Subgroups — #26186411

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253420	C₂	1	1	yes	maximal, characteristic
2	#20252468	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252463	C₂	1	1	yes	maximal, characteristic
4	#20242268	C₂	1	2	yes	maximal, complemented
5	#20242233	C₂	1	2	yes	maximal, complemented
6	#20242202	C₂	1	2	yes	maximal, complemented
7	C₈×#10625325	C₂	1	2	yes	maximal, complemented
8	#20242168	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241836	C₂	1	1	yes	maximal, characteristic, complemented
10	#20241806	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689673	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
14	#15689442	C₄	1	2	yes	—
15	#15689409	C₄	1	2	yes	—
16	#15689324	C₄	1	2	yes	—
17	#15689264	C₄	1	2	yes	—
18	C₈×Dic₉₄	C₂²	1	2	yes	—
19	C₈×Dic₉₄	C₂²	1	2	yes	—
20	#15689247	—	2	2	no	—
21	C₄×#10625325	C₂²	1	2	yes	—
22	C₄×#10625325	C₂²	1	2	yes	—
23	#15689202	C₂²	1	2	yes	—
24	#15689193	—	2	2	no	—
25	C₈×D₁₈₈	C₂²	1	2	yes	—
26	#15689180	C₂²	1	1	yes	characteristic
27	#15689176	C₂²	1	1	yes	characteristic
28	#15689173	—	2	2	no	—
29	#15689088	—	2	2	no	—
30	#15688876	C₂²	1	2	yes	—
31	#15688874	C₂²	1	2	yes	—
32	#15688855	C₂²	1	2	yes	—
33	#15688855	C₂²	1	2	yes	—
34	#15688852	C₂²	1	2	yes	—
35	#15688852	C₂²	1	2	yes	—
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₄₇×#431	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#322	C₂²	1	1	yes	characteristic
40	C₄₇×#322	C₂²	1	1	yes	characteristic
41	C₄₇×#322	C₂²	1	1	yes	characteristic
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625350	—	2	1	no	—
44	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
45	C₂×#10625325	C₂×C₄	1	2	yes	—
46	C₂×#10625325	C₂×C₄	1	2	yes	—
47	C₂×#10625324	—	2	2	no	—
48	C₂×C₈×D₄₇	—	2	2	no	—
49	#13094024	C₂×C₄	1	1	yes	characteristic
50	#13094021	C₂×C₄	1	1	yes	characteristic
51	#13094012	—	2	1	no	—
52	#13093998	—	2	1	no	—
53	#13093998	C₂×C₄	1	2	yes	—
54	C₄×Dic₉₄	—	2	1	no	—
55	C₄×Dic₉₄	—	2	1	no	—
56	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₄	C₂³	1	2	yes	—
58	C₄×D₁₈₈	—	2	1	no	—
59	C₄×D₁₈₈	C₂³	1	2	yes	—
60	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₄₇⋊C₄	—	2	1	no	—
62	C₂×C₄×Dic₄₇	—	2	1	no	—
63	#13093956	C₂×C₄	1	2	yes	—
64	#13093956	C₂×C₄	1	2	yes	—
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	#13093949	C₂×C₄	1	2	yes	—
70	#13093949	C₂×C₄	1	2	yes	—
71	#13093946	C₂×C₄	1	2	yes	—
72	#13093946	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
74	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
75	(C₂×D₄₇)⋊C₈	—	2	2	no	—
76	(C₂×D₄₇)⋊C₈	—	2	2	no	—
77	#13093935	—	2	1	no	—
78	C₄×C₄₇⋊C₈	—	2	1	no	—
79	C₈×Dic₄₇	—	2	2	no	—
80	C₈×Dic₄₇	—	2	2	no	—
81	C₂×M₁₆×C₄₇	—	2	1	no	—
82	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
85	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
87	#97×C₄₇	—	2	1	no	—
88	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
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	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₄₇	—	2	2	no	—
98	C₂²×Dic₄₇	—	2	1	no	—
99	C₂²×Dic₄₇	—	2	1	no	—
100	#10625350	—	8	2	no	—
101	C₂×D₁₈₈	—	2	1	no	—
102	C₂×D₁₈₈	—	2	1	no	—
103	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₄₇	—	2	2	no	—
105	C₂×Dic₉₄	—	2	1	no	—
106	C₂×Dic₉₄	—	2	4	no	—
107	C₂×Dic₉₄	—	2	1	no	—
108	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
109	M₁₆×C₄₇	—	4	2	no	—
110	C₂×C₈×C₄₇	—	2	1	no	—
111	C₂×C₈×C₄₇	—	2	1	no	—
112	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
