Subgroups — #26373221

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255148	C₂	1	1	yes	maximal, characteristic
2	#20245613	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245579	C₂	1	2	yes	maximal, complemented
4	#20245527	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245486	C₂	1	2	yes	maximal, complemented
6	#20245428	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245326	C₂	1	2	yes	maximal, complemented
8	C₄×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245093	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	2	yes	—
13	#15689794	—	2	1	no	—
14	C₂×#13093970	C₂²	1	1	yes	characteristic
15	C₂×#13093967	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
17	#15689500	C₂²	1	1	yes	characteristic
18	#15689359	C₂²	1	2	yes	—
19	#15689324	—	2	1	no	—
20	#15689270	—	2	2	no	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	#15688950	C₂²	1	2	yes	—
23	#15688948	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#15688927	C₂²	1	1	yes	characteristic
26	#15688926	C₂²	1	2	yes	—
27	#15688926	C₂²	1	1	yes	characteristic
28	#15688923	C₂²	1	2	yes	—
29	#15688923	C₂²	1	2	yes	—
30	#15688923	C₂²	1	2	yes	—
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₇×#556	—	2	1	no	—
34	C₄₇×#500	C₂²	1	1	yes	characteristic
35	C₄₇×#485	—	2	2	no	—
36	C₄₇×#475	—	2	1	no	—
37	C₄₇×#467	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#427	C₂²	1	2	yes	—
41	C₂×#10625350	—	2	2	no	—
42	#13094046	—	2	4	no	—
43	#13094046	—	4	2	no	—
44	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
45	C₂×#10625325	—	2	2	no	—
46	C₂×D₃₇₆	—	2	1	no	—
47	C₂×D₃₇₆	C₂³	1	2	yes	—
48	#13094021	—	2	1	no	—
49	#13094020	—	4	1	no	—
50	#13093998	—	2	2	no	—
51	#13093997	—	4	2	no	—
52	C₄×Dic₉₄	—	2	1	no	—
53	#13093986	—	4	1	no	—
54	#13093970	—	2	2	no	—
55	#13093967	—	4	2	no	—
56	C₂×C₄₇⋊C₁₆	—	4	2	no	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13093958	C₂³	1	1	yes	characteristic
61	#13093949	—	2	1	no	—
62	#13093946	—	2	2	no	—
63	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	#126×C₄₇	—	4	2	no	—
67	#124×C₄₇	—	4	1	no	—
68	C₂²⋊Q₈×C₄₇	—	2	2	no	—
69	C₄×Q₈×C₄₇	—	2	1	no	—
70	C₄×Q₈×C₄₇	—	4	1	no	—
71	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
72	#117×C₄₇	—	2	2	no	—
73	#107×C₄₇	—	2	1	no	—
74	#107×C₄₇	C₂³	1	2	yes	—
75	#107×C₄₇	C₂³	1	1	yes	characteristic
76	#106×C₄₇	—	2	2	no	—
77	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
78	#103×C₄₇	—	2	2	no	—
79	#103×C₄₇	—	2	1	no	—
80	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	2	1	no	—
83	C₂×Q₈×C₄₇	—	4	1	no	—
84	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₂²⋊D₄₇	—	4	2	no	—
87	C₂×C₂²⋊D₄₇	—	4	2	no	—
88	#10625350	—	4	4	no	—
89	#10625350	—	8	2	no	—
90	C₂×D₁₈₈	—	2	2	no	—
91	C₂×D₁₈₈	—	4	1	no	—
92	C₂×D₁₈₈	—	4	1	no	—
93	C₂×C₄×D₄₇	—	4	2	no	—
94	C₂×Dic₉₄	—	2	1	no	—
95	Q₁₆×C₄₇	—	2	1	no	—
96	Q₁₆×C₄₇	—	2	1	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₄₇	—	2	1	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	4	1	no	—
105	C₄⋊C₄×C₄₇	—	4	2	no	—
106	C₄⋊C₄×C₄₇	—	4	1	no	—
107	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
111	C₄²×C₄₇	—	4	2	no	—
112	C₄²×C₄₇	—	4	1	no	—
