Subgroups — #26105764

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689175	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689174	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689145	C₂	1	1	yes	maximal, characteristic
4	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241222	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241222	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	—	2	1	no	—
9	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	#15689175	—	2	2	no	—
11	#15689175	C₄	1	4	yes	—
12	#15689174	—	2	2	no	—
13	#15689174	—	2	2	no	—
14	#15689145	—	2	2	no	—
15	#15689145	C₄	1	4	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₇⋊C₈	—	2	1	no	—
19	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
20	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
21	C₂×(C₂×D₄₇)⋊C₈	C₄	1	2	yes	—
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	—	2	1	no	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
30	C₂×#10625324	—	2	4	no	—
31	C₂×#10625324	—	4	2	no	—
32	C₂×#10625324	—	4	2	no	—
33	C₂²×C₄₇⋊C₈	—	2	1	no	—
34	C₂²×C₄₇⋊C₈	—	2	1	no	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
39	C₄²×D₄₇	C₂×C₄	1	2	yes	—
40	C₄²×D₄₇	C₂×C₄	1	2	yes	—
41	C₄²×D₄₇	D₄	1	2	yes	—
42	C₄²×D₄₇	D₄	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
46	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
47	Dic₄₇⋊C₈	—	2	4	no	—
48	Dic₄₇⋊C₈	—	2	4	no	—
49	(C₂×D₄₇)⋊C₈	—	2	4	no	—
50	(C₂×D₄₇)⋊C₈	—	2	4	no	—
51	C₄×C₄₇⋊C₈	—	2	1	no	—
52	C₄×C₄₇⋊C₈	—	2	1	no	—
53	C₈×Dic₄₇	—	2	4	no	—
54	C₂²×C₈×C₄₇	—	2	1	no	—
55	C₂²×C₈×C₄₇	—	2	1	no	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	—	2	1	no	—
64	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×D₄₇	—	2	2	no	—
72	C₂×C₄×D₄₇	—	2	2	no	—
73	C₂×C₄×D₄₇	—	2	1	no	—
74	C₂×C₄×D₄₇	—	2	1	no	—
75	C₂×C₄×D₄₇	—	2	1	no	—
76	C₂×C₄×D₄₇	—	2	1	no	—
77	C₂×C₄×D₄₇	—	2	4	no	—
78	C₂×C₄×D₄₇	C₄²	1	2	yes	—
79	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	4	1	no	—
90	C₂×C₈×C₄₇	—	4	1	no	—
91	C₂×C₈×C₄₇	—	4	1	no	—
92	C₂×C₈×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄×Dic₄₇	C₄²	1	2	yes	—
98	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
99	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
100	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
101	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	—	4	1	no	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	C₂×C₄₇⋊C₈	—	4	1	no	—
111	C₂×C₄₇⋊C₈	—	4	1	no	—
112	#10625324	—	4	8	no	—
113	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₄₇	—	2	2	no	—
115	C₂²×D₄₇	—	2	1	no	—
116	C₂²×D₄₇	—	2	1	no	—
117	C₂²×D₄₇	—	2	2	no	—
118	C₂²×D₄₇	#95	1	2	yes	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#131	1	2	yes	—
127	C₂×C₄×C₄₇	#131	1	2	yes	—
128	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	1	no	—
134	C₂×Dic₄₇	—	2	1	no	—
135	C₂×Dic₄₇	#95	1	2	yes	—
136	C₂×Dic₄₇	#95	1	2	yes	—
137	C₂×Dic₄₇	#95	1	2	yes	—
138	C₂×Dic₄₇	#108	1	4	yes	—
139	C₄×D₄₇	—	2	2	no	—
140	C₄×D₄₇	—	2	2	no	—
141	C₄×D₄₇	—	2	4	no	—
142	C₄×D₄₇	—	2	2	no	—
