Subgroups — #26369636

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093965	C₂	1	1	yes	maximal, characteristic
2	#20252240	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252239	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245261	C₂	1	4	yes	maximal, complemented
5	#20245232	C₂	1	4	yes	maximal, complemented
6	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245105	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	#15689598	—	2	1	no	—
12	#15689597	—	2	1	no	—
13	C₂×#13093965	C₄	1	8	yes	complemented
14	C₂×#13093965	C₂²	1	2	yes	—
15	C₂×#13093965	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#15689235	—	2	2	no	—
18	#15689234	—	2	2	no	—
19	D₈×Dic₄₇	C₂²	1	4	yes	complemented
20	#15689205	C₂²	1	4	yes	complemented
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689128	—	2	1	no	—
23	#15689127	—	2	1	no	—
24	#15689078	C₂²	1	1	yes	characteristic
25	#15688950	C₂²	1	2	yes	complemented
26	#15688948	C₂²	1	2	yes	complemented
27	#15688925	C₂²	1	4	yes	complemented
28	#15688898	C₂²	1	2	yes	complemented
29	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
30	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#427	C₂²	1	1	yes	characteristic
33	C₄₇×#357	C₂²	1	4	yes	complemented
34	C₂²×D₁₈₈	—	2	1	no	—
35	C₂×D₃₇₆	—	2	1	no	—
36	C₂×D₃₇₆	—	2	2	no	—
37	C₂×D₃₇₆	C₂×C₄	1	4	yes	complemented
38	C₂×D₃₇₆	C₂³	1	2	yes	—
39	C₂×#10625329	D₄	1	1	yes	characteristic
40	C₄×D₁₈₈	—	4	2	no	—
41	D₄×Dic₄₇	—	2	4	no	—
42	D₄×Dic₄₇	—	4	2	no	—
43	#13093986	—	2	4	no	—
44	#13093983	—	2	1	no	—
45	#13093983	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊Dic₄₇	—	4	2	no	—
47	C₂²⋊C₄×D₄₇	—	4	2	no	—
48	#13093965	C₂×C₄	1	8	yes	complemented
49	#13093965	C₂×C₄	1	8	yes	complemented
50	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
51	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	#13093958	C₂³	1	2	yes	—
54	#13093956	—	2	1	no	—
55	#13093945	—	2	4	no	—
56	#13093944	—	2	4	no	—
57	#13093934	D₄	1	2	yes	—
58	C₈×Dic₄₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₄₇	—	2	1	no	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	2	no	—
62	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	complemented
63	C₂×D₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	#117×C₄₇	—	2	1	no	—
66	M₃₂×C₄₇	—	2	2	no	—
67	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
68	#107×C₄₇	C₂³	1	2	yes	—
69	#106×C₄₇	—	2	1	no	—
70	C₂⁴×C₄₇	—	4	1	no	—
71	C₂³×D₄₇	—	4	1	no	—
72	C₂×D₄×C₄₇	—	2	4	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	—	4	1	no	—
75	C₂×D₄×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂×D₁₈₈	—	2	4	no	—
79	C₂×D₁₈₈	—	2	2	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×C₄×D₄₇	—	4	4	no	—
83	D₈×C₄₇	—	2	4	no	—
84	D₈×C₄₇	—	2	2	no	—
85	D₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	#41	1	2	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₂²⋊C₄×C₄₇	—	4	1	no	—
92	C₄²×C₄₇	—	4	1	no	—
93	C₂²⋊Dic₄₇	—	4	1	no	—
94	C₂²⋊Dic₄₇	—	4	4	no	—
95	C₂²⋊Dic₄₇	D₈	1	2	yes	—
96	(C₂×D₄₇)⋊C₄	—	4	4	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	D₈	1	2	yes	—
99	C₄×Dic₄₇	—	4	1	no	—
100	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
101	#10625329	—	2	2	no	—
102	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
103	D₃₇₆	—	2	4	no	—
104	D₃₇₆	—	2	2	no	—
105	D₃₇₆	C₂²×C₄	1	4	yes	—
