Subgroups — #26104217

Order 12032.

538 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241211	C₂	1	2	yes	maximal, complemented
4	#20241209	C₂	1	2	yes	maximal
5	C₄₇×#2293	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	C₂²	1	2	yes	—
11	#15688878	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂×#117×C₄₇	—	2	1	no	—
14	C₄₇×#390	C₄	1	1	yes	characteristic
15	C₄₇×#390	C₂²	1	1	yes	characteristic
16	C₄₇×#389	C₄	1	1	yes	characteristic
17	C₄₇×#389	C₂²	1	1	yes	characteristic
18	C₄₇×#388	—	2	1	no	—
19	C₄₇×#388	—	2	1	no	—
20	C₄×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
22	C₄³×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄×C₄₇⋊C₈	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	4	2	no	—
28	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	—	2	1	no	—
30	C₄×D₄×C₄₇	—	2	2	no	—
31	C₄×D₄×C₄₇	C₈	1	4	yes	complemented
32	#117×C₄₇	—	4	2	no	—
33	#117×C₄₇	—	4	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄²×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	#95×C₄₇	—	2	1	no	—
53	#95×C₄₇	—	2	1	no	—
54	#95×C₄₇	—	2	1	no	—
55	C₂⁴×C₄₇	—	2	1	no	—
56	C₂×D₄×C₄₇	—	2	1	no	—
57	C₂×D₄×C₄₇	—	4	1	no	—
58	C₂×D₄×C₄₇	—	4	1	no	—
59	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₄₇	—	2	1	no	—
74	C₄⋊C₄×C₄₇	—	4	1	no	—
75	C₄⋊C₄×C₄₇	—	4	1	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	1	no	—
78	C₄⋊C₄×C₄₇	—	4	1	no	—
79	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄×C₄₇	—	2	4	no	—
81	C₂²⋊C₄×C₄₇	—	4	2	no	—
82	C₂²⋊C₄×C₄₇	—	4	2	no	—
83	C₂²⋊C₄×C₄₇	—	4	1	no	—
84	C₂²⋊C₄×C₄₇	—	4	1	no	—
85	C₂²⋊C₄×C₄₇	—	4	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	4	1	no	—
90	C₄²×C₄₇	—	4	1	no	—
91	C₄²×C₄₇	—	4	2	no	—
92	C₄²×C₄₇	C₂×C₈	1	2	yes	—
93	C₄²×C₄₇	M₁₆	1	2	yes	—
94	C₄²×C₄₇	D₈	1	2	yes	—
95	C₄²×C₄₇	SD₁₆	1	2	yes	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	4	1	no	—
104	C₂³×C₄₇	—	4	1	no	—
105	C₂³×C₄₇	—	4	1	no	—
106	C₂³×C₄₇	—	4	1	no	—
107	C₂³×C₄₇	—	4	1	no	—
108	C₂³×C₄₇	—	4	1	no	—
109	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	D₄×C₄₇	—	4	2	no	—
111	D₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	4	1	no	—
127	C₂×C₄×C₄₇	—	4	8	no	—
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	4	1	no	—
131	C₂×C₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
136	C₂×C₄×C₄₇	#102	1	2	yes	—
137	C₂×C₄×C₄₇	#102	1	2	yes	—
138	C₂×C₄×C₄₇	#104	1	2	yes	—
139	C₂×C₄×C₄₇	#104	1	2	yes	—
140	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#135	1	2	yes	—
145	C₄₇⋊C₈	—	4	4	no	—
146	C₄₇⋊C₈	—	4	4	no	—
147	#7560	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	4	2	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	4	2	no	—
151	C₂²×C₄₇	—	8	1	no	—
152	C₂²×C₄₇	—	8	1	no	—
153	C₂²×C₄₇	—	8	1	no	—
154	C₂²×C₄₇	—	8	1	no	—
155	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₂²×C₄₇	#410	1	1	yes	characteristic
157	C₂²×C₄₇	#413	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
159	C₂²×C₄₇	#416	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	—	8	1	no	—
172	C₄×C₄₇	—	8	1	no	—
173	C₄×C₄₇	D₄⋊C₈	1	4	yes	—
174	#2293	D₄₇	1	1	yes	characteristic
175	C₄×C₄⋊C₈	—	47	1	no	—
176	#1744	—	47	1	no	—
177	#1271	—	47	2	no	—
178	#1270	—	47	2	no	—
179	C₂×C₄₇	—	8	2	no	—
180	C₂×C₄₇	#1458	1	1	yes	characteristic
181	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
