Subgroups — #26186172

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253428	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689247	C₂	1	1	yes	maximal, characteristic
4	#20242237	C₂	1	4	yes	maximal, complemented
5	#20242194	C₂	1	4	yes	maximal, complemented
6	#20241816	C₂	1	1	yes	maximal, characteristic
7	#20241790	C₂	1	1	yes	maximal, characteristic
8	#20241672	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1571	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
12	#15689459	C₄	1	2	yes	—
13	#15689419	C₄	1	2	yes	—
14	#15689334	C₄	1	2	yes	—
15	#15689281	C₄	1	2	yes	—
16	#15689250	—	2	1	no	—
17	#15689250	C₂²	1	2	yes	—
18	#15689247	—	2	1	no	—
19	#15689247	—	2	1	no	—
20	#15689247	—	2	1	no	—
21	#15689247	C₂²	1	2	yes	—
22	#15689247	C₂²	1	2	yes	—
23	#15689247	C₂²	1	2	yes	—
24	C₄×Dic₁₈₈	C₂²	1	4	yes	—
25	#15689176	C₂²	1	1	yes	characteristic
26	#15689151	—	2	1	no	—
27	C₂×Dic₄₇⋊C₈	—	2	1	no	—
28	#15689081	—	2	1	no	—
29	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	—	2	1	no	—
31	#15688853	C₂²	1	4	yes	—
32	#15688853	C₂²	1	4	yes	—
33	#15688845	C₂²	1	2	yes	—
34	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
35	C₄₇×#431	C₂²	1	1	yes	characteristic
36	C₄₇×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#335	C₂²	1	4	yes	—
39	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
40	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
41	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
43	#13094024	C₂×C₄	1	1	yes	characteristic
44	#13094024	C₂×C₄	1	1	yes	characteristic
45	#13094016	—	2	1	no	—
46	#13094003	—	2	1	no	—
47	#13094003	C₂×C₄	1	2	yes	—
48	C₄×Dic₉₄	—	2	1	no	—
49	C₄×Dic₉₄	—	2	1	no	—
50	C₄×Dic₉₄	—	2	1	no	—
51	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
52	C₄×Dic₉₄	C₂³	1	2	yes	—
53	C₄×Dic₉₄	C₂³	1	2	yes	—
54	#13093983	—	2	1	no	—
55	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×Dic₄₇⋊C₄	—	2	1	no	—
57	C₂×C₄×Dic₄₇	—	2	1	no	—
58	#13093958	C₂×C₄	1	4	yes	—
59	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
60	Dic₄₇⋊C₈	—	2	2	no	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	Dic₄₇⋊C₈	—	2	2	no	—
63	Dic₄₇⋊C₈	—	2	2	no	—
64	#13093949	C₂×C₄	1	4	yes	—
65	#13093949	C₂×C₄	1	4	yes	—
66	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
67	#13093934	—	2	2	no	—
68	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
69	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
70	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
71	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×Dic₄₇	—	2	2	no	—
74	C₈×Dic₄₇	—	2	2	no	—
75	C₈×Dic₄₇	—	2	2	no	—
76	C₂×M₁₆×C₄₇	—	2	1	no	—
77	C₂²×C₈×C₄₇	—	2	1	no	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₇	—	2	1	no	—
80	C₄⋊C₈×C₄₇	—	2	1	no	—
81	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
82	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
83	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₄₇	C₂³	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
89	C₂²×Dic₄₇	—	2	1	no	—
90	C₂²×Dic₄₇	—	2	1	no	—
91	C₂×Dic₉₄	—	2	1	no	—
92	C₂×Dic₉₄	—	2	4	no	—
93	C₂×Dic₉₄	—	2	1	no	—
94	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
95	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
96	M₁₆×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	4	2	no	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	#41	1	1	yes	characteristic
