Subgroups — #26213885

Order 12032.

1774 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689078	C₂	1	2	yes	maximal, complemented
3	#20245771	C₂	1	1	yes	maximal, characteristic
4	#20243164	C₂	1	4	yes	maximal, complemented
5	C₈×C₂²⋊Dic₄₇	C₂	1	4	yes	maximal, complemented
6	#20243084	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689087	C₄	1	8	yes	complemented
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	#15689078	—	2	8	no	—
13	C₂×#13093934	C₂²	1	4	yes	complemented
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	complemented
15	#15688986	C₄	1	2	yes	—
16	#15688983	C₄	1	4	yes	—
17	#15688979	C₄	1	2	yes	—
18	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
19	#15688877	C₂²	1	8	yes	complemented
20	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₄₇	—	2	8	no	—
22	C₂³×C₈×C₄₇	C₄	1	8	yes	complemented
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
25	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	#13093983	—	2	4	no	—
32	#13093983	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	8	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
38	#13093934	—	2	8	no	—
39	C₈×Dic₄₇	—	2	8	no	—
40	C₄₇⋊#95	C₂×C₄	1	8	yes	—
41	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	—	2	4	no	—
43	C₂³×C₄×C₄₇	—	2	4	no	—
44	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
47	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	—	2	4	no	—
51	C₂²×C₈×C₄₇	—	2	32	no	—
52	C₂²×C₈×C₄₇	—	2	8	no	—
53	C₂²×C₈×C₄₇	—	2	8	no	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₄₇	D₄	1	8	yes	—
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂⁴×C₄₇	—	2	4	no	—
63	C₂⁴×C₄₇	—	2	8	no	—
64	C₂⁴×C₄₇	C₄²	1	4	yes	—
65	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	8	no	—
69	C₂²×C₄×C₄₇	—	2	8	no	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	—	2	16	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
75	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₇	#41	1	4	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
86	C₂×C₈×C₄₇	—	2	8	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	16	no	—
89	C₂×C₈×C₄₇	—	2	16	no	—
90	C₂×C₈×C₄₇	C₂²⋊C₄	1	16	yes	complemented
91	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
96	C₂²⋊Dic₄₇	—	2	8	no	—
97	C₂²⋊Dic₄₇	—	2	16	no	—
98	C₄⋊Dic₄₇	—	2	8	no	—
99	C₄×Dic₄₇	—	2	4	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	#10625329	—	2	16	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
105	C₂³×C₄₇	—	2	32	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	—	2	4	no	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	—	2	8	no	—
110	C₂³×C₄₇	—	2	16	no	—
111	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
112	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
113	C₂³×C₄₇	C₄×D₄	1	4	yes	—
114	C₂³×C₄₇	#131	1	8	yes	—
115	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	2	16	no	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₇	#117	1	4	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₂×Dic₄₇	—	2	8	no	—
133	C₂×Dic₄₇	—	2	16	no	—
134	C₈×C₄₇	—	2	4	no	—
135	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
136	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
137	C₄₇⋊C₈	—	2	8	no	—
138	#20385	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	32	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	16	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
146	C₂²×C₄₇	#405	1	8	yes	complemented
147	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
149	C₂²×C₄₇	C₂×#131	1	4	yes	—
150	C₄×C₄₇	—	2	4	no	—
151	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
152	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
155	Dic₄₇	—	4	8	no	—
156	C₂⁴×C₈	D₄₇	1	1	yes	characteristic
157	C₂×#407	—	47	1	no	—
158	C₂×#405	—	47	2	no	—
159	#2280	—	47	1	no	—
160	#1752	—	47	4	no	—
161	C₈×C₂²⋊C₄	—	47	4	no	—
162	#1729	—	47	2	no	—
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	—	2	8	no	—
165	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#405	1	2	yes	—
167	C₂×C₄₇	#2929	1	4	yes	—
168	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
169	C₂²×M₁₆	—	47	2	no	—
170	C₂³×C₈	—	2	8	no	—
171	C₂³×C₈	Dic₄₇	1	8	yes	complemented
172	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₂³×C₈	C₂×D₄₇	1	2	yes	—
174	C₂³×C₈	C₂×D₄₇	1	2	yes	—
175	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂×#117	—	47	2	no	—
177	#407	—	47	8	no	—
178	C₂×C₂²⋊C₈	—	47	4	no	—
179	#405	—	94	8	no	—
180	C₂×#97	—	47	4	no	—
181	C₂×C₄×C₈	—	47	4	no	—
182	#381	—	47	2	no	—
183	#380	—	47	4	no	—
184	#379	—	47	2	no	—
185	C₄×C₂²⋊C₄	—	47	4	no	—
186	#336	—	47	8	no	—
