Subgroups — #26201492

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253181	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689325	C₂	1	1	yes	maximal, characteristic
3	#20252411	C₂	1	1	yes	maximal, characteristic
4	#20242808	C₂	1	4	yes	maximal, complemented
5	#20242760	C₂	1	4	yes	maximal, complemented
6	#20242159	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242060	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3229	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625336	—	2	1	no	—
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#15689420	—	2	2	no	—
13	#15689410	C₂²	1	2	yes	—
14	#15689400	C₂²	1	2	yes	—
15	#15689325	—	2	2	no	—
16	#15689325	C₂²	1	4	yes	—
17	C₈×D₁₈₈	C₂²	1	4	yes	—
18	#15689188	—	2	2	no	—
19	#15689176	C₂²	1	1	yes	characteristic
20	#15689166	—	2	1	no	—
21	#15689147	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
23	#15689127	—	2	1	no	—
24	C₂×#13093948	—	2	1	no	—
25	C₂²⋊C₈×D₄₇	—	2	2	no	—
26	#15688874	C₂²	1	4	yes	complemented
27	#15688872	C₂²	1	2	yes	complemented
28	#15688869	C₂²	1	4	yes	complemented
29	#15688869	C₂²	1	2	yes	complemented
30	C₄₇×#533	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#498	C₂²	1	2	yes	complemented
33	C₄₇×#487	C₂²	1	2	yes	complemented
34	C₄₇×#483	—	2	2	no	—
35	C₄₇×#475	—	2	2	no	—
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₄₇	—	2	1	no	—
39	C₂×#10625336	—	2	4	no	—
40	C₂×#10625336	—	4	4	no	—
41	C₂×#10625329	—	2	1	no	—
42	C₂²×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₈×D₄₇	—	2	4	no	—
44	C₄×D₁₈₈	—	2	1	no	—
45	C₄×D₁₈₈	—	2	1	no	—
46	C₄×D₁₈₈	D₄	1	2	yes	—
47	C₄×D₁₈₈	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
49	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
50	#13093957	—	2	2	no	—
51	#13093955	—	2	2	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
55	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
56	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
57	#13093948	—	2	4	no	—
58	#13093944	—	2	2	no	—
59	#13093944	—	2	2	no	—
60	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
61	(C₂×D₄₇)⋊C₈	—	2	4	no	—
62	C₄×C₄₇⋊C₈	—	2	1	no	—
63	C₂²×Q₈×C₄₇	—	2	1	no	—
64	C₄⋊Q₈×C₄₇	—	2	1	no	—
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	C₄⋊Q₈×C₄₇	—	2	1	no	—
67	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
70	#125×C₄₇	—	2	1	no	—
71	C₂²⋊Q₈×C₄₇	—	2	1	no	—
72	C₂²⋊Q₈×C₄₇	—	2	1	no	—
73	C₄×Q₈×C₄₇	—	2	1	no	—
74	C₄×Q₈×C₄₇	—	2	1	no	—
75	#117×C₄₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	#106×C₄₇	—	2	4	no	—
80	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
81	#103×C₄₇	—	2	4	no	—
82	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
83	C₄×C₈×C₄₇	C₂³	1	2	yes	—
84	C₂³×D₄₇	—	2	1	no	—
85	C₂×Q₈×C₄₇	—	2	1	no	—
86	C₂×Q₈×C₄₇	—	2	2	no	—
87	C₂×Q₈×C₄₇	—	2	1	no	—
88	C₂×Q₈×C₄₇	—	4	1	no	—
89	C₂×Q₈×C₄₇	—	4	2	no	—
90	C₂×Q₈×C₄₇	—	4	2	no	—
91	C₂²×C₄×C₄₇	—	4	1	no	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
95	C₂²×Dic₄₇	—	2	1	no	—
96	C₂×D₁₈₈	—	2	1	no	—
97	C₂×D₁₈₈	—	2	2	no	—
98	C₂×D₁₈₈	SD₁₆	1	4	yes	—
99	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
100	C₂×C₄×D₄₇	—	2	2	no	—
101	C₂×C₄×D₄₇	—	2	2	no	—
102	C₂×C₄×D₄₇	—	8	2	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	1	no	—
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	—	2	1	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	1	no	—
110	C₄⋊C₄×C₄₇	—	4	1	no	—
111	C₄⋊C₄×C₄₇	—	4	1	no	—
112	C₄⋊C₄×C₄₇	—	4	1	no	—
113	C₄⋊C₄×C₄₇	—	4	1	no	—
114	C₂²⋊C₄×C₄₇	—	4	1	no	—
115	C₄²×C₄₇	—	4	1	no	—
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₇	#41	1	1	yes	characteristic
119	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
