Subgroups — #26219566

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689149	C₂	1	2	yes	maximal
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20245795	C₂	1	1	yes	maximal, characteristic
4	#20244109	C₂	1	4	yes	maximal, complemented
5	#20244104	C₂	1	4	yes	maximal, complemented
6	#20243084	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	#15689149	—	2	8	no	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
13	C₄×#10625329	—	2	4	no	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688993	C₄	1	2	yes	—
17	#15688989	C₄	1	2	yes	—
18	#15688983	C₄	1	2	yes	—
19	#15688983	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688877	C₂²	1	8	yes	—
23	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	—	2	2	no	—
32	C₂×#10625329	D₄	1	4	yes	—
33	C₂×#10625329	Q₈	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	#13093983	—	2	4	no	—
37	#13093983	—	2	4	no	—
38	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
43	Dic₄₇⋊C₈	—	2	16	no	—
44	#13093935	—	2	4	no	—
45	C₄×C₄₇⋊C₈	—	2	4	no	—
46	C₄₇⋊#95	C₂×C₄	1	4	yes	—
47	C₄₇⋊#95	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	—	2	4	no	—
52	C₂²×C₈×C₄₇	—	2	4	no	—
53	C₂²×C₈×C₄₇	—	2	4	no	—
54	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	D₄	1	4	yes	—
58	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	—	2	4	no	—
62	C₄⋊C₈×C₄₇	—	2	4	no	—
63	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
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	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	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	#41	1	2	yes	—
78	C₂²×C₄×C₄₇	#41	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	4	8	no	—
85	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₂²⋊Dic₄₇	—	2	8	no	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₄⋊Dic₄₇	—	2	8	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₄×Dic₄₇	—	2	4	no	—
97	#10625329	—	2	8	no	—
98	#10625329	—	2	8	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
104	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₄₇	C₄×D₄	1	2	yes	—
109	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#121	1	4	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	#125	1	4	yes	—
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
127	C₂×C₄×C₄₇	#131	1	4	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₄₇	—	2	8	no	—
135	C₂×Dic₄₇	—	2	8	no	—
136	C₂×Dic₄₇	—	2	8	no	—
137	C₈×C₄₇	—	4	4	no	—
138	C₄₇⋊C₈	—	2	4	no	—
139	C₄₇⋊C₈	—	2	4	no	—
140	#21145	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	#389	1	4	yes	—
146	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#131	1	2	yes	—
150	C₂²×C₄₇	#568	1	2	yes	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	#389	1	4	yes	—
155	C₄×C₄₇	#424	1	4	yes	complemented
156	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
157	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
159	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	Dic₄₇	—	4	8	no	—
161	C₂×#424	—	47	2	no	—
162	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
163	C₂×C₄×M₁₆	—	47	1	no	—
164	#2285	—	47	1	no	—
165	#1940	—	47	4	no	—
166	#1939	—	47	4	no	—
167	#1729	—	47	2	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
171	C₂×C₄₇	#2905	1	1	yes	characteristic
172	C₂×C₄₇	#2935	1	2	yes	—
173	C₂×C₄₇	#2962	1	2	yes	—
174	C₂²×M₁₆	—	47	1	no	—
175	C₂²×M₁₆	—	47	1	no	—
176	C₂³×C₈	C₂×D₄₇	1	2	yes	—
177	C₂×#117	—	47	2	no	—
178	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
179	#424	—	94	8	no	—
180	C₂×C₄⋊C₈	—	2	4	no	—
181	C₂×C₄⋊C₈	—	47	8	no	—
182	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
183	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
184	C₄×M₁₆	—	94	4	no	—
185	C₂×#97	—	47	2	no	—
