Subgroups — #26373877

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255736	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245668	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245579	C₂	1	2	yes	maximal, complemented
4	#20245571	C₂	1	2	yes	maximal, complemented
5	#20245532	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245525	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245489	C₂	1	2	yes	maximal, complemented
8	C₄×#13093965	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245220	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245177	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3341	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689777	—	2	1	no	—
13	C₂×#13093965	C₂²	1	1	yes	characteristic
14	C₂×#13093964	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689437	—	2	2	no	—
17	#15689404	—	2	1	no	—
18	#15689359	C₂²	1	2	yes	—
19	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
20	C₂×#13093958	C₂²	1	2	yes	—
21	#15688950	C₂²	1	2	yes	—
22	#15688950	C₂²	1	2	yes	—
23	#15688950	C₂²	1	2	yes	—
24	#15688948	C₂²	1	2	yes	—
25	#15688948	C₂²	1	1	yes	characteristic
26	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
27	#15688924	C₂²	1	2	yes	—
28	#15688924	C₂²	1	1	yes	characteristic
29	#15688923	C₂²	1	2	yes	—
30	#15688923	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₇×#555	—	2	1	no	—
34	C₄₇×#500	C₂²	1	1	yes	characteristic
35	C₄₇×#482	—	2	2	no	—
36	C₄₇×#476	—	2	1	no	—
37	C₄₇×#466	C₂²	1	2	yes	—
38	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#427	C₂²	1	2	yes	—
41	C₂×#10625352	—	4	1	no	—
42	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
43	#13094024	—	2	1	no	—
44	#13094008	—	4	2	no	—
45	#13094004	—	4	1	no	—
46	#13093998	—	2	2	no	—
47	C₄×D₁₈₈	—	2	1	no	—
48	C₂×C₄⋊Dic₄₇	—	2	2	no	—
49	#13093965	—	2	2	no	—
50	#13093964	—	2	4	no	—
51	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13093958	—	2	1	no	—
55	#13093958	—	2	2	no	—
56	#13093958	C₂³	1	2	yes	—
57	#13093958	C₂³	1	1	yes	characteristic
58	#13093946	—	2	2	no	—
59	#13093946	—	2	1	no	—
60	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₄₇	—	2	1	no	—
63	#124×C₄₇	—	4	2	no	—
64	#121×C₄₇	—	2	2	no	—
65	#121×C₄₇	—	4	1	no	—
66	C₄×D₄×C₄₇	—	2	1	no	—
67	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
68	#117×C₄₇	—	2	2	no	—
69	#117×C₄₇	—	4	1	no	—
70	#107×C₄₇	—	2	1	no	—
71	#107×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	C₂³	1	1	yes	characteristic
73	#106×C₄₇	—	2	2	no	—
74	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
75	#102×C₄₇	—	2	2	no	—
76	#102×C₄₇	—	2	1	no	—
77	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	4	1	no	—
80	C₂×D₄×C₄₇	—	2	1	no	—
81	C₂×D₄×C₄₇	—	4	2	no	—
82	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	—	4	1	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₂²⋊D₄₇	—	4	2	no	—
86	C₂²×Dic₄₇	—	4	2	no	—
87	#10625352	—	8	4	no	—
88	C₂×D₁₈₈	—	2	1	no	—
89	C₂×C₄×D₄₇	—	4	1	no	—
90	C₂×Dic₉₄	—	4	1	no	—
91	D₈×C₄₇	—	2	1	no	—
92	D₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₇	—	2	1	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	4	1	no	—
101	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₄₇	—	4	2	no	—
103	C₂²⋊C₄×C₄₇	—	4	2	no	—
104	C₂²⋊C₄×C₄₇	—	4	1	no	—