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	(C₂×D₄₇)⋊C₄	—	4	2	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	Dic₄₇⋊C₄	—	4	1	no	—
117	C₄×Dic₄₇	—	4	1	no	—
118	Dic₁₈₈	—	2	1	no	—
119	Dic₁₈₈	#41	1	2	yes	—
120	D₃₇₆	—	2	2	no	—
121	D₃₇₆	—	2	2	no	—
122	D₃₇₆	—	8	1	no	—
123	#10625325	—	4	4	no	—
124	C₄₇⋊C₁₆	—	2	1	no	—
125	C₄₇⋊C₁₆	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₇	C₂×D₈	1	2	yes	—
128	C₂²×D₄₇	—	4	2	no	—
129	Q₈×C₄₇	—	4	1	no	—
130	Q₈×C₄₇	—	4	1	no	—
131	Q₈×C₄₇	—	8	2	no	—
132	D₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₄₇	—	8	4	no	—
143	C₂²⋊D₄₇	—	16	2	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	—	4	1	no	—
146	D₁₈₈	—	4	2	no	—
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	—	8	2	no	—
149	D₁₈₈	—	16	1	no	—
150	C₄×D₄₇	—	8	4	no	—
151	Dic₉₄	—	2	2	no	—
152	Dic₉₄	—	4	1	no	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	2	1	no	—
155	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
157	#33871	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	—	8	2	no	—
164	C₂×D₄₇	—	8	2	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	—	8	1	no	—
169	C₄×C₄₇	—	8	1	no	—
170	C₄×C₄₇	#508	1	2	yes	—
171	C₄×C₄₇	#546	1	1	yes	characteristic
172	C₄×C₄₇	#577	1	1	yes	characteristic
173	Dic₄₇	—	4	2	no	—
174	Dic₄₇	—	8	1	no	—
175	#3342	D₄₇	1	1	yes	characteristic, complemented
176	#3287	—	47	1	no	—
177	#2240	—	47	1	no	—
178	#2234	—	47	2	no	—
179	#2225	—	47	1	no	—
180	#2217	—	47	2	no	—
181	#2207	—	47	1	no	—
182	#2189	—	47	2	no	—
183	C₄×Q₃₂	—	47	1	no	—
184	D₄×C₁₆	—	47	1	no	—
185	#2138	—	47	2	no	—
186	C₂×C₄₇	—	4	2	no	—
187	C₂×C₄₇	#3294	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
189	C₂×C₄₇	#3416	1	1	yes	characteristic
190	D₄₇	—	16	2	no	—
191	C₂×#135	—	47	2	no	—
192	#556	—	2	1	no	—
193	#550	—	94	1	no	—
194	C₂×Q₃₂	—	47	1	no	—
195	C₂×SD₃₂	—	47	2	no	—
196	C₂²×C₁₆	—	47	2	no	—
197	#500	C₂×D₄₇	1	1	yes	characteristic
198	#493	—	47	1	no	—
199	#485	—	2	2	no	—
200	#475	—	2	1	no	—
201	#467	C₂×D₄₇	1	2	yes	—
202	#466	—	47	2	no	—
203	#460	—	94	1	no	—
204	#449	—	94	2	no	—
205	C₄×Q₁₆	—	47	1	no	—
206	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
207	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
208	#427	C₂×D₄₇	1	2	yes	—
209	#367	—	47	2	no	—
210	#366	—	47	1	no	—
211	C₄⋊C₁₆	—	47	1	no	—
212	#358	—	47	1	no	—
213	#358	—	47	1	no	—
214	#358	—	47	2	no	—
215	#357	—	47	2	no	—
216	#357	—	47	2	no	—
217	#357	—	47	2	no	—
218	C₂²⋊C₁₆	—	47	2	no	—
219	C₄×C₁₆	—	47	1	no	—
220	C₄₇	#33871	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	94	2	no	—
222	#135	—	94	4	no	—
223	#135	—	188	2	no	—
224	C₂×Q₁₆	—	47	1	no	—
225	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
226	C₂×SD₁₆	—	94	2	no	—
227	C₂×D₈	—	47	2	no	—
228	C₂×D₈	—	94	1	no	—
229	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	#127	—	94	1	no	—
232	#127	—	188	1	no	—
233	#126	—	4	2	no	—
234	#124	—	4	1	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	#121	—	94	2	no	—
237	#121	—	188	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	4	1	no	—
240	C₄×Q₈	—	94	1	no	—
241	C₄×D₄	—	188	1	no	—
242	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
243	#117	—	2	2	no	—
244	Q₃₂	—	94	2	no	—
245	SD₃₂	—	188	2	no	—
246	C₂×C₁₆	—	47	1	no	—
