Subgroups — #26199524

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254333	C₂	1	2	yes	maximal
2	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
3	#20242906	C₂	1	4	yes	maximal, complemented
4	#20242782	C₂	1	4	yes	maximal, complemented
5	#20242136	C₂	1	1	yes	maximal, characteristic
6	#20242074	C₂	1	2	yes	maximal
7	C₄₇×#2960	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094024	C₂²	1	2	yes	—
9	#15689528	—	2	1	no	—
10	#15689528	—	2	1	no	—
11	#15689528	C₂²	1	2	yes	—
12	#15689528	C₂²	1	2	yes	—
13	#15689502	—	2	2	no	—
14	#15689475	—	2	2	no	—
15	#15689333	C₂²	1	8	yes	—
16	#15689278	—	2	4	no	—
17	#15689176	C₂²	1	2	yes	—
18	C₂×#13093958	—	2	1	no	—
19	C₂×#13093958	—	2	1	no	—
20	#15688876	C₂²	1	2	yes	—
21	#15688874	C₂²	1	2	yes	—
22	#15688868	C₂²	1	8	yes	—
23	C₄₇×#533	C₂²	1	1	yes	characteristic
24	C₄₇×#446	—	2	1	no	—
25	C₄₇×#446	C₂²	1	2	yes	—
26	C₈×Q₈×C₄₇	—	2	1	no	—
27	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#424	—	2	1	no	—
29	C₄₇×#423	C₂²	1	1	yes	characteristic
30	C₄₇×#405	—	2	1	no	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×Dic₉₄	—	2	2	no	—
33	C₂×#10625337	—	2	8	no	—
34	#13094024	—	2	2	no	—
35	#13094024	—	2	2	no	—
36	#13094024	—	2	2	no	—
37	#13094024	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	#13093958	—	2	2	no	—
41	#13093958	—	2	2	no	—
42	#13093958	—	2	2	no	—
43	#13093958	—	2	2	no	—
44	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
45	#13093947	—	2	8	no	—
46	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
47	C₄×C₄₇⋊C₈	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	1	no	—
49	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
53	#125×C₄₇	—	2	1	no	—
54	C₂²⋊Q₈×C₄₇	—	2	1	no	—
55	C₂²⋊Q₈×C₄₇	D₄	1	2	yes	—
56	C₄×Q₈×C₄₇	—	2	1	no	—
57	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
58	#117×C₄₇	—	2	1	no	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	—	2	2	no	—
61	C₄⋊C₈×C₄₇	—	2	2	no	—
62	C₄⋊C₈×C₄₇	—	2	1	no	—
63	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
64	#97×C₄₇	—	2	2	no	—
65	C₄×C₈×C₄₇	—	2	2	no	—
66	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
67	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₄₇	—	2	1	no	—
71	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂×Dic₉₄	—	2	4	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×Dic₉₄	—	8	4	no	—
79	M₁₆×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	2	1	no	—
88	C₄⋊C₄×C₄₇	—	2	1	no	—
89	C₄⋊C₄×C₄₇	—	2	1	no	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
93	C₂²⋊C₄×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₇	#41	1	1	yes	characteristic
98	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
99	#10625337	—	8	8	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	4	2	no	—
103	C₄⋊Dic₄₇	—	4	2	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂³×C₄₇	#121	1	1	yes	characteristic
106	Q₈×C₄₇	—	2	4	no	—
107	Q₈×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₄₇	#137	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	8	4	no	—
122	Dic₉₄	—	4	8	no	—
123	Dic₉₄	—	8	4	no	—
124	C₈×C₄₇	—	2	1	no	—
125	C₈×C₄₇	—	2	1	no	—
126	C₈×C₄₇	—	4	2	no	—
127	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
128	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
129	C₄₇⋊C₈	—	4	4	no	—
130	#19208	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	#462	1	2	yes	—
134	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	4	2	no	—
138	C₄×C₄₇	—	4	2	no	—
139	C₄×C₄₇	#462	1	2	yes	—
140	C₄×C₄₇	#473	1	2	yes	—
141	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
142	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
143	C₄×C₄₇	#576	1	2	yes	—
144	Dic₄₇	—	8	4	no	—
145	#3209	—	47	2	no	—
146	C₂×#500	—	47	1	no	—
147	#2960	D₄₇	1	1	yes	characteristic
148	#1694	—	47	4	no	—
149	#1674	—	47	4	no	—
150	#1564	—	47	1	no	—
151	#1549	—	47	2	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
