Subgroups — #26131564

Order 12032.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252190	C₂	1	1	yes	maximal, characteristic
2	#20245125	C₂	1	1	yes	maximal, characteristic
3	#20245117	C₂	1	1	yes	maximal, characteristic
4	#20243272	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241503	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	1	yes	characteristic
9	C₄×#10625350	—	2	1	no	—
10	#15689236	C₄	1	2	yes	complemented
11	#15689231	C₄	1	2	yes	complemented
12	#15689225	—	2	2	no	—
13	C₂×#13093960	C₂²	1	1	yes	characteristic
14	#15689161	C₂²	1	1	yes	characteristic
15	#15689161	C₂²	1	1	yes	characteristic
16	#15689120	—	2	1	no	—
17	#15689078	—	2	1	no	—
18	#15688934	C₄	1	4	yes	complemented
19	#15688931	C₄	1	4	yes	complemented
20	#15688883	—	2	1	no	—
21	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#347	—	2	2	no	—
25	C₂×#10625350	—	2	1	no	—
26	#13094046	—	2	1	no	—
27	#13094046	—	2	1	no	—
28	#13094046	C₂×C₄	1	2	yes	—
29	#13094046	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
31	C₂×#10625325	—	2	1	no	—
32	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
33	C₂×#10625329	—	2	1	no	—
34	C₄×Dic₉₄	—	2	2	no	—
35	C₄×D₁₈₈	—	2	2	no	—
36	C₄×C₂²⋊D₄₇	—	4	2	no	—
37	#13093983	—	2	1	no	—
38	#13093983	—	2	1	no	—
39	#13093981	—	4	1	no	—
40	C₄²×D₄₇	—	4	1	no	—
41	#13093960	—	2	1	no	—
42	#13093960	C₂×C₄	1	2	yes	complemented
43	#13093958	—	2	1	no	—
44	#13093958	—	2	1	no	—
45	#13093956	C₂×C₄	1	2	yes	—
46	#13093956	C₂×C₄	1	2	yes	complemented
47	#13093949	—	2	2	no	—
48	#13093946	—	2	2	no	—
49	#13093934	—	4	1	no	—
50	C₈×Dic₄₇	—	4	1	no	—
51	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	M₃₂×C₄₇	—	2	1	no	—
55	M₃₂×C₄₇	—	2	1	no	—
56	M₃₂×C₄₇	—	2	1	no	—
57	M₃₂×C₄₇	C₂×C₄	1	2	yes	complemented
58	C₂×C₁₆×C₄₇	—	2	1	no	—
59	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
60	#97×C₄₇	C₂×C₄	1	2	yes	—
61	#97×C₄₇	D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	—	4	1	no	—
65	#10625350	—	2	2	no	—
66	#10625350	—	2	2	no	—
67	#10625350	—	4	1	no	—
68	#10625350	—	4	1	no	—
69	C₂×D₁₈₈	—	2	1	no	—
70	C₂×C₄×D₄₇	—	4	1	no	—
71	C₂×C₄×D₄₇	—	4	2	no	—
72	C₂×Dic₉₄	—	2	1	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₂²⋊Dic₄₇	—	4	1	no	—
85	C₂²⋊Dic₄₇	—	4	1	no	—
86	(C₂×D₄₇)⋊C₄	—	4	2	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	Dic₄₇⋊C₄	—	4	2	no	—
90	C₄×Dic₄₇	—	4	2	no	—
91	C₄×Dic₄₇	—	4	1	no	—
92	C₄×Dic₄₇	—	4	1	no	—
93	#10625329	—	2	2	no	—
94	#10625329	—	4	1	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	Dic₁₈₈	—	2	1	no	—
97	Dic₁₈₈	C₄²	1	2	yes	—
98	D₃₇₆	—	2	1	no	—
99	D₃₇₆	C₄²	1	2	yes	—
100	#10625325	—	2	2	no	—
101	#10625325	—	2	2	no	—
102	C₁₆×C₄₇	—	2	2	no	—
103	C₁₆×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×D₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	8	1	no	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
114	C₂²⋊D₄₇	—	4	2	no	—
115	C₂²⋊D₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	1	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	D₁₈₈	—	2	2	no	—
120	D₁₈₈	—	4	1	no	—
121	C₄×D₄₇	—	4	2	no	—
122	C₄×D₄₇	—	4	2	no	—
123	C₄×D₄₇	—	8	4	no	—
124	Dic₉₄	—	2	2	no	—
125	Dic₉₄	—	4	1	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
128	C₈×C₄₇	#117	1	1	yes	characteristic
129	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
130	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
131	C₄₇⋊C₈	—	4	2	no	—
132	#12649	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	#418	1	1	yes	characteristic
135	C₂²×C₄₇	#440	1	1	yes	characteristic
136	C₂²×C₄₇	#442	1	1	yes	characteristic
137	C₂×D₄₇	—	4	2	no	—
138	C₂×D₄₇	—	8	2	no	—
139	C₄×C₄₇	—	4	2	no	—
140	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	#417	1	1	yes	characteristic
142	C₄×C₄₇	#440	1	1	yes	characteristic
143	C₄×C₄₇	#442	1	1	yes	characteristic
144	Dic₄₇	—	4	2	no	—
145	Dic₄₇	—	8	2	no	—
146	Dic₄₇	—	8	2	no	—
147	#2943	—	47	1	no	—
148	#2145	—	47	1	no	—
149	#2143	—	47	1	no	—
150	C₂×#347	D₄₇	1	1	yes	characteristic, complemented
151	#1773	—	47	1	no	—
152	#1389	—	47	1	no	—
153	#1384	—	47	1	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	#1762	1	1	yes	characteristic
158	D₄₇	—	8	2	no	—
159	C₂×#135	—	47	1	no	—
160	C₄×#41	—	94	1	no	—
161	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
162	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
163	#442	—	47	2	no	—
164	#441	—	47	2	no	—
165	#440	—	94	2	no	—
166	C₂×#108	—	47	1	no	—
