Subgroups — #26107330

Order 12032.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252357	C₂	1	1	yes	maximal, characteristic
2	#20243076	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241974	C₂	1	1	yes	maximal, characteristic
4	#20241875	C₂	1	1	yes	maximal, characteristic
5	#20241678	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241280	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1302	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689657	C₂²	1	1	yes	characteristic
9	#15689250	C₄	1	1	yes	characteristic, complemented
10	#15689247	C₄	1	1	yes	characteristic, complemented
11	#15689200	C₄	1	1	yes	characteristic, complemented
12	#15689191	—	2	1	no	—
13	#15689188	C₄	1	1	yes	characteristic, complemented
14	(C₂²⋊D₄₇)⋊C₈	—	2	1	no	—
15	#15689185	—	2	1	no	—
16	#15689080	—	2	1	no	—
17	C₄×#10625329	C₂²	1	1	yes	characteristic
18	C₂×#13093934	—	2	1	no	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₇	—	2	1	no	—
21	#15688969	C₂²	1	1	yes	characteristic
22	#15688864	C₄	1	2	yes	complemented
23	#15688862	C₄	1	2	yes	complemented
24	#15688861	C₄	1	2	yes	—
25	#15688858	C₄	1	2	yes	—
26	#15688845	—	2	1	no	—
27	#15688844	—	2	1	no	—
28	C₄₇×#432	C₂²	1	1	yes	characteristic
29	C₄₇×#431	C₂²	1	1	yes	characteristic
30	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#10625329	—	2	1	no	—
32	C₂×#10625324	—	2	1	no	—
33	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₄₇	—	2	1	no	—
36	#13094018	—	2	1	no	—
37	#13094016	C₂×C₄	1	1	yes	characteristic
38	#13094012	C₂×C₄	1	1	yes	characteristic
39	#13094007	—	2	1	no	—
40	#13094003	C₂×C₄	1	1	yes	characteristic
41	#13093998	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
43	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₂²⋊D₄₇	—	2	1	no	—
45	#13093981	—	2	1	no	—
46	C₂×C₄×Dic₄₇	—	2	1	no	—
47	Dic₄₇⋊C₈	—	2	1	no	—
48	Dic₄₇⋊C₈	—	2	1	no	—
49	(C₂×D₄₇)⋊C₈	—	2	1	no	—
50	(C₂×D₄₇)⋊C₈	—	2	1	no	—
51	#13093935	C₂×C₄	1	2	yes	—
52	#13093934	—	2	4	no	—
53	#13093934	—	2	1	no	—
54	C₄×C₄₇⋊C₈	—	2	1	no	—
55	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
59	C₈×Dic₄₇	—	2	4	no	—
60	C₈×Dic₄₇	—	2	1	no	—
61	C₄₇⋊#95	—	2	1	no	—
62	C₄₇⋊#95	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₄₇	—	2	1	no	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₇	—	2	1	no	—
66	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₄₇	—	2	1	no	—
69	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
70	#97×C₄₇	—	2	1	no	—
71	#97×C₄₇	—	2	1	no	—
72	#97×C₄₇	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₄₇	—	2	1	no	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
77	C₂×C₂²⋊D₄₇	—	2	1	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₄₇	—	2	1	no	—
81	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	—	2	1	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	C₂²⋊Dic₄₇	—	2	1	no	—
93	(C₂×D₄₇)⋊C₄	—	2	1	no	—
94	(C₂×D₄₇)⋊C₄	—	2	1	no	—
95	(C₂×D₄₇)⋊C₄	—	2	1	no	—
96	(C₂×D₄₇)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
98	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
99	Dic₄₇⋊C₄	—	2	1	no	—
