Subgroups — #26371842

Order 12032.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253975	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253968	C₂	1	1	yes	maximal, characteristic
3	#20245644	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245617	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245613	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245485	C₂	1	4	yes	maximal, complemented
7	#20245239	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2254	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689678	—	2	1	no	—
11	#15689674	—	2	1	no	—
12	C₂×#13093969	C₂²	1	4	yes	—
13	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
14	#15689503	C₂²	1	1	yes	characteristic
15	#15689500	C₂²	1	1	yes	characteristic
16	#15689475	—	2	1	no	—
17	#15689471	—	2	1	no	—
18	#15689365	—	2	1	no	—
19	#15689365	C₂²	1	2	yes	—
20	#15689360	—	2	1	no	—
21	#15689360	C₂²	1	2	yes	—
22	#15689217	—	2	1	no	—
23	#15689217	C₂²	1	2	yes	—
24	#15689205	—	2	1	no	—
25	#15689205	C₂²	1	2	yes	—
26	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
27	#15688926	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×#500	C₂²	1	1	yes	characteristic
33	C₄₇×#367	C₂²	1	4	yes	—
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#10625354	—	4	1	no	—
36	C₂×Dic₁₈₈	—	2	1	no	—
37	C₂×Dic₁₈₈	C₂³	1	2	yes	—
38	C₂×D₃₇₆	—	2	1	no	—
39	C₂×D₃₇₆	C₂³	1	2	yes	—
40	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
41	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
42	#13094024	—	2	1	no	—
43	#13094021	—	2	1	no	—
44	#13094015	—	4	1	no	—
45	#13094013	—	4	1	no	—
46	#13094004	—	2	2	no	—
47	#13094000	—	4	1	no	—
48	#13093999	—	2	2	no	—
49	#13093999	—	4	1	no	—
50	Q₈×Dic₄₇	—	4	1	no	—
51	#13093989	—	2	2	no	—
52	#13093986	—	2	2	no	—
53	#13093986	—	4	1	no	—
54	#13093982	—	4	1	no	—
55	C₄²×D₄₇	D₄	1	1	yes	characteristic
56	#13093969	—	2	8	no	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
58	C₁₆×D₄₇	—	2	4	no	—
59	#13093947	—	2	2	no	—
60	#13093947	—	2	2	no	—
61	#13093944	—	2	2	no	—
62	#13093944	—	2	2	no	—
63	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
64	C₈×Dic₄₇	D₄	1	1	yes	characteristic
65	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₄₇	—	2	1	no	—
67	C₄⋊Q₈×C₄₇	—	2	1	no	—
68	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
69	#107×C₄₇	—	2	1	no	—
70	#107×C₄₇	—	2	1	no	—
71	#107×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	C₂³	1	2	yes	—
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₄₇	—	4	1	no	—
75	C₂×Q₈×C₄₇	—	4	1	no	—
76	#10625354	—	8	4	no	—
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×D₁₈₈	—	4	2	no	—
79	C₂×D₁₈₈	—	4	1	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×C₄×D₄₇	—	4	2	no	—
82	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
83	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
84	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₉₄	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	4	1	no	—
91	C₄⋊C₄×C₄₇	—	4	1	no	—
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	(C₂×D₄₇)⋊C₄	—	4	2	no	—
94	(C₂×D₄₇)⋊C₄	—	4	2	no	—
95	(C₂×D₄₇)⋊C₄	—	4	2	no	—
