Subgroups — #26220560

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246031	C₂	1	2	yes	maximal, complemented
2	#20244255	C₂	1	2	yes	maximal
3	#20244122	C₂	1	2	yes	maximal, complemented
4	#20244087	C₂	1	4	yes	maximal, complemented
5	C₄×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243213	C₂	1	2	yes	maximal, complemented
7	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093958	C₂²	1	2	yes	—
10	C₂×#13093945	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	1	yes	characteristic
12	C₂×#13093934	C₂²	1	2	yes	—
13	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
14	#15689026	C₂²	1	4	yes	—
15	#15689022	C₂²	1	4	yes	—
16	#15689016	—	2	4	no	—
17	#15688986	—	2	2	no	—
18	#15688983	—	2	2	no	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	#15688969	C₂²	1	1	yes	characteristic
21	#15688886	C₂²	1	2	yes	—
22	#15688885	C₂²	1	4	yes	—
23	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
25	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	#13093958	—	2	4	no	—
37	#13093945	—	2	8	no	—
38	#13093935	—	2	2	no	—
39	#13093934	—	2	4	no	—
40	C₈×Dic₄₇	—	2	4	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₄₇⋊#95	—	2	4	no	—
43	C₄₇⋊#95	D₄	1	1	yes	characteristic
44	C₄₇⋊#95	D₄	1	1	yes	characteristic
45	C₄₇⋊#95	D₄	1	2	yes	—
46	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₇	—	2	2	no	—
48	C₂×D₈×C₄₇	—	2	4	no	—
49	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₄₇	—	2	4	no	—
53	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#107×C₄₇	C₂×C₄	1	4	yes	—
58	#102×C₄₇	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	—	2	2	no	—
61	C₂×D₄×C₄₇	—	2	4	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	8	no	—
67	C₂²×C₄×C₄₇	—	8	4	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
73	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
74	D₈×C₄₇	—	2	8	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	#41	1	4	yes	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²⋊C₄×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	#41	1	2	yes	—
84	C₂²⋊Dic₄₇	—	8	4	no	—
85	C₂²⋊Dic₄₇	—	8	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	8	4	no	—
89	C₄×Dic₄₇	—	2	2	no	—
90	C₄×Dic₄₇	—	2	2	no	—
91	C₄×Dic₄₇	—	8	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	#41	1	4	yes	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	4	4	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	D₄×C₄₇	—	2	4	no	—
102	D₄×C₄₇	—	2	4	no	—
103	D₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	8	8	no	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	#117	1	4	yes	—
127	C₄₇⋊C₈	—	2	4	no	—
128	#21282	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	4	4	no	—
130	C₂²×C₄₇	—	4	4	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	—	8	4	no	—
133	C₂²×C₄₇	—	8	4	no	—
134	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₄₇	#532	1	2	yes	—
136	C₂²×C₄₇	#576	1	2	yes	—
137	C₂²×C₄₇	#576	1	1	yes	characteristic
138	C₂²×C₄₇	#578	1	1	yes	characteristic
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	4	4	no	—
142	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
143	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
144	C₄×C₄₇	#529	1	2	yes	—
145	Dic₄₇	—	4	2	no	—
146	Dic₄₇	—	4	2	no	—
147	Dic₄₇	—	8	4	no	—
148	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
149	#2319	—	47	2	no	—
150	#1970	—	47	2	no	—
151	#1942	—	47	2	no	—
152	#1936	—	47	4	no	—
153	C₄×#107	—	47	1	no	—
154	#1762	—	47	2	no	—
155	#1726	—	47	1	no	—
156	C₂×C₄₇	—	8	4	no	—
157	C₂×C₄₇	#2293	1	1	yes	characteristic
158	C₂×C₄₇	#2948	1	1	yes	characteristic
159	C₂×C₄₇	#2951	1	1	yes	characteristic
160	C₂×C₄₇	#3175	1	2	yes	—
161	C₂×C₄₇	#3176	1	2	yes	—
