Subgroups — #26371627

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253899	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245628	C₂	1	4	yes	maximal, complemented
4	#20245613	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245612	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245611	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245102	C₂	1	4	yes	maximal, complemented
8	C₄×#13093963	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×D₄₇	—	2	1	no	—
11	C₂×D₈×D₄₇	C₂²	1	2	yes	—
12	#15689678	—	2	1	no	—
13	#127×D₄₇	—	2	1	no	—
14	C₂×#13093963	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic, complemented
16	#15689524	—	2	1	no	—
17	#15689524	C₂²	1	2	yes	complemented
18	#15689519	—	2	1	no	—
19	#15689500	C₂²	1	1	yes	characteristic
20	#15689499	—	2	1	no	—
21	#15689498	—	2	1	no	—
22	#15689498	C₂²	1	2	yes	—
23	#15689160	—	2	1	no	—
24	#15689160	C₂²	1	2	yes	—
25	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
26	#15688925	C₂²	1	4	yes	complemented
27	#15688924	C₂²	1	4	yes	complemented
28	#15688923	C₂²	1	4	yes	complemented
29	#15688898	C₂²	1	2	yes	complemented
30	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
32	C₄₇×#493	C₂²	1	1	yes	characteristic
33	C₄₇×#357	C₂²	1	4	yes	complemented
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₄×D₄₇	—	2	2	no	—
36	C₂×D₄×D₄₇	—	4	1	no	—
37	C₂×D₄×D₄₇	—	4	1	no	—
38	C₂×D₃₇₆	—	2	1	no	—
39	C₂×D₃₇₆	C₂³	1	2	yes	—
40	C₂×#10625334	—	2	2	no	—
41	C₂×#10625334	—	2	2	no	—
42	D₈×D₄₇	—	2	8	no	—
43	D₈×D₄₇	—	4	4	no	—
44	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
45	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
46	#13094024	—	2	1	no	—
47	#13094021	—	2	1	no	—
48	#13094020	—	2	2	no	—
49	#13094020	—	4	1	no	—
50	#13094020	—	4	1	no	—
51	#13094015	—	2	2	no	—
52	#13094004	—	4	1	no	—
53	#13094000	—	4	1	no	—
54	Q₈×Dic₄₇	—	4	1	no	—
55	#13093989	—	4	1	no	—
56	#13093982	—	2	2	no	—
57	C₄²×D₄₇	D₄	1	1	yes	characteristic
58	#13093963	—	2	4	no	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
60	#13093958	—	2	1	no	—
61	#13093958	C₂³	1	2	yes	—
62	#13093955	—	2	2	no	—
63	#13093955	—	2	2	no	—
64	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₄₇	D₄	1	1	yes	characteristic
66	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₄₇	—	2	1	no	—
68	C₂×D₈×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	#127×C₄₇	—	2	1	no	—
71	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	C₂³	1	2	yes	—
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂³×D₄₇	—	4	2	no	—
76	C₂×Q₈×C₄₇	—	4	1	no	—
77	C₂×D₄×C₄₇	—	2	2	no	—
78	C₂×D₄×C₄₇	—	4	1	no	—
79	C₂×D₄×C₄₇	—	4	1	no	—
80	C₂×C₂²⋊D₄₇	—	4	2	no	—
81	C₂×C₂²⋊D₄₇	—	4	2	no	—
82	C₂×C₂²⋊D₄₇	—	4	2	no	—
83	D₄×D₄₇	—	4	4	no	—
84	D₄×D₄₇	—	4	4	no	—
85	D₄×D₄₇	—	8	4	no	—
86	D₄×D₄₇	—	8	4	no	—
87	C₂×D₁₈₈	—	2	2	no	—
88	C₂×D₁₈₈	—	4	1	no	—
89	C₂×D₁₈₈	—	4	1	no	—
90	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
91	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
92	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₉₄	—	4	1	no	—
94	D₈×C₄₇	—	2	4	no	—
95	D₈×C₄₇	—	4	2	no	—
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	#10625334	—	4	4	no	—
102	#10625334	—	4	4	no	—
103	(C₂×D₄₇)⋊C₄	—	4	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	—	4	1	no	—
106	C₄⋊Dic₄₇	—	4	2	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	C₄×Dic₄₇	—	4	2	no	—
109	C₄×Dic₄₇	D₈	1	1	yes	characteristic
110	C₄×Dic₄₇	D₈	1	1	yes	characteristic
111	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
114	D₃₇₆	—	2	4	no	—
115	D₃₇₆	—	4	2	no	—
116	C₈×D₄₇	SD₁₆	1	4	yes	—
