Subgroups — #26181987

Order 12032.

900 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₄×M₁₆×D₄₇	C₂	1	2	yes	maximal
3	#20241691	C₂	1	2	yes	maximal, complemented
4	#20241691	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241676	C₂	1	2	yes	maximal, complemented
6	#20241674	C₂	1	2	yes	maximal, complemented
7	#20241667	C₂	1	2	yes	maximal, complemented
8	C₈×#10625324	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₄₇	C₄	1	4	yes	—
11	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
12	#15689178	C₄	1	4	yes	—
13	#15689174	C₂²	1	2	yes	—
14	#15689174	C₂²	1	2	yes	—
15	#15689146	C₄	1	2	yes	—
16	#15689086	C₄	1	2	yes	—
17	C₄×#10625329	C₂²	1	1	yes	characteristic
18	C₄×#10625329	C₂²	1	2	yes	—
19	#15689075	C₄	1	4	yes	—
20	M₁₆×Dic₄₇	C₄	1	4	yes	—
21	C₄×#10625324	C₂²	1	2	yes	—
22	C₄×#10625324	C₂²	1	2	yes	—
23	#97×D₄₇	C₄	1	4	yes	—
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	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
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	4	yes	—
41	C₂×#10625329	C₂×C₄	1	2	yes	—
42	C₂×#10625324	C₂×C₄	1	4	yes	—
43	C₂×#10625324	C₂×C₄	1	4	yes	—
44	M₁₆×D₄₇	—	2	16	no	—
45	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₄₇	C₂×C₄	1	2	yes	—
50	C₄²×D₄₇	C₂×C₄	1	2	yes	—
51	C₄²×D₄₇	C₂×C₄	1	2	yes	—
52	C₄²×D₄₇	C₂³	1	2	yes	—
53	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
54	Dic₄₇⋊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	#13093935	C₂×C₄	1	4	yes	—
58	#13093935	C₂×C₄	1	2	yes	—
59	#13093934	C₂×C₄	1	4	yes	—
60	#13093934	C₂×C₄	1	4	yes	—
61	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
62	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
63	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
64	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
65	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
66	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
67	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	#97×C₄₇	C₂×C₄	1	4	yes	—
70	#97×C₄₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₄₇	C₂³	1	2	yes	—
72	C₄×C₈×C₄₇	C₂³	1	2	yes	—
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂³×D₄₇	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₄₇	C₄²	1	2	yes	—
79	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₄₇	—	2	8	no	—
81	C₂×C₄×D₄₇	—	4	2	no	—
82	C₂×C₄×D₄₇	C₄²	1	2	yes	—
83	C₂×C₄×D₄₇	C₄²	1	2	yes	—
84	C₂×C₄×D₄₇	C₄²	1	2	yes	—
85	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
88	M₁₆×C₄₇	—	2	8	no	—
89	M₁₆×C₄₇	—	4	2	no	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
94	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
98	C₄×Dic₄₇	C₄²	1	2	yes	—
99	C₄×Dic₄₇	C₄²	1	2	yes	—
100	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
101	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
102	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
103	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
104	#10625329	—	2	8	no	—
105	#10625329	—	4	2	no	—
106	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
109	#10625324	—	2	8	no	—
110	#10625324	M₁₆	1	16	yes	complemented
111	C₈×D₄₇	—	2	8	no	—
112	C₈×D₄₇	—	2	8	no	—
113	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
114	C₂²×D₄₇	—	2	4	no	—
115	C₂²×D₄₇	—	4	1	no	—
116	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	4	1	no	—
125	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
126	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
127	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
128	C₄×D₄₇	—	2	4	no	—
129	C₄×D₄₇	—	2	4	no	—
130	C₄×D₄₇	—	2	4	no	—
131	C₄×D₄₇	—	2	4	no	—
132	C₄×D₄₇	—	4	2	no	—
133	C₄×D₄₇	C₂×M₁₆	1	8	yes	—
134	C₈×C₄₇	—	2	4	no	—
135	C₈×C₄₇	—	2	2	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	C₂×M₁₆	1	8	yes	complemented
138	C₄₇⋊C₈	—	2	4	no	—
139	C₄₇⋊C₈	—	2	4	no	—
140	C₄₇⋊C₈	C₂×M₁₆	1	8	yes	complemented
141	M₁₆²	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
145	C₂×D₄₇	—	2	2	no	—
146	C₂×D₄₇	—	2	2	no	—
147	C₂×D₄₇	—	4	4	no	—
148	C₂×D₄₇	C₄×M₁₆	1	4	yes	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	C₂²×M₁₆	1	4	yes	—
152	C₄×C₄₇	#568	1	2	yes	—
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	2	2	no	—
155	Dic₄₇	C₄×M₁₆	1	4	yes	—
156	#2920	—	47	1	no	—
157	C₂×C₄×M₁₆	—	47	2	no	—
158	#1455	—	47	2	no	—
159	#1455	—	47	1	no	—
160	#1455	D₄₇	1	1	yes	characteristic, complemented
161	#1451	—	47	2	no	—
162	#1451	—	47	2	no	—
163	C₈×M₁₆	—	47	2	no	—
164	C₈×M₁₆	—	47	2	no	—
165	C₂×C₄₇	—	4	1	no	—
166	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
167	C₂×C₄₇	#2873	1	1	yes	characteristic
168	D₄₇	—	2	4	no	—
