Subgroups — #26111789

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252469	C₂	1	1	yes	maximal, characteristic, complemented
2	#20241779	C₂	1	2	yes	maximal
3	#20241712	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241217	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2326	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₇⋊C₈	—	2	4	no	—
7	#15689176	—	2	2	no	—
8	#15689176	C₂²	1	2	yes	—
9	#15689176	C₂²	1	2	yes	—
10	D₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	#15688849	—	2	2	no	—
12	C₂×#124×C₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#391	C₄	1	1	yes	characteristic
16	C₄₇×#390	C₄	1	1	yes	characteristic
17	C₄₇×#388	—	2	2	no	—
18	C₄₇×#385	—	2	2	no	—
19	C₄₇×#384	C₄	1	1	yes	characteristic
20	C₄₇×#376	C₂²	1	1	yes	characteristic
21	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	—	2	1	no	—
32	#124×C₄₇	—	2	2	no	—
33	#124×C₄₇	C₈	1	4	yes	complemented
34	C₄×Q₈×C₄₇	—	2	2	no	—
35	C₄×Q₈×C₄₇	—	2	2	no	—
36	C₄×D₄×C₄₇	—	2	2	no	—
37	C₄×D₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	—	2	1	no	—
51	#95×C₄₇	—	2	1	no	—
52	C₂⁴×C₄₇	—	2	1	no	—
53	C₂×Q₈×C₄₇	—	2	2	no	—
54	C₂×Q₈×C₄₇	—	2	2	no	—
55	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
56	C₂×Q₈×C₄₇	M₁₆	1	2	yes	—
57	C₂×D₄×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	—	2	1	no	—
59	C₂×D₄×C₄₇	—	2	2	no	—
60	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	8	2	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	8	2	no	—
75	C₂²⋊C₄×C₄₇	—	2	8	no	—
76	C₂²⋊C₄×C₄₇	—	4	1	no	—
77	C₂²⋊C₄×C₄₇	—	4	1	no	—
78	C₂²⋊C₄×C₄₇	—	8	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	C₂×C₈	1	2	yes	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	C₄²×C₄₇	#41	1	2	yes	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	—	4	1	no	—
92	C₂³×C₄₇	—	4	1	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₄₇	—	2	4	no	—
95	Q₈×C₄₇	—	2	4	no	—
96	D₄×C₄₇	—	2	4	no	—
97	D₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	4	1	no	—
104	C₂×C₄×C₄₇	—	4	1	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	8	4	no	—
113	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#131	1	2	yes	—
121	C₄₇⋊C₈	—	4	4	no	—
122	C₄₇⋊C₈	—	8	2	no	—
123	#9102	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	—	4	4	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	—	8	2	no	—
129	C₂²×C₄₇	—	8	1	no	—
130	C₂²×C₄₇	—	8	1	no	—
131	C₂²×C₄₇	#386	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₄₇	#408	1	1	yes	characteristic
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	4	2	no	—
139	C₄×C₄₇	—	4	2	no	—
140	C₄×C₄₇	—	8	1	no	—
141	C₄×C₄₇	—	8	1	no	—
142	C₄×C₄₇	C₈×D₄	1	2	yes	—
143	C₄×C₄₇	#435	1	2	yes	—
144	#2982	—	47	1	no	—
145	#2326	D₄₇	1	1	yes	characteristic
146	#1478	—	47	2	no	—
147	#1461	—	47	1	no	—
148	#1274	—	47	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	8	2	no	—
152	C₂×C₄₇	#1869	1	1	yes	characteristic
153	C₂×C₄₇	#1910	1	1	yes	characteristic
154	C₂×C₄₇	#1912	1	1	yes	characteristic
155	C₂×#124	Dic₄₇	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
158	C₈×Q₈	—	94	4	no	—
159	#431	—	47	2	no	—
160	#431	—	47	2	no	—
161	#431	—	94	2	no	—
162	#391	Dic₄₇	1	1	yes	characteristic
163	#390	Dic₄₇	1	1	yes	characteristic
164	#388	—	2	2	no	—
165	#385	—	2	2	no	—
166	#384	Dic₄₇	1	1	yes	characteristic
167	#376	C₂×D₄₇	1	1	yes	characteristic
168	D₄⋊C₈	—	94	4	no	—
169	#324	—	94	2	no	—
170	C₄₇	#9102	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
173	C₂³×C₄	—	2	1	no	—
174	#124	—	2	2	no	—
175	#124	C₄₇⋊C₈	1	4	yes	complemented
176	C₄×Q₈	—	2	2	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	C₄⋊C₈	—	94	4	no	—
192	C₄⋊C₈	—	94	2	no	—
193	C₄⋊C₈	—	94	2	no	—
194	C₂²⋊C₈	—	94	2	no	—
195	C₂²⋊C₈	—	94	2	no	—
196	C₄×C₈	—	47	4	no	—
197	C₄×C₈	—	94	4	no	—
198	C₄×C₈	—	94	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
206	C₂×Q₈	#10625329	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	8	2	no	—
216	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	188	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	8	2	no	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	1	no	—
230	C₂²⋊C₄	—	8	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
236	C₄²	D₄×D₄₇	1	2	yes	—
237	C₄²	#10625352	1	2	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
264	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
265	C₂×C₄	#13093992	1	1	yes	characteristic
266	C₂×C₄	#13093996	1	2	yes	—
267	C₂×C₄	#13094005	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
270	C₂×C₄	#13094031	1	2	yes	—
271	C₈	—	376	2	no	—
272	C₈	—	188	4	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	1	no	—
280	C₂²	#15689003	1	1	yes	characteristic
281	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
282	C₂²	#15689090	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
292	C₄	#15689195	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	#20243681	1	1	yes	characteristic
297	C₂	#20243916	1	1	yes	characteristic
298	C₂	#20243926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689003
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2326	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2326	D₄₇
hypocenter	C₄₇	#9102
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2326

Complements

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

N	H ≅ G/N	type
1. #20252469	C₂	semidirect
3. #20241712	C₂	semidirect
4. #20241217	C₂	semidirect
33. #124×C₄₇	C₈	semidirect
170. C₄₇	#9102	semidirect
175. #124	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20252469	2
#20241779	2
#20241712	2
#20241217	2
C₄₇×#2326	2
#9102	47