Subgroups — #26188317

Order 12032.

644 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253964	C₂	1	1	yes	maximal, characteristic
2	#20252453	C₂	1	2	yes	maximal, complemented
3	#20242301	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242215	C₂	1	2	yes	maximal, complemented
5	#20242214	C₂	1	2	yes	maximal, complemented
6	#20242213	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242212	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242133	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241803	C₂	1	2	yes	maximal, complemented
10	#20241674	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1594	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689670	C₂²	1	2	yes	—
13	#15689527	C₄	1	1	yes	characteristic
14	#15689498	C₄	1	1	yes	characteristic
15	#15689468	C₄	1	2	yes	—
16	#15689360	C₄	1	1	yes	characteristic
17	#15689358	C₄	1	2	yes	—
18	#15689356	C₄	1	1	yes	characteristic
19	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
20	C₄×#10625334	C₂²	1	2	yes	—
21	#15689200	C₂²	1	2	yes	—
22	#15689199	—	2	2	no	—
23	C₈×D₁₈₈	C₂²	1	2	yes	—
24	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	(C₂²⋊D₄₇)⋊C₈	—	2	2	no	—
26	#15689175	C₂²	1	2	yes	—
27	#15689174	C₂²	1	1	yes	characteristic
28	#15689091	—	2	2	no	—
29	#15689088	—	2	2	no	—
30	C₄×#10625324	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688875	C₂²	1	2	yes	—
33	#15688874	C₂²	1	1	yes	characteristic
34	#15688852	C₂²	1	2	yes	—
35	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
36	#15688850	C₂²	1	2	yes	—
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688844	C₂²	1	1	yes	characteristic
39	#15688844	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×#436	C₂²	1	2	yes	—
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
44	C₄₇×#322	C₂²	1	1	yes	characteristic
45	C₂×#10625350	—	2	2	no	—
46	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#10625334	C₂×C₄	1	2	yes	—
48	C₂×#10625324	—	2	1	no	—
49	C₂×#10625324	—	2	2	no	—
50	C₂²×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₈×D₄₇	—	2	2	no	—
52	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
54	#13094023	C₂×C₄	1	2	yes	—
55	#13094020	C₂×C₄	1	2	yes	—
56	#13094010	—	2	2	no	—
57	#13093999	C₂×C₄	1	2	yes	—
58	#13093997	—	2	2	no	—
59	#13093995	C₂×C₄	1	2	yes	—
60	C₄×D₁₈₈	C₂³	1	2	yes	—
61	C₄×C₂²⋊D₄₇	—	2	2	no	—
62	C₄²×D₄₇	C₂³	1	1	yes	characteristic
63	#13093958	C₂×C₄	1	1	yes	characteristic
64	#13093957	C₂×C₄	1	2	yes	—
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₄₇⋊C₈	D₄	1	2	yes	—
67	#13093946	C₂×C₄	1	2	yes	—
68	#13093945	C₂×C₄	1	2	yes	—
69	#13093944	C₂×C₄	1	2	yes	—
70	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	(C₂×D₄₇)⋊C₈	—	2	2	no	—
74	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
75	#13093934	—	2	1	no	—
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₄₇	—	2	2	no	—