247	C₂×C₁₆	—	47	1	no	—
248	C₂×C₁₆	—	47	2	no	—
249	C₂×C₁₆	—	188	2	no	—
250	#107	—	2	1	no	—
251	#107	—	47	1	no	—
252	#107	C₂²×D₄₇	1	1	yes	characteristic
253	#107	C₂²×D₄₇	1	2	yes	—
254	#106	—	2	2	no	—
255	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
256	#103	—	2	1	no	—
257	#103	—	2	2	no	—
258	#103	—	94	1	no	—
259	#102	—	94	2	no	—
260	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
261	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
262	#41	—	376	2	no	—
263	#41	—	188	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	94	1	no	—
267	C₂×D₄	—	94	2	no	—
268	C₂×D₄	—	188	2	no	—
269	C₂×D₄	—	188	2	no	—
270	C₂×D₄	—	188	1	no	—
271	C₂×D₄	—	188	1	no	—
272	C₂×D₄	#10625334	1	1	yes	characteristic
273	C₂²×C₄	—	188	2	no	—
274	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	47	2	no	—
278	Q₁₆	—	94	1	no	—
279	SD₁₆	—	188	4	no	—
280	D₈	—	94	2	no	—
281	D₈	—	94	2	no	—
282	D₈	—	376	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
287	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	94	1	no	—
294	C₄⋊C₄	—	188	1	no	—
295	C₄⋊C₄	#10625334	1	1	yes	characteristic
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	188	2	no	—
299	C₂²⋊C₄	#10625334	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	188	1	no	—
303	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
304	C₁₆	—	94	1	no	—
305	C₁₆	—	94	1	no	—
306	C₁₆	—	94	2	no	—
307	C₂³	—	188	2	no	—
308	C₂³	C₂×#10625334	1	2	yes	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	Q₈	—	94	2	no	—
313	Q₈	—	188	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	376	4	no	—
316	D₄	—	376	2	no	—
317	D₄	—	188	2	no	—
318	D₄	—	188	2	no	—
319	D₄	—	752	1	no	—
320	D₄	—	752	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	376	4	no	—
327	C₂×C₄	—	188	1	no	—
328	C₂×C₄	—	188	1	no	—
329	C₂×C₄	C₂×#10625334	1	2	yes	—
330	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	#41×D₄₇	1	1	yes	characteristic
336	C₈	#13094070	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	376	2	no	—
341	C₂²	—	376	2	no	—
342	C₂²	—	376	2	no	—
343	C₂²	C₂²×#10625334	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	376	1	no	—
350	C₄	—	188	2	no	—
351	C₄	#15689553	1	2	yes	—
352	C₄	#15689761	1	1	yes	characteristic
353	C₄	#15689931	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	752	2	no	—
356	C₂	#20255195	1	1	yes	characteristic
357	C₂	C₂×#15689553	1	1	yes	characteristic
358	C₂	#20256532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3342	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255148	C₂
hypercenter	#3342	D₄₇
hypocenter	C₄₇	#33871
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #20245613	C₂	semidirect
3. #20245579	C₂	semidirect
4. #20245527	C₂	semidirect
5. #20245486	C₂	semidirect
6. #20245428	C₂	semidirect
7. #20245326	C₂	semidirect
8. C₄×#13093970	C₂	semidirect
9. D₄×C₄₇⋊C₁₆	C₂	semidirect
10. #20245093	C₂	semidirect
11. C₄₇×#3342	C₂	semidirect
175. #3342	D₄₇	semidirect
220. C₄₇	#33871	semidirect

Maximal subgroups

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

subgroup	index
#20255148	2
#20245613	2
#20245579	2
#20245527	2
#20245486	2
#20245428	2
#20245326	2
C₄×#13093970	2
D₄×C₄₇⋊C₁₆	2
#20245093	2
C₄₇×#3342	2
#33871	47