Subgroups — #26181993

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252016	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241692	C₂	1	2	yes	maximal, complemented
4	#20241692	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241690	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241685	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241683	C₂	1	2	yes	maximal, complemented
8	#20241679	C₂	1	2	yes	maximal, complemented
9	#20241673	C₂	1	1	yes	maximal, characteristic, complemented
10	#20241670	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₄₇	C₄	1	2	yes	—
13	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
14	#15689179	C₄	1	4	yes	—
15	#15689177	C₄	1	4	yes	—
16	#15689176	C₂²	1	2	yes	—
17	#15689174	C₂²	1	2	yes	—
18	#15689174	C₂²	1	2	yes	—
19	#15689145	C₄	1	4	yes	—
20	#15689085	C₄	1	4	yes	—
21	C₄×#10625329	C₂²	1	1	yes	characteristic
22	#15689075	C₄	1	2	yes	—
23	M₁₆×Dic₄₇	C₄	1	2	yes	—
24	C₄×#10625324	C₂²	1	2	yes	—
25	C₄×#10625324	C₂²	1	1	yes	characteristic
26	C₄×#10625324	C₂²	1	2	yes	—
27	C₄×#10625324	C₂²	1	1	yes	characteristic
28	#97×D₄₇	C₄	1	2	yes	—
29	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
31	#15688845	C₂²	1	2	yes	complemented
32	#15688845	C₂²	1	2	yes	complemented
33	#15688845	C₂²	1	2	yes	complemented
34	#15688844	C₂²	1	1	yes	characteristic, complemented
35	#15688844	C₂²	1	2	yes	complemented
36	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
38	C₄₇×#431	C₂²	1	2	yes	—
39	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#322	C₂²	1	1	yes	characteristic, complemented
41	C₄₇×#322	C₂²	1	2	yes	complemented
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
43	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
45	C₂×#10625329	C₂×C₄	1	2	yes	—
46	C₂×#10625324	C₂×C₄	1	2	yes	—
47	C₂×#10625324	C₂×C₄	1	2	yes	—
48	C₂×#10625324	C₂×C₄	1	4	yes	—
49	C₂×#10625324	C₂×C₄	1	4	yes	—
50	M₁₆×D₄₇	—	2	8	no	—
51	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
55	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₄₇	C₂×C₄	1	2	yes	—
58	C₄²×D₄₇	C₂×C₄	1	2	yes	—
59	C₄²×D₄₇	C₂³	1	1	yes	characteristic
60	C₄²×D₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
62	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
63	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
64	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
65	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
66	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
67	#13093935	C₂×C₄	1	2	yes	—
68	#13093934	C₂×C₄	1	2	yes	—
69	#13093934	C₂×C₄	1	2	yes	—
70	#13093934	C₂×C₄	1	4	yes	—
71	#13093934	C₂×C₄	1	4	yes	—
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
75	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
76	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
77	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
78	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
82	#97×C₄₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	2	yes	—
86	C₄×C₈×C₄₇	C₂³	1	2	yes	—
87	C₂³×D₄₇	C₄²	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
91	C₂²×Dic₄₇	C₄²	1	2	yes	—
92	C₂×C₄×D₄₇	—	2	4	no	—
93	C₂×C₄×D₄₇	—	4	4	no	—
94	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₄₇	C₄²	1	2	yes	—
97	C₂×C₄×D₄₇	C₄²	1	2	yes	—
98	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
102	M₁₆×C₄₇	—	2	4	no	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
107	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₄×Dic₄₇	C₄²	1	2	yes	—
111	C₄×Dic₄₇	C₄²	1	2	yes	—
112	C₄×Dic₄₇	C₄²	1	2	yes	—
113	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
116	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
117	#10625329	—	2	4	no	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
120	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
122	#10625324	—	2	4	no	—
123	#10625324	—	2	4	no	—
124	#10625324	—	4	4	no	—
125	#10625324	—	4	4	no	—
126	C₈×D₄₇	—	2	4	no	—
127	C₈×D₄₇	M₁₆	1	8	yes	complemented
128	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₄₇	—	2	2	no	—
130	C₂²×D₄₇	—	4	2	no	—
131	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
132	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	4	no	—
135	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
143	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
144	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	4	no	—
148	C₄×D₄₇	—	2	4	no	—
149	C₄×D₄₇	—	2	2	no	—
150	C₄×D₄₇	—	4	2	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₄×D₄₇	C₂×M₁₆	1	4	yes	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	2	4	no	—
155	C₈×C₄₇	—	2	4	no	—
156	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
157	C₄₇⋊C₈	—	2	2	no	—
158	C₄₇⋊C₈	—	2	4	no	—
159	C₄₇⋊C₈	—	2	4	no	—
160	C₄₇⋊C₈	C₂×M₁₆	1	4	yes	—
161	#17334	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
165	C₂×D₄₇	—	2	2	no	—
166	C₂×D₄₇	—	2	1	no	—
167	C₂×D₄₇	—	2	2	no	—
168	C₂×D₄₇	—	4	4	no	—