186	C₂×C₄×C₈	—	47	2	no	—
187	#384	—	47	2	no	—
188	#382	—	47	2	no	—
189	#380	—	47	2	no	—
190	#380	—	47	2	no	—
191	C₄×C₄⋊C₄	—	47	2	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	#336	—	47	8	no	—
194	C₄₇	#21145	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×M₁₆	—	47	4	no	—
199	C₂×M₁₆	—	47	4	no	—
200	C₂×M₁₆	—	94	2	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂×M₁₆	—	94	2	no	—
203	C₂×M₁₆	—	94	2	no	—
204	C₂²×C₈	—	2	4	no	—
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	47	2	no	—
208	C₂²×C₈	—	47	2	no	—
209	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
210	#117	—	94	4	no	—
211	#117	—	94	4	no	—
212	C₂×C₄⋊C₄	—	47	4	no	—
213	C₂×C₄⋊C₄	—	47	4	no	—
214	C₂×C₄⋊C₄	—	47	4	no	—
215	C₂×C₂²⋊C₄	—	47	4	no	—
216	C₂×C₄²	—	47	4	no	—
217	C₂×C₄²	Dic₉₄	1	4	yes	—
218	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	94	16	no	—
226	#97	—	94	4	no	—
227	C₄×C₈	—	94	4	no	—
228	#95	—	47	4	no	—
229	#95	—	47	4	no	—
230	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
235	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
236	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₄₇	1	2	yes	—
243	C₂²×C₄	#10625352	1	2	yes	—
244	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
245	C₂²×C₄	#10625354	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	M₁₆	—	94	8	no	—
249	M₁₆	—	94	8	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	8	no	—
254	C₂×C₈	—	47	4	no	—
255	C₂×C₈	—	47	4	no	—
256	C₂×C₈	—	94	2	no	—
257	C₂×C₈	—	94	2	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	—	94	2	no	—
260	C₄⋊C₄	—	94	8	no	—
261	C₄⋊C₄	—	94	8	no	—
262	C₄⋊C₄	—	188	4	no	—
263	C₂²⋊C₄	—	94	8	no	—
264	C₄²	—	94	4	no	—
265	C₄²	—	94	4	no	—
266	C₄²	Dic₄₇⋊C₄	1	4	yes	—
267	C₄²	Dic₄₇⋊C₄	1	4	yes	—
268	C₄²	C₂×Dic₉₄	1	2	yes	—
269	C₄²	C₂×Dic₉₄	1	2	yes	—
270	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
271	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
275	C₂³	#13093986	1	2	yes	—
276	C₂³	#13093989	1	2	yes	—
277	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	—	94	8	no	—
285	C₂×C₄	—	94	8	no	—
286	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
290	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
291	C₂×C₄	#13093986	1	2	yes	—
292	C₂×C₄	#13093989	1	2	yes	—
293	C₂×C₄	#13093995	1	4	yes	—
294	C₂×C₄	#13094000	1	4	yes	—
295	C₂×C₄	#13094015	1	4	yes	—
296	C₂×C₄	#13094023	1	4	yes	—
297	C₂×C₄	#13094030	1	4	yes	—
298	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
304	C₈	—	4	4	no	—
305	C₈	—	94	4	no	—
306	C₈	—	94	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	#15689019	1	4	yes	—
312	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
313	C₂²	C₂×#13093986	1	1	yes	characteristic
314	C₂²	C₂×#13093989	1	1	yes	characteristic
315	C₂²	C₂×#13094030	1	2	yes	—
316	C₂²	#15689889	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	188	8	no	—
321	C₄	#15689019	1	4	yes	—
322	C₄	#15689149	1	4	yes	complemented
323	C₄	C₂×#13093995	1	1	yes	characteristic
324	C₄	C₂×#13094000	1	1	yes	characteristic
325	C₄	C₂×#13094015	1	1	yes	characteristic
326	C₄	C₂×#13094023	1	1	yes	characteristic
327	C₂	—	2	4	no	—
328	C₂	C₂×#15689019	1	1	yes	characteristic
329	C₂	C₂×#15689149	1	1	yes	characteristic
330	C₂	#20251951	1	1	yes	characteristic
331	C₂	#20252131	1	2	yes	—
332	C₂	#20252331	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#21145
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #20244109	C₂	semidirect
5. #20244104	C₂	semidirect
155. C₄×C₄₇	#424	semidirect
194. C₄₇	#21145	semidirect
322. C₄	#15689149	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689149	2
C₂×C₄×#10625329	2
#20245795	2
#20244109	2
#20244104	2
#20243084	2
C₂²×C₄⋊C₈×C₄₇	2
#21145	47