Subgroups — #26116147

Order 12032.

516 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	#20253947	C₂	1	1	yes	maximal, characteristic
2	#20244364	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244215	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241842	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241838	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689680	C₂²	1	1	yes	characteristic
9	#15689673	C₂²	1	1	yes	characteristic
10	#15689521	C₄	1	1	yes	characteristic, complemented
11	#15689496	C₄	1	1	yes	characteristic, complemented
12	#15689474	C₄	1	1	yes	characteristic, complemented
13	#15689470	C₄	1	1	yes	characteristic, complemented
14	#15689352	—	2	2	no	—
15	C₄×#10625325	—	2	2	no	—
16	C₂×#13093950	—	2	2	no	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	—	2	2	no	—
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688884	—	2	2	no	—
21	#15688853	—	2	2	no	—
22	#15688850	—	2	2	no	—
23	C₄₇×#423	C₂²	1	1	yes	characteristic
24	C₄₇×#423	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#334	—	2	2	no	—
27	C₂×#10625350	—	2	1	no	—
28	C₂×#10625325	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	#13094024	C₂×C₄	1	1	yes	characteristic
31	#13094024	C₂×C₄	1	1	yes	characteristic
32	#13094024	C₂×C₄	1	1	yes	characteristic
33	#13094021	C₂×C₄	1	1	yes	characteristic
34	#13094016	—	2	1	no	—
35	#13094012	—	2	1	no	—
36	#13094003	—	2	1	no	—
37	#13094003	—	2	1	no	—
38	#13093998	—	2	1	no	—
39	#13093998	—	2	1	no	—
40	C₄×Dic₉₄	—	2	1	no	—
41	C₄×Dic₉₄	—	2	1	no	—
42	C₄×D₁₈₈	—	2	1	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	#13093983	—	2	1	no	—
45	C₂×C₄×Dic₄₇	—	2	1	no	—
46	C₂×C₄×Dic₄₇	—	2	1	no	—
47	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
48	#13093956	—	2	2	no	—
49	#13093950	—	4	4	no	—
50	#13093949	—	2	2	no	—
51	#13093946	—	2	2	no	—
52	C₄×C₄₇⋊C₈	—	2	1	no	—
53	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
54	C₈×Dic₄₇	—	2	8	no	—
55	C₂×M₁₆×C₄₇	—	2	1	no	—
56	C₂×M₁₆×C₄₇	—	2	1	no	—
57	C₂²×C₈×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	C₄⋊C₈×C₄₇	—	2	1	no	—
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₇	D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	4	1	no	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	#10625350	—	4	2	no	—
72	#10625350	—	4	2	no	—
73	C₂×D₁₈₈	—	2	1	no	—
74	C₂×D₁₈₈	—	2	1	no	—
75	C₂×D₁₈₈	—	2	1	no	—
76	C₂×C₄×D₄₇	—	4	1	no	—
77	C₂×Dic₉₄	—	2	1	no	—
78	C₂×Dic₉₄	—	2	1	no	—
79	C₂×Dic₉₄	—	2	1	no	—
80	M₁₆×C₄₇	—	4	2	no	—
81	M₁₆×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	—	4	1	no	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	#41	1	1	yes	characteristic
91	C₂²⋊Dic₄₇	—	4	1	no	—
92	(C₂×D₄₇)⋊C₄	—	4	1	no	—
93	(C₂×D₄₇)⋊C₄	—	4	1	no	—
94	C₄⋊Dic₄₇	—	2	1	no	—
95	C₄⋊Dic₄₇	—	2	1	no	—
96	C₄⋊Dic₄₇	—	2	1	no	—
97	C₄⋊Dic₄₇	—	2	1	no	—
98	C₄⋊Dic₄₇	—	2	1	no	—
99	C₄⋊Dic₄₇	—	2	1	no	—
100	Dic₄₇⋊C₄	—	4	1	no	—
101	Dic₄₇⋊C₄	—	4	1	no	—
102	C₄×Dic₄₇	—	2	4	no	—
103	C₄×Dic₄₇	—	2	4	no	—
104	C₄×Dic₄₇	—	2	2	no	—
105	C₄×Dic₄₇	—	4	1	no	—
106	C₄×Dic₄₇	SD₁₆	1	4	yes	complemented
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	#10625325	—	4	4	no	—