169	C₂×D₄₇	C₄×M₁₆	1	2	yes	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
173	C₄×C₄₇	#568	1	4	yes	—
174	Dic₄₇	—	2	1	no	—
175	Dic₄₇	—	2	2	no	—
176	Dic₄₇	—	2	2	no	—
177	Dic₄₇	C₄×M₁₆	1	2	yes	—
178	#2920	—	47	2	no	—
179	C₂×C₄×M₁₆	—	47	1	no	—
180	#1455	—	47	2	no	—
181	#1455	—	47	1	no	—
182	#1455	—	47	1	no	—
183	#1453	—	47	2	no	—
184	#1453	—	47	1	no	—
185	#1453	D₄₇	1	1	yes	characteristic, complemented
186	#1452	—	47	2	no	—
187	#1450	—	47	1	no	—
188	#1450	—	47	1	no	—
189	C₂×C₄₇	—	4	1	no	—
190	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
191	C₂×C₄₇	#2873	1	2	yes	—
192	D₄₇	—	2	2	no	—
193	D₄₇	—	4	1	no	—
194	C₂²×M₁₆	—	47	2	no	—
195	C₂²×C₄²	—	47	1	no	—
196	#432	—	47	4	no	—
197	#432	—	47	4	no	—
198	#431	—	47	2	no	—
199	#431	—	47	2	no	—
200	#431	—	47	2	no	—
201	#431	C₂×D₄₇	1	2	yes	—
202	#423	—	47	4	no	—
203	#407	—	47	4	no	—
204	C₄×M₁₆	—	47	2	no	—
205	C₄×M₁₆	—	47	1	no	—
206	C₄×M₁₆	—	47	2	no	—
207	C₄×M₁₆	—	47	1	no	—
208	C₄×M₁₆	—	47	1	no	—
209	C₄×M₁₆	—	47	2	no	—
210	C₄×M₁₆	—	47	2	no	—
211	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
212	C₂×#97	—	47	2	no	—
213	C₂×C₄×C₈	—	47	1	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	#322	—	47	1	no	—
216	#322	—	47	2	no	—
217	#322	—	47	2	no	—
218	#322	—	47	2	no	—
219	#322	—	47	2	no	—
220	#322	C₂×D₄₇	1	1	yes	characteristic, complemented
221	#322	C₂×D₄₇	1	2	yes	complemented
222	C₈²	—	47	2	no	—
223	C₈²	—	47	1	no	—
224	C₈²	C₂×D₄₇	1	1	yes	characteristic, complemented
225	C₄₇	#17334	1	1	yes	characteristic, complemented, Sylow
226	C₂³×C₄	—	47	1	no	—
227	C₂³×C₄	—	47	2	no	—
228	C₂×M₁₆	—	47	2	no	—
229	C₂×M₁₆	—	47	2	no	—
230	C₂×M₁₆	—	47	2	no	—
231	C₂×M₁₆	—	47	4	no	—
232	C₂×M₁₆	—	47	4	no	—
233	C₂×M₁₆	—	94	8	no	—
234	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
235	C₂²×C₈	—	47	2	no	—
236	C₂²×C₈	—	47	2	no	—
237	C₂×C₄²	—	47	1	no	—
238	C₂×C₄²	—	47	2	no	—
239	C₂×C₄²	—	47	1	no	—
240	C₂×C₄²	—	47	1	no	—
241	C₂×C₄²	—	47	2	no	—
242	C₂×C₄²	—	47	2	no	—
243	C₂×C₄²	—	47	1	no	—
244	C₂×C₄²	—	47	1	no	—
245	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
246	C₄⋊C₈	—	47	4	no	—
247	C₄⋊C₈	—	47	4	no	—
248	C₄⋊C₈	—	47	4	no	—
249	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
250	C₂²⋊C₈	—	47	4	no	—
251	C₂²⋊C₈	—	47	4	no	—
252	C₂²⋊C₈	—	47	4	no	—
253	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
254	#97	—	47	2	no	—
255	#97	—	47	2	no	—
256	#97	—	47	4	no	—
257	#97	—	47	4	no	—
258	#97	—	47	2	no	—
259	#97	C₄×D₄₇	1	2	yes	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	—	47	2	no	—
262	C₄×C₈	—	47	1	no	—
263	C₄×C₈	—	47	2	no	—
264	C₄×C₈	—	47	2	no	—
265	C₄×C₈	—	47	1	no	—
266	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₇	1	2	yes	—
268	C₄×C₈	C₂²×D₄₇	1	2	yes	—
269	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
270	C₂⁴	—	47	1	no	—
271	C₂²×C₄	—	47	1	no	—
272	C₂²×C₄	—	47	1	no	—
273	C₂²×C₄	—	47	2	no	—
274	C₂²×C₄	—	47	1	no	—
275	C₂²×C₄	—	47	1	no	—
276	C₂²×C₄	—	47	2	no	—
277	C₂²×C₄	—	47	1	no	—
278	C₂²×C₄	—	47	2	no	—
279	C₂²×C₄	—	47	2	no	—
280	C₂²×C₄	—	47	2	no	—
281	C₂²×C₄	—	94	4	no	—
282	C₂²×C₄	—	188	4	no	—
283	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
284	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
285	M₁₆	—	2	4	no	—
286	M₁₆	—	94	4	no	—
287	M₁₆	—	94	4	no	—
288	M₁₆	—	94	4	no	—
289	M₁₆	—	188	4	no	—
290	M₁₆	—	188	4	no	—
291	C₂×C₈	—	47	2	no	—
292	C₂×C₈	—	47	8	no	—
293	C₂×C₈	—	47	2	no	—
294	C₂×C₈	—	47	4	no	—
295	C₂×C₈	—	47	4	no	—
296	C₂×C₈	—	94	4	no	—
297	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
299	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
300	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
301	C₄²	—	47	2	no	—
302	C₄²	—	47	2	no	—
303	C₄²	—	47	2	no	—
304	C₄²	—	47	1	no	—
305	C₄²	—	47	1	no	—
306	C₄²	—	47	2	no	—
307	C₄²	—	47	2	no	—
308	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₄₇	1	2	yes	—
310	C₄²	C₂³×D₄₇	1	1	yes	characteristic
311	C₂³	—	47	1	no	—
312	C₂³	—	47	1	no	—
313	C₂³	—	94	2	no	—
314	C₂³	—	188	2	no	—
315	C₂³	C₄²×D₄₇	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	47	1	no	—
319	C₂×C₄	—	47	4	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	2	no	—
322	C₂×C₄	—	47	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	4	no	—
328	C₂×C₄	—	94	4	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₄²×D₄₇	1	2	yes	—
334	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	47	4	no	—
340	C₈	—	94	2	no	—
341	C₈	—	94	4	no	—
342	C₈	—	94	4	no	—
343	C₈	M₁₆×D₄₇	1	4	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	47	2	no	—
347	C₂²	—	94	2	no	—
348	C₂²	—	94	1	no	—
349	C₂²	—	94	2	no	—
350	C₂²	—	188	4	no	—
351	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	47	2	no	—
355	C₄	—	94	1	no	—
356	C₄	—	94	2	no	—
357	C₄	—	94	2	no	—
358	C₄	C₂×M₁₆×D₄₇	1	2	yes	—
359	C₄	#15689888	1	4	yes	—
360	C₂	—	4	1	no	—
361	C₂	—	94	2	no	—
362	C₂	—	188	1	no	—
363	C₂	C₄×M₁₆×D₄₇	1	1	yes	characteristic
364	C₂	#20251804	1	2	yes	—