117	C₈×D₄₇	#41	1	4	yes	—
118	C₁₆×C₄₇	—	2	2	no	—
119	C₄₇⋊C₁₆	—	2	2	no	—
120	C₂³×C₄₇	—	4	2	no	—
121	C₂²×D₄₇	—	4	2	no	—
122	C₂²×D₄₇	—	8	4	no	—
123	C₂²×D₄₇	—	8	4	no	—
124	C₂²×D₄₇	—	8	4	no	—
125	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₄₇	—	8	2	no	—
127	D₄×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	4	2	no	—
129	D₄×C₄₇	—	8	2	no	—
130	D₄×C₄₇	—	8	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₄₇	—	8	4	no	—
136	C₂²⋊D₄₇	—	8	4	no	—
137	C₂²⋊D₄₇	—	8	4	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
142	D₁₈₈	—	4	2	no	—
143	D₁₈₈	—	4	2	no	—
144	D₁₈₈	—	8	2	no	—
145	D₁₈₈	—	8	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	#124	1	2	yes	—
149	C₄×D₄₇	C₂×SD₁₆	1	2	yes	—
150	Dic₉₄	—	8	2	no	—
151	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
152	C₈×C₄₇	C₂×#41	1	2	yes	—
153	C₄₇⋊C₈	#124	1	2	yes	—
154	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
155	#33704	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	—	8	2	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₄₇	—	2	1	no	—
161	C₂×D₄₇	—	2	1	no	—
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	—	8	2	no	—
164	C₂×D₄₇	—	8	2	no	—
165	C₂×D₄₇	—	16	8	no	—
166	C₂×D₄₇	#486	1	2	yes	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	#509	1	2	yes	—
169	C₄×C₄₇	#509	1	2	yes	—
170	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
171	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
172	Dic₄₇	—	2	1	no	—
173	Dic₄₇	—	2	1	no	—
174	Dic₄₇	—	8	2	no	—
175	Dic₄₇	#486	1	2	yes	—
176	#3160	—	47	1	no	—
177	C₂×#493	—	47	1	no	—
178	#2244	—	47	4	no	—
179	#2240	—	47	1	no	—
180	#2240	—	47	1	no	—
181	#2240	—	47	1	no	—
182	#2240	D₄₇	1	1	yes	characteristic, complemented
183	#2140	—	47	4	no	—
184	C₄×M₃₂	—	47	1	no	—
185	C₂×C₄₇	—	16	2	no	—
186	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
187	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
189	D₄₇	—	2	2	no	—
190	D₄₇	—	16	2	no	—
191	C₂²×D₈	—	47	2	no	—
192	C₂²×D₈	—	94	1	no	—
193	#533	—	94	1	no	—
194	C₂×#127	—	94	1	no	—
195	C₂×M₃₂	—	47	2	no	—
196	#500	—	47	1	no	—
197	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
198	#499	—	47	2	no	—
199	#499	—	94	1	no	—
200	#498	—	94	1	no	—
201	#493	—	47	2	no	—
202	#493	—	47	1	no	—
203	#493	—	94	1	no	—
204	#493	—	94	1	no	—
205	#493	C₂×D₄₇	1	1	yes	characteristic
206	#427	—	47	2	no	—
207	#427	—	94	1	no	—
208	C₂×C₄×C₈	—	47	1	no	—
209	#357	—	47	4	no	—
210	#357	—	47	4	no	—
211	#357	—	47	4	no	—
212	#357	C₂×D₄₇	1	4	yes	complemented
213	#346	—	47	2	no	—
214	C₄×C₁₆	—	47	1	no	—
215	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
216	C₄₇	#33704	1	1	yes	characteristic, complemented, Sylow
217	C₂²×D₄	—	94	2	no	—
218	C₂²×D₄	—	188	1	no	—
219	C₂²×D₄	—	188	1	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	47	2	no	—
222	C₂×D₈	—	94	8	no	—
223	C₂×D₈	—	94	2	no	—
224	C₂×D₈	—	94	2	no	—
225	C₂×D₈	—	94	1	no	—
226	C₂×D₈	—	188	4	no	—
227	C₂×D₈	C₂²×D₄₇	1	2	yes	—
228	C₂²×C₈	—	47	1	no	—
229	C₂²×C₈	—	47	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	94	1	no	—
232	#127	—	2	1	no	—
233	#127	—	94	2	no	—
234	#127	—	94	1	no	—
235	#127	—	188	1	no	—
236	#127	—	188	1	no	—
237	#125	—	94	2	no	—
238	C₂²⋊Q₈	—	188	1	no	—
239	C₂²⋊Q₈	—	188	1	no	—
240	C₄×Q₈	—	188	1	no	—
241	C₄×Q₈	—	188	1	no	—
242	#117	—	94	2	no	—
243	C₂×C₄²	—	47	1	no	—
244	M₃₂	—	94	4	no	—
245	C₂×C₁₆	—	47	2	no	—
246	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
247	#107	—	2	1	no	—
248	#107	—	47	2	no	—
249	#107	—	94	1	no	—
250	#107	C₂²×D₄₇	1	2	yes	—
251	#106	—	94	2	no	—
252	#106	—	94	2	no	—
253	C₄×C₈	—	47	1	no	—