110	C₂³×C₄₇	#121	1	1	yes	characteristic
111	C₂²×D₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₄₇	#135	1	2	yes	—
122	C₂²⋊D₄₇	—	8	2	no	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	1	no	—
127	C₂×Dic₄₇	—	4	1	no	—
128	C₂×Dic₄₇	—	4	1	no	—
129	D₁₈₈	—	4	2	no	—
130	D₁₈₈	—	4	2	no	—
131	D₁₈₈	—	4	2	no	—
132	C₄×D₄₇	—	8	1	no	—
133	C₄×D₄₇	—	8	1	no	—
134	Dic₉₄	—	4	2	no	—
135	Dic₉₄	—	4	2	no	—
136	Dic₉₄	—	4	2	no	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	—	8	1	no	—
139	C₈×C₄₇	—	8	1	no	—
140	C₈×C₄₇	#104	1	4	yes	complemented
141	C₄₇⋊C₈	—	2	4	no	—
142	#9843	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	#390	1	1	yes	characteristic
146	C₂²×C₄₇	#465	1	2	yes	—
147	C₂×D₄₇	—	8	1	no	—
148	C₂×D₄₇	—	8	1	no	—
149	C₂×D₄₇	—	8	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	C₂×#104	1	2	yes	—
153	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
154	C₄×C₄₇	#443	1	2	yes	—
155	C₄×C₄₇	#492	1	1	yes	characteristic
156	C₄×C₄₇	#495	1	1	yes	characteristic
157	Dic₄₇	—	2	8	no	—
158	Dic₄₇	—	8	1	no	—
159	Dic₄₇	—	8	1	no	—
160	Dic₄₇	—	8	1	no	—
161	#3167	—	47	1	no	—
162	#1994	—	47	1	no	—
163	#1962	—	47	1	no	—
164	C₄²×C₈	—	47	1	no	—
165	#1566	D₄₇	1	1	yes	characteristic, complemented
166	#1493	—	47	1	no	—
167	#1492	—	47	1	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	#1962	1	1	yes	characteristic
170	C₂×C₄₇	#1968	1	1	yes	characteristic
171	C₂×C₄₇	#1974	1	1	yes	characteristic
172	D₄₇	—	16	1	no	—
173	#533	—	47	1	no	—
174	#532	—	47	1	no	—
175	#498	—	47	1	no	—
176	#492	—	47	1	no	—
177	#487	—	47	1	no	—
178	#486	—	47	1	no	—
179	#465	—	94	2	no	—
180	C₄×SD₁₆	—	94	2	no	—
181	#423	C₂×D₄₇	1	1	yes	characteristic
182	#423	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#104	—	94	2	no	—
184	C₂×C₄×C₈	—	47	1	no	—
185	C₂×C₄×C₈	—	94	2	no	—
186	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
187	C₄³	—	47	1	no	—
188	#339	—	94	2	no	—
189	#334	—	2	2	no	—
190	Q₈⋊C₈	—	94	2	no	—
191	D₄⋊C₈	—	94	2	no	—
192	C₄₇	#9843	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	94	1	no	—
194	C₂×SD₁₆	—	94	2	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂²×C₈	—	94	1	no	—
198	C₂²×C₈	D₁₈₈	1	2	yes	—
199	C₄⋊Q₈	—	47	1	no	—
200	C₄⋊Q₈	—	47	1	no	—
201	C₄⋊Q₈	—	47	1	no	—
202	#127	—	47	1	no	—
203	#125	—	94	1	no	—
204	#124	—	94	1	no	—
205	C₂²⋊Q₈	—	94	1	no	—
206	C₂²⋊Q₈	—	94	1	no	—
207	#121	—	94	1	no	—
208	#121	—	94	1	no	—
209	C₄×Q₈	—	94	1	no	—
210	C₄×Q₈	—	94	1	no	—
211	C₄×D₄	—	94	1	no	—
212	C₄×D₄	—	94	1	no	—
213	#117	—	94	1	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	—	94	1	no	—
216	C₂×C₄²	—	94	1	no	—
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	#106	—	94	2	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	#104	—	188	4	no	—
224	#103	—	94	2	no	—
225	#102	—	94	2	no	—
226	C₄×C₈	—	47	2	no	—
227	C₄×C₈	—	94	8	no	—
228	C₄×C₈	—	94	1	no	—
229	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
230	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
232	#41	—	188	2	no	—
233	#41	—	188	2	no	—
234	C₂×Q₈	—	94	1	no	—
235	C₂×Q₈	—	94	1	no	—
236	C₂×Q₈	—	94	1	no	—
237	C₂×D₄	—	94	1	no	—