	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	#17334
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. #20241692	C₂	semidirect
4. #20241692	C₂	semidirect
5. #20241690	C₂	semidirect
6. #20241685	C₂	semidirect
7. #20241683	C₂	semidirect
8. #20241679	C₂	semidirect
9. #20241673	C₂	semidirect
10. #20241670	C₂	semidirect
11. C₄₇×#1453	C₂	semidirect
31. #15688845	C₂²	semidirect
32. #15688845	C₂²	semidirect
33. #15688845	C₂²	semidirect
34. #15688844	C₂²	semidirect
35. #15688844	C₂²	semidirect
36. C₈×C₄₇⋊C₈	C₂²	semidirect
37. C₈×C₄₇⋊C₈	C₂²	semidirect
40. C₄₇×#322	C₂²	semidirect
41. C₄₇×#322	C₂²	semidirect
42. C₈²×C₄₇	C₂²	semidirect
127. C₈×D₄₇	M₁₆	semidirect
185. #1453	D₄₇	semidirect
220. #322	C₂×D₄₇	semidirect
221. #322	C₂×D₄₇	semidirect
224. C₈²	C₂×D₄₇	semidirect
225. C₄₇	#17334	semidirect

Maximal subgroups

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

subgroup	index
#20252016	2
C₄×M₁₆×D₄₇	2
#20241692	2
#20241690	2
#20241685	2
#20241683	2
#20241679	2
#20241673	2
#20241670	2
C₄₇×#1453	2
#17334	47