143	C₄×D₄₇	—	2	2	no	—
144	C₄×D₄₇	—	2	2	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	4	no	—
147	C₄×D₄₇	—	2	4	no	—
148	C₈×C₄₇	—	4	2	no	—
149	C₈×C₄₇	—	4	2	no	—
150	C₈×C₄₇	—	4	2	no	—
151	C₄₇⋊C₈	—	4	2	no	—
152	C₄₇⋊C₈	—	4	2	no	—
153	C₄₇⋊C₈	—	4	2	no	—
154	#7880	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
156	C₂²×C₄₇	#424	1	2	yes	—
157	C₂²×C₄₇	#428	1	1	yes	characteristic
158	C₂²×C₄₇	#428	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#108	1	2	yes	—
160	C₂×D₄₇	—	2	2	no	—
161	C₂×D₄₇	—	2	2	no	—
162	C₂×D₄₇	—	2	2	no	—
163	C₂×D₄₇	—	4	2	no	—
164	C₂×D₄₇	—	4	2	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	#405	1	1	yes	characteristic
168	C₄×C₄₇	#405	1	1	yes	characteristic
169	C₄×C₄₇	#408	1	2	yes	—
170	C₄×C₄₇	#417	1	1	yes	characteristic
171	C₄×C₄₇	#417	1	1	yes	characteristic
172	C₄×C₄₇	#418	1	1	yes	characteristic
173	C₄×C₄₇	#419	1	1	yes	characteristic
174	Dic₄₇	—	2	2	no	—
175	Dic₄₇	—	2	2	no	—
176	Dic₄₇	#341	1	4	yes	—
177	C₂×#431	—	47	1	no	—
178	C₂×#431	—	47	1	no	—
179	C₂×#423	—	47	1	no	—
180	#1277	—	47	1	no	—
181	#1277	—	47	1	no	—
182	#1277	—	47	1	no	—
183	#1277	D₄₇	1	1	yes	characteristic, complemented
184	C₂×C₄₇	#1574	1	2	yes	—
185	C₂×C₄₇	#1574	1	2	yes	—
186	C₂×C₄₇	#1729	1	1	yes	characteristic
187	C₂×C₄₇	#1736	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
189	D₄₇	—	4	2	no	—
190	C₂²×M₁₆	—	94	1	no	—
191	C₂²×C₄²	—	47	1	no	—
192	#431	—	47	4	no	—
193	#431	—	94	2	no	—
194	#431	—	94	2	no	—
195	#431	—	94	2	no	—
196	#423	—	47	4	no	—
197	#423	—	94	2	no	—
198	C₂×C₄⋊C₈	—	47	1	no	—
199	C₂×C₄⋊C₈	—	47	1	no	—
200	C₂×C₄⋊C₈	—	47	2	no	—
201	C₂×C₄⋊C₈	—	94	1	no	—
202	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
203	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
204	C₂×C₂²⋊C₈	—	47	2	no	—
205	C₂×C₂²⋊C₈	—	94	1	no	—
206	C₂×C₄×C₈	—	47	1	no	—
207	C₂×C₄×C₈	—	94	1	no	—
208	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
209	C₄₇	#7880	1	1	yes	characteristic, complemented, Sylow
210	C₂³×C₄	—	47	1	no	—
211	C₂³×C₄	—	47	1	no	—
212	C₂³×C₄	—	47	1	no	—
213	C₂×M₁₆	—	94	4	no	—
214	C₂×M₁₆	—	188	2	no	—
215	C₂×M₁₆	—	188	2	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	47	2	no	—
219	C₂²×C₈	—	94	1	no	—
220	C₂²×C₈	—	94	1	no	—
221	C₂²×C₈	C₄×D₄₇	1	2	yes	—
222	C₂×C₄²	—	47	1	no	—
223	C₂×C₄²	—	47	1	no	—
224	C₂×C₄²	—	47	1	no	—
225	C₂×C₄²	—	47	2	no	—
226	C₂×C₄²	—	47	2	no	—
227	C₂×C₄²	—	47	2	no	—
228	C₂×C₄²	—	47	2	no	—
229	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	47	2	no	—
233	C₄⋊C₈	—	47	2	no	—
234	C₄⋊C₈	—	94	4	no	—
235	C₄⋊C₈	—	94	4	no	—
236	C₄⋊C₈	—	94	1	no	—
237	C₄⋊C₈	—	94	1	no	—
238	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
239	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
240	C₂²⋊C₈	—	94	4	no	—
241	C₂²⋊C₈	—	94	4	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	94	4	no	—
245	C₄×C₈	—	94	1	no	—
246	C₄×C₈	—	94	1	no	—
247	C₂⁴	—	47	1	no	—
248	C₂²×C₄	—	47	1	no	—
249	C₂²×C₄	—	47	1	no	—
250	C₂²×C₄	—	47	1	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	2	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	47	2	no	—