105	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
106	C₄²×C₄₇	—	4	2	no	—
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	(C₂×D₄₇)⋊C₄	—	4	1	no	—
109	(C₂×D₄₇)⋊C₄	—	4	2	no	—
110	C₄⋊Dic₄₇	—	2	1	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	4	2	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	D₃₇₆	—	2	1	no	—
115	D₃₇₆	—	2	1	no	—
116	C₄₇⋊C₁₆	—	2	2	no	—
117	C₄₇⋊C₁₆	—	2	1	no	—
118	C₄₇⋊C₁₆	#41	1	2	yes	complemented
119	C₂³×C₄₇	—	4	1	no	—
120	C₂³×C₄₇	C₂×D₈	1	2	yes	—
121	C₂²×D₄₇	—	4	1	no	—
122	Q₈×C₄₇	—	16	1	no	—
123	D₄×C₄₇	—	2	2	no	—
124	D₄×C₄₇	—	4	1	no	—
125	D₄×C₄₇	—	4	1	no	—
126	D₄×C₄₇	—	16	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	8	2	no	—
132	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₄₇	—	8	4	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	8	4	no	—
140	D₁₈₈	—	4	1	no	—
141	D₁₈₈	—	4	1	no	—
142	C₄×D₄₇	—	8	2	no	—
143	Dic₉₄	—	8	2	no	—
144	C₈×C₄₇	—	2	2	no	—
145	C₈×C₄₇	—	2	1	no	—
146	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
148	#33921	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	8	1	no	—
153	C₂²×C₄₇	—	8	1	no	—
154	C₂²×C₄₇	—	8	1	no	—
155	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₇	—	8	1	no	—
157	C₂×D₄₇	—	8	1	no	—
158	C₂×D₄₇	—	8	1	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	8	1	no	—
162	C₄×C₄₇	—	8	2	no	—
163	C₄×C₄₇	#509	1	2	yes	—
164	C₄×C₄₇	#547	1	1	yes	characteristic
165	C₄×C₄₇	#578	1	1	yes	characteristic
166	Dic₄₇	—	8	2	no	—
167	Dic₄₇	—	8	1	no	—
168	#3341	D₄₇	1	1	yes	characteristic, complemented
169	#3334	—	47	1	no	—
170	#2254	—	47	1	no	—
171	#2234	—	47	2	no	—
172	#2233	—	47	2	no	—
173	#2226	—	47	1	no	—
174	#2224	—	47	1	no	—
175	#2218	—	47	2	no	—
176	C₄×D₁₆	—	47	1	no	—
177	#2169	—	47	1	no	—
178	#2159	—	47	2	no	—
179	C₂×C₄₇	—	4	2	no	—
180	C₂×C₄₇	—	16	1	no	—
181	C₂×C₄₇	#3334	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
183	C₂×C₄₇	#3416	1	1	yes	characteristic
184	D₄₇	—	16	1	no	—
185	#555	—	2	1	no	—
186	#547	—	94	1	no	—
187	C₂×D₁₆	—	47	1	no	—
188	C₂×M₃₂	—	47	2	no	—
189	#500	—	47	1	no	—
190	#500	C₂×D₄₇	1	1	yes	characteristic
191	#482	—	2	2	no	—
192	#480	—	94	2	no	—
193	#476	—	2	1	no	—
194	#474	—	94	1	no	—
195	#466	—	47	2	no	—
196	#466	C₂×D₄₇	1	2	yes	—
197	C₄×D₈	—	47	1	no	—
198	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
199	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
200	#427	C₂×D₄₇	1	2	yes	—
201	C₂×#107	—	47	2	no	—
202	#367	—	47	2	no	—
203	#367	—	47	2	no	—
204	#367	—	47	2	no	—
205	#366	—	47	1	no	—
206	#366	—	47	2	no	—
207	C₄⋊C₁₆	—	47	1	no	—
208	#357	—	47	1	no	—
209	#357	—	47	2	no	—
210	#357	—	47	1	no	—
211	#357	—	47	2	no	—
212	C₂²⋊C₁₆	—	47	2	no	—
213	C₄×C₁₆	—	47	1	no	—
214	C₄₇	#33921	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	188	1	no	—
216	C₂×D₈	—	47	1	no	—
217	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	94	1	no	—
221	#124	—	4	2	no	—
222	#123	—	188	2	no	—
223	C₂²⋊Q₈	—	188	1	no	—
224	#121	—	2	2	no	—
225	#121	—	4	1	no	—
226	#121	—	94	2	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	94	1	no	—
229	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
230	#117	—	2	2	no	—
231	#117	—	4	1	no	—
232	C₂×C₄⋊C₄	—	94	2	no	—
233	D₁₆	—	94	2	no	—
234	M₃₂	—	94	4	no	—
235	C₂×C₁₆	—	47	1	no	—
236	C₂×C₁₆	—	47	1	no	—
237	C₂×C₁₆	—	47	2	no	—
238	#107	—	2	1	no	—