80	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₂×M₁₆×C₄₇	—	2	2	no	—
84	C₄×D₄×C₄₇	C₂³	1	2	yes	—
85	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	#102×C₄₇	C₂×C₄	1	2	yes	—
88	C₂²⋊C₈×C₄₇	—	2	2	no	—
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₄₇	—	2	2	no	—
94	C₂×C₂²⋊D₄₇	—	2	2	no	—
95	C₂×C₂²⋊D₄₇	—	2	2	no	—
96	C₂×C₂²⋊D₄₇	—	2	2	no	—
97	#10625350	—	8	2	no	—
98	#10625350	—	8	2	no	—
99	C₂×D₁₈₈	—	2	2	no	—
100	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₄₇	—	2	2	no	—
102	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
105	C₂×Dic₉₄	—	2	2	no	—
106	D₈×C₄₇	—	2	2	no	—
107	M₁₆×C₄₇	—	8	2	no	—
108	C₂×C₈×C₄₇	—	2	2	no	—
109	C₂×C₈×C₄₇	—	2	1	no	—
110	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₄₇	—	2	2	no	—
115	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
116	C₂²⋊Dic₄₇	—	2	2	no	—
117	#10625334	—	4	2	no	—
118	(C₂×D₄₇)⋊C₄	—	2	2	no	—
119	(C₂×D₄₇)⋊C₄	—	2	2	no	—
120	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
121	Dic₄₇⋊C₄	—	2	2	no	—
122	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₄₇	#41	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	—	8	2	no	—
128	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
131	D₃₇₆	—	2	2	no	—
132	#10625324	—	4	2	no	—
133	#10625324	—	8	2	no	—
134	C₈×D₄₇	—	2	2	no	—
135	C₈×D₄₇	—	4	1	no	—
136	C₈×D₄₇	—	8	2	no	—
137	C₂³×C₄₇	—	2	2	no	—
138	C₂²×D₄₇	—	2	2	no	—
139	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
140	D₄×C₄₇	—	2	2	no	—
141	D₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	4	4	no	—
149	C₂²⋊D₄₇	—	8	2	no	—
150	C₂²⋊D₄₇	—	8	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
155	D₁₈₈	—	2	2	no	—
156	D₁₈₈	—	2	2	no	—
157	D₁₈₈	—	8	2	no	—
158	C₄×D₄₇	—	2	1	no	—
159	C₄×D₄₇	—	2	1	no	—
160	C₄×D₄₇	—	2	2	no	—
161	C₄×D₄₇	—	2	2	no	—
162	C₄×D₄₇	—	4	2	no	—
163	C₄×D₄₇	—	4	2	no	—
164	Dic₉₄	—	8	2	no	—
165	C₈×C₄₇	—	2	1	no	—
166	C₈×C₄₇	—	2	2	no	—
167	C₈×C₄₇	—	4	1	no	—
168	C₈×C₄₇	—	4	2	no	—
169	C₈×C₄₇	#131	1	2	yes	—
170	C₄₇⋊C₈	—	2	1	no	—
171	C₄₇⋊C₈	—	2	1	no	—
172	C₄₇⋊C₈	—	2	2	no	—
173	C₄₇⋊C₈	—	4	1	no	—
174	C₄₇⋊C₈	—	4	2	no	—
175	#18091	—	47	1	no	maximal, Sylow
176	C₂²×C₄₇	—	4	2	no	—
177	C₂²×C₄₇	—	4	2	no	—
178	C₂²×C₄₇	—	4	2	no	—
179	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×D₄₇	—	2	1	no	—
181	C₂×D₄₇	—	2	1	no	—
182	C₂×D₄₇	—	2	1	no	—
183	C₂×D₄₇	—	4	2	no	—
184	C₂×D₄₇	—	4	2	no	—
185	C₂×D₄₇	—	4	2	no	—
186	C₄×C₄₇	—	2	1	no	—
187	C₄×C₄₇	—	4	2	no	—
188	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
189	C₄×C₄₇	#568	1	1	yes	characteristic
190	C₄×C₄₇	#576	1	2	yes	—
191	Dic₄₇	—	2	1	no	—
192	Dic₄₇	—	2	1	no	—
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	#444	1	2	yes	—
195	#3169	—	47	1	no	—
196	#2979	—	47	2	no	—
197	#1594	—	47	1	no	—
198	#1594	D₄₇	1	1	yes	characteristic, complemented
199	#1581	—	47	2	no	—
200	#1581	—	47	2	no	—
