Subgroups — #26115210

Order 12032.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20242136	C₂	1	1	yes	maximal, characteristic
2	#20241783	C₂	1	2	yes	maximal
3	#20241712	C₂	1	2	yes	maximal
4	#20241671	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689176	C₂²	1	1	yes	characteristic
8	#15689176	C₂²	1	1	yes	characteristic
9	#15688876	—	2	1	no	—
10	#15688874	—	2	1	no	—
11	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
12	#15688849	—	2	4	no	—
13	#15688844	—	2	1	no	—
14	C₈×C₄₇⋊C₈	—	2	1	no	—
15	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₄₇	—	2	1	no	—
18	C₄×Q₁₆×C₄₇	—	2	1	no	—
19	C₄×Q₁₆×C₄₇	—	2	1	no	—
20	C₄×Q₁₆×C₄₇	—	2	1	no	—
21	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
22	C₂×#103×C₄₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
30	C₄×C₄₇⋊C₈	—	2	1	no	—
31	C₄×C₄₇⋊C₈	—	2	1	no	—
32	C₄×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₄₇	—	2	1	no	—
35	C₂×Q₁₆×C₄₇	—	2	1	no	—
36	C₂×Q₁₆×C₄₇	—	2	2	no	—
37	C₂×Q₁₆×C₄₇	C₈	1	4	yes	complemented
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₄₇	—	2	2	no	—
41	C₄×Q₈×C₄₇	—	2	2	no	—
42	C₄×Q₈×C₄₇	—	2	2	no	—
43	C₄×Q₈×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#107×C₄₇	—	2	1	no	—
49	#107×C₄₇	—	2	1	no	—
50	#103×C₄₇	—	2	2	no	—
51	#103×C₄₇	—	2	2	no	—
52	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₄₇	—	2	2	no	—
57	C₂×Q₈×C₄₇	—	2	2	no	—
58	C₂×Q₈×C₄₇	—	2	2	no	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	Q₁₆×C₄₇	—	2	4	no	—
68	Q₁₆×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	—	4	2	no	—
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	#41	1	1	yes	characteristic
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₄₇	—	2	4	no	—
92	Q₈×C₄₇	—	2	4	no	—
93	Q₈×C₄₇	—	4	2	no	—
94	Q₈×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	1	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	8	2	no	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
112	C₈×C₄₇	—	2	1	no	—
113	C₈×C₄₇	—	2	1	no	—
114	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
115	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
116	C₄₇⋊C₈	—	4	2	no	—
117	C₄₇⋊C₈	—	4	2	no	—
118	C₄₇⋊C₈	—	8	2	no	—
119	#9669	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	#390	1	1	yes	characteristic
123	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	#436	1	1	yes	characteristic
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	4	4	no	—
129	C₄×C₄₇	—	8	2	no	—
130	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₄₇	#408	1	1	yes	characteristic
132	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
133	C₄×C₄₇	#436	1	1	yes	characteristic
134	C₄×C₄₇	#473	1	2	yes	—
135	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
136	#1564	—	47	1	no	—
137	#1479	—	47	2	no	—
138	#1461	—	47	2	no	—
139	#1450	—	47	1	no	—
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	#1954	1	1	yes	characteristic
143	C₂×C₄₇	#1976	1	1	yes	characteristic
144	C₂×C₄₇	#1979	1	1	yes	characteristic
145	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
147	C₄×Q₁₆	—	2	1	no	—
148	C₄×Q₁₆	—	2	1	no	—
149	C₄×Q₁₆	—	2	1	no	—
150	C₄×Q₁₆	—	2	1	no	—
151	#431	—	47	1	no	—
152	#431	—	47	1	no	—
153	#431	—	47	2	no	—
154	C₂×#107	Dic₄₇	1	1	yes	characteristic
155	C₂×#103	Dic₄₇	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	#335	—	94	1	no	—
158	#334	—	94	1	no	—
159	Q₈⋊C₈	—	94	4	no	—
160	#324	—	94	4	no	—
161	#322	—	94	1	no	—
162	C₈²	—	94	1	no	—
163	C₄₇	#9669	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	C₄₇⋊C₈	1	4	yes	complemented
169	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	#107	—	2	1	no	—
180	#107	—	2	1	no	—
181	C₄⋊C₈	—	47	2	no	—
182	C₄⋊C₈	—	94	1	no	—
183	C₄⋊C₈	—	94	2	no	—
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	C₂²⋊C₈	—	47	2	no	—
187	C₂²⋊C₈	—	94	1	no	—
188	C₂²⋊C₈	—	94	2	no	—
189	C₄×C₈	—	94	2	no	—
190	C₄×C₈	—	94	1	no	—
191	C₄×C₈	—	94	1	no	—
192	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	#10625352	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	188	2	no	—
214	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
215	C₂×C₈	#10625329	1	2	yes	—
216	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	D₄×D₄₇	1	1	yes	characteristic
227	C₄²	#10625352	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
244	C₂×C₄	#13093995	1	1	yes	characteristic
245	C₂×C₄	#13093995	1	1	yes	characteristic
246	C₂×C₄	#13094011	1	1	yes	characteristic
247	C₂×C₄	#13094020	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
250	C₂×C₄	#13094031	1	1	yes	characteristic
251	C₂×C₄	#13094031	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	376	2	no	—
255	C₈	—	188	2	no	—
256	C₈	—	188	2	no	—
257	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
258	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#15689026	1	1	yes	characteristic
262	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
263	C₂²	#15689201	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
270	C₄	#15689090	1	1	yes	characteristic
271	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
272	C₄	#15689201	1	1	yes	characteristic
273	C₄	#15689395	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#20244183	1	1	yes	characteristic
277	C₂	#20244287	1	1	yes	characteristic
278	C₂	#20244304	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093995
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#9669
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
37. C₂×Q₁₆×C₄₇	C₈	semidirect
163. C₄₇	#9669	semidirect
168. C₂×Q₁₆	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20242136	2
#20241783	2
#20241712	2
#20241671	2
C₂×C₄×Q₁₆×C₄₇	2
#9669	47