Subgroups — #26106914

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245849	C₂	1	1	yes	maximal, characteristic
2	#20243243	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688852	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688850	C₂	1	1	yes	maximal, characteristic
6	#20241229	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×#13093981	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×#13093934	—	2	1	no	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	—	2	2	no	—
14	#15689024	C₄	1	2	yes	—
15	#15689019	C₄	1	2	yes	—
16	#15689012	—	2	2	no	—
17	C₄×(C₂×D₄₇)⋊C₄	—	2	1	no	—
18	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688877	—	2	1	no	—
20	#15688852	C₄	1	4	yes	complemented
21	#15688850	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₄×D₁₈₈	—	2	1	no	—
30	C₄×D₁₈₈	—	2	1	no	—
31	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
32	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
33	#13093981	—	4	2	no	—
34	#13093981	—	4	2	no	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₄₇⋊C₄	—	2	2	no	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
38	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
39	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
46	#13093934	—	4	2	no	—
47	C₄×C₄₇⋊C₈	—	2	1	no	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₈×Dic₄₇	—	2	8	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₂²×C₈×C₄₇	—	2	1	no	—
52	C₂²×C₈×C₄₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	—	2	1	no	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	—	2	1	no	—
57	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₇	—	2	1	no	—
59	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
60	#95×C₄₇	—	2	1	no	—
61	C₂³×D₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	1	no	—
67	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×D₁₈₈	—	2	2	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×D₁₈₈	—	2	1	no	—
72	C₂×D₁₈₈	C₄²	1	2	yes	—
73	C₂×C₄×D₄₇	—	2	2	no	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	4	1	no	—
76	C₂×C₄×D₄₇	—	4	1	no	—
77	C₂×C₄×D₄₇	—	4	1	no	—
78	C₂×C₄×D₄₇	—	4	1	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₇	#41	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	2	2	no	—
93	(C₂×D₄₇)⋊C₄	—	2	2	no	—
94	(C₂×D₄₇)⋊C₄	—	4	4	no	—
95	(C₂×D₄₇)⋊C₄	—	4	4	no	—
96	C₄⋊Dic₄₇	—	2	1	no	—
97	C₄⋊Dic₄₇	C₄²	1	2	yes	—
98	Dic₄₇⋊C₄	—	4	4	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	2	1	no	—
102	C₄×Dic₄₇	—	2	1	no	—
103	C₄×Dic₄₇	—	2	4	no	—
104	C₄×Dic₄₇	D₈	1	2	yes	—
105	C₄×Dic₄₇	SD₁₆	1	2	yes	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	C₂×C₄₇⋊C₈	—	4	1	no	—
111	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×D₄₇	—	2	2	no	—
113	C₂²×D₄₇	—	4	2	no	—
114	C₂²×D₄₇	—	4	1	no	—
115	C₂²×D₄₇	—	4	1	no	—
116	C₂²×D₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#131	1	2	yes	—
128	C₂×C₄×C₄₇	#131	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	#102	1	4	yes	—
141	C₂×Dic₄₇	#104	1	4	yes	—
142	D₁₈₈	—	2	2	no	—
143	D₁₈₈	—	2	2	no	—
144	D₁₈₈	—	4	2	no	—
145	C₄×D₄₇	—	4	2	no	—
146	C₄×D₄₇	—	4	2	no	—
147	C₄×D₄₇	—	4	2	no	—
148	C₄×D₄₇	—	4	2	no	—
149	C₈×C₄₇	—	2	4	no	—
