Subgroups — #26371580

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253899	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253817	C₂	1	1	yes	maximal, characteristic
3	#20245621	C₂	1	4	yes	maximal, complemented
4	#20245615	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245613	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245612	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245094	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094047	—	2	1	no	—
11	C₂×#13094047	C₂²	1	2	yes	—
12	#15689678	—	2	1	no	—
13	#15689674	—	2	1	no	—
14	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689524	—	2	1	no	—
17	#15689524	C₂²	1	2	yes	—
18	#15689519	—	2	1	no	—
19	#15689508	—	2	1	no	—
20	#15689508	C₂²	1	2	yes	—
21	#15689500	C₂²	1	1	yes	characteristic
22	#15689497	—	2	1	no	—
23	#15689160	—	2	1	no	—
24	#15689160	C₂²	1	2	yes	—
25	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
26	#15688927	C₂²	1	4	yes	—
27	#15688925	C₂²	1	4	yes	—
28	#15688923	C₂²	1	4	yes	—
29	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₄₇	C₂²	1	2	yes	—
31	C₄₇×#500	C₂²	1	1	yes	characteristic
32	C₄₇×#494	C₂²	1	1	yes	characteristic
33	C₄₇×#358	C₂²	1	4	yes	—
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#10625354	—	2	2	no	—
36	#13094047	—	2	8	no	—
37	#13094047	—	4	4	no	—
38	C₂×#10625336	—	2	2	no	—
39	C₂×#10625336	—	2	2	no	—
40	C₂×D₃₇₆	—	2	1	no	—
41	C₂×D₃₇₆	C₂³	1	2	yes	—
42	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
44	#13094024	—	2	1	no	—
45	#13094021	—	2	1	no	—
46	#13094015	—	2	2	no	—
47	#13094013	—	2	2	no	—
48	#13094004	—	4	1	no	—
49	#13094000	—	4	1	no	—
50	#13093999	—	4	1	no	—
51	#13093999	—	4	1	no	—
52	Q₈×Dic₄₇	—	4	1	no	—
53	#13093989	—	4	1	no	—
54	#13093986	—	4	1	no	—
55	#13093986	—	4	1	no	—
56	#13093982	—	2	2	no	—
57	C₄²×D₄₇	D₄	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
59	C₁₆×D₄₇	—	2	4	no	—
60	#13093958	—	2	1	no	—
61	#13093958	C₂³	1	2	yes	—
62	#13093955	—	2	2	no	—
63	#13093955	—	2	2	no	—
64	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₄₇	D₄	1	1	yes	characteristic
66	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₄₇	—	2	1	no	—
68	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	—	2	1	no	—
71	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	C₂³	1	2	yes	—
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂×Q₈×C₄₇	—	4	1	no	—
77	#10625354	—	4	4	no	—
78	#10625354	—	4	4	no	—
79	C₂×D₁₈₈	—	2	2	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×D₁₈₈	—	4	1	no	—
82	C₂×D₁₈₈	—	4	1	no	—
83	C₂×C₄×D₄₇	—	4	2	no	—
84	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
85	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
86	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₉₄	—	4	1	no	—
88	Q₁₆×C₄₇	—	2	4	no	—
89	Q₁₆×C₄₇	—	4	2	no	—
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	4	1	no	—
94	C₄⋊C₄×C₄₇	—	4	1	no	—
95	C₄⋊C₄×C₄₇	—	4	1	no	—
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	#10625336	—	4	4	no	—
98	#10625336	—	4	4	no	—
99	(C₂×D₄₇)⋊C₄	—	4	2	no	—
100	(C₂×D₄₇)⋊C₄	—	4	2	no	—