182	C₂×C₄₇	#1476	1	1	yes	characteristic
183	C₂×C₄₇	#1500	1	1	yes	characteristic
184	C₂×C₄₇	#1502	1	1	yes	characteristic
185	C₂×C₄₇	#1524	1	1	yes	characteristic
186	C₂×C₄₇	#1540	1	1	yes	characteristic
187	C₂×C₄×D₄	Dic₄₇	1	1	yes	characteristic
188	C₂×#117	—	2	1	no	—
189	C₂×C₄⋊C₈	—	94	1	no	—
190	C₂×C₄×C₈	—	94	1	no	—
191	C₂×C₄×C₈	—	94	1	no	—
192	#390	Dic₄₇	1	1	yes	characteristic
193	#390	C₂×D₄₇	1	1	yes	characteristic
194	#389	Dic₄₇	1	1	yes	characteristic
195	#389	C₂×D₄₇	1	1	yes	characteristic
196	#388	—	2	1	no	—
197	#388	—	2	1	no	—
198	C₄×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
199	C₄×C₂²⋊C₄	—	2	1	no	—
200	C₄³	C₂×D₄₇	1	1	yes	characteristic
201	#336	—	47	2	no	—
202	#336	—	47	2	no	—
203	#336	—	94	1	no	—
204	C₄₇	#7560	1	1	yes	characteristic, complemented, Sylow
205	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂³×C₄	—	2	1	no	—
207	C₂²×C₈	—	94	2	no	—
208	C₂²×C₈	—	94	2	no	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	C₄₇⋊C₈	1	4	yes	complemented
211	#117	—	4	2	no	—
212	#117	—	4	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₄⋊C₈	—	94	4	no	—
232	C₄×C₈	—	94	4	no	—
233	C₄×C₈	—	188	2	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	C₂⁴	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₂×C₈	—	94	4	no	—
256	C₂×C₈	—	94	4	no	—
257	C₂×C₈	—	188	2	no	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	—	188	2	no	—
260	C₂×C₈	—	188	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
281	C₄²	#10625329	1	2	yes	—
282	C₄²	#10625334	1	2	yes	—
283	C₄²	#10625335	1	2	yes	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	8	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
317	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
318	C₂×C₄	#13093945	1	2	yes	—
319	C₂×C₄	#13093945	1	2	yes	—
320	C₂×C₄	#13093951	1	2	yes	—
321	C₂×C₄	#13093951	1	2	yes	—
322	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
323	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
324	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
325	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
326	C₂×C₄	#13094044	1	2	yes	—
327	C₈	—	188	4	no	—
328	C₈	—	188	4	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
337	C₂²	#15689097	1	1	yes	characteristic
338	C₂²	#15689107	1	1	yes	characteristic
339	C₂²	C₂×#13093945	1	1	yes	characteristic
340	C₂²	#15689119	1	1	yes	characteristic
341	C₂²	C₂×#13093951	1	1	yes	characteristic
342	C₂²	C₂×#13093951	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	1	no	—
353	C₄	—	8	1	no	—
354	C₄	D₄⋊(C₄₇⋊C₈)	1	4	yes	—
355	C₂	—	8	2	no	—
356	C₂	#20241702	1	1	yes	characteristic
357	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
358	C₂	#20241770	1	1	yes	characteristic
359	C₂	#20241874	1	1	yes	characteristic
360	C₂	#20241883	1	1	yes	characteristic
361	C₂	#20241976	1	1	yes	characteristic
362	C₂	#20242037	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2293	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2293	D₄₇
hypocenter	C₄₇	#7560
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2293

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. #20243134	C₂	semidirect
3. #20241211	C₂	semidirect
31. C₄×D₄×C₄₇	C₈	semidirect
204. C₄₇	#7560	semidirect
210. C₄×D₄	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243134	2
#20241211	2
#20241209	2
C₄₇×#2293	2
#7560	47