104	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
105	C₂²⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	1	no	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	2	1	no	—
109	C₄⋊Dic₄₇	—	2	1	no	—
110	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
112	Dic₄₇⋊C₄	—	2	2	no	—
113	Dic₄₇⋊C₄	—	2	2	no	—
114	Dic₄₇⋊C₄	—	2	2	no	—
115	Dic₄₇⋊C₄	—	2	2	no	—
116	C₄×Dic₄₇	—	2	2	no	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₄×Dic₄₇	—	2	2	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	Q₁₆	1	4	yes	complemented
122	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
126	Dic₁₈₈	—	2	8	no	—
127	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₄₇	#135	1	2	yes	—
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	Dic₉₄	—	2	4	no	—
145	Dic₉₄	—	2	2	no	—
146	Dic₉₄	—	2	4	no	—
147	Dic₉₄	—	4	2	no	—
148	Dic₉₄	#131	1	4	yes	—
149	C₈×C₄₇	—	2	4	no	—
150	C₈×C₄₇	—	4	2	no	—
151	C₈×C₄₇	—	4	2	no	—
152	C₄₇⋊C₈	—	2	1	no	—
153	C₄₇⋊C₈	—	2	1	no	—
154	C₄₇⋊C₈	—	4	2	no	—
155	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
156	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
157	#17903	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
161	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
166	C₄×C₄₇	C₂×#131	1	2	yes	—
167	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
168	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
169	Dic₄₇	—	2	4	no	—
170	Dic₄₇	—	4	2	no	—
171	Dic₄₇	—	4	2	no	—
172	Dic₄₇	—	4	2	no	—
173	#3104	—	47	1	no	—
174	#2960	—	47	1	no	—
175	C₂×C₈×Q₈	—	47	1	no	—
176	#1584	—	47	4	no	—
177	C₈×Q₁₆	—	47	4	no	—
178	#1571	D₄₇	1	1	yes	characteristic
179	#1487	—	47	1	no	—
180	#1481	—	47	1	no	—
181	#1450	—	47	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	#2929	1	1	yes	characteristic
184	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
185	C₂×C₄₇	#2955	1	1	yes	characteristic
186	#533	—	47	1	no	—
187	C₂×C₄×Q₈	—	47	1	no	—
188	#484	—	47	2	no	—
189	#477	—	47	2	no	—
190	#462	—	47	2	no	—
191	#451	—	47	2	no	—
192	#446	—	47	2	no	—
193	#446	—	94	1	no	—
194	C₈×Q₈	—	47	2	no	—
195	C₈×Q₈	—	47	2	no	—
196	C₈×Q₈	—	47	2	no	—
197	C₈×Q₈	—	94	1	no	—
198	C₈×Q₈	—	94	1	no	—
199	C₈×Q₈	—	94	1	no	—
200	C₄×Q₁₆	—	47	4	no	—
201	#431	—	47	1	no	—
202	#431	C₂×D₄₇	1	1	yes	characteristic
203	#424	—	94	1	no	—
204	#423	C₂×D₄₇	1	1	yes	characteristic
205	C₂×C₄⋊C₈	—	94	1	no	—
206	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
207	#405	—	94	1	no	—
208	C₂×C₄×C₈	—	47	1	no	—
209	C₂×C₄×C₈	—	94	1	no	—
210	#335	C₂×D₄₇	1	4	yes	—
211	Q₈⋊C₈	—	47	4	no	—
212	Q₈⋊C₈	—	47	4	no	—
213	#322	—	47	2	no	—
214	C₈²	—	47	2	no	—
215	C₄₇	#17903	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	47	1	no	—
217	C₂×Q₁₆	—	47	4	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	—	47	1	no	—
221	C₂²×C₈	—	47	1	no	—
222	C₄⋊Q₈	—	47	1	no	—
223	C₄⋊Q₈	—	47	1	no	—
224	#125	—	94	1	no	—
225	C₂²⋊Q₈	—	47	2	no	—
226	C₂²⋊Q₈	—	94	1	no	—
227	C₄×Q₈	—	47	2	no	—
228	C₄×Q₈	—	47	2	no	—
229	C₄×Q₈	—	47	2	no	—
230	C₄×Q₈	—	94	1	no	—
231	C₄×Q₈	—	94	1	no	—
232	C₄×Q₈	—	94	1	no	—
233	#117	—	94	1	no	—
234	C₂×C₄⋊C₄	—	47	1	no	—
235	C₂×C₄⋊C₄	—	94	1	no	—
236	C₂×C₄²	—	94	1	no	—
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	#107	—	47	4	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	94	2	no	—
242	C₄⋊C₈	—	94	2	no	—
243	C₄⋊C₈	—	94	2	no	—
244	C₄⋊C₈	—	94	2	no	—
