Subgroups — #26216721

Order 12032.

1070 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×#15689165	C₂	1	2	yes	maximal
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
4	#20246011	C₂	1	2	yes	maximal
5	#20243556	C₂	1	4	yes	maximal, complemented
6	#20243552	C₂	1	2	yes	maximal, complemented
7	#20243545	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	#15689165	—	2	8	no	—
12	#15689161	—	2	4	no	—
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093958	C₂²	1	4	yes	—
15	C₂×#13093956	—	2	4	no	—
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093956	C₂²	1	2	yes	—
18	C₂×#13093956	C₂²	1	1	yes	characteristic
19	C₂×#13093956	C₂²	1	1	yes	characteristic
20	C₂×#13093956	C₂²	1	4	yes	—
21	#15689022	—	2	4	no	—
22	#15688998	—	2	4	no	—
23	#15688993	C₂²	1	4	yes	—
24	#15688989	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#405	C₄	1	8	yes	complemented
29	C₂×#97×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	C₂³×Dic₄₇	—	2	2	no	—
32	#13093983	—	2	4	no	—
33	#13093983	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	4	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	8	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	#13093958	—	2	4	no	—
44	#13093958	—	2	8	no	—
45	#13093956	—	2	4	no	—
46	#13093956	—	2	4	no	—
47	#13093956	—	2	2	no	—
48	#13093956	—	2	2	no	—
49	#13093956	—	2	8	no	—
50	C₄₇⋊#95	—	2	8	no	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
54	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
56	C₂²×C₈×C₄₇	D₄	1	4	yes	—
57	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	#117×C₄₇	C₂×C₄	1	4	yes	—
62	#117×C₄₇	Q₈	1	4	yes	—
63	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
66	#97×C₄₇	C₂×C₄	1	8	yes	complemented
67	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
68	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	2	2	no	—
81	C₂²×Dic₄₇	—	4	4	no	—
82	C₂²×Dic₄₇	—	8	4	no	—
83	M₁₆×C₄₇	C₄⋊C₄	1	16	yes	complemented
84	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
95	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
97	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
98	C₂²⋊Dic₄₇	—	4	4	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	—	4	4	no	—
106	C₄⋊Dic₄₇	—	8	8	no	—
107	C₄×Dic₄₇	—	4	4	no	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₄₇	#127	1	2	yes	—
111	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	#127	1	2	yes	—
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	8	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	8	8	no	—
135	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
137	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
138	#20802	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	#384	1	4	yes	—
143	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₄₇	#577	1	4	yes	—
147	C₄×C₄₇	—	2	8	no	—
148	C₄×C₄₇	#384	1	4	yes	—
149	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₄₇	—	8	2	no	—
154	Dic₄₇	—	8	2	no	—
155	Dic₄₇	—	8	4	no	—
156	C₂×#428	—	47	2	no	—
157	C₂×#427	—	47	1	no	—
158	C₂²×#106	—	47	1	no	—
159	C₂×#405	D₄₇	1	1	yes	characteristic
160	#2317	—	47	2	no	—
161	#1847	—	47	4	no	—
162	#1846	—	47	2	no	—
163	#1844	—	47	2	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
166	C₂×C₄₇	#2912	1	2	yes	—
167	C₂×C₄₇	#3154	1	2	yes	—
168	C₂×C₄₇	#3163	1	2	yes	—
169	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
170	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#117	—	47	2	no	—
172	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	—	47	2	no	—
174	#428	—	94	8	no	—
175	#427	—	94	4	no	—
176	C₂×#107	—	47	4	no	—
177	C₂×#107	—	47	2	no	—
178	C₂×#106	—	47	1	no	—
179	C₂×#106	—	47	2	no	—
180	C₂×#106	—	47	4	no	—
181	C₂×#106	—	47	1	no	—
182	C₂×#106	—	47	2	no	—
183	C₂×#106	—	94	4	no	—
184	#405	Dic₄₇	1	8	yes	complemented
185	C₂×#97	C₂×D₄₇	1	2	yes	—
186	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
187	#389	—	94	4	no	—
188	#385	—	94	4	no	—
189	#384	—	47	4	no	—
190	#382	—	47	4	no	—
191	C₄₇	#20802	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	94	2	no	—
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂×M₁₆	Dic₉₄	1	4	yes	—
195	C₂×M₁₆	D₁₈₈	1	4	yes	—
196	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
197	C₂²×C₈	Dic₉₄	1	4	yes	—
198	C₂²×C₈	D₁₈₈	1	4	yes	—
199	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
