Subgroups — #26371644

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253900	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689508	C₂	1	1	yes	maximal, characteristic
3	#20245626	C₂	1	4	yes	maximal, complemented
4	#20245621	C₂	1	4	yes	maximal, complemented
5	#20245108	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245105	C₂	1	1	yes	maximal, characteristic
7	#20245103	C₂	1	1	yes	maximal, characteristic
8	M₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625336	—	2	1	no	—
11	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
12	#15689676	—	2	1	no	—
13	C₂×#13094013	—	2	1	no	—
14	C₂×#13093964	C₂²	1	1	yes	characteristic
15	#15689527	C₂²	1	2	yes	complemented
16	#15689524	C₂²	1	2	yes	complemented
17	#15689523	—	2	1	no	—
18	#15689520	—	2	1	no	—
19	#15689508	—	2	1	no	—
20	#15689508	—	2	1	no	—
21	#15689508	C₂²	1	2	yes	—
22	#15689508	C₂²	1	2	yes	—
23	#15689161	C₂²	1	1	yes	characteristic
24	#15689159	—	2	1	no	—
25	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
26	#15688927	C₂²	1	4	yes	complemented
27	#15688925	C₂²	1	4	yes	—
28	#15688923	C₂²	1	4	yes	—
29	#15688898	C₂²	1	2	yes	complemented
30	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
31	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#427	C₂²	1	1	yes	characteristic
34	C₄₇×#358	C₂²	1	4	yes	complemented
35	C₂²×D₁₈₈	—	2	1	no	—
36	C₂×#10625336	—	2	2	no	—
37	C₂×#10625336	—	2	2	no	—
38	C₂×#10625336	—	4	4	no	—
39	C₂×D₃₇₆	—	2	2	no	—
40	C₂×D₃₇₆	—	2	2	no	—
41	C₂×D₃₇₆	—	2	1	no	—
42	C₂×D₃₇₆	C₂³	1	2	yes	—
43	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
44	#13094023	—	2	2	no	—
45	#13094015	—	2	2	no	—
46	#13094013	—	2	2	no	—
47	#13094013	—	2	2	no	—
48	#13094013	—	4	2	no	—
49	#13094001	—	4	2	no	—
50	C₄×Dic₉₄	—	4	2	no	—
51	#13093983	—	2	1	no	—
52	C₂×(C₂×D₄₇)⋊C₄	—	4	2	no	—
53	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
54	#13093964	—	2	2	no	—
55	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
56	#13093958	C₂³	1	2	yes	—
57	#13093957	—	2	2	no	—
58	#13093956	—	2	1	no	—
59	#13093955	—	2	2	no	—
60	C₈×Dic₄₇	D₄	1	1	yes	characteristic
61	C₈×Dic₄₇	D₄	1	1	yes	characteristic
62	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
63	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₄₇	—	2	1	no	—
65	C₂×Q₁₆×C₄₇	—	2	1	no	—
66	C₂×Q₁₆×C₄₇	—	2	2	no	—
67	C₂×Q₁₆×C₄₇	—	2	2	no	—
68	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	#117×C₄₇	—	2	1	no	—
71	M₃₂×C₄₇	—	2	2	no	—
72	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
73	#107×C₄₇	C₂³	1	2	yes	—
74	#106×C₄₇	—	2	1	no	—
75	C₂³×D₄₇	—	4	1	no	—
76	C₂×Q₈×C₄₇	—	2	2	no	—
77	C₂×Q₈×C₄₇	—	4	1	no	—
78	C₂×Q₈×C₄₇	—	4	2	no	—
79	C₂×Q₈×C₄₇	—	4	2	no	—
80	C₂²×C₄×C₄₇	—	4	1	no	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	D₈	1	2	yes	—
83	C₂×D₁₈₈	—	2	2	no	—
84	C₂×D₁₈₈	—	4	2	no	—
85	C₂×D₁₈₈	—	4	2	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×Dic₉₄	—	4	2	no	—
88	Q₁₆×C₄₇	—	2	2	no	—
89	Q₁₆×C₄₇	—	2	2	no	—
90	Q₁₆×C₄₇	—	4	2	no	—
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	#41	1	2	yes	—
93	C₂×C₈×C₄₇	#41	1	2	yes	—
94	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₄₇	—	2	2	no	—
96	C₂²⋊C₄×C₄₇	—	4	1	no	—
97	C₄²×C₄₇	—	4	1	no	—
98	C₂²⋊Dic₄₇	—	4	1	no	—