245	C₄⋊C₈	—	94	1	no	—
246	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
247	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
248	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
249	#103	—	47	4	no	—
250	#103	—	47	4	no	—
251	C₂²⋊C₈	—	94	1	no	—
252	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
253	#97	—	94	2	no	—
254	C₄×C₈	—	47	1	no	—
255	C₄×C₈	—	47	1	no	—
256	C₄×C₈	—	47	1	no	—
257	C₄×C₈	—	47	1	no	—
258	C₄×C₈	—	47	2	no	—
259	C₄×C₈	—	94	2	no	—
260	C₄×C₈	—	94	2	no	—
261	C₄×C₈	—	94	2	no	—
262	C₄×C₈	C₂²×D₄₇	1	2	yes	—
263	C₂×Q₈	—	47	2	no	—
264	C₂×Q₈	—	47	2	no	—
265	C₂×Q₈	—	94	4	no	—
266	C₂×Q₈	—	94	1	no	—
267	C₂×Q₈	—	94	1	no	—
268	C₂²×C₄	—	94	1	no	—
269	C₂²×C₄	—	94	1	no	—
270	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	#10625354	1	1	yes	characteristic
273	Q₁₆	—	94	8	no	—
274	M₁₆	—	4	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	47	4	no	—
279	C₂×C₈	—	47	1	no	—
280	C₂×C₈	—	47	1	no	—
281	C₂×C₈	—	47	1	no	—
282	C₂×C₈	—	47	1	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
286	C₄⋊C₄	—	47	2	no	—
287	C₄⋊C₄	—	47	2	no	—
288	C₄⋊C₄	—	94	2	no	—
289	C₄⋊C₄	—	94	2	no	—
290	C₄⋊C₄	—	94	1	no	—
291	C₄⋊C₄	—	94	1	no	—
292	C₄⋊C₄	—	94	1	no	—
293	C₄⋊C₄	—	94	1	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₂²⋊C₄	—	94	2	no	—
297	C₄²	—	94	2	no	—
298	C₄²	—	94	2	no	—
299	C₄²	—	94	2	no	—
300	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
301	C₄²	D₄×D₄₇	1	1	yes	characteristic
302	C₄²	#10625354	1	1	yes	characteristic
303	C₄²	C₂³×D₄₇	1	1	yes	characteristic
304	C₂³	#13093986	1	1	yes	characteristic
305	Q₈	—	47	4	no	—
306	Q₈	—	94	4	no	—
307	Q₈	—	94	2	no	—
308	Q₈	—	94	4	no	—
309	Q₈	—	188	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	94	4	no	—
313	C₂×C₄	—	94	2	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	#13093986	1	1	yes	characteristic
319	C₂×C₄	#13093986	1	1	yes	characteristic
320	C₂×C₄	#13093986	1	1	yes	characteristic
321	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
322	C₂×C₄	#13094043	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
326	C₈	—	2	4	no	—
327	C₈	—	4	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	47	2	no	—
330	C₈	—	47	2	no	—
331	C₈	—	94	1	no	—
332	C₈	—	94	1	no	—
333	C₈	—	188	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	#15689217	1	2	yes	—
337	C₂²	C₂×#13093986	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	94	4	no	—
342	C₄	—	188	2	no	—
343	C₄	—	188	2	no	—
344	C₄	—	188	2	no	—
345	C₄	#15689217	1	2	yes	—
346	C₄	#131×D₄₇	1	2	yes	—
347	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
348	C₄	C₂×#13094043	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	#20252081	1	1	yes	characteristic
351	C₂	C₂×#15689217	1	1	yes	characteristic
352	C₂	#20252290	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×#1571	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1571	D₄₇
hypocenter	C₄₇	#17903
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1571

Complements

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

N	H ≅ G/N	type
4. #20242237	C₂	semidirect
5. #20242194	C₂	semidirect
121. C₂×C₄₇⋊C₈	Q₁₆	semidirect
215. C₄₇	#17903	semidirect

Maximal subgroups

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

subgroup	index
#20253428	2
#20252318	2
C₂×#15689247	2
#20242237	2
#20242194	2
#20241816	2
#20241790	2
#20241672	2
C₄₇×#1571	2
#17903	47