Subgroups — #26216708

Order 12032.

1190 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689166	C₂	1	4	yes	maximal
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
4	#20243557	C₂	1	4	yes	maximal, complemented
5	#20243547	C₂	1	2	yes	maximal, complemented
6	#20243544	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	#15689166	—	2	16	no	—
12	C₂×#13093957	C₂²	1	8	yes	—
13	C₂×#13093955	C₂²	1	8	yes	—
14	C₄×#10625329	C₄	1	8	yes	complemented
15	C₂×#13093935	C₂²	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	#15688993	—	2	2	no	—
18	#15688993	C₂²	1	2	yes	—
19	#15688993	C₂²	1	2	yes	—
20	C₂×#117×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#384	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	1	yes	characteristic
24	C₄₇×#382	C₂²	1	2	yes	—
25	C₄₇×#380	—	2	2	no	—
26	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₂³×Dic₄₇	—	2	2	no	—
28	C₂×#10625329	C₂×C₄	1	8	yes	—
29	C₂×#10625329	D₄	1	8	yes	—
30	C₂×#10625329	Q₈	1	8	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	4	no	—
36	C₂×C₄⋊Dic₄₇	—	4	8	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	#13093957	—	2	16	no	—
39	#13093955	—	2	16	no	—
40	#13093935	C₂×C₄	1	8	yes	complemented
41	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
42	C₂³×C₄×C₄₇	D₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	#117×C₄₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	D₄	1	2	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	#95×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	4	8	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₂²⋊C₄×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	—	4	4	no	—
80	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₄₇	—	2	8	no	—
86	C₄⋊Dic₄₇	—	4	8	no	—
87	C₄⋊Dic₄₇	—	4	4	no	—
88	C₄⋊Dic₄₇	—	4	8	no	—
89	#10625329	C₄⋊C₄	1	32	yes	complemented
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
91	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
92	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₄₇	#127	1	2	yes	—
97	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
98	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	16	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	4	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	#127	1	2	yes	—
111	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
113	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₄₇	#137	1	4	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	8	no	—
121	C₂×Dic₄₇	—	8	8	no	—
122	C₄₇⋊C₈	C₂×C₄⋊C₄	1	16	yes	complemented
123	#20800	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	#384	1	4	yes	—
129	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
133	C₂²×C₄₇	C₂×#137	1	2	yes	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	—	8	2	no	—
137	C₄×C₄₇	#384	1	4	yes	—
138	C₄×C₄₇	#428	1	8	yes	complemented
139	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
140	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
141	Dic₄₇	—	8	4	no	—
142	C₂×#428	—	47	4	no	—
143	C₂×C₄×M₁₆	—	47	1	no	—
144	#2285	D₄₇	1	1	yes	characteristic
145	C₂×#384	—	47	1	no	—
146	#1847	—	47	4	no	—
147	#1845	—	47	2	no	—
148	#1844	—	47	2	no	—
149	C₂×C₄₇	—	2	4	no	—
150	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#428	1	2	yes	—
152	C₂×C₄₇	#3151	1	1	yes	characteristic
153	C₂×C₄₇	#3152	1	1	yes	characteristic
154	C₂×C₄₇	#3165	1	2	yes	—
155	C₂²×M₁₆	—	47	2	no	—
156	C₂×#117	C₂×D₄₇	1	2	yes	—
157	C₂²×C₄⋊C₄	—	47	2	no	—
158	C₂²×C₄⋊C₄	—	94	2	no	—
159	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
160	#428	—	94	16	no	—
161	C₂×#107	—	47	8	no	—
162	C₂×#106	—	47	8	no	—
163	C₄×M₁₆	—	47	8	no	—
164	C₂×#97	—	47	2	no	—
165	C₂×C₄×C₈	—	47	2	no	—
166	#384	—	47	2	no	—
167	#384	—	47	2	no	—
168	#384	—	94	2	no	—
