Subgroups — #26106159

Order 12032.

986 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245771	C₂	1	2	yes	maximal
2	#20241256	C₂	1	4	yes	maximal, complemented
3	C₂²×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	—	2	2	no	—
5	#15688986	C₄	1	4	yes	complemented
6	#15688983	—	2	2	no	—
7	#15688983	C₂²	1	4	yes	—
8	#15688979	—	2	2	no	—
9	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
10	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	complemented
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₄₇	C₄	1	2	yes	complemented
13	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
14	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	complemented
16	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
17	#13093983	—	4	8	no	—
18	C₂×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
22	C₂×C₄×Dic₄₇	—	2	4	no	—
23	C₄₇⋊#95	—	2	4	no	—
24	C₄₇⋊#95	C₂×C₄	1	8	yes	—
25	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	—	4	4	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	—	4	4	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
41	C₂²⋊C₈×C₄₇	—	2	4	no	—
42	C₂²⋊C₈×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	—	2	4	no	—
44	C₂⁴×C₄₇	—	4	2	no	—
45	C₂⁴×C₄₇	—	4	2	no	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	8	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	4	4	no	—
82	C₂×C₈×C₄₇	—	4	4	no	—
83	C₂×C₈×C₄₇	—	4	4	no	—
84	C₂²⋊Dic₄₇	—	4	8	no	—
85	C₂²⋊Dic₄₇	—	4	8	no	—
86	C₂²⋊Dic₄₇	—	4	8	no	—
87	C₄⋊Dic₄₇	—	4	8	no	—
88	C₄×Dic₄₇	—	4	8	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	1	no	—
99	C₂³×C₄₇	—	4	1	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	—	4	1	no	—
103	C₂³×C₄₇	—	4	16	no	—
104	C₂³×C₄₇	#95	1	2	yes	—
105	C₂³×C₄₇	#95	1	1	yes	characteristic
106	C₂³×C₄₇	#95	1	1	yes	characteristic
107	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₄₇	#135	1	4	yes	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	1	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	1	no	—
127	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#95	1	2	yes	—
129	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#99	1	4	yes	—
131	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₄₇	—	4	8	no	—
135	C₂×Dic₄₇	—	4	8	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₈×C₄₇	—	4	4	no	—
138	#7982	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	8	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	4	4	no	—
152	C₂²×C₄₇	—	4	8	no	—
153	C₂²×C₄₇	—	4	8	no	—
154	C₂²×C₄₇	—	4	4	no	—
155	C₂²×C₄₇	H(3,ℤ/4)	1	2	yes	—
156	C₂²×C₄₇	#342	1	2	yes	—
157	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
158	C₂²×C₄₇	#416	1	4	yes	—
159	C₂²×C₄₇	#427	1	2	yes	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	#342	1	2	yes	—
165	C₄×C₄₇	C₂×#99	1	2	yes	—
166	Dic₄₇	—	8	8	no	—
167	C₂²×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
168	#2280	—	47	2	no	—
169	#1289	—	47	4	no	—
170	C₂×C₄₇	—	2	4	no	—
171	C₂×C₄₇	—	2	2	no	—
172	C₂×C₄₇	—	2	2	no	—
173	C₂×C₄₇	—	2	2	no	—
174	C₂×C₄₇	—	4	2	no	—
175	C₂×C₄₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
176	C₂×C₄₇	#1735	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
178	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
179	C₂³×C₈	Dic₄₇	1	2	yes	complemented
180	C₂×#117	—	94	2	no	—
181	C₂×C₂²⋊C₈	—	2	2	no	—
182	C₂×C₂²⋊C₈	—	2	2	no	—
183	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	complemented
184	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
185	#381	—	47	4	no	—
186	#380	—	47	4	no	—
187	#380	—	94	2	no	—
188	#379	—	94	2	no	—
189	C₄×C₂²⋊C₄	—	47	4	no	—
190	C₄×C₂²⋊C₄	—	94	2	no	—
191	C₄₇	#7982	1	1	yes	characteristic, complemented, Sylow
192	C₂⁵	D₁₈₈	1	1	yes	characteristic
193	C₂³×C₄	—	4	4	no	—
194	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
195	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
196	C₂³×C₄	C₄×D₄₇	1	2	yes	—
197	C₂³×C₄	C₄×D₄₇	1	2	yes	—
198	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
199	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
202	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
203	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	4	4	no	—
207	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
208	#117	—	188	8	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₂²⋊C₄	—	94	4	no	—
211	C₂×C₂²⋊C₄	—	94	4	no	—
212	C₂×C₂²⋊C₄	—	94	4	no	—
213	C₂×C₄²	—	94	4	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	#95	—	47	8	no	—
217	#95	—	94	4	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	—	4	2	no	—
220	C₂⁴	—	4	2	no	—
221	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
222	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
223	C₂⁴	(C₂×D₄₇)⋊C₄	1	2	yes	—
224	C₂⁴	(C₂×D₄₇)⋊C₄	1	2	yes	—
225	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	94	8	no	—
237	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
240	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
241	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
242	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
243	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
244	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
247	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
248	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
249	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	4	no	—
257	C₂×C₈	—	4	4	no	—
258	C₂×C₈	—	4	4	no	—
259	C₄⋊C₄	—	188	8	no	—
260	C₂²⋊C₄	—	188	8	no	—
261	C₂²⋊C₄	—	188	8	no	—
262	C₂²⋊C₄	—	188	8	no	—
263	C₄²	—	188	8	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	16	no	—
279	C₂³	C₄₇⋊#95	1	2	yes	—
280	C₂³	C₄₇⋊#95	1	1	yes	characteristic
281	C₂³	C₄₇⋊#95	1	1	yes	characteristic
282	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
283	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
284	C₂³	#13094046	1	4	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	188	8	no	—
303	C₂×C₄	—	188	8	no	—
304	C₂×C₄	—	188	8	no	—
305	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
306	C₂×C₄	C₄₇⋊#95	1	2	yes	—
307	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
308	C₂×C₄	#13093939	1	4	yes	—
309	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₂²	—	2	8	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	8	no	—
327	C₂²	—	4	8	no	—
328	C₂²	—	4	4	no	—
329	C₂²	#15688880	1	2	yes	—
330	C₂²	#15688890	1	2	yes	—
331	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
332	C₂²	#15689120	1	4	yes	—
333	C₂²	#15689161	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	376	8	no	—
339	C₄	#15688890	1	2	yes	—
340	C₄	C₂×#13093939	1	2	yes	—
341	C₂	—	2	4	no	—
342	C₂	—	2	2	no	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	2	no	—
346	C₂	C₂×#15688880	1	1	yes	characteristic
347	C₂	#20243105	1	1	yes	characteristic
348	C₂	C₂×#15688890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×C₂²⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#7982
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20241256	C₂	semidirect
5. #15688986	C₄	semidirect
10. C₄×C₂²⋊Dic₄₇	C₄	semidirect
12. C₂³×C₈×C₄₇	C₄	semidirect
15. C₂×C₂²⋊C₈×C₄₇	C₄	semidirect
179. C₂³×C₈	Dic₄₇	semidirect
183. C₂×C₂²⋊C₈	Dic₄₇	semidirect
191. C₄₇	#7982	semidirect

Maximal subgroups

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

subgroup	index
#20245771	2
#20241256	2
C₂²×C₂²⋊C₈×C₄₇	2
#7982	47