Subgroups — #26373872

Order 12032.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255819	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245661	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245590	C₂	1	2	yes	maximal, complemented
4	#20245576	C₂	1	2	yes	maximal, complemented
5	#20245527	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245525	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245499	C₂	1	2	yes	maximal, complemented
8	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245220	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245174	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3334	C₂	1	1	yes	maximal, characteristic
12	#15689843	—	2	1	no	—
13	C₂×#13093968	C₂²	1	1	yes	characteristic
14	C₂×#13093964	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689465	—	2	2	no	—
17	#15689409	—	2	1	no	—
18	#15689364	C₂²	1	2	yes	—
19	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
20	#15689161	C₂²	1	2	yes	—
21	#15688950	C₂²	1	2	yes	—
22	#15688950	C₂²	1	1	yes	characteristic
23	#15688948	C₂²	1	2	yes	—
24	#15688948	C₂²	1	2	yes	—
25	#15688948	C₂²	1	2	yes	—
26	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
27	#15688926	C₂²	1	2	yes	—
28	#15688926	C₂²	1	1	yes	characteristic
29	#15688924	C₂²	1	2	yes	—
30	#15688924	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₇×#547	—	2	1	no	—
34	C₄₇×#500	C₂²	1	1	yes	characteristic
35	C₄₇×#480	—	2	2	no	—
36	C₄₇×#474	—	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	C₂×#107×C₄₇	C₂²	1	2	yes	—
41	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
42	#13094024	—	2	1	no	—
43	#13094017	—	4	2	no	—
44	#13094011	—	4	1	no	—
45	#13094003	—	2	2	no	—
46	C₄×Dic₉₄	—	2	1	no	—
47	#13093989	—	4	1	no	—
48	#13093983	—	2	2	no	—
49	#13093968	—	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	C₂³	1	1	yes	characteristic
56	#13093958	C₂³	1	2	yes	—
57	#13093956	—	2	2	no	—
58	#13093949	—	2	1	no	—
59	#13093949	—	2	2	no	—
60	C₂×#41×C₄₇	—	4	1	no	—
61	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₄₇	—	2	1	no	—
64	#123×C₄₇	—	4	2	no	—
65	C₂²⋊Q₈×C₄₇	—	4	1	no	—
66	#121×C₄₇	—	2	2	no	—
67	C₄×D₄×C₄₇	—	2	1	no	—
68	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
70	#107×C₄₇	—	2	2	no	—
71	#107×C₄₇	—	2	1	no	—
72	#107×C₄₇	C₂³	1	2	yes	—
73	#107×C₄₇	C₂³	1	1	yes	characteristic
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	#41×C₄₇	—	8	4	no	—
80	C₂×Q₈×C₄₇	—	4	1	no	—
81	C₂×D₄×C₄₇	—	2	1	no	—
82	C₂×D₄×C₄₇	—	4	2	no	—
83	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	—	4	2	no	—
85	C₂²×C₄×C₄₇	—	4	1	no	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×Dic₉₄	—	2	1	no	—
88	C₂×Dic₉₄	—	4	1	no	—
89	D₈×C₄₇	—	2	1	no	—
90	D₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₄₇	—	2	1	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	4	2	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
101	C₂²⋊C₄×C₄₇	—	4	1	no	—
102	C₂²⋊C₄×C₄₇	—	4	2	no	—
103	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂²⋊Dic₄₇	—	4	2	no	—
106	C₂²⋊Dic₄₇	—	4	2	no	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	4	1	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	Dic₄₇⋊C₄	—	4	1	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₄×Dic₄₇	—	4	1	no	—
114	Dic₁₈₈	—	2	1	no	—
115	Dic₁₈₈	—	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	Q₈×C₄₇	—	8	2	no	—
122	D₄×C₄₇	—	2	2	no	—
123	D₄×C₄₇	—	4	1	no	—
124	D₄×C₄₇	—	4	1	no	—
125	D₄×C₄₇	—	8	4	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	8	4	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	C₂×Dic₄₇	—	4	1	no	—
137	C₂×Dic₄₇	—	4	1	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	Dic₉₄	—	4	1	no	—
140	Dic₉₄	—	4	1	no	—
141	Dic₉₄	—	8	2	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	—	2	1	no	—
144	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
146	#33920	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	8	1	no	—
151	C₂²×C₄₇	—	8	1	no	—
152	C₂²×C₄₇	—	8	1	no	—
153	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	8	1	no	—
157	C₄×C₄₇	—	8	2	no	—
158	C₄×C₄₇	#509	1	2	yes	—
159	C₄×C₄₇	#547	1	1	yes	characteristic
160	C₄×C₄₇	#578	1	1	yes	characteristic
161	Dic₄₇	—	8	1	no	—
162	Dic₄₇	—	8	2	no	—
163	Dic₄₇	—	8	1	no	—
164	#3342	—	47	1	no	—
165	#3334	D₄₇	1	1	yes	characteristic
166	#2252	—	47	1	no	—
167	#2236	—	47	2	no	—
168	#2234	—	47	2	no	—
169	#2225	—	47	1	no	—
170	#2224	—	47	1	no	—
171	#2220	—	47	2	no	—
172	C₄×SD₃₂	—	47	1	no	—
173	#2169	—	47	1	no	—
174	#2159	—	47	2	no	—
