Subgroups — #26181918

Order 12032.

772 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252019	C₂	1	1	yes	maximal, characteristic
2	#431×D₄₇	C₂	1	2	yes	maximal
3	#20241685	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241683	C₂	1	2	yes	maximal, complemented
5	#20241679	C₂	1	2	yes	maximal, complemented
6	#20241679	C₂	1	2	yes	maximal, complemented
7	#20241673	C₂	1	2	yes	maximal, complemented
8	#20241670	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	#15689178	C₄	1	4	yes	—
12	#15689176	C₂²	1	2	yes	—
13	#15689175	C₄	1	4	yes	—
14	#15689174	C₂²	1	2	yes	—
15	#15689174	C₂²	1	2	yes	—
16	#15689174	C₂²	1	2	yes	—
17	#15689174	C₂²	1	2	yes	—
18	#15689173	C₄	1	4	yes	—
19	#15689146	C₄	1	2	yes	—
20	C₄⋊C₈×D₄₇	C₄	1	4	yes	—
21	#15689086	C₄	1	2	yes	—
22	C₂²⋊C₈×D₄₇	C₄	1	4	yes	—
23	C₄×#10625329	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₄₇	C₂²	1	2	yes	—
25	C₄×C₈×D₄₇	C₂²	1	2	yes	—
26	#15688845	C₂²	1	2	yes	complemented
27	#15688845	C₂²	1	2	yes	complemented
28	#15688845	C₂²	1	2	yes	complemented
29	#15688844	C₂²	1	2	yes	complemented
30	#15688844	C₂²	1	1	yes	characteristic, complemented
31	C₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
32	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₄₇×#431	C₂²	1	2	yes	—
34	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#322	C₂²	1	2	yes	complemented
36	C₄₇×#322	C₂²	1	1	yes	characteristic, complemented
37	C₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
38	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×#10625329	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
45	C₄²×D₄₇	C₂×C₄	1	2	yes	—
46	C₄²×D₄₇	C₂×C₄	1	2	yes	—
47	C₄²×D₄₇	C₂×C₄	1	2	yes	—
48	C₄²×D₄₇	C₂³	1	2	yes	—
49	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
50	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
51	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
52	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
53	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
54	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
55	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
56	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
57	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
58	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
59	#13093935	C₂×C₄	1	2	yes	—
60	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
61	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
62	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
63	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
64	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
65	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
69	#97×C₄₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂³	1	2	yes	—
71	C₄×C₈×C₄₇	C₂³	1	2	yes	—
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂³×D₄₇	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₄₇	C₄²	1	2	yes	—
78	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
79	C₂×C₄×D₄₇	—	2	8	no	—
80	C₂×C₄×D₄₇	—	4	2	no	—
81	C₂×C₄×D₄₇	C₄²	1	2	yes	—
82	C₂×C₄×D₄₇	C₄²	1	2	yes	—
83	C₂×C₄×D₄₇	C₄²	1	2	yes	—
84	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
87	M₁₆×C₄₇	—	4	2	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
96	C₄×Dic₄₇	C₄²	1	2	yes	—
97	C₄×Dic₄₇	C₄²	1	2	yes	—
98	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
99	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
100	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
101	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
102	#10625329	—	4	2	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₈×D₄₇	—	2	8	no	—
107	C₈×D₄₇	—	2	8	no	—
108	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂²×D₄₇	—	2	4	no	—
110	C₂²×D₄₇	—	4	1	no	—
111	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	4	1	no	—
120	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
121	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
122	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
123	C₄×D₄₇	—	2	4	no	—
124	C₄×D₄₇	—	2	4	no	—
125	C₄×D₄₇	—	2	4	no	—
126	C₄×D₄₇	—	2	4	no	—
127	C₄×D₄₇	—	4	2	no	—
128	C₄×D₄₇	#131	1	8	yes	—
129	C₈×C₄₇	—	2	4	no	—
130	C₈×C₄₇	—	2	4	no	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	C₄₇⋊C₈	—	2	4	no	—
135	C₄₇⋊C₈	—	2	4	no	—
136	#17325	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
140	C₂×D₄₇	—	2	2	no	—
141	C₂×D₄₇	—	2	2	no	—
142	C₂×D₄₇	—	4	4	no	—
143	C₂×D₄₇	#405	1	4	yes	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	C₂×#131	1	4	yes	—
147	C₄×C₄₇	#568	1	2	yes	—
148	Dic₄₇	—	2	2	no	—
149	Dic₄₇	—	2	2	no	—
150	Dic₄₇	#405	1	4	yes	—
151	#2920	—	47	1	no	—
152	C₂×#431	—	47	2	no	—
153	#1453	—	47	2	no	—
154	#1453	—	47	1	no	—
155	#1453	D₄₇	1	1	yes	characteristic, complemented
156	#1452	—	47	2	no	—
157	#1452	—	47	2	no	—
158	#1450	—	47	2	no	—
159	#1450	—	47	2	no	—
160	C₂×C₄₇	—	4	1	no	—
161	C₂×C₄₇	C₂×#405	1	2	yes	—
162	C₂×C₄₇	#2873	1	1	yes	characteristic
163	D₄₇	—	2	4	no	—
164	C₂²×C₄²	—	47	1	no	—
165	#432	—	47	4	no	—
166	#431	—	47	2	no	—
167	#431	—	47	4	no	—
168	#431	—	47	4	no	—
169	#431	—	47	2	no	—
170	#431	—	47	2	no	—
171	#431	—	47	2	no	—
172	#431	—	47	2	no	—
173	#431	C₂×D₄₇	1	2	yes	—
174	#423	—	47	2	no	—
175	C₂×C₄⋊C₈	—	47	4	no	—
176	#407	—	47	2	no	—
177	C₂×C₂²⋊C₈	—	47	4	no	—
178	C₄×M₁₆	—	47	1	no	—
179	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
180	C₂×C₄×C₈	—	47	2	no	—
181	C₂×C₄×C₈	—	47	2	no	—
182	#322	—	47	2	no	—
183	#322	—	47	1	no	—
184	#322	—	47	2	no	—
185	#322	—	47	2	no	—
186	#322	—	47	2	no	—
187	#322	C₂×D₄₇	1	2	yes	complemented
188	#322	C₂×D₄₇	1	1	yes	characteristic, complemented
189	C₈²	—	47	1	no	—
190	C₈²	—	47	2	no	—
191	C₈²	C₂×D₄₇	1	1	yes	characteristic, complemented
192	C₄₇	#17325	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	—	47	2	no	—
194	C₂³×C₄	—	47	1	no	—
195	C₂×M₁₆	—	47	2	no	—
196	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
197	C₂²×C₈	—	47	4	no	—
198	C₂²×C₈	—	47	4	no	—
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	47	2	no	—
204	C₂×C₄²	—	47	2	no	—
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₄⋊C₈	—	47	4	no	—
207	C₄⋊C₈	—	47	4	no	—
208	C₄⋊C₈	—	47	4	no	—
209	C₄⋊C₈	—	47	4	no	—
210	C₄⋊C₈	—	47	4	no	—
211	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
212	C₂²⋊C₈	—	47	4	no	—
213	C₂²⋊C₈	—	47	4	no	—
214	C₂²⋊C₈	—	47	4	no	—
215	C₂²⋊C₈	—	47	4	no	—
216	C₂²⋊C₈	—	47	4	no	—
217	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
218	#97	—	47	2	no	—
219	#97	C₄×D₄₇	1	2	yes	—
220	C₄×C₈	—	47	4	no	—
221	C₄×C₈	—	47	4	no	—
222	C₄×C₈	—	47	2	no	—
223	C₄×C₈	—	47	2	no	—
224	C₄×C₈	—	47	2	no	—
225	C₄×C₈	C₂²×D₄₇	1	2	yes	—
226	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₄₇	1	2	yes	—
229	C₂⁴	—	47	1	no	—
230	C₂²×C₄	—	47	2	no	—
231	C₂²×C₄	—	47	2	no	—
232	C₂²×C₄	—	47	1	no	—
233	C₂²×C₄	—	47	2	no	—
234	C₂²×C₄	—	47	2	no	—
235	C₂²×C₄	—	47	2	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	94	8	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	M₁₆	—	4	2	no	—
243	M₁₆	—	188	2	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	94	8	no	—
248	C₂×C₈	—	94	8	no	—
249	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
250	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
251	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
252	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
253	C₄²	—	47	1	no	—
254	C₄²	—	47	2	no	—
255	C₄²	—	47	1	no	—
256	C₄²	—	47	2	no	—
257	C₄²	—	47	2	no	—
258	C₄²	—	47	2	no	—
259	C₄²	—	47	2	no	—
260	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₄₇	1	2	yes	—
262	C₄²	C₂³×D₄₇	1	1	yes	characteristic
263	C₂³	—	47	2	no	—
264	C₂³	—	94	4	no	—
265	C₂³	—	188	1	no	—
266	C₂³	C₄²×D₄₇	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	47	2	no	—
270	C₂×C₄	—	47	8	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	47	2	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	94	4	no	—
277	C₂×C₄	—	94	4	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	1	no	—
280	C₂×C₄	C₄²×D₄₇	1	2	yes	—
281	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
283	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	2	2	no	—
287	C₈	—	2	2	no	—
288	C₈	—	94	4	no	—
289	C₈	—	94	4	no	—
290	C₈	—	94	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	47	4	no	—
294	C₂²	—	94	2	no	—
295	C₂²	—	94	2	no	—
296	C₂²	—	188	4	no	—
297	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	47	4	no	—
301	C₄	—	94	2	no	—
302	C₄	—	94	2	no	—
303	C₄	#131×D₄₇	1	4	yes	—
304	C₄	#15689889	1	2	yes	—
305	C₂	—	4	1	no	—
306	C₂	—	94	4	no	—
307	C₂	#405×D₄₇	1	2	yes	—
308	C₂	#20251805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1453	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1453	D₄₇
hypocenter	C₄₇	#17325
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #1453