96	C₄⋊Dic₄₇	—	4	1	no	—
97	C₄⋊Dic₄₇	—	4	1	no	—
98	C₄⋊Dic₄₇	—	4	2	no	—
99	Dic₄₇⋊C₄	—	4	2	no	—
100	C₄×Dic₄₇	—	4	2	no	—
101	C₄×Dic₄₇	D₈	1	1	yes	characteristic
102	C₄×Dic₄₇	D₈	1	1	yes	characteristic
103	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
106	Dic₁₈₈	—	2	4	no	—
107	Dic₁₈₈	—	4	2	no	—
108	D₃₇₆	—	2	4	no	—
109	D₃₇₆	—	4	2	no	—
110	C₈×D₄₇	—	2	2	no	—
111	C₈×D₄₇	—	2	2	no	—
112	C₁₆×C₄₇	#41	1	4	yes	complemented
113	C₄₇⋊C₁₆	—	2	2	no	—
114	C₂²×D₄₇	—	4	2	no	—
115	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₄₇	—	8	2	no	—
117	Q₈×C₄₇	—	8	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
127	D₁₈₈	—	4	2	no	—
128	D₁₈₈	—	4	2	no	—
129	D₁₈₈	—	8	4	no	—
130	D₁₈₈	—	8	2	no	—
131	D₁₈₈	—	8	2	no	—
132	C₄×D₄₇	—	2	1	no	—
133	C₄×D₄₇	—	2	1	no	—
134	C₄×D₄₇	—	2	2	no	—
135	C₄×D₄₇	—	8	4	no	—
136	C₄×D₄₇	SD₃₂	1	4	yes	complemented
137	Dic₉₄	—	4	2	no	—
138	Dic₉₄	—	4	2	no	—
139	C₈×C₄₇	#135	1	2	yes	—
140	C₈×C₄₇	C₂×#41	1	2	yes	—
141	C₄₇⋊C₈	—	2	1	no	—
142	C₄₇⋊C₈	—	2	1	no	—
143	#33736	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₄₇	—	2	1	no	—
146	C₂×D₄₇	—	2	1	no	—
147	C₂×D₄₇	—	8	2	no	—
148	C₂×D₄₇	—	8	2	no	—
149	C₂×D₄₇	—	8	2	no	—
150	C₂×D₄₇	C₂×SD₃₂	1	2	yes	—
151	C₄×C₄₇	—	8	2	no	—
152	C₄×C₄₇	—	8	2	no	—
153	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
154	C₄×C₄₇	#508	1	2	yes	—
155	C₄×C₄₇	#532	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
157	Dic₄₇	—	2	1	no	—
158	Dic₄₇	—	2	1	no	—
159	Dic₄₇	—	8	2	no	—
160	Dic₄₇	C₂×SD₃₂	1	2	yes	—
161	#3170	—	47	1	no	—
162	#3169	—	47	1	no	—
163	#2254	D₄₇	1	1	yes	characteristic, complemented
164	#2248	—	47	1	no	—
165	#2241	—	47	1	no	—
166	#2240	—	47	1	no	—
167	#2217	—	47	4	no	—
168	C₄×SD₃₂	—	47	4	no	—
169	C₂×C₄×C₁₆	—	47	1	no	—
170	C₂×C₄₇	#3169	1	1	yes	characteristic
171	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
173	D₄₇	—	2	2	no	—
174	D₄₇	—	16	2	no	—
175	#533	—	94	1	no	—
176	#532	—	94	1	no	—
177	C₂×SD₃₂	—	47	4	no	—
178	C₂²×C₁₆	—	47	2	no	—
179	#500	C₂×D₄₇	1	1	yes	characteristic
180	#500	C₂×D₄₇	1	1	yes	characteristic
181	#494	—	47	1	no	—
182	#493	—	47	1	no	—
183	#487	—	94	1	no	—
184	#486	—	94	1	no	—
185	#467	—	47	2	no	—
186	#467	—	94	1	no	—
187	#466	—	47	2	no	—
188	#466	—	94	1	no	—
189	C₄×Q₁₆	—	47	2	no	—
190	C₄×Q₁₆	—	94	1	no	—
191	C₄×D₈	—	47	2	no	—
192	C₄×D₈	—	94	1	no	—
193	C₂×C₄×C₈	—	47	1	no	—
194	#367	C₂×D₄₇	1	4	yes	—
195	#358	—	47	4	no	—
196	#357	—	47	4	no	—
197	C₄×C₁₆	—	47	2	no	—
198	C₄×C₁₆	—	47	1	no	—
199	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
200	C₄₇	#33736	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	188	1	no	—
202	C₂×Q₁₆	—	47	2	no	—
203	C₂×Q₁₆	—	94	1	no	—
204	C₂×D₈	—	47	2	no	—
205	C₂×D₈	—	94	1	no	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	94	1	no	—
211	#127	—	94	1	no	—
212	#125	—	188	1	no	—
213	#124	—	188	1	no	—
214	C₂²⋊Q₈	—	94	2	no	—
215	C₂²⋊Q₈	—	188	1	no	—
216	#121	—	94	2	no	—
217	#121	—	188	1	no	—
218	C₄×Q₈	—	94	2	no	—
219	C₄×Q₈	—	188	1	no	—
220	C₄×D₄	—	94	2	no	—
221	C₄×D₄	—	188	1	no	—
222	#117	—	188	1	no	—
223	C₂×C₄²	—	47	1	no	—
224	SD₃₂	—	94	8	no	—
225	C₂×C₁₆	—	47	2	no	—
226	C₂×C₁₆	—	94	4	no	—
