Subgroups — #26182004

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252352	C₂	1	2	yes	maximal
2	C₄×#13094032	C₂	1	1	yes	maximal, characteristic
3	#20241692	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241691	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241682	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241676	C₂	1	2	yes	maximal, complemented
7	#20241673	C₂	1	1	yes	maximal, characteristic, complemented
8	#20241669	C₂	1	2	yes	maximal, complemented
9	#20241667	C₂	1	2	yes	maximal, complemented
10	C₈²×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094032	C₄	1	2	yes	—
13	C₄×#10625350	C₂²	1	1	yes	characteristic
14	#15689252	C₄	1	4	yes	—
15	#15689198	C₄	1	4	yes	—
16	#15689197	C₄	1	4	yes	—
17	#15689193	C₄	1	4	yes	—
18	#15689174	C₂²	1	2	yes	—
19	#15689174	C₂²	1	2	yes	—
20	#15689081	C₄	1	2	yes	—
21	#15689080	C₄	1	2	yes	—
22	#15689080	C₄	1	2	yes	—
23	C₄×#10625329	C₂²	1	2	yes	—
24	C₄×#10625324	C₂²	1	2	yes	—
25	C₄×#10625324	C₂²	1	1	yes	characteristic
26	C₄×#10625324	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₇	C₂²	1	2	yes	—
31	#15688845	C₂²	1	2	yes	—
32	#15688845	C₂²	1	2	yes	—
33	#15688845	C₂²	1	2	yes	—
34	#15688844	C₂²	1	1	yes	characteristic
35	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
36	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#431	C₂²	1	2	yes	—
39	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#322	C₂²	1	2	yes	—
41	C₄₇×#322	C₂²	1	1	yes	characteristic
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
44	#13094032	—	2	8	no	—
45	C₂×#10625329	C₂×C₄	1	4	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	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
54	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
55	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
56	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
57	#13093983	C₂×C₄	1	1	yes	characteristic
58	#13093981	C₂×C₄	1	1	yes	characteristic
59	#13093981	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₄₇	C₂³	1	1	yes	characteristic
61	C₄²×D₄₇	C₂³	1	1	yes	characteristic
62	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
63	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
64	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
65	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
66	#13093935	C₂×C₄	1	4	yes	—
67	#13093934	C₂×C₄	1	4	yes	—
68	#13093934	C₂×C₄	1	2	yes	—
69	#13093934	C₂×C₄	1	2	yes	—
70	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
75	C₈×Dic₄₇	C₂×C₄	1	4	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	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₄₇	C₄²	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	C₄²	1	1	yes	characteristic
91	#10625350	—	2	4	no	—
92	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
93	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
98	M₁₆×C₄₇	—	2	4	no	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊Dic₄₇	C₄²	1	2	yes	—
107	(C₂×D₄₇)⋊C₄	C₄²	1	2	yes	—
108	(C₂×D₄₇)⋊C₄	C₄²	1	2	yes	—
109	C₄⋊Dic₄₇	C₄²	1	2	yes	—
110	Dic₄₇⋊C₄	C₄²	1	2	yes	—
111	Dic₄₇⋊C₄	C₄²	1	2	yes	—
112	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
115	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
116	#10625329	—	2	8	no	—
117	C₂×C₄₇⋊C₈	—	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	2	yes	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