100	Dic₄₇⋊C₄	—	2	1	no	—
101	Dic₄₇⋊C₄	—	2	1	no	—
102	Dic₄₇⋊C₄	—	2	1	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	2	2	no	—
105	C₄×Dic₄₇	—	2	1	no	—
106	#10625329	—	4	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	#10625324	—	4	2	no	—
115	C₈×D₄₇	—	4	2	no	—
116	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂²⋊D₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	1	no	—
131	C₂×Dic₄₇	—	2	1	no	—
132	C₂×Dic₄₇	—	2	1	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	D₁₈₈	—	4	1	no	—
136	C₄×D₄₇	—	4	1	no	—
137	C₄×D₄₇	—	4	1	no	—
138	Dic₉₄	—	4	1	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₈×C₄₇	—	4	2	no	—
141	C₈×C₄₇	—	4	1	no	—
142	C₈×C₄₇	—	4	1	no	—
143	C₄₇⋊C₈	—	2	1	no	—
144	C₄₇⋊C₈	—	2	1	no	—
145	C₄₇⋊C₈	—	4	2	no	—
146	C₄₇⋊C₈	#104	1	2	yes	—
147	C₄₇⋊C₈	#104	1	2	yes	—
148	#8249	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₇	#444	1	2	yes	—
153	C₂×D₄₇	—	4	1	no	—
154	C₂×D₄₇	—	4	1	no	—
155	C₂×D₄₇	—	4	1	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
162	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
163	Dic₄₇	—	4	4	no	—
164	Dic₄₇	—	4	1	no	—
165	Dic₄₇	—	4	1	no	—
166	Dic₄₇	—	4	1	no	—
167	#2966	—	47	1	no	—
168	#1726	—	47	1	no	—
169	#1524	—	47	1	no	—
170	#1500	—	47	1	no	—
171	#1451	—	47	1	no	—
172	#1302	D₄₇	1	1	yes	characteristic, complemented
173	#1300	—	47	1	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	#1757	1	1	yes	characteristic
176	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
177	C₂×C₄₇	#1765	1	1	yes	characteristic
178	D₄₇	—	8	1	no	—
179	#529	—	47	1	no	—
180	#446	—	47	1	no	—
181	C₈×Q₈	—	47	1	no	—
182	#436	—	47	1	no	—
183	#435	—	94	1	no	—
184	C₈×D₄	—	47	1	no	—
185	C₈×D₄	—	94	1	no	—
186	#433	—	94	1	no	—
187	#432	C₂×D₄₇	1	1	yes	characteristic
188	#431	C₂×D₄₇	1	1	yes	characteristic
189	#405	—	94	1	no	—
190	C₄×M₁₆	—	47	1	no	—
191	C₂×#97	—	94	1	no	—
192	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
193	C₂×C₄×C₈	—	47	1	no	—
194	C₂×C₄×C₈	—	94	1	no	—
195	#376	—	47	1	no	—
196	#330	—	47	2	no	—
197	#329	—	47	2	no	—
198	#328	—	47	2	no	—
199	#327	—	47	2	no	—
200	#322	—	94	1	no	—
201	#322	—	94	1	no	—
202	C₄₇	#8249	1	1	yes	characteristic, complemented, Sylow
203	C₂×M₁₆	—	94	1	no	—
204	C₂×M₁₆	—	94	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₂²×C₈	—	94	1	no	—
209	#126	—	94	1	no	—
210	#125	—	47	1	no	—
211	#124	—	47	1	no	—
212	#123	—	94	1	no	—
213	C₂²⋊Q₈	—	47	1	no	—
214	#121	—	47	1	no	—
215	C₄×Q₈	—	47	1	no	—
216	C₄×D₄	—	47	1	no	—
217	C₄×D₄	—	94	1	no	—
218	#117	—	94	1	no	—
219	C₂×C₄²	—	94	1	no	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	94	1	no	—
223	C₄⋊C₈	—	94	1	no	—
224	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
225	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	94	1	no	—
228	C₂²⋊C₈	—	94	1	no	—
229	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
230	#97	—	2	1	no	—
231	#97	—	2	1	no	—
232	#97	—	47	2	no	—
233	#97	—	94	1	no	—
234	#97	—	94	4	no	—
235	#97	C₄×D₄₇	1	2	yes	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	—	47	1	no	—
240	C₄×C₈	—	47	1	no	—