169	#384	C₂×D₄₇	1	1	yes	characteristic
170	#384	C₂×D₄₇	1	1	yes	characteristic
171	#382	C₂×D₄₇	1	2	yes	—
172	#380	—	2	2	no	—
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄₇	#20800	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	94	2	no	—
176	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	47	8	no	—
179	C₂×M₁₆	—	47	8	no	—
180	C₂×M₁₆	—	47	8	no	—
181	C₂²×C₈	—	47	4	no	—
182	#117	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	47	4	no	—
186	C₂×C₄⋊C₄	—	94	8	no	—
187	C₂×C₄⋊C₄	—	94	2	no	—
188	C₂×C₄⋊C₄	—	94	4	no	—
189	C₂×C₄⋊C₄	—	94	4	no	—
190	C₂×C₄⋊C₄	—	188	8	no	—
191	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
192	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	Dic₉₄	1	4	yes	—
196	C₂×C₄²	C₄×D₄₇	1	4	yes	—
197	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#107	—	94	16	no	—
201	#106	—	94	16	no	—
202	#97	—	47	8	no	—
203	C₄×C₈	—	47	8	no	—
204	#95	—	2	4	no	—
205	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	188	8	no	—
212	C₂²×C₄	—	188	4	no	—
213	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
214	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
216	C₂²×C₄	D₄×D₄₇	1	4	yes	—
217	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
222	M₁₆	—	47	32	no	—
223	C₂×C₈	—	47	8	no	—
224	C₂×C₈	—	47	8	no	—
225	C₂×C₈	—	47	8	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	94	8	no	—
231	C₄⋊C₄	—	188	8	no	—
232	C₄⋊C₄	—	188	4	no	—
233	C₄⋊C₄	—	188	8	no	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₄²	—	4	4	no	—
236	C₄²	Dic₄₇⋊C₄	1	8	yes	—
237	C₄²	C₂×Dic₉₄	1	2	yes	—
238	C₄²	C₂×Dic₉₄	1	2	yes	—
239	C₄²	C₂×C₄×D₄₇	1	2	yes	—
240	C₄²	C₂×C₄×D₄₇	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
244	C₂³	#13094020	1	2	yes	—
245	C₂³	#13094022	1	2	yes	—
246	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	376	8	no	—
255	C₂×C₄	—	188	8	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	C₄⋊C₄×D₄₇	1	8	yes	—
258	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
259	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
260	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
261	C₂×C₄	#13094020	1	2	yes	—
262	C₂×C₄	#13094022	1	2	yes	—
263	C₂×C₄	#13094023	1	8	yes	—
264	C₂×C₄	#13094049	1	4	yes	—
265	C₂×C₄	#13094054	1	4	yes	—
266	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
268	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
269	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
270	C₈	—	47	16	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	#15688992	1	4	yes	—
276	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
277	C₂²	C₂×#13094020	1	1	yes	characteristic
278	C₂²	C₂×#13094022	1	1	yes	characteristic
279	C₂²	C₂×#13094049	1	2	yes	—
280	C₂²	C₂×#13094054	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	376	4	no	—
285	C₄	#15688992	1	4	yes	—
286	C₄	#15689166	1	8	yes	complemented
287	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
288	C₄	C₂×#13094023	1	2	yes	—
289	C₂	—	2	4	no	—
290	C₂	C₂×#15688992	1	1	yes	characteristic
291	C₂	C₂×#15689166	1	2	yes	—
292	C₂	#20253841	1	1	yes	characteristic
293	C₂	#20253848	1	1	yes	characteristic
294	C₂	#20253935	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#20800
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
4. #20243557	C₂	semidirect
5. #20243547	C₂	semidirect
6. #20243544	C₂	semidirect
14. C₄×#10625329	C₄	semidirect
40. #13093935	C₂×C₄	semidirect
41. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
89. #10625329	C₄⋊C₄	semidirect
122. C₄₇⋊C₈	C₂×C₄⋊C₄	semidirect
138. C₄×C₄₇	#428	semidirect
174. C₄₇	#20800	semidirect
286. C₄	#15689166	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689166	2
C₂×C₄×#10625329	2
C₂×#15688993	2
#20243557	2
#20243547	2
#20243544	2
C₄₇×#2285	2
#20800	47