122	#10625324	—	2	4	no	—
123	#10625324	—	2	4	no	—
124	#10625324	—	2	8	no	—
125	C₈×D₄₇	—	2	4	no	—
126	C₈×D₄₇	—	2	8	no	—
127	C₈×D₄₇	M₁₆	1	8	yes	complemented
128	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
130	C₂²×D₄₇	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
137	C₂²⋊D₄₇	—	2	2	no	—
138	C₂²⋊D₄₇	—	2	2	no	—
139	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
140	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
141	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
142	C₂×Dic₄₇	C₂×C₄²	1	1	yes	characteristic
143	D₁₈₈	—	2	2	no	—
144	C₄×D₄₇	—	2	2	no	—
145	C₄×D₄₇	—	2	4	no	—
146	C₄×D₄₇	—	2	4	no	—
147	C₄×D₄₇	C₂×M₁₆	1	4	yes	—
148	Dic₉₄	—	2	2	no	—
149	C₈×C₄₇	—	2	2	no	—
150	C₈×C₄₇	—	2	4	no	—
151	C₈×C₄₇	—	2	4	no	—
152	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
153	C₄₇⋊C₈	—	2	2	no	—
154	C₄₇⋊C₈	—	2	4	no	—
155	C₄₇⋊C₈	C₂×M₁₆	1	4	yes	—
156	C₄₇⋊C₈	#131	1	8	yes	complemented
157	#17335	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
161	C₂×D₄₇	—	2	2	no	—
162	C₂×D₄₇	—	2	1	no	—
163	C₂×D₄₇	—	2	2	no	—
164	C₂×D₄₇	C₄×M₁₆	1	2	yes	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
168	C₄×C₄₇	C₂×#131	1	4	yes	—
169	Dic₄₇	—	2	1	no	—
170	Dic₄₇	—	2	2	no	—
171	Dic₄₇	—	2	2	no	—
172	Dic₄₇	C₄×M₁₆	1	2	yes	—
173	#2965	—	47	2	no	—
174	C₄×#131	—	47	1	no	—
175	#1455	—	47	1	no	—
176	#1455	—	47	1	no	—
177	#1453	—	47	1	no	—
178	#1453	D₄₇	1	1	yes	characteristic, complemented
179	#1451	—	47	2	no	—
180	#1450	—	47	1	no	—
181	C₈×M₁₆	—	47	2	no	—
182	C₈×M₁₆	—	47	2	no	—
183	C₂×C₈²	—	47	1	no	—
184	C₂×C₄₇	—	4	1	no	—
185	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
186	C₂×C₄₇	C₄×#131	1	2	yes	—
187	D₄₇	—	2	2	no	—
188	D₄₇	—	4	1	no	—
189	C₂×#131	—	47	2	no	—
190	C₄×#41	—	47	1	no	—
191	#446	—	47	4	no	—
192	#436	—	47	4	no	—
193	#435	—	47	4	no	—
194	#435	—	47	4	no	—
195	#431	—	47	2	no	—
196	#431	—	47	2	no	—
197	#431	C₂×D₄₇	1	2	yes	—
198	#405	—	47	2	no	—
199	#405	—	47	2	no	—
200	#405	—	47	2	no	—
201	C₄×M₁₆	—	47	2	no	—
202	C₄×M₁₆	—	47	1	no	—
203	C₄×M₁₆	—	47	1	no	—
204	C₄×M₁₆	—	47	2	no	—
205	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
206	C₂×C₄×C₈	—	47	1	no	—
207	C₂×C₄×C₈	—	47	1	no	—
208	C₂×C₄×C₈	—	47	1	no	—
209	C₂×C₄×C₈	—	47	2	no	—
210	#322	—	47	1	no	—
211	#322	—	47	2	no	—
212	#322	—	47	2	no	—
213	#322	—	47	2	no	—
214	#322	C₂×D₄₇	1	2	yes	—
215	#322	C₂×D₄₇	1	1	yes	characteristic
216	C₈²	—	47	2	no	—
217	C₈²	—	47	1	no	—
218	C₈²	—	47	2	no	—
219	C₈²	C₂×D₄₇	1	1	yes	characteristic
220	C₄₇	#17335	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	47	1	no	—
222	#131	—	94	8	no	—
223	C₂×M₁₆	—	47	4	no	—
224	C₂×M₁₆	—	47	2	no	—
225	C₂×M₁₆	—	47	2	no	—
226	C₂×M₁₆	—	47	4	no	—
227	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
228	C₂²×C₈	—	47	2	no	—
229	C₂²×C₈	—	47	2	no	—
230	C₂²×C₈	—	47	2	no	—
231	C₂²×C₈	—	47	4	no	—
232	C₄×Q₈	—	47	2	no	—
233	C₄×D₄	—	47	2	no	—
234	C₄×D₄	—	47	2	no	—
235	C₄×D₄	—	47	2	no	—
236	#117	—	47	1	no	—
237	#117	—	47	1	no	—
238	#117	—	47	1	no	—
239	C₂×C₄²	—	47	1	no	—
240	C₂×C₄²	—	47	1	no	—
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	C₄⋊C₈	—	47	4	no	—
243	C₄⋊C₈	—	47	4	no	—
244	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
245	C₂²⋊C₈	—	47	4	no	—
246	C₂²⋊C₈	—	47	4	no	—
247	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
248	#97	—	47	2	no	—
249	#97	—	47	2	no	—
250	#97	—	47	4	no	—
