Subgroups — #26106375

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂×#13093983	C₄	1	2	yes	—
7	#15689147	—	2	2	no	—
8	#15689147	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#15688993	C₂²	1	2	yes	—
12	#15688989	C₄	1	2	yes	—
13	#15688983	—	2	1	no	—
14	#15688983	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	—	2	1	no	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	C₂²×M₁₆×C₄₇	—	2	1	no	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#423	—	2	2	no	—
20	C₄₇×#423	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	4	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	#13093983	—	4	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	—	4	4	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	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	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₇	—	2	4	no	—
52	C₄⋊C₈×C₄₇	—	2	2	no	—
53	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₄²	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	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
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	C₂²×Dic₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	4	8	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	C₄²	1	2	yes	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
87	C₂²⋊Dic₄₇	—	4	4	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	4	4	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	#10625329	—	4	8	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	4	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	#95	1	2	yes	—
102	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	#95	1	2	yes	—
116	C₂×C₄×C₄₇	#95	1	2	yes	—
117	C₂×C₄×C₄₇	#95	1	2	yes	—
118	C₂×C₄×C₄₇	#100	1	2	yes	—
119	C₂×C₄×C₄₇	#101	1	2	yes	—
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×C₄⋊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	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₄₇	#137	1	2	yes	—
129	C₂×C₄×C₄₇	#137	1	2	yes	—
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	C₄₇⋊C₈	—	4	4	no	—
135	#8041	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
144	C₂²×C₄₇	#417	1	2	yes	—
145	C₂²×C₄₇	#417	1	2	yes	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	#343	1	4	yes	—
151	C₄×C₄₇	#418	1	1	yes	characteristic
152	C₄×C₄₇	#418	1	1	yes	characteristic
153	C₄×C₄₇	#419	1	1	yes	characteristic
154	C₄×C₄₇	#419	1	1	yes	characteristic
155	C₄×C₄₇	#428	1	2	yes	—
156	C₄×C₄₇	#428	1	2	yes	—
157	Dic₄₇	—	8	4	no	—
158	C₂×#423	—	47	1	no	—
159	C₂×#423	D₄₇	1	1	yes	characteristic
160	#2285	—	47	1	no	—
161	#1296	—	47	4	no	—
162	C₂×C₄₇	—	4	2	no	—
163	C₂×C₄₇	#1560	1	2	yes	—
164	C₂×C₄₇	#1561	1	2	yes	—
165	C₂×C₄₇	#1736	1	1	yes	characteristic
166	C₂×C₄₇	#1736	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
168	C₂²×M₁₆	—	2	1	no	—
169	C₂²×M₁₆	—	94	1	no	—
170	C₂×#117	—	47	2	no	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#423	—	2	2	no	—
173	#423	—	47	4	no	—
174	#423	—	94	2	no	—
175	#423	Dic₄₇	1	4	yes	complemented
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	47	2	no	—
178	C₂×C₄⋊C₈	—	94	1	no	—
179	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
180	#384	—	47	2	no	—
181	#382	—	47	2	no	—
182	#380	—	94	1	no	—
183	#380	—	94	1	no	—
184	C₄×C₄⋊C₄	—	94	1	no	—
185	C₄×C₄⋊C₄	—	94	1	no	—
186	C₄₇	#8041	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
188	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
189	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	4	4	no	—
192	C₂×M₁₆	—	94	4	no	—
193	C₂×M₁₆	—	188	4	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	94	2	no	—
196	#117	—	188	4	no	—
197	C₂×C₄⋊C₄	—	47	4	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₂²⋊C₄	—	94	2	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	Dic₉₄	1	2	yes	—
203	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₄₇	1	2	yes	—
205	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	47	4	no	—
213	C₄⋊C₈	—	94	4	no	—
214	C₄⋊C₈	—	94	2	no	—
215	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
216	#95	—	94	2	no	—
217	#95	—	94	2	no	—
218	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
227	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
228	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	M₁₆	—	4	8	no	—
239	M₁₆	—	188	8	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	188	4	no	—
244	C₄⋊C₄	—	188	4	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	2	no	—
247	C₄⋊C₄	—	188	2	no	—
248	C₄⋊C₄	—	188	2	no	—
249	C₂²⋊C₄	—	188	4	no	—
250	C₄²	—	188	4	no	—
251	C₄²	C₄×Dic₄₇	1	2	yes	—
252	C₄²	Dic₄₇⋊C₄	1	2	yes	—
253	C₄²	Dic₄₇⋊C₄	1	2	yes	—
254	C₄²	C₄⋊Dic₄₇	1	2	yes	—
255	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
256	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
257	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	C₄₇⋊#95	1	2	yes	—
266	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	C₄₇⋊#95	1	2	yes	—
283	C₂×C₄	C₄₇⋊#95	1	2	yes	—
284	C₂×C₄	C₄₇⋊#95	1	2	yes	—
285	C₂×C₄	#13093940	1	2	yes	—
286	C₂×C₄	#13093942	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	#13094050	1	2	yes	—
294	C₂×C₄	#13094052	1	2	yes	—
295	C₂×C₄	#13094055	1	2	yes	—
296	C₂×C₄	#13094057	1	2	yes	—
297	C₈	—	4	4	no	—
298	C₈	—	188	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
305	C₂²	C₂×#13093940	1	1	yes	characteristic
306	C₂²	C₂×#13093942	1	1	yes	characteristic
307	C₂²	#15689122	1	2	yes	—
308	C₂²	#15689125	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	376	4	no	—
314	C₄	#15688891	1	4	yes	—
315	C₄	#15689128	1	1	yes	characteristic
316	C₄	#15689129	1	1	yes	characteristic
317	C₄	#15689132	1	1	yes	characteristic
318	C₄	#15689133	1	1	yes	characteristic
319	C₄	#15689163	1	2	yes	—
320	C₄	#15689165	1	2	yes	—
321	C₂	—	4	2	no	—
322	C₂	#20242121	1	2	yes	—
323	C₂	#20242125	1	2	yes	—
324	C₂	#20243107	1	1	yes	characteristic
325	C₂	#20243108	1	1	yes	characteristic
326	C₂	C₂×#15688891	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#8041
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20241272	C₂	semidirect
8. #15689147	C₄	semidirect
20. C₄₇×#423	C₄	semidirect
175. #423	Dic₄₇	semidirect
186. C₄₇	#8041	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
#20245795	2
#20241272	2
C₂×C₄₇×#423	2
#8041	47