Subgroups — #26106307

Order 12032.

786 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
3	#20241268	C₂	1	4	yes	maximal
4	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	1	no	—
6	C₂²×C₄×Dic₄₇	—	2	1	no	—
7	#15688993	C₄	1	2	yes	complemented
8	#15688989	—	2	1	no	—
9	#15688983	—	2	1	no	—
10	#15688983	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₄₇	—	2	1	no	—
12	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
13	#15688969	—	2	2	no	—
14	#15688969	C₄	1	4	yes	complemented
15	C₂×C₄₇⋊#95	—	2	1	no	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#432	—	2	2	no	—
19	C₄₇×#432	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	#13093983	—	4	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	8	no	—
30	C₂×C₄×Dic₄₇	—	2	4	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₄₇⋊#95	—	2	4	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
52	#97×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	8	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	4	4	no	—
80	C₄²×C₄₇	C₄²	1	2	yes	—
81	C₄²×C₄₇	C₄²	1	2	yes	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
88	C₂²⋊Dic₄₇	—	4	4	no	—
89	C₄⋊Dic₄₇	—	4	4	no	—
90	C₄⋊Dic₄₇	—	4	4	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄×Dic₄₇	—	2	16	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
100	C₂³×C₄₇	#137	1	2	yes	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	#95	1	2	yes	—
117	C₂×C₄×C₄₇	#95	1	2	yes	—
118	C₂×C₄×C₄₇	#95	1	2	yes	—
119	C₂×C₄×C₄₇	#95	1	2	yes	—
120	C₂×C₄×C₄₇	#104	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	8	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	8	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₈×C₄₇	—	4	4	no	—
134	#8022	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
142	C₂²×C₄₇	#410	1	2	yes	—
143	C₂²×C₄₇	#417	1	2	yes	—
144	C₂²×C₄₇	#418	1	1	yes	characteristic
145	C₂²×C₄₇	#419	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#104	1	2	yes	—
147	C₂²×C₄₇	#428	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	#339	1	4	yes	—
156	Dic₄₇	—	4	8	no	—
157	Dic₄₇	—	8	4	no	—
158	C₂×#432	D₄₇	1	1	yes	characteristic
159	#2285	—	47	1	no	—
160	C₂×#376	—	47	1	no	—
161	#1294	—	47	4	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	#1505	1	2	yes	—
165	C₂×C₄₇	#1506	1	2	yes	—
166	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
167	C₂×C₄₇	#1736	1	1	yes	characteristic
168	C₂×C₄₇	#1740	1	1	yes	characteristic
169	C₂×#117	—	94	1	no	—
170	C₂²×C₄²	—	94	1	no	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#432	—	2	2	no	—
173	#432	Dic₄₇	1	4	yes	complemented
174	C₂×C₄⋊C₈	—	2	1	no	—
175	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
176	C₂×#97	—	2	1	no	—
177	#384	—	47	2	no	—
178	#382	—	94	1	no	—
179	#380	—	47	2	no	—
180	#380	—	94	1	no	—
181	C₄×C₄⋊C₄	—	47	2	no	—
182	C₄×C₄⋊C₄	—	94	1	no	—
183	#376	—	47	4	no	—
184	#376	—	94	2	no	—
185	C₂×#95	—	47	2	no	—
186	C₂×#95	—	94	1	no	—
187	C₄₇	#8022	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	94	2	no	—
189	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
190	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
191	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂²×C₈	—	2	2	no	—
193	#117	—	188	4	no	—
194	C₂×C₄⋊C₄	—	47	4	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	—	94	2	no	—
197	C₂×C₂²⋊C₄	—	94	2	no	—
198	C₂×C₄²	—	94	8	no	—
199	C₂×C₄²	—	94	4	no	—
200	C₂×C₄²	—	94	2	no	—
201	C₂×C₄²	Dic₉₄	1	2	yes	—
202	C₂×C₄²	C₄×D₄₇	1	2	yes	—
203	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
206	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
212	#97	—	2	4	no	—
213	#95	—	47	4	no	—
214	#95	—	94	4	no	—
215	#95	—	94	2	no	—
216	#95	—	94	2	no	—
217	#95	—	94	2	no	—
218	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	94	8	no	—
224	C₂²×C₄	—	94	4	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	—	188	4	no	—
227	C₂²×C₄	—	188	2	no	—
228	C₂²×C₄	—	188	2	no	—
229	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
230	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₄⋊C₄	—	188	4	no	—
246	C₄⋊C₄	—	188	4	no	—
247	C₄⋊C₄	—	188	4	no	—
248	C₂²⋊C₄	—	188	4	no	—
249	C₄²	—	94	16	no	—
250	C₄²	—	188	4	no	—
251	C₄²	C₄×Dic₄₇	1	2	yes	—
252	C₄²	C₄×Dic₄₇	1	2	yes	—
253	C₄²	Dic₄₇⋊C₄	1	2	yes	—
254	C₄²	Dic₄₇⋊C₄	1	2	yes	—
255	C₄²	Dic₄₇⋊C₄	1	2	yes	—
256	C₄²	Dic₄₇⋊C₄	1	2	yes	—
257	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
258	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
262	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
264	C₂³	#13094052	1	2	yes	—
265	C₂³	#13094057	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	94	8	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	8	no	—
287	C₂×C₄	—	188	8	no	—
288	C₂×C₄	—	188	4	no	—
289	C₂×C₄	C₄₇⋊#95	1	2	yes	—
290	C₂×C₄	C₄₇⋊#95	1	2	yes	—
291	C₂×C₄	C₄₇⋊#95	1	2	yes	—
292	C₂×C₄	C₄₇⋊#95	1	2	yes	—
293	C₂×C₄	#13093950	1	4	yes	—
294	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
298	C₈	—	4	4	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
306	C₂²	#15689096	1	2	yes	—
307	C₂²	#15689125	1	2	yes	—
308	C₂²	#15689129	1	1	yes	characteristic
309	C₂²	#15689133	1	1	yes	characteristic
310	C₂²	C₂×#13093950	1	2	yes	—
311	C₂²	#15689165	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	376	4	no	—
320	C₄	—	188	8	no	—
321	C₄	#15688884	1	4	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	#20241897	1	2	yes	—
325	C₂	#20241901	1	2	yes	—
326	C₂	C₂×#15688884	1	1	yes	characteristic
327	C₂	#20243108	1	1	yes	characteristic
328	C₂	#20243120	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#8022
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
7. #15688993	C₄	semidirect
12. C₄×C₄⋊Dic₄₇	C₄	semidirect
14. #15688969	C₄	semidirect
19. C₄₇×#432	C₄	semidirect
173. #432	Dic₄₇	semidirect
187. C₄₇	#8022	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
C₂×#15688969	2
#20241268	2
C₂×C₄₇×#432	2
#8022	47