Subgroups — #499×Dic₄₇ #26225023

Order 12032.

774 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×Dic₄₇	C₂	1	2	yes	maximal
2	#20244354	C₂	1	2	yes	maximal
3	#20244122	C₂	1	2	yes	maximal
4	#20244117	C₂	1	2	yes	maximal
5	#20243552	C₂	1	1	yes	maximal, characteristic
6	#107×Dic₄₇	C₂	1	2	yes	maximal
7	#106×Dic₄₇	C₂	1	2	yes	maximal
8	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#499	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093956	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
17	#15689022	C₂²	1	4	yes	—
18	#15689009	—	2	4	no	—
19	#15688989	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₄₇	—	2	4	no	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688885	C₂²	1	4	yes	—
24	#15688885	C₂²	1	4	yes	—
25	C₂×#125×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#499	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	2	yes	—
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	#13093958	—	2	4	no	—
40	#13093956	—	2	4	no	—
41	C₄×C₄₇⋊C₈	—	2	2	no	—
42	C₈×Dic₄₇	—	2	2	no	—
43	C₈×Dic₄₇	—	2	2	no	—
44	C₈×Dic₄₇	Q₈	1	4	yes	—
45	C₈×Dic₄₇	Q₈	1	4	yes	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₄₇⋊#95	—	2	4	no	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	#125×C₄₇	—	2	4	no	—
52	#125×C₄₇	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#107×C₄₇	C₂×C₄	1	8	yes	—
57	#106×C₄₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
65	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₂×C₈×C₄₇	#41	1	2	yes	—
72	C₄⋊C₄×C₄₇	—	2	16	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	#41	1	2	yes	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	8	no	—
80	C₄×Dic₄₇	—	2	1	no	—
81	C₄×Dic₄₇	—	2	1	no	—
82	C₄×Dic₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	4	8	no	—
85	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
86	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	—	2	1	no	—
89	C₂×C₄₇⋊C₈	—	2	1	no	—
90	C₂×C₄₇⋊C₈	—	2	1	no	—
91	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
113	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
114	C₂×Dic₄₇	#135	1	4	yes	—
115	C₂×Dic₄₇	#135	1	4	yes	—
116	C₈×C₄₇	C₄×Q₈	1	4	yes	—
117	C₈×C₄₇	C₄×Q₈	1	4	yes	—
118	C₄₇⋊C₈	—	2	2	no	—
119	C₄₇⋊C₈	—	2	2	no	—
120	C₄×#499	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₄₇	#532	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#135	1	2	yes	—
125	C₂²×C₄₇	C₂×#135	1	2	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	4	8	no	—
128	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₄₇	#533	1	1	yes	characteristic
131	C₄×C₄₇	#533	1	1	yes	characteristic
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	—	8	4	no	—
135	Dic₄₇	#499	1	4	yes	direct product factor
136	C₂×#499	D₄₇	1	1	yes	characteristic
137	C₄×#125	—	47	2	no	—
138	#1991	—	47	2	no	—
139	#1942	—	47	2	no	—
140	#1941	—	47	2	no	—
141	#1846	—	47	1	no	—
142	C₄×#107	—	47	2	no	—
143	C₄×#106	—	47	2	no	—
144	C₄²×C₈	—	47	1	no	—
145	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
146	C₂×C₄₇	C₄×#135	1	2	yes	—
147	C₂×C₄₇	C₂×#499	1	1	yes	characteristic
148	C₂×C₄₇	#3160	1	1	yes	characteristic
149	C₂×C₄₇	#3168	1	2	yes	—
150	C₂×#125	C₂×D₄₇	1	2	yes	—
151	#499	Dic₄₇	1	8	yes	direct product factor
152	C₂×#107	—	47	2	no	—
153	C₂×#107	C₂×D₄₇	1	2	yes	—
154	C₂×#106	—	47	2	no	—
155	C₂×#106	C₂×D₄₇	1	2	yes	—
156	C₂×C₄×C₈	—	47	1	no	—
157	C₂×C₄×C₈	—	47	1	no	—
158	C₂×C₄×C₈	—	47	1	no	—
159	C₂×C₄×C₈	—	47	1	no	—
160	C₂×C₄×C₈	—	47	1	no	—
161	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
162	#389	—	47	4	no	—
163	#387	—	94	4	no	—
164	#382	—	47	2	no	—
165	C₄×C₄⋊C₄	—	47	4	no	—
166	C₄×C₄⋊C₄	—	94	4	no	—
167	C₄³	—	47	1	no	—
168	#340	—	47	4	no	—
169	#340	—	47	4	no	—
170	C₄₇	C₄×#499	1	1	yes	characteristic, complemented, Sylow