101	(C₂×D₄₇)⋊C₄	—	4	2	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	4	1	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	Dic₄₇⋊C₄	—	4	2	no	—
106	C₄×Dic₄₇	—	4	2	no	—
107	C₄×Dic₄₇	D₈	1	1	yes	characteristic
108	C₄×Dic₄₇	D₈	1	1	yes	characteristic
109	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	D₃₇₆	—	2	4	no	—
113	D₃₇₆	—	4	2	no	—
114	C₈×D₄₇	SD₁₆	1	4	yes	—
115	C₈×D₄₇	#41	1	4	yes	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₄₇⋊C₁₆	—	2	2	no	—
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	4	2	no	—
122	Q₈×C₄₇	—	8	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
132	D₁₈₈	—	4	2	no	—
133	D₁₈₈	—	4	2	no	—
134	D₁₈₈	—	8	4	no	—
135	D₁₈₈	—	8	2	no	—
136	D₁₈₈	—	8	2	no	—
137	C₄×D₄₇	—	2	2	no	—
138	C₄×D₄₇	—	2	2	no	—
139	C₄×D₄₇	—	8	4	no	—
140	C₄×D₄₇	#124	1	2	yes	—
141	C₄×D₄₇	C₂×SD₁₆	1	2	yes	—
142	Dic₉₄	—	8	2	no	—
143	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
144	C₈×C₄₇	C₂×#41	1	2	yes	—
145	C₄₇⋊C₈	#124	1	2	yes	—
146	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
147	#33697	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₄₇	—	2	1	no	—
150	C₂×D₄₇	—	2	1	no	—
151	C₂×D₄₇	—	8	2	no	—
152	C₂×D₄₇	—	8	2	no	—
153	C₂×D₄₇	—	8	2	no	—
154	C₂×D₄₇	#486	1	2	yes	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	#508	1	2	yes	—
158	C₄×C₄₇	#508	1	2	yes	—
159	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
160	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
161	Dic₄₇	—	2	1	no	—
162	Dic₄₇	—	2	1	no	—
163	Dic₄₇	—	8	2	no	—
164	Dic₄₇	#486	1	2	yes	—
165	#3160	—	47	1	no	—
166	#3147	—	47	1	no	—
167	#2242	—	47	4	no	—
168	#2241	—	47	1	no	—
169	#2241	D₄₇	1	1	yes	characteristic, complemented
170	#2240	—	47	1	no	—
171	#2240	—	47	1	no	—
172	#2138	—	47	4	no	—
173	C₂×C₄×C₁₆	—	47	1	no	—
174	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
177	D₄₇	—	2	2	no	—
178	D₄₇	—	16	2	no	—
179	C₂×#135	—	47	2	no	—
180	C₂×#135	—	94	1	no	—
181	#533	—	94	1	no	—
182	#532	—	94	1	no	—
183	C₂²×C₁₆	—	47	2	no	—
184	#500	—	47	1	no	—
185	#500	C₂×D₄₇	1	1	yes	characteristic
186	#499	—	47	2	no	—
187	#499	—	94	1	no	—
188	#498	—	94	1	no	—
189	#495	—	47	2	no	—
190	#495	—	94	1	no	—
191	#494	C₂×D₄₇	1	1	yes	characteristic
192	#493	—	47	1	no	—
193	#492	—	94	1	no	—
194	#427	—	47	2	no	—
195	#427	—	94	1	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	#358	—	47	4	no	—
198	#358	C₂×D₄₇	1	4	yes	—
199	#357	—	47	4	no	—
200	#357	—	47	4	no	—
201	C₄×C₁₆	—	47	2	no	—
202	C₄×C₁₆	—	47	1	no	—
203	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
204	C₄₇	#33697	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	94	2	no	—
206	#135	—	94	8	no	—
207	#135	—	188	4	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
210	C₂×SD₁₆	—	94	2	no	—
211	C₂×SD₁₆	—	94	2	no	—
212	C₂×D₈	—	47	2	no	—
213	C₂×D₈	—	94	1	no	—
214	C₂²×C₈	—	47	1	no	—
215	C₂²×C₈	—	47	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	94	1	no	—
219	#127	—	94	1	no	—
220	#125	—	94	2	no	—
221	#124	—	94	2	no	—
222	C₂²⋊Q₈	—	188	1	no	—
223	C₂²⋊Q₈	—	188	1	no	—
224	#121	—	188	1	no	—
225	#121	—	188	1	no	—
226	C₄×Q₈	—	188	1	no	—
227	C₄×Q₈	—	188	1	no	—
228	C₄×D₄	—	188	1	no	—