120	#10625336	—	4	8	no	—
121	(C₂×D₄₇)⋊C₄	—	2	2	no	—
122	(C₂×D₄₇)⋊C₄	—	2	2	no	—
123	C₄⋊Dic₄₇	—	2	1	no	—
124	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
125	#10625329	—	4	2	no	—
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	—	4	2	no	—
130	C₈×D₄₇	—	8	4	no	—
131	C₂³×C₄₇	#121	1	1	yes	characteristic
132	C₂²×D₄₇	—	2	2	no	—
133	C₂²×D₄₇	—	2	4	no	—
134	C₂²×D₄₇	—	8	1	no	—
135	C₂²×D₄₇	—	8	1	no	—
136	Q₈×C₄₇	—	4	2	no	—
137	Q₈×C₄₇	—	4	2	no	—
138	Q₈×C₄₇	—	4	2	no	—
139	Q₈×C₄₇	—	8	1	no	—
140	Q₈×C₄₇	—	8	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	4	1	no	—
143	C₂×C₄×C₄₇	—	4	1	no	—
144	C₂×C₄×C₄₇	—	4	1	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	8	2	no	—
147	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₄₇	#135	1	2	yes	—
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₄₇	—	2	2	no	—
156	C₂×Dic₄₇	—	8	1	no	—
157	D₁₈₈	—	2	2	no	—
158	D₁₈₈	—	2	4	no	—
159	D₁₈₈	C₂×SD₁₆	1	4	yes	—
160	C₄×D₄₇	—	4	4	no	—
161	C₄×D₄₇	—	8	2	no	—
162	C₈×C₄₇	—	2	4	no	—
163	C₄₇⋊C₈	—	4	2	no	—
164	C₄₇⋊C₈	—	4	2	no	—
165	C₄₇⋊C₈	—	4	2	no	—
166	#19390	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	—	2	1	no	—
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	#465	1	2	yes	—
170	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
171	C₂×D₄₇	—	2	4	no	—
172	C₂×D₄₇	—	4	2	no	—
173	C₂×D₄₇	—	4	2	no	—
174	C₂×D₄₇	—	8	4	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	8	1	no	—
179	C₄×C₄₇	—	8	2	no	—
180	C₄×C₄₇	—	8	1	no	—
181	C₄×C₄₇	#465	1	2	yes	—
182	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
184	C₄×C₄₇	#577	1	2	yes	—
185	Dic₄₇	—	4	2	no	—
186	#3229	D₄₇	1	1	yes	characteristic, complemented
187	#3077	—	47	1	no	—
188	C₂×#461	—	47	1	no	—
189	#2972	—	47	1	no	—
190	#1679	—	47	4	no	—
191	#1671	—	47	4	no	—
192	#1570	—	47	1	no	—
193	#1546	—	47	1	no	—
194	#1542	—	47	1	no	—
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	#3042	1	1	yes	characteristic
197	C₂×C₄₇	C₂×#465	1	1	yes	characteristic
198	C₂×C₄₇	#3069	1	1	yes	characteristic
199	D₄₇	—	4	4	no	—
200	C₂²×SD₁₆	—	94	1	no	—
201	#533	C₂×D₄₇	1	1	yes	characteristic
202	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
203	C₂×C₄×D₄	—	47	1	no	—
204	#498	C₂×D₄₇	1	2	yes	complemented
205	#487	C₂×D₄₇	1	2	yes	complemented
206	#483	—	2	2	no	—
207	#478	—	94	2	no	—
208	#476	—	47	2	no	—
209	#475	—	2	2	no	—
210	#474	—	47	2	no	—
211	#461	—	47	4	no	—
212	#461	—	94	2	no	—
213	C₈×D₄	—	47	4	no	—
214	C₈×D₄	—	94	2	no	—
215	#431	—	47	1	no	—
216	#431	C₂×D₄₇	1	1	yes	characteristic
217	#428	—	94	1	no	—
218	#423	—	47	1	no	—
219	C₂×C₄⋊C₈	—	47	1	no	—
220	#418	—	94	1	no	—
221	C₂×#103	—	94	1	no	—
222	C₂×C₂²⋊C₈	—	94	2	no	—
223	#334	—	47	4	no	—
224	#333	—	47	2	no	—
225	#332	—	47	2	no	—
226	#332	—	47	4	no	—
227	C₄₇	#19390	1	1	yes	characteristic, complemented, Sylow
228	C₂²×Q₈	—	2	1	no	—
229	C₂²×D₄	—	47	1	no	—
230	C₂³×C₄	—	94	1	no	—
231	C₂×SD₁₆	—	94	4	no	—
232	C₂×SD₁₆	—	188	4	no	—
233	C₂×M₁₆	—	94	1	no	—
234	C₂²×C₈	—	94	1	no	—
235	C₂²×C₈	—	94	4	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
240	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
242	#125	—	2	1	no	—
243	C₂²⋊Q₈	—	2	1	no	—
244	C₂²⋊Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	—	2	1	no	—
247	C₄×D₄	—	47	2	no	—
248	C₄×D₄	—	47	2	no	—
249	C₄×D₄	—	94	1	no	—
250	C₄×D₄	—	94	1	no	—
251	#117	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	47	1	no	—
255	C₂×C₂²⋊C₄	—	94	1	no	—
256	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
257	#107	—	94	2	no	—
258	#106	—	2	4	no	—
259	#106	—	94	2	no	—
260	C₄⋊C₈	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	47	2	no	—
263	C₄⋊C₈	—	94	1	no	—
264	C₄⋊C₈	—	94	1	no	—
