Subgroups — #26115440

Order 12032.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244308	C₂	1	2	yes	maximal, complemented
2	C₄×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241883	C₂	1	2	yes	maximal, complemented
4	C₈×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#3169	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	1	yes	characteristic
7	C₂×#13093958	—	2	1	no	—
8	C₂×#13093951	—	2	4	no	—
9	C₄×#10625329	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	#15688886	—	2	1	no	—
12	#15688857	—	2	4	no	—
13	#15688845	—	2	1	no	—
14	C₈×C₄₇⋊C₈	—	2	1	no	—
15	C₄₇×#532	C₂²	1	2	yes	—
16	C₄₇×#500	C₄	1	1	yes	characteristic, complemented
17	C₄₇×#493	C₄	1	1	yes	characteristic, complemented
18	C₄₇×#486	C₄	1	2	yes	complemented
19	C₄₇×#466	—	2	1	no	—
20	C₄₇×#466	—	2	1	no	—
21	C₄×D₈×C₄₇	—	2	1	no	—
22	C₄×D₈×C₄₇	—	2	1	no	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#10625329	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	#13093958	—	2	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
29	#13093951	—	4	8	no	—
30	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
31	#13093935	—	2	1	no	—
32	C₄×C₄₇⋊C₈	—	2	1	no	—
33	C₄×C₄₇⋊C₈	—	2	1	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₂×#41×C₄₇	—	2	2	no	—
36	C₂×D₈×C₄₇	—	2	1	no	—
37	C₂×D₈×C₄₇	—	2	1	no	—
38	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
41	#127×C₄₇	C₂×C₄	1	2	yes	—
42	#124×C₄₇	—	2	2	no	—
43	#121×C₄₇	—	2	2	no	—
44	#121×C₄₇	—	2	2	no	—
45	C₄×D₄×C₄₇	—	2	2	no	—
46	C₄×D₄×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	#102×C₄₇	—	2	2	no	—
51	#102×C₄₇	—	2	2	no	—
52	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
56	#41×C₄₇	—	4	4	no	—
57	#41×C₄₇	—	4	4	no	—
58	C₂×Q₈×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	—	2	2	no	—
60	C₂×D₄×C₄₇	—	2	2	no	—
61	C₂×D₄×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	4	2	no	—
68	D₈×C₄₇	—	4	2	no	—
69	D₈×C₄₇	—	4	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₂²⋊C₄×C₄₇	—	4	2	no	—
79	C₂²⋊C₄×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₇	#41	1	1	yes	characteristic
83	C₄²×C₄₇	#41	1	1	yes	characteristic
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	4	2	no	—
86	C₄×Dic₄₇	—	4	4	no	—
87	#10625329	—	2	2	no	—
88	#10625329	D₈	1	4	yes	complemented
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	#127	1	1	yes	characteristic
93	Q₈×C₄₇	—	4	4	no	—
94	D₄×C₄₇	—	4	2	no	—
95	D₄×C₄₇	—	4	2	no	—
96	D₄×C₄₇	—	4	4	no	—
97	D₄×C₄₇	—	8	2	no	—
98	D₄×C₄₇	—	8	2	no	—
99	D₄×C₄₇	—	8	2	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	8	4	no	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₄₇	#135	1	2	yes	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	8	2	no	—
115	C₂×Dic₄₇	—	8	2	no	—
116	C₈×C₄₇	—	2	1	no	—
117	C₈×C₄₇	—	2	1	no	—
118	C₈×C₄₇	—	2	1	no	—
119	C₈×C₄₇	#104	1	2	yes	complemented
120	C₄₇⋊C₈	—	2	4	no	—
121	C₄₇⋊C₈	—	4	2	no	—
122	#9719	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	8	2	no	—
126	C₂²×C₄₇	—	8	2	no	—
127	C₂²×C₄₇	—	8	2	no	—
128	C₂²×C₄₇	#390	1	1	yes	characteristic
129	C₂²×C₄₇	#493	1	1	yes	characteristic
130	C₂²×C₄₇	#495	1	1	yes	characteristic
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	8	2	no	—
135	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
136	C₄×C₄₇	#421	1	1	yes	characteristic
137	C₄×C₄₇	C₄×D₈	1	2	yes	—
138	C₄×C₄₇	#466	1	2	yes	—
139	Dic₄₇	—	8	4	no	—
140	#3169	D₄₇	1	1	yes	characteristic
141	#1980	—	47	2	no	—
142	C₄×#107	—	47	1	no	—
143	#1502	—	47	2	no	—
144	C₈×M₁₆	—	47	1	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	16	2	no	—
148	C₂×C₄₇	#1970	1	1	yes	characteristic
149	C₂×C₄₇	#1976	1	1	yes	characteristic
150	C₂×C₄₇	#1980	1	1	yes	characteristic