238	C₂×D₄	—	94	1	no	—
239	C₂×D₄	—	94	1	no	—
240	C₂×D₄	—	188	1	no	—
241	C₂²×C₄	—	94	2	no	—
242	C₂²×C₄	—	188	1	no	—
243	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
245	C₂²×C₄	#10625354	1	1	yes	characteristic
246	SD₁₆	—	188	4	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
255	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
256	C₄⋊C₄	—	94	1	no	—
257	C₄⋊C₄	—	94	1	no	—
258	C₄⋊C₄	—	94	1	no	—
259	C₄⋊C₄	—	94	1	no	—
260	C₄⋊C₄	—	94	1	no	—
261	C₄⋊C₄	—	94	1	no	—
262	C₄⋊C₄	—	188	1	no	—
263	C₄⋊C₄	—	188	1	no	—
264	C₂²⋊C₄	—	188	1	no	—
265	C₂²⋊C₄	—	188	1	no	—
266	C₂²⋊C₄	—	188	1	no	—
267	C₄²	—	47	4	no	—
268	C₄²	—	94	4	no	—
269	C₄²	—	94	4	no	—
270	C₄²	—	94	2	no	—
271	C₄²	—	188	1	no	—
272	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₄²	D₄×D₄₇	1	1	yes	characteristic
274	C₄²	#10625354	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂³	—	188	1	no	—
277	C₂³	#13093999	1	1	yes	characteristic
278	Q₈	—	188	2	no	—
279	Q₈	—	188	2	no	—
280	Q₈	—	188	2	no	—
281	D₄	—	376	2	no	—
282	D₄	—	188	2	no	—
283	D₄	—	188	2	no	—
284	D₄	—	188	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	376	1	no	—
291	C₂×C₄	—	376	1	no	—
292	C₂×C₄	—	188	1	no	—
293	C₂×C₄	—	188	1	no	—
294	C₂×C₄	—	188	1	no	—
295	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	#13093986	1	1	yes	characteristic
297	C₂×C₄	#13093986	1	1	yes	characteristic
298	C₂×C₄	#13093999	1	1	yes	characteristic
299	C₂×C₄	#13094013	1	1	yes	characteristic
300	C₂×C₄	#13094020	1	1	yes	characteristic
301	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
302	C₂×C₄	#13094045	1	2	yes	—
303	C₈	—	2	2	no	—
304	C₈	—	8	1	no	—
305	C₈	—	8	1	no	—
306	C₈	—	94	4	no	—
307	C₈	#13093950	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	376	1	no	—
311	C₂²	—	376	1	no	—
312	C₂²	—	376	1	no	—
313	C₂²	#15689024	1	1	yes	characteristic
314	C₂²	#15689353	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	94	8	no	—
318	C₄	—	376	1	no	—
319	C₄	—	376	1	no	—
320	C₄	—	376	1	no	—
321	C₄	C₂×#13093950	1	2	yes	—
322	C₄	#15689210	1	2	yes	—
323	C₄	#15689239	1	2	yes	—
324	C₄	#15689494	1	1	yes	characteristic
325	C₄	#15689504	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	752	1	no	—
328	C₂	#20244215	1	1	yes	characteristic
329	C₂	#20244240	1	1	yes	characteristic
330	C₂	#20244274	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1566	C₂
socle	C₂²×C₄₇	#390
prime radical	#15689673	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1566	D₄₇
hypocenter	C₄₇	#9843
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
2. #20244364	C₂	semidirect
4. C₄×C₈×Dic₄₇	C₂	semidirect
5. #20241842	C₂	semidirect
7. C₄₇×#1566	C₂	semidirect
10. #15689521	C₄	semidirect
11. #15689496	C₄	semidirect
12. #15689474	C₄	semidirect
13. #15689470	C₄	semidirect
106. C₄×Dic₄₇	SD₁₆	semidirect
140. C₈×C₄₇	#104	semidirect
165. #1566	D₄₇	semidirect
192. C₄₇	#9843	semidirect
307. C₈	#13093950	semidirect

Maximal subgroups

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

subgroup	index
#20253947	2
#20244364	2
#20244215	2
C₄×C₈×Dic₄₇	2
#20241842	2
#20241838	2
C₄₇×#1566	2
#9843	47