
321	C₂³	—	752	1	no	—
322	C₂³	D₈×D₄₇	1	1	yes	characteristic
323	Q₈	—	94	2	no	—
324	D₄	—	94	4	no	—
325	D₄	—	94	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	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	47	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	94	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	752	2	no	—
340	C₂×C₄	—	752	1	no	—
341	C₂×C₄	#13094009	1	2	yes	—
342	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
343	C₂×C₄	#13094056	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
345	C₈	—	94	2	no	—
346	C₈	#13094009	1	2	yes	—
347	C₈	C₂×#10625350	1	1	yes	characteristic
348	C₈	C₂×#10625354	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	94	2	no	—
352	C₂²	—	94	2	no	—
353	C₂²	—	94	2	no	—
354	C₂²	—	376	2	no	—
355	C₂²	—	376	2	no	—
356	C₂²	—	376	2	no	—
357	C₂²	—	752	4	no	—
358	C₂²	#15689438	1	2	yes	—
359	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	94	2	no	—
365	C₄	—	376	2	no	—
366	C₄	#15689438	1	2	yes	—
367	C₄	C₂×#13094009	1	1	yes	characteristic
368	C₄	C₂×#13094056	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	188	2	no	—
371	C₂	—	752	2	no	—
372	C₂	C₂×#15689438	1	1	yes	characteristic
373	C₂	#20256526	1	1	yes	characteristic
374	C₂	#20256531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2158	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20253037	C₂
hypercenter	#2158	D₄₇
hypocenter	C₄₇	#33650
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #20251985	C₂	semidirect
3. #20245582	C₂	semidirect
4. #20245581	C₂	semidirect
5. #20245575	C₂	semidirect
6. #20245570	C₂	semidirect
7. #20245177	C₂	semidirect
8. #20245103	C₂	semidirect
9. C₂×#15688925	C₂	semidirect
10. #20245001	C₂	semidirect
11. C₄₇×#2158	C₂	semidirect
22. #15689160	C₂²	semidirect
24. #107×D₄₇	C₂²	semidirect
26. #15688950	C₂²	semidirect
27. #15688948	C₂²	semidirect
32. (C₂×D₄₇)⋊C₁₆	C₂²	semidirect
33. (C₂×D₄₇)⋊C₁₆	C₂²	semidirect
37. C₄₇×#367	C₂²	semidirect
38. C₄₇×#367	C₂²	semidirect
39. C₄₇×#366	C₂²	semidirect
40. C₄₇×#366	C₂²	semidirect
189. #2158	D₄₇	semidirect
217. #367	C₂×D₄₇	semidirect
218. #367	C₂×D₄₇	semidirect
220. #366	C₂×D₄₇	semidirect
221. #366	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
#20253037	2
#20251985	2
#20245582	2
#20245581	2
#20245575	2
#20245570	2
#20245177	2
#20245103	2
C₂×#15688925	2
#20245001	2
C₄₇×#2158	2
#33650	47