Subgroups — #26102827

Order 12032.

468 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	#20241615	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
5	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄₇⋊#95	—	2	1	no	—
7	C₂×C₄₇⋊#95	—	2	1	no	—
8	C₂×C₄₇⋊#95	C₄	1	2	yes	—
9	#15688878	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	complemented
13	C₄₇×#336	—	2	1	no	—
14	C₂³×Dic₄₇	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
18	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
19	C₄₇⋊#95	—	2	2	no	—
20	C₄₇⋊#95	—	4	2	no	—
21	C₄₇⋊#95	—	4	1	no	—
22	C₄₇⋊#95	—	4	1	no	—
23	C₄₇⋊#95	C₈	1	4	yes	complemented
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	—	4	2	no	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²⋊C₈×C₄₇	—	2	4	no	—
34	C₄×C₈×C₄₇	—	4	2	no	—
35	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	4	1	no	—
39	C₂²×C₄×C₄₇	—	4	1	no	—
40	C₂²×C₄×C₄₇	—	4	1	no	—
41	C₂²×C₄×C₄₇	—	4	1	no	—
42	C₂²×C₄×C₄₇	—	4	1	no	—
43	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	4	no	—
56	C₂²×Dic₄₇	—	4	2	no	—
57	C₂²×Dic₄₇	—	4	2	no	—
58	C₂²×Dic₄₇	—	4	2	no	—
59	C₂²×Dic₄₇	—	4	2	no	—
60	C₂²×Dic₄₇	—	4	2	no	—
61	C₂²×Dic₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	—	4	1	no	—
68	C₄²×C₄₇	—	4	1	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂³×C₄₇	—	2	1	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	1	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	#95	1	1	yes	characteristic
77	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
78	C₂³×C₄₇	#102	1	1	yes	characteristic
79	C₂³×C₄₇	#103	1	1	yes	characteristic
80	C₂³×C₄₇	#104	1	2	yes	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	1	no	—
97	C₂×C₄×C₄₇	—	4	1	no	—
98	C₂×C₄×C₄₇	—	4	1	no	—
99	C₂×C₄×C₄₇	—	4	1	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#99	1	2	yes	—
104	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	8	2	no	—
113	C₂×Dic₄₇	—	8	2	no	—
114	C₂×Dic₄₇	—	8	2	no	—
115	C₂×Dic₄₇	—	8	2	no	—
116	C₈×C₄₇	—	4	4	no	—
117	C₄₇⋊C₈	—	8	2	no	—
118	#7024	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	—	2	1	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	D₄⋊C₈	1	2	yes	—
128	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
129	C₂²×C₄₇	#336	1	1	yes	characteristic
130	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
131	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
132	C₂²×C₄₇	#339	1	1	yes	characteristic
133	C₂²×C₄₇	#340	1	1	yes	characteristic
134	C₂²×C₄₇	#342	1	1	yes	characteristic
135	C₂²×C₄₇	#344	1	1	yes	characteristic
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	4	1	no	—
145	Dic₄₇	—	8	4	no	—
146	#1797	—	47	1	no	—
147	C₄×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
148	#1434	—	47	1	no	—
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	#1274	1	1	yes	characteristic
152	C₂×C₄₇	#1279	1	1	yes	characteristic
153	C₂×C₄₇	#1282	1	1	yes	characteristic
154	C₂×C₄₇	#1284	1	1	yes	characteristic
155	C₂×C₄₇	#1286	1	1	yes	characteristic
156	C₂×C₄₇	#1288	1	1	yes	characteristic
157	C₂×C₄₇	#1289	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	94	1	no	—
160	C₂×C₂²⋊C₈	—	2	1	no	—
161	C₂×C₂²⋊C₈	—	94	1	no	—
162	C₂×C₄×C₈	Dic₄₇	1	2	yes	complemented
163	C₂×#95	—	47	2	no	—
164	C₂×#95	—	94	1	no	—
165	C₂×#95	—	94	1	no	—
166	#336	—	2	1	no	—
167	#336	—	47	2	no	—
168	C₄₇	#7024	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	94	2	no	—
170	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
171	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
172	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	94	2	no	—
175	C₂×C₄²	—	4	2	no	—
176	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₄⋊C₈	—	188	2	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	188	1	no	—
183	C₂²⋊C₈	—	188	1	no	—
184	C₄×C₈	—	4	2	no	—
185	#95	—	47	4	no	—
186	#95	—	94	2	no	—
187	#95	—	188	2	no	—
188	#95	—	188	1	no	—
189	#95	—	188	1	no	—
190	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	94	4	no	—
199	C₂²×C₄	—	188	2	no	—
200	C₂²×C₄	—	188	2	no	—
201	C₂²×C₄	—	188	2	no	—
202	C₂²×C₄	—	188	2	no	—
203	C₂²×C₄	—	188	2	no	—
204	C₂²×C₄	—	188	2	no	—
205	C₂²×C₄	C₈×D₄₇	1	2	yes	—
206	C₂²×C₄	#10625324	1	2	yes	—
207	C₂²×C₄	#10625325	1	1	yes	characteristic
208	C₂²×C₄	#10625325	1	1	yes	characteristic
209	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
210	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
211	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
215	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	188	2	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	C₄₇⋊#95	1	1	yes	characteristic
232	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
233	C₂³	#13093946	1	1	yes	characteristic
234	C₂³	#13093949	1	1	yes	characteristic
235	C₂³	#13093952	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	376	2	no	—
257	C₂×C₄	—	376	2	no	—
258	C₂×C₄	—	376	2	no	—
259	C₂×C₄	—	376	2	no	—
260	C₂×C₄	—	188	4	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
264	C₂×C₄	#13093934	1	1	yes	characteristic
265	C₂×C₄	#13093939	1	2	yes	—
266	C₂×C₄	#13093946	1	1	yes	characteristic
267	C₂×C₄	#13093949	1	1	yes	characteristic
268	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
269	C₂×C₄	#13093956	1	1	yes	characteristic
270	C₂×C₄	#13093958	1	1	yes	characteristic
271	C₈	—	4	4	no	—
272	C₈	—	376	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#15688852	1	2	yes	—
282	C₂²	#15688855	1	2	yes	—
283	C₂²	#15688877	1	1	yes	characteristic
284	C₂²	#15688879	1	1	yes	characteristic
285	C₂²	#15688880	1	1	yes	characteristic
286	C₂²	#15688883	1	1	yes	characteristic
287	C₂²	#15688886	1	1	yes	characteristic
288	C₂²	#15688890	1	1	yes	characteristic
289	C₂²	#15688894	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	376	4	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	#20241216	1	1	yes	characteristic
303	C₂	#20241228	1	1	yes	characteristic
304	C₂	#20241235	1	1	yes	characteristic
305	C₂	#20241241	1	1	yes	characteristic
306	C₂	#20241247	1	1	yes	characteristic
307	C₂	#20241253	1	1	yes	characteristic
308	C₂	#20241256	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	C₄×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#7024
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
9. #15688878	C₄	semidirect
12. C₂×C₄×C₈×C₄₇	C₄	semidirect
23. C₄₇⋊#95	C₈	semidirect
162. C₂×C₄×C₈	Dic₄₇	semidirect
168. C₄₇	#7024	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
#20241615	2
C₄×C₂²⋊C₈×C₄₇	2
#7024	47