227	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂²×D₄₇	1	2	yes	—
231	#107	C₂²×D₄₇	1	2	yes	—
232	#103	—	94	2	no	—
233	#103	—	94	2	no	—
234	#102	—	94	2	no	—
235	#102	—	94	2	no	—
236	C₄×C₈	—	47	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
240	#41	—	376	4	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	94	2	no	—
244	C₂×D₄	—	94	2	no	—
245	C₂×D₄	—	188	2	no	—
246	C₂×D₄	—	188	1	no	—
247	C₂×D₄	—	188	1	no	—
248	C₂²×C₄	—	47	1	no	—
249	C₂²×C₄	—	47	1	no	—
250	C₂²×C₄	—	47	1	no	—
251	C₂²×C₄	—	188	2	no	—
252	Q₁₆	—	94	4	no	—
253	Q₁₆	—	188	2	no	—
254	D₈	—	94	4	no	—
255	D₈	—	188	2	no	—
256	C₂×C₈	—	47	1	no	—
257	C₂×C₈	—	47	1	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	—	94	2	no	—
260	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₄⋊C₄	—	188	2	no	—
268	C₄⋊C₄	—	188	1	no	—
269	C₄⋊C₄	—	188	1	no	—
270	C₂²⋊C₄	—	188	2	no	—
271	C₂²⋊C₄	—	188	2	no	—
272	C₂²⋊C₄	—	188	2	no	—
273	C₄²	—	47	1	no	—
274	C₄²	—	47	1	no	—
275	C₄²	—	47	1	no	—
276	C₄²	—	188	2	no	—
277	C₄²	D₄×D₄₇	1	1	yes	characteristic
278	C₁₆	—	94	2	no	—
279	C₁₆	#10625354	1	4	yes	complemented
280	C₂³	—	47	1	no	—
281	C₂³	—	188	2	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	188	2	no	—
285	Q₈	—	188	2	no	—
286	D₄	—	376	4	no	—
287	D₄	—	376	2	no	—
288	D₄	—	376	2	no	—
289	D₄	—	188	2	no	—
290	D₄	—	188	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	47	1	no	—
294	C₂×C₄	—	47	4	no	—
295	C₂×C₄	—	47	1	no	—
296	C₂×C₄	—	47	1	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	1	no	—
299	C₂×C₄	—	94	1	no	—
300	C₂×C₄	—	376	4	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
303	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
305	C₈	—	94	1	no	—
306	C₈	—	94	1	no	—
307	C₈	#13094047	1	2	yes	—
308	C₈	C₂×#10625354	1	2	yes	—
309	C₂²	—	47	2	no	—
310	C₂²	—	94	1	no	—
311	C₂²	—	94	1	no	—
312	C₂²	—	376	2	no	—
313	C₂²	—	376	2	no	—
314	C₂²	—	376	2	no	—
315	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	47	2	no	—
319	C₄	—	94	1	no	—
320	C₄	—	94	1	no	—
321	C₄	—	376	2	no	—
322	C₄	SD₃₂×D₄₇	1	2	yes	—
323	C₄	#15689554	1	2	yes	—
324	C₄	#15689674	1	1	yes	characteristic
325	C₄	C₂×#13094047	1	1	yes	characteristic
326	C₂	—	94	2	no	—
327	C₂	—	752	2	no	—
328	C₂	#20253968	1	1	yes	characteristic
329	C₂	C₂×SD₃₂×D₄₇	1	1	yes	characteristic
330	C₂	C₂×#15689554	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×#2254	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20253968	C₂
hypercenter	#2254	D₄₇
hypocenter	C₄₇	#33736
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
1. #20253975	C₂	semidirect
3. #20245644	C₂	semidirect
4. #20245617	C₂	semidirect
5. #20245613	C₂	semidirect
6. #20245485	C₂	semidirect
7. #20245239	C₂	semidirect
8. C₄×C₁₆×D₄₇	C₂	semidirect
9. C₄₇×#2254	C₂	semidirect
112. C₁₆×C₄₇	#41	semidirect
136. C₄×D₄₇	SD₃₂	semidirect
163. #2254	D₄₇	semidirect
200. C₄₇	#33736	semidirect
279. C₁₆	#10625354	semidirect

Maximal subgroups

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

subgroup	index
#20253975	2
#20253968	2
#20245644	2
#20245617	2
#20245613	2
#20245485	2
#20245239	2
C₄×C₁₆×D₄₇	2
C₄₇×#2254	2
#33736	47