167	#427	—	47	1	no	—
168	#427	—	47	1	no	—
169	#416	—	94	1	no	—
170	#405	—	94	1	no	—
171	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
172	#360	—	47	4	no	—
173	#359	—	47	4	no	—
174	#347	—	2	2	no	—
175	#339	—	94	1	no	—
176	C₄₇	#12649	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	94	1	no	—
178	#135	—	47	2	no	—
179	#135	—	47	2	no	—
180	#135	—	94	1	no	—
181	#135	—	94	1	no	—
182	C₂×Q₁₆	—	47	1	no	—
183	C₂×SD₁₆	—	94	1	no	—
184	C₂×D₈	—	47	1	no	—
185	C₂×M₁₆	—	94	1	no	—
186	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₄×Q₈	—	94	2	no	—
189	C₄×D₄	—	94	2	no	—
190	C₄×D₄	—	188	2	no	—
191	#117	—	94	1	no	—
192	#117	—	94	1	no	—
193	#117	—	188	1	no	—
194	C₂×C₄²	—	188	1	no	—
195	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
196	M₃₂	—	2	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	C₄×D₄₇	1	2	yes	complemented
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	C₄×D₄₇	1	2	yes	complemented
202	#108	—	47	2	no	—
203	#108	—	94	1	no	—
204	#107	—	94	1	no	—
205	#107	—	94	1	no	—
206	#106	—	47	2	no	—
207	#106	—	47	2	no	—
208	#103	—	94	2	no	—
209	#102	—	94	2	no	—
210	#97	—	188	1	no	—
211	#97	C₄×D₄₇	1	2	yes	—
212	#97	C₂²⋊D₄₇	1	2	yes	—
213	C₄×C₈	—	188	1	no	—
214	#41	—	94	2	no	—
215	#41	—	94	2	no	—
216	#41	—	188	1	no	—
217	#41	—	188	1	no	—
218	C₂×Q₈	—	94	1	no	—
219	C₂×D₄	—	94	1	no	—
220	C₂×D₄	—	188	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	188	2	no	—
223	C₂²×C₄	—	188	1	no	—
224	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
225	Q₁₆	—	47	2	no	—
226	Q₁₆	—	94	1	no	—
227	SD₁₆	—	94	2	no	—
228	SD₁₆	—	94	2	no	—
229	D₈	—	47	2	no	—
230	D₈	—	94	1	no	—
231	M₁₆	—	94	2	no	—
232	M₁₆	—	188	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	188	1	no	—
236	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂×C₈	#10625350	1	1	yes	characteristic
241	C₂×C₈	#10625350	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄⋊C₄	—	94	2	no	—
244	C₄⋊C₄	—	94	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₂²⋊C₄	—	188	2	no	—
247	C₂²⋊C₄	—	188	1	no	—
248	C₂²⋊C₄	—	188	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	188	2	no	—
251	C₄²	—	188	1	no	—
252	C₄²	—	188	1	no	—
253	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₂³	—	188	1	no	—
257	C₂³	#13094052	1	1	yes	characteristic
258	Q₈	—	94	2	no	—
259	Q₈	—	188	1	no	—
260	D₄	—	94	2	no	—
261	D₄	—	188	2	no	—
262	D₄	—	188	2	no	—
263	D₄	—	188	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	—	376	4	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	1	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
274	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
275	C₂×C₄	#13094052	1	1	yes	characteristic
276	C₂×C₄	#13094052	1	1	yes	characteristic
277	C₂×C₄	#13094057	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	188	2	no	—
280	C₈	C₄²×D₄₇	1	1	yes	characteristic
281	C₈	#13093981	1	1	yes	characteristic
282	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
283	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	376	2	no	—
286	C₂²	—	188	2	no	—
287	C₂²	#15689129	1	1	yes	characteristic
288	C₂²	#15689225	1	1	yes	characteristic
289	C₂²	#15689236	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	376	2	no	—
292	C₄	—	376	2	no	—
293	C₄	—	188	2	no	—
294	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
295	C₄	#15689125	1	1	yes	characteristic
296	C₄	#15689225	1	1	yes	characteristic
297	C₄	#15689236	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	376	2	no	—
302	C₂	#20243214	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689236
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#347	C₂
socle	C₂×C₄₇	#1762
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#347	D₄₇
hypocenter	C₄₇	#12649
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #20243272	C₂	semidirect
5. #20241503	C₂	semidirect
6. #20241490	C₂	semidirect
7. C₂×C₄₇×#347	C₂	semidirect
10. #15689236	C₄	semidirect
11. #15689231	C₄	semidirect
18. #15688934	C₄	semidirect
19. #15688931	C₄	semidirect
42. #13093960	C₂×C₄	semidirect
46. #13093956	C₂×C₄	semidirect
57. M₃₂×C₄₇	C₂×C₄	semidirect
59. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
150. C₂×#347	D₄₇	semidirect
176. C₄₇	#12649	semidirect
199. M₃₂	C₄×D₄₇	semidirect
201. C₂×C₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252190	2
#20245125	2
#20245117	2
#20243272	2
#20241503	2
#20241490	2
C₂×C₄₇×#347	2
#12649	47