187	C₄₇	#20385	1	1	yes	characteristic, complemented, Sylow
188	C₂⁵	C₄×D₄₇	1	1	yes	characteristic
189	C₂³×C₄	—	2	4	no	—
190	C₂³×C₄	—	2	4	no	—
191	C₂³×C₄	C₄×D₄₇	1	2	yes	—
192	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₄₇	1	4	yes	—
194	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×M₁₆	—	47	8	no	—
198	C₂×M₁₆	—	94	4	no	—
199	C₂×M₁₆	—	94	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	—	2	32	no	—
202	C₂²×C₈	—	2	8	no	—
203	C₂²×C₈	—	2	8	no	—
204	C₂²×C₈	—	47	4	no	—
205	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
206	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
207	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
208	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
209	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
210	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
211	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	#117	—	94	4	no	—
214	#117	—	94	4	no	—
215	C₂×C₄⋊C₄	—	47	4	no	—
216	C₂×C₂²⋊C₄	—	47	4	no	—
217	C₂×C₂²⋊C₄	—	47	8	no	—
218	C₂×C₄²	—	47	4	no	—
219	C₂²⋊C₈	—	47	16	no	—
220	#97	—	94	8	no	—
221	C₄×C₈	—	94	8	no	—
222	#95	—	47	8	no	—
223	C₂⁴	—	2	4	no	—
224	C₂⁴	—	2	8	no	—
225	C₂⁴	C₄×Dic₄₇	1	4	yes	—
226	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂⁴	C₂×C₄×D₄₇	1	2	yes	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	16	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	47	8	no	—
236	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
237	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
242	C₂²×C₄	#10625350	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	M₁₆	—	94	16	no	—
248	C₂×C₈	—	2	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	16	no	—
251	C₂×C₈	—	2	16	no	—
252	C₂×C₈	—	47	8	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	94	4	no	—
255	C₂×C₈	C₂²⋊Dic₄₇	1	16	yes	complemented
256	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
257	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
258	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
260	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
261	C₄⋊C₄	—	94	8	no	—
262	C₂²⋊C₄	—	94	8	no	—
263	C₂²⋊C₄	—	94	16	no	—
264	C₄²	—	94	4	no	—
265	C₄²	—	94	4	no	—
266	C₂³	—	2	32	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	16	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	4	no	—
272	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
273	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
274	C₂³	C₄×C₂²⋊D₄₇	1	4	yes	—
275	C₂³	#13094030	1	8	yes	—
276	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	16	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	—	94	16	no	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
288	C₂×C₄	#13093983	1	4	yes	—
289	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
290	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
291	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
294	C₂×C₄	C₂×#10625350	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	94	8	no	—
297	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
298	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
299	C₂²	—	2	32	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	16	no	—
304	C₂²	—	2	4	no	—
305	C₂²	C₄×C₂²⋊Dic₄₇	1	4	yes	—
306	C₂²	#15689078	1	8	yes	complemented
307	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
308	C₂²	C₂×C₄×C₂²⋊D₄₇	1	2	yes	—
309	C₂²	C₂×#13094030	1	4	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	188	8	no	—
312	C₄	C₄×C₂²⋊Dic₄₇	1	4	yes	—
313	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
314	C₄	C₂×#13093983	1	1	yes	characteristic
315	C₄	C₂×C₄×C₂²⋊D₄₇	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	2	8	no	—
318	C₂	C₂×C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
319	C₂	C₂×#15689078	1	2	yes	—
320	C₂	#20252082	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂⁴×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂⁴×C₈	D₄₇
hypocenter	C₄₇	#20385
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15689078	C₂	semidirect
4. #20243164	C₂	semidirect
5. C₈×C₂²⋊Dic₄₇	C₂	semidirect
6. #20243084	C₂	semidirect
10. #15689087	C₄	semidirect
13. C₂×#13093934	C₂²	semidirect
14. C₂×C₈×Dic₄₇	C₂²	semidirect
19. #15688877	C₂²	semidirect
22. C₂³×C₈×C₄₇	C₄	semidirect
90. C₂×C₈×C₄₇	C₂²⋊C₄	semidirect
146. C₂²×C₄₇	#405	semidirect
171. C₂³×C₈	Dic₄₇	semidirect
187. C₄₇	#20385	semidirect
255. C₂×C₈	C₂²⋊Dic₄₇	semidirect
306. C₂²	#15689078	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
C₂×#15689078	2
#20245771	2
#20243164	2
C₈×C₂²⋊Dic₄₇	2
#20243084	2
C₂⁴×C₈×C₄₇	2
#20385	47