254	C₄×C₈	—	47	1	no	—
255	C₄×C₈	—	47	1	no	—
256	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
257	C₂⁴	—	188	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	188	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	94	2	no	—
264	C₂×D₄	—	376	4	no	—
265	C₂×D₄	—	376	4	no	—
266	C₂×D₄	—	188	4	no	—
267	C₂×D₄	—	188	4	no	—
268	C₂×D₄	—	188	2	no	—
269	C₂×D₄	—	188	2	no	—
270	C₂×D₄	—	188	2	no	—
271	C₂×D₄	—	188	1	no	—
272	C₂×D₄	—	188	1	no	—
273	C₂²×C₄	—	47	1	no	—
274	C₂²×C₄	—	47	1	no	—
275	C₂²×C₄	—	47	1	no	—
276	D₈	—	2	4	no	—
277	D₈	—	4	2	no	—
278	D₈	—	94	4	no	—
279	D₈	—	188	4	no	—
280	D₈	—	188	4	no	—
281	D₈	—	188	2	no	—
282	C₂×C₈	—	47	1	no	—
283	C₂×C₈	—	47	1	no	—
284	C₂×C₈	—	47	4	no	—
285	C₂×C₈	—	47	4	no	—
286	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	94	2	no	—
291	C₄⋊C₄	—	188	2	no	—
292	C₄⋊C₄	—	188	2	no	—
293	C₄⋊C₄	—	188	1	no	—
294	C₂²⋊C₄	—	188	2	no	—
295	C₄²	—	47	1	no	—
296	C₄²	—	47	1	no	—
297	C₄²	—	47	1	no	—
298	C₄²	—	188	2	no	—
299	C₄²	D₄×D₄₇	1	1	yes	characteristic
300	C₁₆	—	2	2	no	—
301	C₁₆	—	94	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	47	1	no	—
304	C₂³	—	376	4	no	—
305	C₂³	—	376	4	no	—
306	C₂³	—	376	4	no	—
307	C₂³	—	188	2	no	—
308	Q₈	—	8	2	no	—
309	Q₈	—	376	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	2	no	—
313	D₄	—	8	2	no	—
314	D₄	—	376	4	no	—
315	D₄	—	376	4	no	—
316	D₄	—	376	4	no	—
317	D₄	—	376	2	no	—
318	D₄	—	376	2	no	—
319	D₄	—	188	2	no	—
320	D₄	—	188	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	47	1	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	47	1	no	—
326	C₂×C₄	—	47	1	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
331	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
333	C₈	—	47	2	no	—
334	C₈	—	47	2	no	—
335	C₈	SD₁₆×D₄₇	1	2	yes	—
336	C₈	#41×D₄₇	1	2	yes	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	47	2	no	—
341	C₂²	—	94	1	no	—
342	C₂²	—	94	1	no	—
343	C₂²	—	376	2	no	—
344	C₂²	—	376	2	no	—
345	C₂²	—	376	2	no	—
346	C₂²	—	752	8	no	—
347	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
348	C₄	—	8	2	no	—
349	C₄	—	47	2	no	—
350	C₄	—	94	1	no	—
351	C₄	—	94	1	no	—
352	C₄	—	376	2	no	—
353	C₄	#15689557	1	2	yes	—
354	C₄	#15689560	1	2	yes	—
355	C₄	#124×D₄₇	1	1	yes	characteristic
356	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
357	C₂	—	16	2	no	—
358	C₂	—	94	2	no	—
359	C₂	—	752	2	no	—
360	C₂	#486×D₄₇	1	1	yes	characteristic
361	C₂	C₂×#15689557	1	1	yes	characteristic
362	C₂	C₂×#15689560	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×#2240	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#493×D₄₇	C₂
hypercenter	#2240	D₄₇
hypocenter	C₄₇	#33704
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #20253899	C₂	semidirect
3. #20245628	C₂	semidirect
4. #20245613	C₂	semidirect
5. #20245612	C₂	semidirect
6. #20245611	C₂	semidirect
7. #20245102	C₂	semidirect
8. C₄×#13093963	C₂	semidirect
9. C₄₇×#2240	C₂	semidirect
15. #15689528	C₂²	semidirect
17. #15689524	C₂²	semidirect
26. #15688925	C₂²	semidirect
27. #15688924	C₂²	semidirect
28. #15688923	C₂²	semidirect
29. #15688898	C₂²	semidirect
30. C₄×C₄₇⋊C₁₆	C₂²	semidirect
31. C₄₇×#500	C₂²	semidirect
33. C₄₇×#357	C₂²	semidirect
34. C₄×C₁₆×C₄₇	C₂²	semidirect
182. #2240	D₄₇	semidirect
197. #500	C₂×D₄₇	semidirect
212. #357	C₂×D₄₇	semidirect
215. C₄×C₁₆	C₂×D₄₇	semidirect
216. C₄₇	#33704	semidirect

Maximal subgroups

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

subgroup	index
#20253899	2
#493×D₄₇	2
#20245628	2
#20245613	2
#20245612	2
#20245611	2
#20245102	2
C₄×#13093963	2
C₄₇×#2240	2
#33704	47