151	#532	C₂×D₄₇	1	2	yes	—
152	#500	Dic₄₇	1	1	yes	characteristic, complemented
153	#493	Dic₄₇	1	1	yes	characteristic, complemented
154	#486	Dic₄₇	1	2	yes	complemented
155	#466	—	2	1	no	—
156	#466	—	2	1	no	—
157	C₄×D₈	—	2	1	no	—
158	C₄×D₈	—	2	1	no	—
159	#431	—	47	1	no	—
160	C₂×#107	—	94	1	no	—
161	C₂×#104	—	94	4	no	—
162	C₄×M₁₆	—	47	1	no	—
163	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	47	2	no	—
165	#340	—	94	1	no	—
166	#327	—	94	4	no	—
167	#322	—	94	1	no	—
168	C₈²	—	94	1	no	—
169	C₄₇	#9719	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×M₁₆	—	47	2	no	—
174	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
177	#127	C₂×Dic₄₇	1	2	yes	—
178	#124	—	2	2	no	—
179	#121	—	2	2	no	—
180	#121	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	94	2	no	—
184	C₂×C₄²	—	94	2	no	—
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#107	—	94	4	no	—
189	C₄⋊C₈	—	94	1	no	—
190	#104	—	188	8	no	—
191	#102	—	2	2	no	—
192	#102	—	2	2	no	—
193	C₂²⋊C₈	—	47	2	no	—
194	#97	—	94	1	no	—
195	C₄×C₈	—	94	1	no	—
196	C₄×C₈	—	94	1	no	—
197	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
198	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
201	#95	—	94	2	no	—
202	#41	—	4	4	no	—
203	#41	—	4	4	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	188	2	no	—
211	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
214	D₈	—	4	2	no	—
215	D₈	—	4	2	no	—
216	M₁₆	—	47	4	no	—
217	M₁₆	—	94	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	94	2	no	—
221	C₂×C₈	—	94	2	no	—
222	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	94	4	no	—
229	C₄⋊C₄	—	188	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	188	4	no	—
233	C₄²	#10625352	1	1	yes	characteristic
234	C₄²	#10625352	1	1	yes	characteristic
235	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂³	—	4	2	no	—
238	C₂³	#13094020	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	376	2	no	—
252	C₂×C₄	—	376	2	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	#13093995	1	1	yes	characteristic
258	C₂×C₄	#13093995	1	1	yes	characteristic
259	C₂×C₄	#13094011	1	1	yes	characteristic
260	C₂×C₄	D₈×D₄₇	1	2	yes	—
261	C₂×C₄	#13094043	1	2	yes	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	94	4	no	—
266	C₈	—	188	2	no	—
267	C₈	#13093951	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	#15689026	1	1	yes	characteristic
274	C₂²	#15689498	1	1	yes	characteristic
275	C₂²	#15689506	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	376	4	no	—
281	C₄	C₂×#13093951	1	1	yes	characteristic
282	C₄	#15689140	1	1	yes	characteristic
283	C₄	D₈×Dic₄₇	1	2	yes	—
284	C₄	#15689356	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	16	2	no	—
288	C₂	#20244255	1	1	yes	characteristic
289	C₂	#20244287	1	1	yes	characteristic
290	C₂	#20244308	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093995
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting, prime radical	C₄₇×#3169	C₂
socle	C₂²×C₄₇	#390
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3169	D₄₇
hypocenter	C₄₇	#9719
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20244308	C₂	semidirect
2. C₄×#13093958	C₂	semidirect
3. #20241883	C₂	semidirect
4. C₈×#10625329	C₂	semidirect
16. C₄₇×#500	C₄	semidirect
17. C₄₇×#493	C₄	semidirect
18. C₄₇×#486	C₄	semidirect
88. #10625329	D₈	semidirect
119. C₈×C₄₇	#104	semidirect
152. #500	Dic₄₇	semidirect
153. #493	Dic₄₇	semidirect
154. #486	Dic₄₇	semidirect
169. C₄₇	#9719	semidirect
267. C₈	#13093951	semidirect

Maximal subgroups

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

subgroup	index
#20244308	2
C₄×#13093958	2
#20241883	2
C₈×#10625329	2
C₄₇×#3169	2
#9719	47