150	C₈×C₄₇	—	4	2	no	—
151	C₄₇⋊C₈	—	2	4	no	—
152	C₄₇⋊C₈	—	4	2	no	—
153	#8177	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄₇	#408	1	2	yes	—
156	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
157	C₂²×C₄₇	#416	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#104	1	2	yes	—
159	C₂×D₄₇	—	4	2	no	—
160	C₂×D₄₇	—	4	2	no	—
161	C₂×D₄₇	—	4	2	no	—
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	—	8	2	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	8	2	no	—
167	C₄×C₄₇	#405	1	1	yes	characteristic
168	C₄×C₄₇	#405	1	1	yes	characteristic
169	C₄×C₄₇	#433	1	2	yes	—
170	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
171	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
172	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
173	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
174	Dic₄₇	—	2	4	no	—
175	Dic₄₇	—	2	4	no	—
176	Dic₄₇	—	4	2	no	—
177	#2293	—	47	1	no	—
178	#1767	—	47	1	no	—
179	C₄²×C₈	—	47	1	no	—
180	C₂×D₄⋊C₈	—	47	1	no	—
181	C₂×D₄⋊C₈	—	47	1	no	—
182	#1279	—	47	1	no	—
183	#1279	D₄₇	1	1	yes	characteristic, complemented
184	C₂×C₄₇	#1492	1	2	yes	—
185	C₂×C₄₇	#1494	1	2	yes	—
186	C₂×C₄₇	#1750	1	1	yes	characteristic
187	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
188	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
189	D₄₇	—	8	2	no	—
190	C₂×C₄×D₄	—	47	1	no	—
191	C₂×#117	—	94	1	no	—
192	C₂×C₄⋊C₈	—	47	1	no	—
193	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
194	C₂×#97	—	94	1	no	—
195	C₂×C₄×C₈	—	47	1	no	—
196	C₂×C₄×C₈	—	94	2	no	—
197	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
198	#390	—	47	2	no	—
199	#389	—	47	2	no	—
200	#388	—	94	2	no	—
201	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
202	C₄×C₂²⋊C₄	—	94	1	no	—
203	C₄³	—	47	1	no	—
204	#336	—	94	1	no	—
205	D₄⋊C₈	—	47	4	no	—
206	D₄⋊C₈	—	47	4	no	—
207	C₄₇	#8177	1	1	yes	characteristic, complemented, Sylow
208	C₂²×D₄	—	47	1	no	—
209	C₂³×C₄	—	94	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	94	1	no	—
213	C₂²×C₈	—	94	1	no	—
214	C₄×D₄	—	47	2	no	—
215	C₄×D₄	—	47	2	no	—
216	C₄×D₄	—	94	1	no	—
217	C₄×D₄	—	94	1	no	—
218	#117	—	188	2	no	—
219	#117	—	188	2	no	—
220	C₂×C₄⋊C₄	—	47	1	no	—
221	C₂×C₄⋊C₄	—	94	2	no	—
222	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
223	C₂×C₂²⋊C₄	—	94	1	no	—
224	C₂×C₂²⋊C₄	—	94	2	no	—
225	C₂×C₂²⋊C₄	—	94	2	no	—
226	C₂×C₄²	—	2	1	no	—
227	C₂×C₄²	—	47	1	no	—
228	C₂×C₄²	—	47	2	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	47	2	no	—
234	C₄⋊C₈	—	94	1	no	—
235	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
236	#97	—	188	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	47	2	no	—
239	C₄×C₈	—	94	8	no	—
240	C₄×C₈	—	94	1	no	—
241	C₄×C₈	C₄×D₄₇	1	2	yes	—
242	#95	—	2	1	no	—
243	#95	—	94	2	no	—
244	C₂⁴	—	94	1	no	—
245	C₂×D₄	—	47	2	no	—
246	C₂×D₄	—	94	1	no	—
247	C₂×D₄	—	94	2	no	—
248	C₂×D₄	—	94	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	47	2	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	94	1	no	—
253	C₂²×C₄	—	94	2	no	—
254	C₂²×C₄	—	94	2	no	—
255	C₂²×C₄	—	188	1	no	—
256	C₂²×C₄	—	188	1	no	—
257	C₂²×C₄	—	188	1	no	—
258	C₂²×C₄	—	188	1	no	—
259	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