265	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
266	#103	—	2	4	no	—
267	#103	—	94	4	no	—
268	#102	—	94	2	no	—
269	#102	—	94	2	no	—
270	C₂²⋊C₈	—	94	1	no	—
271	C₂²⋊C₈	—	94	4	no	—
272	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
273	C₄×C₈	—	94	1	no	—
274	C₄×C₈	C₂²×D₄₇	1	2	yes	—
275	C₂⁴	—	94	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×D₄	—	47	4	no	—
283	C₂×D₄	—	47	2	no	—
284	C₂×D₄	—	94	1	no	—
285	C₂×D₄	—	94	2	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	94	1	no	—
288	C₂²×C₄	—	94	2	no	—
289	C₂²×C₄	—	94	2	no	—
290	C₂²×C₄	—	376	2	no	—
291	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
292	C₂²×C₄	#10625352	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	SD₁₆	—	188	8	no	—
295	M₁₆	—	188	2	no	—
296	C₂×C₈	—	94	2	no	—
297	C₂×C₈	—	94	2	no	—
298	C₂×C₈	—	94	2	no	—
299	C₂×C₈	—	376	4	no	—
300	C₂×C₈	—	188	2	no	—
301	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	47	2	no	—
313	C₄⋊C₄	—	94	1	no	—
314	C₂²⋊C₄	—	4	1	no	—
315	C₂²⋊C₄	—	94	2	no	—
316	C₂²⋊C₄	—	94	2	no	—
317	C₄²	—	4	1	no	—
318	C₄²	D₄×D₄₇	1	1	yes	characteristic
319	C₄²	#10625352	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₄²	C₂³×D₄₇	1	1	yes	characteristic
322	C₂³	—	94	4	no	—
323	C₂³	—	94	2	no	—
324	C₂³	—	376	1	no	—
325	C₂³	—	376	1	no	—
326	C₂³	#13093995	1	1	yes	characteristic
327	Q₈	—	4	2	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	Q₈	—	8	1	no	—
331	Q₈	—	8	2	no	—
332	D₄	—	47	4	no	—
333	D₄	—	94	4	no	—
334	D₄	—	94	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	8	2	no	—
341	C₂×C₄	—	94	2	no	—
342	C₂×C₄	—	376	2	no	—
343	C₂×C₄	—	376	1	no	—
344	C₂×C₄	—	188	4	no	—
345	C₂×C₄	#13093995	1	1	yes	characteristic
346	C₂×C₄	#13093995	1	1	yes	characteristic
347	C₂×C₄	#13093995	1	1	yes	characteristic
348	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
349	C₂×C₄	#13094045	1	2	yes	—
350	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
351	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
352	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	188	2	no	—
355	C₈	—	188	2	no	—
356	C₈	—	188	2	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	94	4	no	—
360	C₂²	—	376	4	no	—
361	C₂²	—	188	2	no	—
362	C₂²	—	188	2	no	—
363	C₂²	#15689349	1	2	yes	—
364	C₂²	C₂×#13093995	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	1	no	—
370	C₄	—	8	1	no	—
371	C₄	—	188	2	no	—
372	C₄	#15689349	1	2	yes	—
373	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
374	C₄	C₂×#13094045	1	1	yes	characteristic
375	C₄	#15689931	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	188	4	no	—
378	C₂	#20252896	1	1	yes	characteristic
379	C₂	C₂×#15689349	1	1	yes	characteristic
380	C₂	#20253121	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3229	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#3229	D₄₇
hypocenter	C₄₇	#19390
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3229

Complements

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

N	H ≅ G/N	type
4. #20242808	C₂	semidirect
5. #20242760	C₂	semidirect
6. #20242159	C₂	semidirect
7. #20242060	C₂	semidirect
8. #20242045	C₂	semidirect
9. C₄₇×#3229	C₂	semidirect
26. #15688874	C₂²	semidirect
27. #15688872	C₂²	semidirect
28. #15688869	C₂²	semidirect
29. #15688869	C₂²	semidirect
32. C₄₇×#498	C₂²	semidirect
33. C₄₇×#487	C₂²	semidirect
186. #3229	D₄₇	semidirect
204. #498	C₂×D₄₇	semidirect
205. #487	C₂×D₄₇	semidirect
227. C₄₇	#19390	semidirect

Maximal subgroups

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

subgroup	index
#20253181	2
C₂×#15689325	2
#20252411	2
#20242808	2
#20242760	2
#20242159	2
#20242060	2
#20242045	2
C₄₇×#3229	2
#19390	47