175	C₂×C₄₇	—	4	2	no	—
176	C₂×C₄₇	—	16	1	no	—
177	C₂×C₄₇	#3334	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
179	C₂×C₄₇	#3417	1	1	yes	characteristic
180	#556	—	94	1	no	—
181	#547	—	2	1	no	—
182	C₂×SD₃₂	—	47	1	no	—
183	C₂×M₃₂	—	47	2	no	—
184	#500	—	47	1	no	—
185	#500	C₂×D₄₇	1	1	yes	characteristic
186	#485	—	94	2	no	—
187	#480	—	2	2	no	—
188	#475	—	94	1	no	—
189	#474	—	2	1	no	—
190	#467	—	47	2	no	—
191	#466	C₂×D₄₇	1	2	yes	—
192	C₄×Q₁₆	—	47	1	no	—
193	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
194	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
195	#427	—	47	2	no	—
196	C₂×#107	C₂×D₄₇	1	2	yes	—
197	#367	—	47	1	no	—
198	#367	—	47	2	no	—
199	#366	—	47	2	no	—
200	#366	—	47	2	no	—
201	#366	—	47	2	no	—
202	C₄⋊C₁₆	—	47	1	no	—
203	#358	—	47	1	no	—
204	#358	—	47	2	no	—
205	#357	—	47	1	no	—
206	#357	—	47	2	no	—
207	C₂²⋊C₁₆	—	47	2	no	—
208	C₄×C₁₆	—	47	1	no	—
209	C₄₇	#33920	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	4	1	no	—
211	C₂×Q₁₆	—	47	1	no	—
212	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	94	1	no	—
216	#126	—	188	2	no	—
217	#124	—	188	1	no	—
218	#123	—	4	2	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	C₂²⋊Q₈	—	94	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	94	1	no	—
223	C₄×Q₈	—	188	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
226	#117	—	94	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	SD₃₂	—	94	2	no	—
229	M₃₂	—	94	4	no	—
230	C₂×C₁₆	—	47	1	no	—
231	C₂×C₁₆	—	47	1	no	—
232	C₂×C₁₆	—	47	2	no	—
233	#107	—	2	2	no	—
234	#107	—	2	1	no	—
235	#107	—	47	2	no	—
236	#107	—	47	1	no	—
237	#107	—	94	1	no	—
238	#107	C₂²×D₄₇	1	1	yes	characteristic
239	#107	C₂²×D₄₇	1	2	yes	—
240	#106	—	94	2	no	—
241	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
242	#103	—	94	1	no	—
243	#103	—	94	2	no	—
244	#102	—	2	1	no	—
245	#102	—	2	2	no	—
246	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
247	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	#41	—	8	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	94	1	no	—
251	C₂×Q₈	—	188	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	#10625334	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
258	Q₁₆	—	94	1	no	—
259	Q₁₆	—	94	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
266	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	94	1	no	—
272	C₄⋊C₄	—	94	2	no	—
273	C₄⋊C₄	—	188	1	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₄⋊C₄	—	188	1	no	—
276	C₄⋊C₄	#10625334	1	1	yes	characteristic
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	188	2	no	—
280	C₂²⋊C₄	—	188	2	no	—
281	C₂²⋊C₄	#10625334	1	2	yes	—
282	C₄²	—	188	2	no	—
283	C₄²	—	188	1	no	—
284	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
285	C₁₆	—	47	2	no	—
286	C₁₆	—	94	2	no	—
287	C₁₆	—	94	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	C₂×#10625334	1	2	yes	—
290	Q₈	—	8	2	no	—
291	Q₈	—	376	2	no	—
292	Q₈	—	188	1	no	—
293	Q₈	—	188	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	188	1	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	—	188	1	no	—
307	C₂×C₄	C₂×#10625334	1	2	yes	—
308	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
309	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	#41×D₄₇	1	1	yes	characteristic
314	C₈	#13094071	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×#10625334	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	376	1	no	—
327	C₄	—	376	1	no	—
328	C₄	—	376	2	no	—
329	C₄	#15689559	1	2	yes	—
330	C₄	#15689772	1	1	yes	characteristic
331	C₄	#15689938	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	16	1	no	—
334	C₂	#20255731	1	1	yes	characteristic
335	C₂	C₂×#15689559	1	1	yes	characteristic
336	C₂	#20256539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3334	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3334	D₄₇
hypocenter	C₄₇	#33920
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #20255819	C₂	semidirect
2. #20245661	C₂	semidirect
3. #20245590	C₂	semidirect
4. #20245576	C₂	semidirect
5. #20245527	C₂	semidirect
6. #20245525	C₂	semidirect
7. #20245499	C₂	semidirect
8. C₄×#13093968	C₂	semidirect
9. #20245220	C₂	semidirect
10. #20245174	C₂	semidirect
118. C₄₇⋊C₁₆	#41	semidirect
209. C₄₇	#33920	semidirect

Maximal subgroups

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

subgroup	index
#20255819	2
#20245661	2
#20245590	2
#20245576	2
#20245527	2
#20245525	2
#20245499	2
C₄×#13093968	2
#20245220	2
#20245174	2
C₄₇×#3334	2
#33920	47