116	C₄²×C₄₇	C₂²×C₄	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	(C₂×D₄₇)⋊C₄	—	2	2	no	—
121	(C₂×D₄₇)⋊C₄	—	2	2	no	—
122	C₄⋊Dic₄₇	—	2	1	no	—
123	C₄⋊Dic₄₇	—	2	1	no	—
124	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
125	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
126	Dic₄₇⋊C₄	—	2	2	no	—
127	Dic₄₇⋊C₄	—	2	2	no	—
128	C₄×Dic₄₇	—	2	2	no	—
129	C₄×Dic₄₇	—	2	2	no	—
130	C₂×C₄₇⋊C₈	—	2	2	no	—
131	C₂×C₄₇⋊C₈	—	2	2	no	—
132	#10625325	—	2	4	no	—
133	#10625325	—	2	4	no	—
134	#10625324	—	8	2	no	—
135	C₈×D₄₇	—	8	2	no	—
136	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	4	1	no	—
141	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₄₇	—	8	2	no	—
148	C₂×Dic₄₇	—	2	4	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	8	1	no	—
153	D₁₈₈	—	2	2	no	—
154	D₁₈₈	—	2	2	no	—
155	D₁₈₈	—	8	1	no	—
156	D₁₈₈	—	8	1	no	—
157	C₄×D₄₇	—	4	2	no	—
158	C₄×D₄₇	—	4	2	no	—
159	Dic₉₄	—	2	4	no	—
160	Dic₉₄	—	2	2	no	—
161	Dic₉₄	—	8	1	no	—
162	Dic₉₄	#131	1	4	yes	—
163	C₈×C₄₇	—	2	2	no	—
164	C₈×C₄₇	—	2	2	no	—
165	C₈×C₄₇	—	2	2	no	—
166	C₈×C₄₇	—	2	2	no	—
167	C₈×C₄₇	—	2	2	no	—
168	C₈×C₄₇	—	4	1	no	—
169	C₄₇⋊C₈	—	4	2	no	—
170	C₄₇⋊C₈	—	4	2	no	—
171	#17923	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	—	2	1	no	—
174	C₂²×C₄₇	—	2	1	no	—
175	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₂×D₄₇	—	4	2	no	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	C₂×#131	1	2	yes	—
184	C₄×C₄₇	#577	1	2	yes	—
185	Dic₄₇	—	2	4	no	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	—	4	2	no	—
188	#3103	—	47	1	no	—
189	#2982	—	47	1	no	—
190	#2981	—	47	1	no	—
191	#1588	—	47	2	no	—
192	#1583	—	47	2	no	—
193	SD₁₆⋊C₈	—	47	2	no	—
194	C₈×SD₁₆	—	47	2	no	—
195	#1572	—	47	1	no	—
196	#1491	—	47	1	no	—
197	#1484	—	47	1	no	—
198	#1453	D₄₇	1	1	yes	characteristic, complemented
199	C₂×C₄₇	—	4	1	no	—
200	C₂×C₄₇	#2929	1	1	yes	characteristic
201	C₂×C₄₇	#2949	1	1	yes	characteristic
202	C₂×C₄₇	#2956	1	1	yes	characteristic
203	D₄₇	—	8	2	no	—
204	#532	—	47	1	no	—
205	C₂×C₄×Q₈	—	47	1	no	—
206	#481	—	47	2	no	—
207	#475	—	47	2	no	—
208	#460	—	47	2	no	—
209	#448	—	47	2	no	—
210	C₈×Q₈	—	47	2	no	—
211	C₈×Q₈	—	47	2	no	—
212	C₈×Q₈	—	94	2	no	—
213	C₄×SD₁₆	—	47	2	no	—
214	C₄×SD₁₆	—	47	2	no	—
215	#436	—	47	2	no	—
216	#435	—	94	2	no	—
217	C₈×D₄	—	47	2	no	—
218	#432	—	47	1	no	—
219	#431	—	47	1	no	—
220	#431	—	94	2	no	—
221	#431	C₂×D₄₇	1	1	yes	characteristic
222	#431	C₂×D₄₇	1	1	yes	characteristic
223	#408	—	94	2	no	—
224	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
225	#335	—	47	2	no	—
226	#334	—	47	2	no	—
227	Q₈⋊C₈	—	47	2	no	—
228	Q₈⋊C₈	—	47	2	no	—
229	D₄⋊C₈	—	47	2	no	—
230	D₄⋊C₈	—	47	2	no	—
231	#322	C₂×D₄₇	1	1	yes	characteristic
232	#322	C₂×D₄₇	1	1	yes	characteristic
233	#322	C₂×D₄₇	1	1	yes	characteristic
234	C₈²	C₂×D₄₇	1	1	yes	characteristic
235	C₄₇	#17923	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	94	1	no	—
237	C₂²×Q₈	—	47	1	no	—
238	C₂×SD₁₆	—	47	2	no	—
239	C₂×SD₁₆	—	47	2	no	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	94	2	no	—
242	C₂²×C₈	—	94	2	no	—
243	C₄⋊Q₈	—	47	1	no	—
244	#127	—	47	1	no	—
245	#124	—	94	1	no	—
246	#121	—	47	2	no	—
247	#121	—	94	1	no	—
248	C₄×Q₈	—	47	2	no	—
249	C₄×Q₈	—	47	2	no	—
250	C₄×Q₈	—	94	1	no	—
251	C₄×Q₈	—	94	1	no	—
252	C₄×D₄	—	47	2	no	—
253	C₄×D₄	—	94	1	no	—
254	C₂×C₄⋊C₄	—	47	1	no	—
255	C₂×C₄⋊C₄	—	94	1	no	—
256	C₂×C₄²	—	94	1	no	—
257	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
258	#106	—	47	2	no	—
259	#106	—	47	2	no	—
260	C₄⋊C₈	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	94	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