106	C₁₆×C₄₇	—	2	2	no	—
107	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₇	—	4	2	no	—
110	C₂³×C₄₇	—	4	4	no	—
111	C₂³×C₄₇	—	16	2	no	—
112	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₄₇	—	4	2	no	—
114	C₂²×D₄₇	—	4	4	no	—
115	C₂²×D₄₇	—	16	2	no	—
116	D₄×C₄₇	—	2	4	no	—
117	D₄×C₄₇	—	4	2	no	—
118	D₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#135	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	16	4	no	—
126	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
127	D₁₈₈	—	2	4	no	—
128	D₁₈₈	—	4	2	no	—
129	D₁₈₈	—	4	4	no	—
130	C₄×D₄₇	—	16	4	no	—
131	C₈×C₄₇	C₄×D₄	1	2	yes	—
132	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₄₇⋊C₈	—	2	2	no	—
135	#33456	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	8	4	no	—
139	C₂²×C₄₇	—	16	4	no	—
140	C₂²×C₄₇	C₄×D₈	1	2	yes	—
141	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₄₇	—	4	4	no	—
143	C₂×D₄₇	—	8	4	no	—
144	C₂×D₄₇	—	16	4	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	C₄×D₈	1	2	yes	—
147	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
149	Dic₄₇	—	2	4	no	—
150	Dic₄₇	—	8	2	no	—
151	C₂²×D₁₆	—	47	1	no	—
152	#2948	—	47	1	no	—
153	#2948	—	47	1	no	—
154	#2177	—	47	4	no	—
155	C₄×D₁₆	—	47	4	no	—
156	#2158	—	47	1	no	—
157	#2140	—	47	1	no	—
158	#2140	D₄₇	1	1	yes	characteristic, complemented
159	#2108	—	47	1	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	8	4	no	—
162	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₄₇	#3415	1	2	yes	—
164	D₄₇	—	8	4	no	—
165	C₂²×D₈	—	47	1	no	—
166	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
167	#518	—	94	1	no	—
168	#518	—	94	1	no	—
169	C₂×D₁₆	—	47	8	no	—
170	C₂×D₁₆	—	47	2	no	—
171	C₂×D₁₆	—	47	2	no	—
172	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×C₁₆	—	47	1	no	—
174	#442	—	94	2	no	—
175	#442	—	94	2	no	—
176	C₄×D₈	—	47	4	no	—
177	C₄×D₈	—	47	4	no	—
178	#427	—	47	1	no	—
179	#427	C₂×D₄₇	1	1	yes	characteristic
180	#418	—	94	1	no	—
181	#418	—	94	1	no	—
182	#405	—	47	1	no	—
183	#367	—	47	2	no	—
184	#366	—	47	2	no	—
185	#357	—	47	4	no	—
186	#357	C₂×D₄₇	1	4	yes	complemented
187	#346	—	47	2	no	—
188	C₄×C₁₆	—	47	2	no	—
189	C₄₇	#33456	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₄	—	2	1	no	—
191	C₂²×D₄	—	94	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	47	4	no	—
195	C₂×D₈	—	47	2	no	—
196	C₂×D₈	—	94	1	no	—
197	C₂×D₈	—	94	2	no	—
198	C₂×D₈	C₄×D₄₇	1	4	yes	complemented
199	C₂×D₈	C₂²×D₄₇	1	2	yes	—
200	C₂×M₁₆	—	47	1	no	—
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₄×D₄	—	94	4	no	—
203	C₄×D₄	—	94	4	no	—
204	C₄×D₄	—	188	2	no	—
205	C₄×D₄	—	188	2	no	—
206	#117	—	2	1	no	—
207	#117	—	47	1	no	—
208	#117	—	94	1	no	—
209	C₂×C₂²⋊C₄	—	188	2	no	—
210	C₂×C₂²⋊C₄	—	188	2	no	—
211	D₁₆	—	47	8	no	—
212	D₁₆	—	47	8	no	—
213	M₃₂	—	2	2	no	—
214	C₂×C₁₆	—	47	4	no	—
215	C₂×C₁₆	—	47	1	no	—
216	C₂×C₁₆	—	47	1	no	—
217	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
218	#107	—	47	2	no	—
219	#107	C₂²×D₄₇	1	2	yes	—
220	#106	—	2	1	no	—
221	#106	—	94	1	no	—
222	#102	—	94	4	no	—
223	#102	—	94	4	no	—
224	#97	—	47	2	no	—
225	C₄×C₈	—	47	2	no	—
226	C₂⁴	—	4	1	no	—
227	C₂⁴	—	188	1	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	94	4	no	—
233	C₂×D₄	—	94	2	no	—
234	C₂×D₄	—	188	1	no	—
235	C₂×D₄	—	188	2	no	—
236	C₂²×C₄	—	188	4	no	—
237	C₂²×C₄	—	188	4	no	—
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	D₈	—	2	4	no	—