203	#117	—	94	4	no	—
204	#117	—	94	4	no	—
205	#117	Dic₉₄	1	4	yes	—
206	#117	C₂×Dic₄₇	1	4	yes	—
207	C₂×C₄⋊C₄	—	47	4	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	—	47	2	no	—
210	C₂×C₄⋊C₄	—	94	4	no	—
211	C₂×C₄⋊C₄	—	94	4	no	—
212	C₂×C₄⋊C₄	—	94	8	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
215	C₂×C₂²⋊C₄	—	94	2	no	—
216	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
217	C₂×C₄²	—	94	2	no	—
218	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
219	#107	—	94	4	no	—
220	#107	—	94	8	no	—
221	#106	—	94	4	no	—
222	#106	—	94	4	no	—
223	#106	—	94	2	no	—
224	#106	—	94	2	no	—
225	#106	—	94	8	no	—
226	#97	C₂×Dic₄₇	1	8	yes	complemented
227	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
228	#95	—	94	8	no	—
229	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	—	94	2	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	376	4	no	—
234	C₂²×C₄	—	188	4	no	—
235	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
236	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
237	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
238	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
241	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	M₁₆	C₄⋊Dic₄₇	1	16	yes	complemented
245	C₂×C₈	C₄⋊Dic₄₇	1	16	yes	complemented
246	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
247	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
248	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
249	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
250	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
251	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
252	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
253	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
254	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	—	94	2	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	94	2	no	—
259	C₄⋊C₄	—	94	2	no	—
260	C₄⋊C₄	—	94	2	no	—
261	C₄⋊C₄	—	376	8	no	—
262	C₄⋊C₄	—	188	4	no	—
263	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
264	C₂²⋊C₄	—	188	4	no	—
265	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
266	C₄²	—	188	4	no	—
267	C₄²	C₂×Dic₉₄	1	4	yes	—
268	C₄²	C₂²×Dic₄₇	1	4	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
271	C₂³	#13094021	1	2	yes	—
272	C₂³	#13094024	1	2	yes	—
273	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	376	8	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	8	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
283	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
286	C₂×C₄	#13094021	1	2	yes	—
287	C₂×C₄	#13094024	1	2	yes	—
288	C₂×C₄	#13094024	1	4	yes	—
289	C₂×C₄	#13094024	1	4	yes	—
290	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
296	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
297	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
298	C₈	C₂×C₄⋊Dic₄₇	1	8	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#15688993	1	4	yes	—
303	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
304	C₂²	C₂×#13094021	1	1	yes	characteristic
305	C₂²	C₂×#13094024	1	1	yes	characteristic
306	C₂²	#15689935	1	4	yes	—
307	C₄	—	2	8	no	—
308	C₄	—	376	4	no	—
309	C₄	—	376	2	no	—
310	C₄	—	376	2	no	—
311	C₄	#15688993	1	4	yes	—
312	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
313	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
314	C₄	C₂×#13094024	1	1	yes	characteristic
315	C₄	C₂×#13094024	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×#15688993	1	1	yes	characteristic
318	C₂	#20251983	1	2	yes	—
319	C₂	#20253864	1	2	yes	—
320	C₂	#20253919	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#20802
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #20243556	C₂	semidirect
6. #20243552	C₂	semidirect
7. #20243545	C₂	semidirect
28. C₄₇×#405	C₄	semidirect
66. #97×C₄₇	C₂×C₄	semidirect
67. C₄×C₈×C₄₇	C₂×C₄	semidirect
83. M₁₆×C₄₇	C₄⋊C₄	semidirect
84. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
184. #405	Dic₄₇	semidirect
191. C₄₇	#20802	semidirect
226. #97	C₂×Dic₄₇	semidirect
227. C₄×C₈	C₂×Dic₄₇	semidirect
244. M₁₆	C₄⋊Dic₄₇	semidirect
245. C₂×C₈	C₄⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689165	2
C₂×#15689161	2
C₂²×#13093956	2
#20246011	2
#20243556	2
#20243552	2
#20243545	2
C₂×C₄₇×#405	2
#20802	47