261	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	94	4	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	—	188	1	no	—
271	C₂×C₈	—	188	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	47	2	no	—
275	C₄⋊C₄	—	94	1	no	—
276	C₄⋊C₄	—	188	4	no	—
277	C₂²⋊C₄	—	94	2	no	—
278	C₂²⋊C₄	—	94	2	no	—
279	C₂²⋊C₄	—	188	4	no	—
280	C₂²⋊C₄	—	188	4	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	47	2	no	—
284	C₄²	—	47	2	no	—
285	C₄²	—	94	4	no	—
286	C₄²	—	94	2	no	—
287	C₄²	—	94	2	no	—
288	C₄²	—	94	1	no	—
289	C₄²	—	94	1	no	—
290	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₄²	#10625352	1	1	yes	characteristic
293	C₄²	#10625354	1	1	yes	characteristic
294	C₂³	—	94	2	no	—
295	C₂³	—	188	2	no	—
296	C₂³	—	188	1	no	—
297	C₂³	—	188	1	no	—
298	C₂³	—	188	2	no	—
299	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
300	D₄	—	94	2	no	—
301	D₄	—	94	2	no	—
302	D₄	—	188	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	47	4	no	—
308	C₂×C₄	—	47	4	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	2	no	—
314	C₂×C₄	—	188	1	no	—
315	C₂×C₄	—	188	1	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
321	C₂×C₄	#13093982	1	1	yes	characteristic
322	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
323	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
324	C₂×C₄	#13093986	1	1	yes	characteristic
325	C₂×C₄	#13093986	1	1	yes	characteristic
326	C₂×C₄	#13094030	1	2	yes	—
327	C₂×C₄	#13094032	1	2	yes	—
328	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
330	C₂×C₄	#13094045	1	1	yes	characteristic
331	C₂×C₄	#13094047	1	1	yes	characteristic
332	C₈	—	2	4	no	—
333	C₈	—	4	2	no	—
334	C₈	—	94	4	no	—
335	C₈	—	188	2	no	—
336	C₂²	—	376	2	no	—
337	C₂²	—	376	2	no	—
338	C₂²	—	188	2	no	—
339	C₂²	—	188	2	no	—
340	C₂²	—	188	2	no	—
341	C₂²	#15688970	1	1	yes	characteristic
342	C₂²	#15689088	1	2	yes	—
343	C₂²	#102×D₄₇	1	1	yes	characteristic
344	C₂²	#15689118	1	1	yes	characteristic
345	C₂²	#104×D₄₇	1	2	yes	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	94	4	no	—
350	C₄	—	94	4	no	—
351	C₄	—	188	2	no	—
352	C₄	#15689077	1	1	yes	characteristic
353	C₄	#15689080	1	1	yes	characteristic
354	C₄	#15689183	1	2	yes	—
355	C₄	#15689203	1	1	yes	characteristic
356	C₄	#15689205	1	1	yes	characteristic
357	C₄	#15689208	1	1	yes	characteristic
358	C₄	#15689210	1	1	yes	characteristic
359	C₂	—	376	2	no	—
360	C₂	#20241837	1	2	yes	—
361	C₂	#20241847	1	2	yes	—
362	C₂	#20243153	1	1	yes	characteristic
363	C₂	#20243192	1	1	yes	characteristic
364	C₂	#20243202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1279	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1279	D₄₇
hypocenter	C₄₇	#8177
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
2. #20243243	C₂	semidirect
3. C₄×C₈×Dic₄₇	C₂	semidirect
6. #20241229	C₂	semidirect
7. C₄₇×#1279	C₂	semidirect
20. #15688852	C₄	semidirect
21. #15688850	C₄	semidirect
183. #1279	D₄₇	semidirect
207. C₄₇	#8177	semidirect

Maximal subgroups

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

subgroup	index
#20245849	2
#20243243	2
C₄×C₈×Dic₄₇	2
C₂×#15688852	2
C₂×#15688850	2
#20241229	2
C₄₇×#1279	2
#8177	47