201	#1580	—	47	1	no	—
202	#1580	—	47	1	no	—
203	#1564	—	47	1	no	—
204	#1484	—	47	2	no	—
205	#1451	—	47	1	no	—
206	C₂×C₄₇	—	8	2	no	—
207	C₂×C₄₇	#2935	1	1	yes	characteristic
208	C₂×C₄₇	#2948	1	1	yes	characteristic
209	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
210	D₄₇	—	4	1	no	—
211	D₄₇	—	8	2	no	—
212	#532	—	47	2	no	—
213	#500	—	47	1	no	—
214	#493	—	47	1	no	—
215	#486	—	47	2	no	—
216	#466	—	47	1	no	—
217	#466	—	47	1	no	—
218	#466	—	47	2	no	—
219	C₄×D₈	—	47	2	no	—
220	C₄×D₈	—	47	1	no	—
221	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
222	#436	—	47	2	no	—
223	#436	—	94	2	no	—
224	#436	C₂×D₄₇	1	2	yes	—
225	C₈×D₄	—	47	2	no	—
226	C₈×D₄	—	47	2	no	—
227	C₈×D₄	—	94	2	no	—
228	#431	—	47	1	no	—
229	#431	—	47	2	no	—
230	#408	—	94	2	no	—
231	#408	—	94	2	no	—
232	C₄×M₁₆	—	47	1	no	—
233	C₂×C₄×C₈	—	47	1	no	—
234	#335	—	47	2	no	—
235	#334	—	47	1	no	—
236	#334	C₂×D₄₇	1	1	yes	characteristic
237	D₄⋊C₈	—	47	2	no	—
238	D₄⋊C₈	—	47	2	no	—
239	D₄⋊C₈	—	47	2	no	—
240	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
241	#322	—	47	1	no	—
242	#322	—	47	1	no	—
243	#322	—	47	1	no	—
244	#322	C₂×D₄₇	1	1	yes	characteristic
245	C₄₇	#18091	1	1	yes	characteristic, complemented, Sylow
246	C₂×#41	—	94	2	no	—
247	C₂×D₈	—	47	2	no	—
248	C₂×D₈	—	47	1	no	—
249	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	94	2	no	—
252	C₂×M₁₆	—	94	1	no	—
253	C₂²×C₈	—	47	1	no	—
254	C₂²×C₈	—	47	1	no	—
255	C₂²×C₈	—	94	2	no	—
256	C₂²×C₈	—	94	2	no	—
257	C₄⋊Q₈	—	47	2	no	—
258	#127	—	47	2	no	—
259	#124	—	94	2	no	—
260	#121	—	47	2	no	—
261	#121	—	47	2	no	—
262	#121	—	94	2	no	—
263	C₄×D₄	—	47	2	no	—
264	C₄×D₄	—	94	2	no	—
265	C₄×D₄	C₂²×D₄₇	1	2	yes	—
266	C₂×C₄²	—	47	1	no	—
267	#107	—	47	1	no	—
268	#107	—	47	2	no	—
269	#107	C₄×D₄₇	1	1	yes	characteristic
270	C₄⋊C₈	—	47	2	no	—
271	C₄⋊C₈	—	47	2	no	—
272	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
273	#102	—	47	2	no	—
274	#102	—	47	2	no	—
275	#102	—	47	2	no	—
276	#102	C₄×D₄₇	1	2	yes	—
277	C₂²⋊C₈	—	2	2	no	—
278	C₂²⋊C₈	—	47	2	no	—
279	C₂²⋊C₈	—	94	2	no	—
280	C₂²⋊C₈	—	94	2	no	—
281	C₂²⋊C₈	—	94	2	no	—
282	C₂²⋊C₈	—	94	2	no	—
283	#97	—	94	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	—	47	1	no	—
286	C₄×C₈	—	47	1	no	—
287	C₄×C₈	—	47	1	no	—
288	C₄×C₈	—	47	1	no	—
289	C₄×C₈	—	94	2	no	—
290	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
293	#41	—	376	2	no	—
294	#41	—	376	2	no	—
295	C₂×Q₈	—	94	2	no	—
296	C₂×D₄	—	47	2	no	—
297	C₂×D₄	—	94	2	no	—
298	C₂×D₄	—	94	2	no	—
299	C₂×D₄	—	94	2	no	—
300	C₂×D₄	—	94	2	no	—
301	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	47	1	no	—
304	C₂²×C₄	—	47	1	no	—
305	C₂²×C₄	—	47	1	no	—
306	C₂²×C₄	—	94	2	no	—
307	D₈	—	2	2	no	—
308	D₈	—	94	2	no	—
309	D₈	—	188	2	no	—
310	M₁₆	—	8	2	no	—
311	M₁₆	—	376	2	no	—
312	M₁₆	—	188	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	47	1	no	—