256	C₂²×C₄	—	47	2	no	—
257	C₂²×C₄	—	94	4	no	—
258	C₂²×C₄	—	94	2	no	—
259	C₂²×C₄	—	94	2	no	—
260	C₂²×C₄	—	94	1	no	—
261	C₂²×C₄	—	94	1	no	—
262	C₂²×C₄	—	94	1	no	—
263	C₂²×C₄	—	94	1	no	—
264	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
266	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
267	M₁₆	—	188	8	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	94	2	no	—
281	C₂×C₈	—	94	2	no	—
282	C₂×C₈	—	188	1	no	—
283	C₂×C₈	—	188	1	no	—
284	C₂×C₈	—	188	1	no	—
285	C₂×C₈	—	188	1	no	—
286	C₄²	—	47	2	no	—
287	C₄²	—	47	2	no	—
288	C₄²	—	47	2	no	—
289	C₄²	—	47	2	no	—
290	C₄²	—	47	2	no	—
291	C₄²	—	47	2	no	—
292	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
293	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
294	C₄²	D₄×D₄₇	1	1	yes	characteristic
295	C₄²	D₄×D₄₇	1	1	yes	characteristic
296	C₂³	—	47	2	no	—
297	C₂³	—	94	2	no	—
298	C₂³	—	94	1	no	—
299	C₂³	—	94	1	no	—
300	C₂³	—	94	2	no	—
301	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	47	2	no	—
304	C₂×C₄	—	47	4	no	—
305	C₂×C₄	—	47	2	no	—
306	C₂×C₄	—	47	2	no	—
307	C₂×C₄	—	94	4	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	1	no	—
311	C₂×C₄	—	94	1	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	2	no	—
314	C₂×C₄	—	94	4	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
322	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
323	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
324	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
325	C₂×C₄	#13094030	1	2	yes	—
326	C₂×C₄	#13094032	1	2	yes	—
327	C₂×C₄	#13094048	1	1	yes	characteristic
328	C₂×C₄	#13094051	1	1	yes	characteristic
329	C₂×C₄	#13094053	1	1	yes	characteristic
330	C₂×C₄	#13094056	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	188	2	no	—
335	C₈	—	188	2	no	—
336	C₈	—	188	2	no	—
337	C₂²	—	94	2	no	—
338	C₂²	—	94	2	no	—
339	C₂²	—	94	2	no	—
340	C₂²	—	188	2	no	—
341	C₂²	—	188	2	no	—
342	C₂²	#95×D₄₇	1	1	yes	characteristic
343	C₂²	#15689148	1	2	yes	—
344	C₂²	#15689162	1	1	yes	characteristic
345	C₂²	#15689164	1	1	yes	characteristic
346	C₂²	#108×D₄₇	1	2	yes	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	47	4	no	—
350	C₄	—	94	2	no	—
351	C₄	—	94	2	no	—
352	C₄	#15689077	1	1	yes	characteristic
353	C₄	#15689080	1	1	yes	characteristic
354	C₄	#15689088	1	2	yes	—
355	C₄	#15689121	1	1	yes	characteristic
356	C₄	#15689123	1	1	yes	characteristic
357	C₄	#15689126	1	1	yes	characteristic
358	C₄	#15689130	1	1	yes	characteristic
359	C₂	—	188	2	no	—
360	C₂	#20242175	1	2	yes	—
361	C₂	#20242178	1	2	yes	—
362	C₂	#20243083	1	1	yes	characteristic
363	C₂	#20243106	1	1	yes	characteristic
364	C₂	#341×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#7880
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #20241223	C₂	semidirect
5. #20241222	C₂	semidirect
6. #20241222	C₂	semidirect
7. C₄₇×#1277	C₂	semidirect
183. #1277	D₄₇	semidirect
209. C₄₇	#7880	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689175	2
C₂×#15689174	2
C₂×#15689145	2
#20241223	2
#20241222	2
C₄₇×#1277	2
#7880	47