154	C₂×C₄₇	#3058	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
156	#533	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	—	47	2	no	—
158	#500	—	47	2	no	—
159	#500	—	47	2	no	—
160	#500	—	94	1	no	—
161	#500	—	94	1	no	—
162	#494	—	94	2	no	—
163	#487	—	94	2	no	—
164	#462	—	47	8	no	—
165	#451	—	94	4	no	—
166	#446	—	2	1	no	—
167	#446	C₂×D₄₇	1	2	yes	—
168	C₈×Q₈	—	2	1	no	—
169	C₈×Q₈	C₂×D₄₇	1	2	yes	—
170	#431	—	47	2	no	—
171	C₂×#107	—	94	1	no	—
172	C₂×#107	—	94	1	no	—
173	#424	—	2	1	no	—
174	#423	C₂×D₄₇	1	1	yes	characteristic
175	#405	—	2	1	no	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	#335	—	47	2	no	—
178	#334	—	47	2	no	—
179	#332	—	47	8	no	—
180	C₄₇	#19208	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	94	2	no	—
182	C₂×Q₁₆	—	94	8	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₄⋊Q₈	—	47	4	no	—
187	C₄⋊Q₈	—	94	2	no	—
188	C₄⋊Q₈	—	94	2	no	—
189	C₄⋊Q₈	—	94	2	no	—
190	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
191	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
192	#125	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	C₂²⋊Q₈	D₁₈₈	1	2	yes	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
197	#117	—	2	1	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	#107	—	94	2	no	—
202	#107	—	94	2	no	—
203	#107	—	94	2	no	—
204	#107	—	94	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	47	4	no	—
209	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
210	#103	—	94	8	no	—
211	C₂²⋊C₈	—	94	2	no	—
212	#97	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	—	94	2	no	—
215	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	94	4	no	—
221	C₂×Q₈	—	376	4	no	—
222	C₂×Q₈	—	188	2	no	—
223	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
226	C₂²×C₄	#10625350	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	Q₁₆	—	376	8	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	#10625337	1	4	yes	complemented
234	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	94	4	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	2	no	—
247	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
251	C₄²	#10625350	1	1	yes	characteristic
252	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₄²	C₂³×D₄₇	1	1	yes	characteristic
254	C₂³	#13093998	1	1	yes	characteristic
255	Q₈	—	2	4	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	376	4	no	—
258	Q₈	—	188	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	376	4	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	#13093998	1	1	yes	characteristic
266	C₂×C₄	#13093998	1	1	yes	characteristic
267	C₂×C₄	#13093998	1	1	yes	characteristic
268	C₂×C₄	C₂×#10625337	1	2	yes	—
269	C₂×C₄	#13094054	1	2	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	2	no	—
276	C₈	—	188	4	no	—
277	C₈	C₂×#10625337	1	2	yes	—
278	C₈	C₂×#10625337	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#15689333	1	2	yes	—
282	C₂²	C₂×#13093998	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	376	4	no	—
288	C₄	#15689333	1	2	yes	—
289	C₄	#15689398	1	2	yes	—
290	C₄	C₂²×#10625337	1	1	yes	characteristic
291	C₄	C₂×#13094054	1	1	yes	characteristic
292	C₄	#15689928	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	C₂×#15689333	1	1	yes	characteristic
295	C₂	#20253036	1	1	yes	characteristic
296	C₂	C₂×#15689398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#2960	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂³×C₄₇	#121
hypercenter	#2960	D₄₇
hypocenter	C₄₇	#19208
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #20242906	C₂	semidirect
4. #20242782	C₂	semidirect
82. C₂×C₈×C₄₇	Q₁₆	semidirect
180. C₄₇	#19208	semidirect
233. C₂×C₈	#10625337	semidirect

Maximal subgroups

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

subgroup	index
#20254333	2
C₂×#15689528	2
#20242906	2
#20242782	2
#20242136	2
#20242074	2
C₄₇×#2960	2
#19208	47