240	D₈	—	2	2	no	—
241	D₈	—	47	4	no	—
242	D₈	—	94	4	no	—
243	D₈	—	94	2	no	—
244	D₈	C₂×C₄×D₄₇	1	4	yes	—
245	M₁₆	—	94	2	no	—
246	C₂×C₈	—	47	2	no	—
247	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
248	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
249	C₂×C₈	#10625352	1	2	yes	—
250	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	47	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	47	2	no	—
256	C₂²⋊C₄	—	188	4	no	—
257	C₂²⋊C₄	—	188	4	no	—
258	C₂²⋊C₄	—	188	1	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	47	2	no	—
261	C₄²	—	188	1	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	47	2	no	—
264	C₁₆	—	47	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	16	2	no	—
268	C₂³	—	188	2	no	—
269	C₂³	—	188	4	no	—
270	C₂³	—	752	2	no	—
271	C₂³	D₈×D₄₇	1	1	yes	characteristic
272	D₄	—	2	4	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	4	no	—
275	D₄	—	94	4	no	—
276	D₄	—	188	2	no	—
277	D₄	—	188	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	47	4	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	752	4	no	—
282	C₂×C₄	—	752	4	no	—
283	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
284	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
285	C₂×C₄	#13094045	1	2	yes	—
286	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
287	C₈	—	94	2	no	—
288	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
289	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
290	C₈	C₂×#10625352	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	16	4	no	—
295	C₂²	—	376	4	no	—
296	C₂²	—	188	4	no	—
297	C₂²	—	752	4	no	—
298	C₂²	#15689203	1	2	yes	—
299	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
300	C₄	—	8	2	no	—
301	C₄	—	94	4	no	—
302	C₄	—	376	2	no	—
303	C₄	#15689203	1	2	yes	—
304	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
305	C₄	C₂×#13094045	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	8	4	no	—
308	C₂	—	376	4	no	—
309	C₂	C₂×#15689203	1	1	yes	characteristic
310	C₂	#20256528	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2140	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×#13093965	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2140	D₄₇
hypocenter	C₄₇	#33456
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2140

Complements

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

N	H ≅ G/N	type
2. #20252240	C₂	semidirect
3. #20252239	C₂	semidirect
4. #20245261	C₂	semidirect
5. #20245232	C₂	semidirect
6. #20245170	C₂	semidirect
7. #20245105	C₂	semidirect
8. #20244987	C₂	semidirect
9. C₄₇×#2140	C₂	semidirect
13. C₂×#13093965	C₄	semidirect
19. D₈×Dic₄₇	C₂²	semidirect
20. #15689205	C₂²	semidirect
25. #15688950	C₂²	semidirect
26. #15688948	C₂²	semidirect
27. #15688925	C₂²	semidirect
28. #15688898	C₂²	semidirect
29. C₁₆×Dic₄₇	C₂²	semidirect
33. C₄₇×#357	C₂²	semidirect
37. C₂×D₃₇₆	C₂×C₄	semidirect
48. #13093965	C₂×C₄	semidirect
49. #13093965	C₂×C₄	semidirect
50. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
62. C₂×D₈×C₄₇	C₂×C₄	semidirect
158. #2140	D₄₇	semidirect
186. #357	C₂×D₄₇	semidirect
189. C₄₇	#33456	semidirect
198. C₂×D₈	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093965	2
#20252240	2
#20252239	2
#20245261	2
#20245232	2
#20245170	2
#20245105	2
#20244987	2
C₄₇×#2140	2
#33456	47