Subgroups — #26115235

Order 12032.

736 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	D₄⋊(C₄₇⋊C₈)	—	2	8	no	—
12	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	—	2	1	no	—
15	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#390	—	2	1	no	—
17	C₄₇×#390	—	2	1	no	—
18	C₄₇×#390	C₄	1	2	yes	—
19	C₄₇×#384	C₄	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	complemented
26	C₄×C₄₇⋊C₈	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	4	2	no	—
28	C₂²×D₄×C₄₇	—	2	1	no	—
29	C₂²×D₄×C₄₇	—	2	1	no	—
30	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	—	2	2	no	—
32	#127×C₄₇	—	2	2	no	—
33	#127×C₄₇	C₈	1	4	yes	complemented
34	C₄×D₄×C₄₇	—	2	4	no	—
35	C₄×D₄×C₄₇	—	2	4	no	—
36	C₄×D₄×C₄₇	—	4	4	no	—
37	C₄×D₄×C₄₇	—	4	2	no	—
38	C₄×D₄×C₄₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	2	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	C₂⁴×C₄₇	—	2	2	no	—
53	C₂×D₄×C₄₇	—	2	4	no	—
54	C₂×D₄×C₄₇	—	2	4	no	—
55	C₂×D₄×C₄₇	—	2	4	no	—
56	C₂×D₄×C₄₇	—	2	2	no	—
57	C₂×D₄×C₄₇	—	2	4	no	—
58	C₂×D₄×C₄₇	—	4	2	no	—
59	C₂×D₄×C₄₇	—	4	2	no	—
60	C₂×D₄×C₄₇	—	4	2	no	—
61	C₂×D₄×C₄₇	—	4	2	no	—
62	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₂²⋊C₄×C₄₇	—	4	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	4	no	—
80	C₂²⋊C₄×C₄₇	—	4	2	no	—
81	C₂²⋊C₄×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	4	1	no	—
86	C₄²×C₄₇	C₂×C₈	1	2	yes	—
87	C₄²×C₄₇	M₁₆	1	2	yes	—
88	C₄²×C₄₇	D₈	1	2	yes	—
89	C₄²×C₄₇	SD₁₆	1	2	yes	—
90	C₄²×C₄₇	C₂×D₄	1	2	yes	—
91	C₄²×C₄₇	#41	1	2	yes	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	—	2	4	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	—	4	2	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	D₄×C₄₇	—	2	8	no	—
107	D₄×C₄₇	—	4	8	no	—
108	D₄×C₄₇	—	4	8	no	—
109	D₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	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	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	8	4	no	—
123	C₂×C₄×C₄₇	—	8	4	no	—
124	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₄₇	#102	1	2	yes	—
127	C₂×C₄×C₄₇	#102	1	2	yes	—
128	C₂×C₄×C₄₇	#104	1	2	yes	—
129	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₄₇	#121	1	2	yes	—
131	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#131	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
139	C₄₇⋊C₈	—	4	4	no	—
140	C₄₇⋊C₈	—	4	4	no	—
141	#9674	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	4	4	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	—	8	2	no	—
146	C₂²×C₄₇	—	8	2	no	—
147	C₂²×C₄₇	—	8	2	no	—
148	C₂²×C₄₇	—	8	2	no	—
149	C₂²×C₄₇	#390	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₄₇	#408	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
153	C₂²×C₄₇	#418	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
155	C₂²×C₄₇	#421	1	1	yes	characteristic
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	—	8	2	no	—
162	C₄×C₄₇	D₄⋊C₈	1	4	yes	complemented
163	C₄×C₄₇	C₈×D₄	1	2	yes	—
164	C₄×C₄₇	#436	1	2	yes	—
165	C₄×C₄₇	#459	1	2	yes	—
166	C₄×C₄₇	#460	1	2	yes	—
167	C₄×#127	D₄₇	1	1	yes	characteristic
168	C₄×C₄⋊C₈	—	47	1	no	—
169	C₄×C₄⋊C₈	—	47	1	no	—
170	C₂×D₄⋊C₈	—	47	4	no	—
171	C₂×C₄₇	—	8	4	no	—
172	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
173	C₂×C₄₇	#1478	1	1	yes	characteristic
174	C₂×C₄₇	#1480	1	1	yes	characteristic
175	C₂×C₄₇	#1484	1	1	yes	characteristic
176	C₂×C₄₇	#1954	1	1	yes	characteristic
177	C₂×C₄₇	#1980	1	1	yes	characteristic
178	C₂×C₄₇	#1982	1	1	yes	characteristic