171	C₂²×C₈	—	47	1	no	—
172	C₂²×C₈	—	47	1	no	—
173	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	#125	—	2	4	no	—
176	#125	C₂×Dic₄₇	1	8	yes	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	47	4	no	—
179	C₂×C₄⋊C₄	—	94	4	no	—
180	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
181	C₂×C₄²	—	47	1	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	2	no	—
184	C₂×C₄²	—	47	1	no	—
185	C₂×C₄²	—	94	4	no	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	#107	—	94	4	no	—
188	#107	C₂×Dic₄₇	1	8	yes	—
189	#106	—	94	4	no	—
190	#106	C₂×Dic₄₇	1	8	yes	—
191	C₄×C₈	—	47	4	no	—
192	C₄×C₈	—	47	4	no	—
193	C₄×C₈	—	94	2	no	—
194	C₄×C₈	—	94	2	no	—
195	C₄×C₈	—	94	2	no	—
196	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
197	#95	—	94	4	no	—
198	#95	—	94	4	no	—
199	#95	—	94	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	47	1	no	—
202	C₂²×C₄	—	47	1	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	D₄×D₄₇	1	2	yes	—
206	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
207	C₂×C₈	—	47	2	no	—
208	C₂×C₈	—	47	2	no	—
209	C₂×C₈	—	94	1	no	—
210	C₂×C₈	—	94	1	no	—
211	C₂×C₈	—	94	1	no	—
212	C₂×C₈	—	94	1	no	—
213	C₂×C₈	Q₈×D₄₇	1	2	yes	—
214	C₂×C₈	Q₈×D₄₇	1	2	yes	—
215	C₂×C₈	#10625354	1	2	yes	—
216	C₂×C₈	#10625354	1	2	yes	—
217	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
218	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	94	4	no	—
222	C₄⋊C₄	—	94	4	no	—
223	C₄⋊C₄	—	188	8	no	—
224	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
225	C₄²	—	47	2	no	—
226	C₄²	—	47	2	no	—
227	C₄²	—	94	4	no	—
228	C₄²	—	94	4	no	—
229	C₄²	—	94	1	no	—
230	C₄²	—	94	1	no	—
231	C₄²	—	188	8	no	—
232	C₄²	#10625352	1	2	yes	—
233	C₄²	C₂²×Dic₄₇	1	2	yes	—
234	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	47	4	no	—
241	C₂×C₄	—	47	2	no	—
242	C₂×C₄	—	47	4	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	94	2	no	—
245	C₂×C₄	—	94	2	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	—	188	4	no	—
249	C₂×C₄	—	188	4	no	—
250	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
251	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
255	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
256	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
257	C₈	—	94	2	no	—
258	C₈	—	94	2	no	—
259	C₈	Q₈×Dic₄₇	1	4	yes	—
260	C₈	Q₈×Dic₄₇	1	4	yes	—
261	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
262	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
263	C₂²	#15689674	1	1	yes	characteristic
264	C₂²	#135×D₄₇	1	2	yes	—
265	C₂²	#135×D₄₇	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	47	4	no	—
269	C₄	—	94	4	no	—
270	C₄	—	94	2	no	—
271	C₄	—	376	4	no	—
272	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
273	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
274	C₄	#15689678	1	1	yes	characteristic
275	C₄	#15689678	1	1	yes	characteristic
276	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
277	C₂	#135×Dic₄₇	1	2	yes	—
278	C₂	#499×D₄₇	1	1	yes	characteristic
279	C₂	#20253901	1	1	yes	characteristic
280	C₂	#20253960	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#499	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#499	D₄₇
hypocenter	C₄₇	C₄×#499
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
26. C₄₇×#499	C₄	semidirect
135. Dic₄₇	#499	direct (N × H)
151. #499	Dic₄₇	direct (N × H)
170. C₄₇	C₄×#499	semidirect

Maximal subgroups

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

subgroup	index
#125×Dic₄₇	2
#20244354	2
#20244122	2
#20244117	2
#20243552	2
#107×Dic₄₇	2
#106×Dic₄₇	2
C₄×C₈×Dic₄₇	2
C₂×C₄₇×#499	2
C₄×#499	47

Subgroups — #499×Dic47 #26225023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #499×Dic₄₇ #26225023