169	C₂²×M₁₆	—	47	4	no	—
170	C₂²×C₄²	—	47	1	no	—
171	#432	—	47	4	no	—
172	#431	—	47	2	no	—
173	#431	—	47	2	no	—
174	#423	—	47	2	no	—
175	#407	—	47	2	no	—
176	C₄×M₁₆	—	47	4	no	—
177	C₄×M₁₆	—	47	2	no	—
178	C₄×M₁₆	—	47	1	no	—
179	C₄×M₁₆	—	47	4	no	—
180	C₄×M₁₆	—	47	2	no	—
181	C₄×M₁₆	—	47	2	no	—
182	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
183	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
184	C₂×#97	—	47	4	no	—
185	C₂×C₄×C₈	—	47	2	no	—
186	C₂×C₄×C₈	—	47	2	no	—
187	#322	—	47	2	no	—
188	#322	—	47	1	no	—
189	#322	—	47	2	no	—
190	#322	—	47	2	no	—
191	#322	—	47	2	no	—
192	#322	C₂×D₄₇	1	2	yes	complemented
193	#322	C₂×D₄₇	1	1	yes	characteristic, complemented
194	C₈²	—	47	1	no	—
195	C₈²	—	47	2	no	—
196	C₈²	C₂×D₄₇	1	1	yes	characteristic, complemented
197	C₄₇	M₁₆²	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₄	—	47	2	no	—
199	C₂³×C₄	—	47	1	no	—
200	C₂×M₁₆	—	47	4	no	—
201	C₂×M₁₆	—	47	2	no	—
202	C₂×M₁₆	—	47	4	no	—
203	C₂×M₁₆	—	47	4	no	—
204	C₂×M₁₆	—	94	16	no	—
205	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
206	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
207	C₂²×C₈	—	47	4	no	—
208	C₂²×C₈	—	47	4	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	47	1	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	47	2	no	—
213	C₂×C₄²	—	47	2	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	47	4	no	—
217	C₄⋊C₈	—	47	4	no	—
218	C₂²⋊C₈	—	47	4	no	—
219	C₂²⋊C₈	—	47	4	no	—
220	#97	—	47	4	no	—
221	#97	—	47	4	no	—
222	#97	—	47	4	no	—
223	#97	—	47	2	no	—
224	#97	C₄×D₄₇	1	4	yes	—
225	#97	C₄×D₄₇	1	2	yes	—
226	C₄×C₈	—	47	4	no	—
227	C₄×C₈	—	47	4	no	—
228	C₄×C₈	—	47	2	no	—
229	C₄×C₈	—	47	2	no	—
230	C₄×C₈	—	47	2	no	—
231	C₄×C₈	C₂²×D₄₇	1	2	yes	—
232	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₄₇	1	2	yes	—
235	C₂⁴	—	47	1	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	47	1	no	—
239	C₂²×C₄	—	47	2	no	—
240	C₂²×C₄	—	47	2	no	—
241	C₂²×C₄	—	47	2	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	—	94	8	no	—
245	C₂²×C₄	—	188	2	no	—
246	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	M₁₆	—	2	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	47	16	no	—
251	M₁₆	—	94	8	no	—
252	M₁₆	—	94	8	no	—
253	M₁₆	—	188	2	no	—
254	C₂×C₈	—	47	4	no	—
255	C₂×C₈	—	47	4	no	—
256	C₂×C₈	—	47	4	no	—
257	C₂×C₈	—	94	8	no	—
258	C₂×C₈	—	94	8	no	—
259	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
260	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
262	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
263	C₄²	—	47	1	no	—
264	C₄²	—	47	2	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	1	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	2	no	—
269	C₄²	—	47	2	no	—
270	C₄²	C₂×C₄×D₄₇	1	2	yes	—
271	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₄²	C₂³×D₄₇	1	1	yes	characteristic
273	C₂³	—	47	2	no	—
274	C₂³	—	94	4	no	—
275	C₂³	—	188	1	no	—
276	C₂³	C₄²×D₄₇	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	47	2	no	—
280	C₂×C₄	—	47	8	no	—
281	C₂×C₄	—	47	2	no	—
282	C₂×C₄	—	47	2	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	1	no	—
290	C₂×C₄	C₄²×D₄₇	1	2	yes	—
291	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
293	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	47	8	no	—
298	C₈	—	94	4	no	—
299	C₈	—	94	4	no	—
300	C₈	M₁₆×D₄₇	1	8	yes	complemented
301	C₂²	—	2	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	47	4	no	—
304	C₂²	—	94	2	no	—
305	C₂²	—	94	2	no	—
306	C₂²	—	188	4	no	—
307	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	47	4	no	—
311	C₄	—	94	2	no	—
312	C₄	—	94	2	no	—
313	C₄	C₂×M₁₆×D₄₇	1	4	yes	—
314	C₄	#15689889	1	2	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	94	4	no	—
317	C₂	C₄×M₁₆×D₄₇	1	2	yes	—
318	C₂	#20251805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1455	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1455	D₄₇
hypocenter	C₄₇	M₁₆²
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
3. #20241691	C₂	semidirect
4. #20241691	C₂	semidirect
5. #20241676	C₂	semidirect
6. #20241674	C₂	semidirect
7. #20241667	C₂	semidirect
8. C₈×#10625324	C₂	semidirect
9. C₄₇×#1455	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
110. #10625324	M₁₆	semidirect
137. C₈×C₄₇	C₂×M₁₆	semidirect
140. C₄₇⋊C₈	C₂×M₁₆	semidirect
160. #1455	D₄₇	semidirect
192. #322	C₂×D₄₇	semidirect
193. #322	C₂×D₄₇	semidirect
196. C₈²	C₂×D₄₇	semidirect
197. C₄₇	M₁₆²	semidirect
300. C₈	M₁₆×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252019	2
C₄×M₁₆×D₄₇	2
#20241691	2
#20241676	2
#20241674	2
#20241667	2
C₈×#10625324	2
C₄₇×#1455	2
M₁₆²	47