316	C₂×C₈	—	47	1	no	—
317	C₂×C₈	—	47	2	no	—
318	C₂×C₈	—	94	2	no	—
319	C₂×C₈	—	94	2	no	—
320	C₂×C₈	—	94	1	no	—
321	C₂×C₈	—	376	2	no	—
322	C₂×C₈	—	376	2	no	—
323	C₂×C₈	—	188	1	no	—
324	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
326	C₂×C₈	D₄×D₄₇	1	2	yes	—
327	C₄⋊C₄	—	47	2	no	—
328	C₄⋊C₄	—	94	2	no	—
329	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	94	2	no	—
332	C₂²⋊C₄	—	94	2	no	—
333	C₂²⋊C₄	—	94	2	no	—
334	C₄²	—	47	1	no	—
335	C₄²	—	47	1	no	—
336	C₄²	—	47	1	no	—
337	C₄²	C₂³×D₄₇	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	—	47	1	no	—
340	C₂³	—	94	2	no	—
341	Q₈	—	376	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	94	2	no	—
345	D₄	—	94	2	no	—
346	D₄	—	376	2	no	—
347	D₄	—	376	2	no	—
348	D₄	—	376	2	no	—
349	D₄	—	188	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	47	1	no	—
354	C₂×C₄	—	47	1	no	—
355	C₂×C₄	—	47	1	no	—
356	C₂×C₄	—	94	2	no	—
357	C₂×C₄	—	94	1	no	—
358	C₂×C₄	—	94	1	no	—
359	C₂×C₄	—	94	2	no	—
360	C₂×C₄	—	94	2	no	—
361	C₂×C₄	—	188	2	no	—
362	C₂×C₄	—	188	2	no	—
363	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
365	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	2	no	—
368	C₈	—	4	1	no	—
369	C₈	—	4	2	no	—
370	C₈	—	94	1	no	—
371	C₈	—	94	1	no	—
372	C₈	—	94	2	no	—
373	C₈	—	188	1	no	—
374	C₈	—	188	2	no	—
375	C₈	#13094032	1	2	yes	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	94	1	no	—
380	C₂²	—	94	1	no	—
381	C₂²	—	94	1	no	—
382	C₂²	—	188	2	no	—
383	C₂²	—	188	2	no	—
384	C₂²	—	188	2	no	—
385	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	2	no	—
388	C₄	—	47	2	no	—
389	C₄	—	94	1	no	—
390	C₄	—	94	1	no	—
391	C₄	—	188	2	no	—
392	C₄	C₂×#13094032	1	1	yes	characteristic
393	C₄	#15689888	1	1	yes	characteristic
394	C₄	#15689927	1	2	yes	—
395	C₂	—	8	2	no	—
396	C₂	—	376	2	no	—
397	C₂	—	188	1	no	—
398	C₂	#20252133	1	1	yes	characteristic
399	C₂	#20252237	1	1	yes	characteristic
400	C₂	#444×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1594	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#20253964	C₂
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1594	D₄₇
hypocenter	C₄₇	#18091
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1594

Complements

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

N	H ≅ G/N	type
2. #20252453	C₂	semidirect
3. #20242301	C₂	semidirect
4. #20242215	C₂	semidirect
5. #20242214	C₂	semidirect
6. #20242213	C₂	semidirect
7. #20242212	C₂	semidirect
8. #20242133	C₂	semidirect
9. #20241803	C₂	semidirect
10. #20241674	C₂	semidirect
11. C₄₇×#1594	C₂	semidirect
198. #1594	D₄₇	semidirect
245. C₄₇	#18091	semidirect

Maximal subgroups

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

subgroup	index
#20253964	2
#20252453	2
#20242301	2
#20242215	2
#20242214	2
#20242213	2
#20242212	2
#20242133	2
#20241803	2
#20241674	2
C₄₇×#1594	2
#18091	47