99	#10625336	—	4	4	no	—
100	#10625336	—	4	4	no	—
101	(C₂×D₄₇)⋊C₄	—	4	4	no	—
102	(C₂×D₄₇)⋊C₄	—	4	4	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	Dic₄₇⋊C₄	—	4	2	no	—
106	Dic₄₇⋊C₄	—	4	2	no	—
107	C₄×Dic₄₇	—	4	1	no	—
108	C₄×Dic₄₇	D₈	1	2	yes	—
109	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
112	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
114	D₃₇₆	—	2	2	no	—
115	D₃₇₆	—	2	2	no	—
116	D₃₇₆	—	4	2	no	—
117	C₁₆×C₄₇	—	2	2	no	—
118	C₄₇⋊C₁₆	—	2	2	no	—
119	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
120	C₂²×D₄₇	—	4	2	no	—
121	C₂²×D₄₇	—	8	4	no	—
122	C₂²×D₄₇	—	8	2	no	—
123	Q₈×C₄₇	—	4	2	no	—
124	Q₈×C₄₇	—	4	2	no	—
125	Q₈×C₄₇	—	8	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	#124	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
136	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
137	D₁₈₈	—	4	2	no	—
138	D₁₈₈	—	4	2	no	—
139	D₁₈₈	—	8	2	no	—
140	Dic₉₄	—	8	4	no	—
141	C₈×C₄₇	#124	1	2	yes	—
142	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
145	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
146	#33706	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	#486	1	2	yes	—
149	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₄₇	—	8	2	no	—
151	C₂×D₄₇	—	8	4	no	—
152	C₂×D₄₇	—	16	4	no	—
153	C₄×C₄₇	—	8	2	no	—
154	C₄×C₄₇	—	8	2	no	—
155	C₄×C₄₇	#486	1	2	yes	—
156	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
157	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
158	Dic₄₇	—	2	2	no	—
159	Dic₄₇	—	2	2	no	—
160	Dic₄₇	—	8	2	no	—
161	#3160	—	47	1	no	—
162	C₂×#495	—	47	1	no	—
163	#2243	—	47	4	no	—
164	#2242	—	47	4	no	—
165	#2141	—	47	1	no	—
166	#2141	D₄₇	1	1	yes	characteristic, complemented
167	#2140	—	47	1	no	—
168	#2140	—	47	1	no	—
169	C₄×M₃₂	—	47	1	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
172	C₂×C₄₇	#3414	1	2	yes	—
173	D₄₇	—	16	2	no	—
174	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
175	C₂²×SD₁₆	—	94	1	no	—
176	C₂²×D₈	—	47	1	no	—
177	#533	—	94	1	no	—
178	C₂×#124	—	94	1	no	—
179	C₂×M₃₂	—	47	1	no	—
180	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
181	#500	—	47	2	no	—
182	#499	—	47	2	no	—
183	#499	—	94	1	no	—
184	#498	—	94	1	no	—
185	#495	—	47	2	no	—
186	#495	—	47	2	no	—
187	#495	—	94	1	no	—
188	#495	—	94	1	no	—
189	#427	—	47	1	no	—
190	#427	—	94	1	no	—
191	#427	C₂×D₄₇	1	1	yes	characteristic
192	C₂×C₄×C₈	—	47	1	no	—
193	#358	—	47	4	no	—
194	#358	C₂×D₄₇	1	4	yes	complemented
195	#357	—	47	4	no	—
196	#357	—	47	4	no	—
197	#346	—	47	2	no	—
198	C₄×C₁₆	—	47	2	no	—
199	C₄₇	#33706	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×D₄	—	94	1	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
206	C₂×SD₁₆	—	94	2	no	—
207	C₂×SD₁₆	—	94	2	no	—
208	C₂×SD₁₆	—	188	4	no	—
209	C₂×D₈	—	47	2	no	—
210	C₂×D₈	—	94	2	no	—
211	C₂×D₈	—	94	1	no	—
212	C₂×D₈	—	94	2	no	—
213	C₂²×C₈	—	47	1	no	—
214	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	94	2	no	—
216	#125	—	94	2	no	—
217	#124	—	94	2	no	—
218	#124	—	94	2	no	—
219	#124	—	188	2	no	—
220	C₂²⋊Q₈	—	188	2	no	—
221	C₄×Q₈	—	188	2	no	—
222	#117	—	2	1	no	—
223	#117	—	94	1	no	—
224	C₂×C₂²⋊C₄	—	188	2	no	—
225	C₂×C₄²	—	47	1	no	—
226	M₃₂	—	2	2	no	—
227	M₃₂	—	94	2	no	—
228	C₂×C₁₆	—	47	2	no	—