239	#107	—	47	1	no	—
240	#107	—	47	2	no	—
241	#107	—	94	2	no	—
242	#107	—	94	1	no	—
243	#107	C₂²×D₄₇	1	1	yes	characteristic
244	#107	C₂²×D₄₇	1	2	yes	—
245	#106	—	2	2	no	—
246	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
247	#102	—	2	1	no	—
248	#102	—	2	2	no	—
249	#102	—	94	1	no	—
250	#102	—	94	2	no	—
251	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
252	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
253	#41	—	376	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	188	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	94	1	no	—
259	C₂×D₄	—	188	2	no	—
260	C₂×D₄	#10625334	1	1	yes	characteristic
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	188	2	no	—
263	C₂²×C₄	—	188	1	no	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	94	1	no	—
268	D₈	—	94	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	94	1	no	—
278	C₄⋊C₄	—	94	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	—	188	1	no	—
281	C₄⋊C₄	#10625334	1	1	yes	characteristic
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	188	1	no	—
286	C₂²⋊C₄	—	188	2	no	—
287	C₂²⋊C₄	#10625334	1	2	yes	—
288	C₄²	—	4	2	no	—
289	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
290	C₁₆	—	47	2	no	—
291	C₁₆	—	94	2	no	—
292	C₁₆	—	94	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	188	1	no	—
295	C₂³	C₂×#10625334	1	2	yes	—
296	Q₈	—	16	1	no	—
297	Q₈	—	376	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	16	2	no	—
302	D₄	—	376	4	no	—
303	D₄	—	188	1	no	—
304	D₄	—	188	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	376	4	no	—
311	C₂×C₄	—	376	2	no	—
312	C₂×C₄	—	188	1	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	C₂×#10625334	1	2	yes	—
315	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
316	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	#41×D₄₇	1	1	yes	characteristic
321	C₈	#13094071	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	376	1	no	—
329	C₂²	—	376	1	no	—
330	C₂²	—	376	1	no	—
331	C₂²	C₂²×#10625334	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	376	1	no	—
337	C₄	—	376	2	no	—
338	C₄	#15689558	1	2	yes	—
339	C₄	#15689772	1	1	yes	characteristic
340	C₄	#15689938	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	16	1	no	—
343	C₂	—	752	1	no	—
344	C₂	#20255731	1	1	yes	characteristic
345	C₂	C₂×#15689558	1	1	yes	characteristic
346	C₂	#20256535	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3341	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3341	D₄₇
hypocenter	C₄₇	#33921
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3341

Complements

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

N	H ≅ G/N	type
1. #20255736	C₂	semidirect
2. #20245668	C₂	semidirect
3. #20245579	C₂	semidirect
4. #20245571	C₂	semidirect
5. #20245532	C₂	semidirect
6. #20245525	C₂	semidirect
7. #20245489	C₂	semidirect
8. C₄×#13093965	C₂	semidirect
9. #20245220	C₂	semidirect
10. #20245177	C₂	semidirect
11. C₄₇×#3341	C₂	semidirect
118. C₄₇⋊C₁₆	#41	semidirect
168. #3341	D₄₇	semidirect
214. C₄₇	#33921	semidirect

Maximal subgroups

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

subgroup	index
#20255736	2
#20245668	2
#20245579	2
#20245571	2
#20245532	2
#20245525	2
#20245489	2
C₄×#13093965	2
#20245220	2
#20245177	2
C₄₇×#3341	2
#33921	47