Complements

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

N	H ≅ G/N	type
3. #20241685	C₂	semidirect
4. #20241683	C₂	semidirect
5. #20241679	C₂	semidirect
6. #20241679	C₂	semidirect
7. #20241673	C₂	semidirect
8. #20241670	C₂	semidirect
9. C₄₇×#1453	C₂	semidirect
26. #15688845	C₂²	semidirect
27. #15688845	C₂²	semidirect
28. #15688845	C₂²	semidirect
29. #15688844	C₂²	semidirect
30. #15688844	C₂²	semidirect
31. C₈×C₄₇⋊C₈	C₂²	semidirect
32. C₈×C₄₇⋊C₈	C₂²	semidirect
35. C₄₇×#322	C₂²	semidirect
36. C₄₇×#322	C₂²	semidirect
37. C₈²×C₄₇	C₂²	semidirect
155. #1453	D₄₇	semidirect
187. #322	C₂×D₄₇	semidirect
188. #322	C₂×D₄₇	semidirect
191. C₈²	C₂×D₄₇	semidirect
192. C₄₇	#17325	semidirect

Maximal subgroups

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

subgroup	index
#20252019	2
#431×D₄₇	2
#20241685	2
#20241683	2
#20241679	2
#20241673	2
#20241670	2
C₄₇×#1453	2
#17325	47