162	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
164	C₄×D₈	Dic₄₇	1	8	yes	complemented
165	C₂×#107	—	47	2	no	—
166	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
167	C₂×#102	—	47	4	no	—
168	C₂×#102	C₂×D₄₇	1	2	yes	—
169	C₂×#97	—	47	2	no	—
170	C₂×#97	—	47	1	no	—
171	C₂×C₄×C₈	—	47	2	no	—
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	#390	—	47	4	no	—
174	#389	—	47	4	no	—
175	#388	—	94	4	no	—
176	#381	—	94	2	no	—
177	#380	—	94	2	no	—
178	C₄×C₄⋊C₄	—	47	2	no	—
179	#376	—	47	1	no	—
180	#340	—	47	2	no	—
181	#340	—	47	4	no	—
182	C₄₇	#21282	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	C₄×D₄	—	2	4	no	—
191	C₄×D₄	C₂×Dic₄₇	1	8	yes	—
192	C₂×C₄⋊C₄	—	47	4	no	—
193	C₂×C₄⋊C₄	—	94	4	no	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	94	4	no	—
197	C₂×C₂²⋊C₄	—	94	4	no	—
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	94	2	no	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	#107	—	94	4	no	—
202	#107	C₂×Dic₄₇	1	4	yes	—
203	#102	—	94	8	no	—
204	#102	C₂×Dic₄₇	1	8	yes	—
205	#97	—	94	4	no	—
206	#97	—	94	2	no	—
207	C₄×C₈	—	94	4	no	—
208	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
209	#95	—	47	1	no	—
210	#95	—	47	1	no	—
211	#95	—	47	2	no	—
212	#95	—	94	4	no	—
213	#95	—	94	2	no	—
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	8	4	no	—
222	C₂²×C₄	—	47	2	no	—
223	C₂²×C₄	—	47	2	no	—
224	C₂²×C₄	—	94	4	no	—
225	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
228	D₈	—	2	8	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	47	4	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	#10625350	1	4	yes	—
234	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
235	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	94	4	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	376	4	no	—
240	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	376	4	no	—
243	C₂²⋊C₄	—	376	4	no	—
244	C₄²	—	94	2	no	—
245	C₄²	—	94	2	no	—
246	C₄²	—	376	2	no	—
247	C₄²	#10625352	1	2	yes	—
248	C₄²	C₂²×Dic₄₇	1	2	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	—	94	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	94	2	no	—
267	C₂×C₄	—	94	2	no	—
268	C₂×C₄	—	94	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
274	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
275	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	#41×D₄₇	1	2	yes	—
277	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625350	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	94	4	no	—
281	C₈	#13093983	1	4	yes	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
288	C₂²	#15689670	1	2	yes	—
289	C₂²	#15689924	1	2	yes	—
290	C₂²	#15689925	1	1	yes	characteristic
291	C₂²	#15689937	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	376	4	no	—
296	C₄	—	188	2	no	—
297	C₄	—	188	2	no	—
298	C₄	C₂×#13093983	1	1	yes	characteristic
299	C₄	#41×Dic₄₇	1	1	yes	characteristic
300	C₄	#15689655	1	2	yes	—
301	C₂	—	8	4	no	—
302	C₂	#20245853	1	1	yes	characteristic
303	C₂	#20252240	1	1	yes	characteristic
304	C₂	#20252264	1	1	yes	characteristic
305	C₂	#20254023	1	2	yes	—
306	C₂	#20254034	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#21282
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20246031	C₂	semidirect
3. #20244122	C₂	semidirect
4. #20244087	C₂	semidirect
5. C₄×#13093958	C₂	semidirect
6. #20243213	C₂	semidirect
7. #20243074	C₂	semidirect
25. C₄×D₈×C₄₇	C₄	semidirect
164. C₄×D₈	Dic₄₇	semidirect
182. C₄₇	#21282	semidirect

Maximal subgroups

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

subgroup	index
#20246031	2
#20244255	2
#20244122	2
#20244087	2
C₄×#13093958	2
#20243213	2
#20243074	2
C₂×C₄×D₈×C₄₇	2
#21282	47