229	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
230	#107	—	47	2	no	—
231	#107	—	94	2	no	—
232	#107	C₂²×D₄₇	1	2	yes	—
233	#106	—	2	1	no	—
234	#106	—	94	2	no	—
235	#106	—	94	1	no	—
236	C₄×C₈	—	47	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	—	47	1	no	—
240	C₂⁴	—	188	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	188	2	no	—
246	C₂×D₄	—	94	2	no	—
247	C₂×D₄	—	188	2	no	—
248	C₂×D₄	—	188	1	no	—
249	C₂×D₄	—	188	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	2	no	—
256	SD₁₆	—	188	4	no	—
257	SD₁₆	—	188	4	no	—
258	D₈	—	94	2	no	—
259	D₈	—	94	2	no	—
260	D₈	—	188	2	no	—
261	C₂×C₈	—	47	4	no	—
262	C₂×C₈	—	47	1	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	#10625350	1	2	yes	—
265	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
266	C₂×C₈	#10625352	1	2	yes	—
267	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	94	2	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₄⋊C₄	—	188	2	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	188	4	no	—
275	C₂²⋊C₄	—	188	4	no	—
276	C₂²⋊C₄	—	188	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	47	2	no	—
279	C₄²	—	47	1	no	—
280	C₄²	—	47	1	no	—
281	C₄²	—	188	1	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	94	2	no	—
284	C₂³	—	376	4	no	—
285	C₂³	—	376	2	no	—
286	C₂³	—	188	2	no	—
287	C₂³	D₈×D₄₇	1	1	yes	characteristic
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	376	4	no	—
292	D₄	—	376	2	no	—
293	D₄	—	188	2	no	—
294	D₄	—	188	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	47	2	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	#13094010	1	2	yes	—
303	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
305	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
306	C₈	—	47	2	no	—
307	C₈	—	47	2	no	—
308	C₈	#13094010	1	2	yes	—
309	C₈	C₂×#10625352	1	1	yes	characteristic
310	C₈	C₂×#10625350	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	376	4	no	—
313	C₂²	—	376	2	no	—
314	C₂²	—	752	4	no	—
315	C₂²	#15689468	1	2	yes	—
316	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	94	2	no	—
320	C₄	—	94	2	no	—
321	C₄	—	376	2	no	—
322	C₄	#15689468	1	2	yes	—
323	C₄	C₂×#13094010	1	1	yes	characteristic
324	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	—	752	2	no	—
327	C₂	C₂×#15689468	1	1	yes	characteristic
328	C₂	#20256523	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2141	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	#2141	D₄₇
hypocenter	C₄₇	#33706
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2141

Complements

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

N	H ≅ G/N	type
1. #20253900	C₂	semidirect
3. #20245626	C₂	semidirect
4. #20245621	C₂	semidirect
5. #20245108	C₂	semidirect
8. M₃₂×Dic₄₇	C₂	semidirect
9. C₄₇×#2141	C₂	semidirect
15. #15689527	C₂²	semidirect
16. #15689524	C₂²	semidirect
26. #15688927	C₂²	semidirect
29. #15688898	C₂²	semidirect
30. C₁₆×Dic₄₇	C₂²	semidirect
34. C₄₇×#358	C₂²	semidirect
166. #2141	D₄₇	semidirect
194. #358	C₂×D₄₇	semidirect
199. C₄₇	#33706	semidirect

Maximal subgroups

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

subgroup	index
#20253900	2
C₂×#15689508	2
#20245626	2
#20245621	2
#20245108	2
#20245105	2
#20245103	2
M₃₂×Dic₄₇	2
C₄₇×#2141	2
#33706	47