179	C₂×#127	Dic₄₇	1	1	yes	characteristic
180	C₂×C₄×D₄	—	2	1	no	—
181	C₂×C₄×D₄	—	2	1	no	—
182	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
183	C₂×C₄⋊C₈	—	47	2	no	—
184	C₂×C₄⋊C₈	—	94	1	no	—
185	C₂×C₄×C₈	—	47	2	no	—
186	C₂×C₄×C₈	—	94	1	no	—
187	#390	—	2	1	no	—
188	#390	—	2	1	no	—
189	#390	Dic₄₇	1	2	yes	—
190	#384	Dic₄₇	1	1	yes	characteristic
191	C₄³	C₂×D₄₇	1	1	yes	characteristic
192	#336	—	94	1	no	—
193	#336	—	94	1	no	—
194	D₄⋊C₈	—	94	8	no	—
195	C₄₇	#9674	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	2	1	no	—
198	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂²×C₈	—	94	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	#127	—	2	2	no	—
203	#127	C₄₇⋊C₈	1	4	yes	complemented
204	C₄×D₄	—	2	4	no	—
205	C₄×D₄	—	2	4	no	—
206	C₄×D₄	—	4	4	no	—
207	C₄×D₄	—	4	2	no	—
208	C₄×D₄	—	4	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₄⋊C₈	—	47	4	no	—
223	C₄⋊C₈	—	94	4	no	—
224	C₄⋊C₈	—	94	2	no	—
225	C₄×C₈	—	94	4	no	—
226	C₄×C₈	—	188	2	no	—
227	C₂⁴	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	188	2	no	—
253	C₂×C₈	—	188	2	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	1	no	—
267	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
268	C₄²	#10625329	1	2	yes	—
269	C₄²	#10625334	1	2	yes	—
270	C₄²	#10625335	1	2	yes	—
271	C₄²	D₄×D₄₇	1	2	yes	—
272	C₄²	#10625352	1	2	yes	—
273	C₂³	—	2	4	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
281	D₄	—	2	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	4	8	no	—
284	D₄	—	4	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
300	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
301	C₂×C₄	#13093945	1	2	yes	—
302	C₂×C₄	#13093945	1	2	yes	—
303	C₂×C₄	#13093951	1	2	yes	—
304	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
305	C₂×C₄	#13093995	1	2	yes	—
306	C₂×C₄	#13094011	1	1	yes	characteristic
307	C₂×C₄	#13094020	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
309	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
310	C₂×C₄	#13094031	1	2	yes	—
311	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
312	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
313	C₂×C₄	#13094050	1	2	yes	—
314	C₈	—	188	4	no	—
315	C₈	—	188	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	#15689026	1	1	yes	characteristic
324	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
325	C₂²	#15689090	1	1	yes	characteristic
326	C₂²	C₂×#13093945	1	1	yes	characteristic
327	C₂²	#15689128	1	1	yes	characteristic
328	C₂²	C₂×#13093951	1	1	yes	characteristic
329	C₂²	#15689140	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	1	no	—
335	C₄	—	8	2	no	—
336	C₄	D₄⋊(C₄₇⋊C₈)	1	4	yes	complemented
337	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
338	C₄	#15689201	1	2	yes	—
339	C₄	#15689315	1	2	yes	—
340	C₄	#15689320	1	2	yes	—
341	C₂	—	8	4	no	—
342	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
343	C₂	#20241779	1	1	yes	characteristic
344	C₂	#20241787	1	1	yes	characteristic
345	C₂	#20241805	1	1	yes	characteristic
346	C₂	#20244183	1	1	yes	characteristic
347	C₂	#20244308	1	1	yes	characteristic
348	C₂	#20244316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#9674
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. C₂×D₄⋊(C₄₇⋊C₈)	C₂	semidirect
25. C₄⋊(C₄₇⋊C₈)	D₄	semidirect
33. #127×C₄₇	C₈	semidirect
162. C₄×C₄₇	D₄⋊C₈	semidirect
195. C₄₇	#9674	semidirect
203. #127	C₄₇⋊C₈	semidirect
336. 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
C₂×D₄⋊(C₄₇⋊C₈)	2
C₄×#127×C₄₇	2
#9674	47