265	C₄⋊C₈	D₁₈₈	1	2	yes	—
266	#103	—	47	2	no	—
267	#103	—	47	2	no	—
268	#102	—	47	2	no	—
269	#102	—	47	2	no	—
270	C₂²⋊C₈	—	94	1	no	—
271	C₂²⋊C₈	—	94	1	no	—
272	C₂²⋊C₈	—	94	2	no	—
273	C₂²⋊C₈	—	94	2	no	—
274	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
275	C₂²⋊C₈	D₁₈₈	1	2	yes	—
276	#97	—	2	1	no	—
277	#97	—	94	1	no	—
278	C₄×C₈	—	94	2	no	—
279	C₄×C₈	—	94	2	no	—
280	C₄×C₈	—	94	1	no	—
281	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	#41	—	376	2	no	—
288	C₂×Q₈	—	47	2	no	—
289	C₂×Q₈	—	94	4	no	—
290	C₂×Q₈	—	94	1	no	—
291	C₂×Q₈	—	94	1	no	—
292	C₂×D₄	—	47	2	no	—
293	C₂×D₄	—	94	2	no	—
294	C₂×D₄	—	94	1	no	—
295	C₂×D₄	—	94	1	no	—
296	C₂²×C₄	—	94	1	no	—
297	C₂²×C₄	—	94	1	no	—
298	C₂²×C₄	—	94	2	no	—
299	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
301	C₂²×C₄	#10625350	1	1	yes	characteristic
302	SD₁₆	—	94	4	no	—
303	SD₁₆	—	94	4	no	—
304	M₁₆	—	4	2	no	—
305	M₁₆	—	376	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	94	2	no	—
309	C₂×C₈	—	94	2	no	—
310	C₂×C₈	—	376	2	no	—
311	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
312	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
313	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
314	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
315	C₄⋊C₄	—	47	2	no	—
316	C₄⋊C₄	—	47	2	no	—
317	C₄⋊C₄	—	94	2	no	—
318	C₄⋊C₄	—	94	1	no	—
319	C₄⋊C₄	—	94	1	no	—
320	C₄⋊C₄	—	94	2	no	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₄²	—	94	2	no	—
324	C₄²	—	94	2	no	—
325	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
326	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
327	C₄²	#10625350	1	1	yes	characteristic
328	C₄²	C₂³×D₄₇	1	1	yes	characteristic
329	C₂³	—	94	2	no	—
330	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
331	Q₈	—	47	4	no	—
332	Q₈	—	94	4	no	—
333	Q₈	—	94	2	no	—
334	Q₈	—	376	1	no	—
335	D₄	—	94	2	no	—
336	D₄	—	94	2	no	—
337	D₄	—	376	1	no	—
338	D₄	—	376	1	no	—
339	D₄	—	376	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	94	4	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	376	1	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
351	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
352	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
355	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	4	1	no	—
362	C₈	—	188	2	no	—
363	C₈	—	188	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	188	2	no	—
368	C₂²	—	188	2	no	—
369	C₂²	—	188	2	no	—
370	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	94	4	no	—
376	C₄	—	188	2	no	—
377	C₄	—	188	2	no	—
378	C₄	#131×D₄₇	1	2	yes	—
379	C₄	#15689935	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	376	2	no	—
382	C₂	#20252084	1	1	yes	characteristic
383	C₂	#20252249	1	1	yes	characteristic
384	C₂	#20252299	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1453	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1453	D₄₇
hypocenter	C₄₇	#17923
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
2. #20252468	C₂	semidirect
4. #20242268	C₂	semidirect
5. #20242233	C₂	semidirect
6. #20242202	C₂	semidirect
7. C₈×#10625325	C₂	semidirect
8. #20242168	C₂	semidirect
9. #20241836	C₂	semidirect
11. C₄₇×#1453	C₂	semidirect
198. #1453	D₄₇	semidirect
235. C₄₇	#17923	semidirect

Maximal subgroups

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

subgroup	index
#20253420	2
#20252468	2
#20252463	2
#20242268	2
#20242233	2
#20242202	2
C₈×#10625325	2
#20242168	2
#20241836	2
#20241806	2
C₄₇×#1453	2
#17923	47