241	C₄×C₈	—	94	1	no	—
242	C₄×C₈	—	94	4	no	—
243	C₄×C₈	—	94	1	no	—
244	#95	—	47	2	no	—
245	#95	—	94	1	no	—
246	C₂×Q₈	—	47	1	no	—
247	C₂×D₄	—	47	1	no	—
248	C₂×D₄	—	94	1	no	—
249	C₂²×C₄	—	94	2	no	—
250	C₂²×C₄	—	94	1	no	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	#10625354	1	1	yes	characteristic
254	M₁₆	—	188	2	no	—
255	M₁₆	—	188	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	47	1	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	—	47	1	no	—
265	C₂×C₈	—	47	1	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	—	188	2	no	—
270	C₄⋊C₄	—	47	1	no	—
271	C₄⋊C₄	—	47	1	no	—
272	C₄⋊C₄	—	94	1	no	—
273	C₄⋊C₄	—	94	1	no	—
274	C₄⋊C₄	—	94	1	no	—
275	C₄⋊C₄	—	94	1	no	—
276	C₂²⋊C₄	—	94	1	no	—
277	C₂²⋊C₄	—	94	1	no	—
278	C₂²⋊C₄	—	94	1	no	—
279	C₂²⋊C₄	—	94	1	no	—
280	C₂²⋊C₄	—	94	1	no	—
281	C₄²	—	94	2	no	—
282	C₄²	—	94	2	no	—
283	C₄²	—	94	1	no	—
284	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
286	C₄²	D₄×D₄₇	1	1	yes	characteristic
287	C₄²	#10625352	1	1	yes	characteristic
288	C₂³	—	94	1	no	—
289	C₂³	#13093986	1	1	yes	characteristic
290	Q₈	—	188	1	no	—
291	D₄	—	188	1	no	—
292	D₄	—	188	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	94	4	no	—
298	C₂×C₄	—	94	1	no	—
299	C₂×C₄	—	94	1	no	—
300	C₂×C₄	—	94	1	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	—	188	1	no	—
304	C₂×C₄	—	188	1	no	—
305	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	#13093982	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	#13093986	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	1	no	—
314	C₈	—	4	1	no	—
315	C₈	—	47	2	no	—
316	C₈	—	47	2	no	—
317	C₈	—	94	1	no	—
318	C₈	—	94	1	no	—
319	C₈	—	188	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	188	1	no	—
323	C₂²	—	188	1	no	—
324	C₂²	—	188	1	no	—
325	C₂²	#15688970	1	1	yes	characteristic
326	C₂²	#15689244	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	188	4	no	—
333	C₄	—	188	1	no	—
334	C₄	—	188	1	no	—
335	C₄	—	188	1	no	—
336	C₄	#104×D₄₇	1	1	yes	characteristic
337	C₄	#104×D₄₇	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	376	1	no	—
340	C₂	#20243187	1	1	yes	characteristic
341	C₂	#20243192	1	1	yes	characteristic
342	C₂	#20243231	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1302	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	#13093998	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1302	D₄₇
hypocenter	C₄₇	#8249
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1302

Complements

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

N	H ≅ G/N	type
2. #20243076	C₂	semidirect
5. #20241678	C₂	semidirect
6. #20241280	C₂	semidirect
7. C₄₇×#1302	C₂	semidirect
9. #15689250	C₄	semidirect
10. #15689247	C₄	semidirect
11. #15689200	C₄	semidirect
13. #15689188	C₄	semidirect
22. #15688864	C₄	semidirect
23. #15688862	C₄	semidirect
172. #1302	D₄₇	semidirect
202. C₄₇	#8249	semidirect

Maximal subgroups

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

subgroup	index
#20252357	2
#20243076	2
#20241974	2
#20241875	2
#20241678	2
#20241280	2
C₄₇×#1302	2
#8249	47