229	C₄×D₄	—	188	1	no	—
230	#117	—	94	2	no	—
231	C₂×C₄²	—	47	1	no	—
232	C₂×C₁₆	—	47	2	no	—
233	C₂×C₁₆	—	94	4	no	—
234	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	47	2	no	—
237	#107	—	94	1	no	—
238	#107	C₂²×D₄₇	1	2	yes	—
239	#106	—	94	2	no	—
240	#106	—	94	2	no	—
241	C₄×C₈	—	47	1	no	—
242	C₄×C₈	—	47	1	no	—
243	C₄×C₈	—	47	1	no	—
244	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	#41	—	188	4	no	—
246	#41	—	188	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×D₄	—	94	2	no	—
251	C₂×D₄	—	188	2	no	—
252	C₂×D₄	—	188	1	no	—
253	C₂×D₄	—	188	1	no	—
254	C₂²×C₄	—	47	1	no	—
255	C₂²×C₄	—	47	1	no	—
256	C₂²×C₄	—	47	1	no	—
257	C₂²×C₄	—	188	2	no	—
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	SD₁₆	—	188	4	no	—
261	SD₁₆	—	188	4	no	—
262	D₈	—	94	4	no	—
263	D₈	—	188	2	no	—
264	C₂×C₈	—	47	1	no	—
265	C₂×C₈	—	47	1	no	—
266	C₂×C₈	—	47	4	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	94	2	no	—
275	C₄⋊C₄	—	188	2	no	—
276	C₄⋊C₄	—	188	2	no	—
277	C₄⋊C₄	—	188	1	no	—
278	C₂²⋊C₄	—	188	2	no	—
279	C₂²⋊C₄	—	188	2	no	—
280	C₂²⋊C₄	—	188	2	no	—
281	C₄²	—	47	1	no	—
282	C₄²	—	47	1	no	—
283	C₄²	—	47	1	no	—
284	C₄²	—	188	2	no	—
285	C₄²	D₄×D₄₇	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	94	2	no	—
288	C₂³	—	47	1	no	—
289	C₂³	—	188	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	376	2	no	—
294	D₄	—	376	4	no	—
295	D₄	—	376	2	no	—
296	D₄	—	376	2	no	—
297	D₄	—	188	2	no	—
298	D₄	—	188	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	47	1	no	—
302	C₂×C₄	—	47	2	no	—
303	C₂×C₄	—	47	2	no	—
304	C₂×C₄	—	47	1	no	—
305	C₂×C₄	—	47	1	no	—
306	C₂×C₄	—	94	2	no	—
307	C₂×C₄	—	94	2	no	—
308	C₂×C₄	—	376	4	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
311	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
313	C₈	—	47	2	no	—
314	C₈	—	47	2	no	—
315	C₈	SD₁₆×D₄₇	1	2	yes	—
316	C₈	#41×D₄₇	1	2	yes	—
317	C₂²	—	47	2	no	—
318	C₂²	—	94	1	no	—
319	C₂²	—	94	1	no	—
320	C₂²	—	376	2	no	—
321	C₂²	—	376	2	no	—
322	C₂²	—	376	2	no	—
323	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	47	2	no	—
327	C₄	—	94	1	no	—
328	C₄	—	94	1	no	—
329	C₄	—	376	2	no	—
330	C₄	#15689554	1	2	yes	—
331	C₄	#15689556	1	2	yes	—
332	C₄	#124×D₄₇	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
334	C₂	—	94	2	no	—
335	C₂	—	752	2	no	—
336	C₂	#486×D₄₇	1	1	yes	characteristic
337	C₂	C₂×#15689554	1	1	yes	characteristic
338	C₂	C₂×#15689556	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2241	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20253817	C₂
hypercenter	#2241	D₄₇
hypocenter	C₄₇	#33697
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #20253899	C₂	semidirect
3. #20245621	C₂	semidirect
4. #20245615	C₂	semidirect
5. #20245613	C₂	semidirect
6. #20245612	C₂	semidirect
7. #20245094	C₂	semidirect
8. C₄×C₁₆×D₄₇	C₂	semidirect
9. C₄₇×#2241	C₂	semidirect
169. #2241	D₄₇	semidirect
204. C₄₇	#33697	semidirect

Maximal subgroups

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

subgroup	index
#20253899	2
#20253817	2
#20245621	2
#20245615	2
#20245613	2
#20245612	2
#20245094	2
C₄×C₁₆×D₄₇	2
C₄₇×#2241	2
#33697	47