251	#97	—	47	4	no	—
252	#97	C₄×D₄₇	1	2	yes	—
253	C₄×C₈	—	47	2	no	—
254	C₄×C₈	—	47	2	no	—
255	C₄×C₈	—	47	4	no	—
256	C₄×C₈	—	47	2	no	—
257	C₄×C₈	—	47	1	no	—
258	C₄×C₈	—	47	2	no	—
259	C₄×C₈	—	47	2	no	—
260	C₄×C₈	—	47	1	no	—
261	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₄₇	1	2	yes	—
263	C₄×C₈	C₂²×D₄₇	1	2	yes	—
264	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
265	#41	—	94	4	no	—
266	C₂×Q₈	—	47	1	no	—
267	C₂×D₄	—	47	1	no	—
268	C₂×D₄	—	47	1	no	—
269	C₂×D₄	—	47	1	no	—
270	C₂²×C₄	—	47	1	no	—
271	C₂²×C₄	—	47	2	no	—
272	C₂²×C₄	—	47	2	no	—
273	C₂²×C₄	—	47	1	no	—
274	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
275	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	94	8	no	—
278	M₁₆	—	94	4	no	—
279	M₁₆	—	94	8	no	—
280	M₁₆	—	94	4	no	—
281	C₂×C₈	—	47	2	no	—
282	C₂×C₈	—	47	4	no	—
283	C₂×C₈	—	47	8	no	—
284	C₂×C₈	—	47	2	no	—
285	C₂×C₈	—	47	4	no	—
286	C₂×C₈	—	94	8	no	—
287	C₂×C₈	—	94	4	no	—
288	C₂×C₈	—	94	4	no	—
289	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
290	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
291	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
292	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
293	C₄⋊C₄	—	47	2	no	—
294	C₄⋊C₄	—	47	2	no	—
295	C₄⋊C₄	—	47	2	no	—
296	C₂²⋊C₄	—	47	2	no	—
297	C₂²⋊C₄	—	47	2	no	—
298	C₂²⋊C₄	—	47	2	no	—
299	C₄²	—	47	1	no	—
300	C₄²	—	47	1	no	—
301	C₄²	—	47	2	no	—
302	C₄²	—	47	2	no	—
303	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
304	C₄²	C₂×C₄×D₄₇	1	2	yes	—
305	C₄²	C₂³×D₄₇	1	1	yes	characteristic
306	C₂³	—	47	1	no	—
307	C₂³	—	47	1	no	—
308	C₂³	C₄²×D₄₇	1	1	yes	characteristic
309	Q₈	—	94	2	no	—
310	D₄	—	94	2	no	—
311	D₄	—	94	2	no	—
312	D₄	—	94	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	47	1	no	—
316	C₂×C₄	—	47	4	no	—
317	C₂×C₄	—	47	2	no	—
318	C₂×C₄	—	47	2	no	—
319	C₂×C₄	—	47	1	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	4	no	—
322	C₂×C₄	—	94	4	no	—
323	C₂×C₄	C₄²×D₄₇	1	2	yes	—
324	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
327	C₈	—	2	2	no	—
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	—	47	4	no	—
331	C₈	—	47	8	no	—
332	C₈	—	94	2	no	—
333	C₈	—	94	4	no	—
334	C₈	M₁₆×D₄₇	1	4	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	47	2	no	—
338	C₂²	—	94	2	no	—
339	C₂²	—	94	1	no	—
340	C₂²	—	94	2	no	—
341	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	47	2	no	—
345	C₄	—	94	1	no	—
346	C₄	—	94	2	no	—
347	C₄	—	94	2	no	—
348	C₄	C₂×M₁₆×D₄₇	1	2	yes	—
349	C₄	#131×D₄₇	1	4	yes	—
350	C₂	—	4	1	no	—
351	C₂	—	94	2	no	—
352	C₂	—	188	1	no	—
353	C₂	C₄×M₁₆×D₄₇	1	1	yes	characteristic
354	C₂	#20251810	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1453	C₂
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1453	D₄₇
hypocenter	C₄₇	#17335
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. #20241691	C₂	semidirect
5. #20241682	C₂	semidirect
6. #20241676	C₂	semidirect
7. #20241673	C₂	semidirect
8. #20241669	C₂	semidirect
9. #20241667	C₂	semidirect
10. C₈²×D₄₇	C₂	semidirect
11. C₄₇×#1453	C₂	semidirect
127. C₈×D₄₇	M₁₆	semidirect
156. C₄₇⋊C₈	#131	semidirect
178. #1453	D₄₇	semidirect
220. C₄₇	#17335	semidirect

Maximal subgroups

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

subgroup	index
#20252352	2
C₄×#13094032	2
#20241692	2
#20241691	2
#20241682	2
#20241676	2
#20241673	2
#20241669	2
#20241667	2
C₈²×D₄₇	2
C₄₇×#1453	2
#17335	47