Subgroups — #26375127

Order 12032.

2126 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	#20246137	C₂	1	4	yes	maximal, complemented
2	#20246130	C₂	1	4	yes	maximal, complemented
3	#20246123	C₂	1	8	yes	maximal
4	#20245872	C₂	1	2	yes	maximal
5	#20245866	C₂	1	8	yes	maximal, complemented
6	#20245795	C₂	1	1	yes	maximal, characteristic
7	#117×Dic₄₇	C₂	1	2	yes	maximal
8	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	#15689030	C₂²	1	16	yes	—
14	#15689022	C₂²	1	16	yes	—
15	#15689022	C₂²	1	8	yes	—
16	#15689009	C₂²	1	16	yes	—
17	#15689009	C₂²	1	16	yes	—
18	#15688998	C₂²	1	8	yes	—
19	#15688993	C₂²	1	2	yes	—
20	#15688989	C₂²	1	2	yes	—
21	#15688983	C₂²	1	4	yes	—
22	C₄⋊C₄×Dic₄₇	C₂²	1	8	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
25	C₂²⋊C₄×Dic₄₇	C₂²	1	8	yes	—
26	#15688969	C₂²	1	4	yes	—
27	#15688969	C₂²	1	4	yes	—
28	#15688969	C₂²	1	2	yes	—
29	#15688969	C₂²	1	2	yes	—
30	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
31	C₄²×Dic₄₇	C₂²	1	4	yes	—
32	C₄₇×#533	C₄	1	16	yes	complemented
33	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
34	C₂×#125×C₄₇	C₂²	1	2	yes	—
35	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
36	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
37	C₂×#117×C₄₇	C₂²	1	2	yes	—
38	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂³×Dic₄₇	C₂³	1	2	yes	—
40	C₂³×Dic₄₇	C₂³	1	2	yes	—
41	#13093983	—	2	8	no	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
45	C₂×C₂²⋊Dic₄₇	C₂³	1	4	yes	—
46	C₂×C₄×Dic₄₇	—	2	4	no	—
47	C₂×C₄×Dic₄₇	Q₈	1	8	yes	complemented
48	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
54	C₄₇⋊#95	C₂³	1	8	yes	—
55	C₄₇⋊#95	C₂³	1	8	yes	—
56	C₄₇⋊#95	C₂³	1	16	yes	—
57	C₄₇⋊#95	C₂³	1	8	yes	—
58	C₄₇⋊#95	C₂³	1	16	yes	—
59	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	complemented
63	#125×C₄₇	C₂×C₄	1	16	yes	complemented
64	C₂²⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
65	C₄×Q₈×C₄₇	C₂×C₄	1	32	yes	complemented
66	#117×C₄₇	C₂×C₄	1	16	yes	—
67	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
70	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
71	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
73	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂×Q₈×C₄₇	—	4	8	no	—
78	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
79	C₂²×C₄×C₄₇	—	4	4	no	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₄₇	#41	1	4	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₇	—	2	4	no	—
90	C₂²×Dic₄₇	—	4	4	no	—
91	C₂²×Dic₄₇	—	4	4	no	—
92	C₂²×Dic₄₇	C₂×Q₈	1	8	yes	—
93	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
94	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
95	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
97	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
98	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
99	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
102	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
103	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
104	C₂²⋊Dic₄₇	—	2	8	no	—
105	C₄⋊Dic₄₇	—	2	8	no	—
106	C₄⋊Dic₄₇	—	4	8	no	—
107	C₄⋊Dic₄₇	—	4	16	no	—
108	C₄×Dic₄₇	—	2	4	no	—
109	C₄×Dic₄₇	—	2	4	no	—
110	C₄×Dic₄₇	—	2	8	no	—
111	C₄×Dic₄₇	—	4	16	no	—
112	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
113	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
114	C₂³×C₄₇	—	2	4	no	—
115	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
116	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
117	C₂³×C₄₇	C₂×#41	1	2	yes	—
118	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
119	Q₈×C₄₇	—	4	16	no	—
120	C₂×C₄×C₄₇	—	2	16	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	4	8	no	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	16	yes	—
125	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×Dic₄₇	—	2	8	no	—
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	2	8	no	—
137	C₂×Dic₄₇	—	2	16	no	—
138	C₂×Dic₄₇	—	2	8	no	—
139	C₂×Dic₄₇	—	4	8	no	—
140	C₂×Dic₄₇	—	4	8	no	—
141	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
142	#34074	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	2	4	no	—
146	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
147	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
150	C₂²×C₄₇	#585	1	4	yes	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	—	2	16	no	—
154	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
155	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
156	Dic₄₇	—	2	8	no	—
157	Dic₄₇	—	4	8	no	—
158	Dic₄₇	—	4	4	no	—
159	C₂×#533	D₄₇	1	1	yes	characteristic
160	#2331	—	47	4	no	—
161	#2330	—	47	4	no	—
162	#2329	—	47	8	no	—
163	#2296	—	47	2	no	—
164	#2295	—	47	8	no	—
165	#2285	—	47	1	no	—
166	C₄×#117	—	47	2	no	—
167	C₂×#376	—	47	1	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
170	C₂×C₄₇	#3428	1	2	yes	—
171	C₂×C₄₇	#3455	1	2	yes	—
172	C₂×C₄₇	#3458	1	2	yes	—
173	#533	Dic₄₇	1	16	yes	complemented
174	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
175	C₂×#125	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
177	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
178	C₂×#117	—	47	2	no	—
179	C₂×#117	C₂×D₄₇	1	2	yes	—
180	C₂²×C₄²	—	47	1	no	—
181	C₂²×C₄²	—	47	1	no	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	#391	—	47	16	no	—
184	#389	—	47	16	no	—
185	#389	—	47	8	no	—
186	#387	—	47	16	no	—
187	#387	—	47	16	no	—
188	#385	—	47	8	no	—
189	#384	—	47	2	no	—
190	#382	—	47	2	no	—
191	#380	—	47	4	no	—
192	C₄×C₄⋊C₄	—	47	8	no	—
193	C₄×C₄⋊C₄	—	47	4	no	—
194	C₄×C₄⋊C₄	—	47	8	no	—
195	C₄×C₂²⋊C₄	—	47	8	no	—
196	#376	—	47	2	no	—
197	#376	—	47	2	no	—
198	#376	—	47	4	no	—
199	#376	—	47	4	no	—
200	C₂×#95	—	47	4	no	—
201	C₄³	—	47	4	no	—
202	C₄₇	#34074	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
204	C₂³×C₄	—	47	2	no	—
205	C₂³×C₄	—	47	2	no	—
206	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	complemented
209	#125	C₂×Dic₄₇	1	16	yes	complemented
210	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
211	C₄×Q₈	C₂×Dic₄₇	1	32	yes	complemented
212	#117	—	94	8	no	—
213	#117	C₂×Dic₄₇	1	16	yes	—
214	C₂×C₄⋊C₄	—	47	4	no	—
215	C₂×C₄⋊C₄	—	47	8	no	—
216	C₂×C₄⋊C₄	—	47	16	no	—
217	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
218	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
219	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
220	C₂×C₂²⋊C₄	—	47	4	no	—
221	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
222	C₂×C₄²	—	47	8	no	—
223	C₂×C₄²	—	47	2	no	—
224	C₂×C₄²	—	47	2	no	—
225	C₂×C₄²	—	47	2	no	—
226	C₂×C₄²	—	47	4	no	—
227	C₂×C₄²	—	47	2	no	—
228	C₂×C₄²	—	47	16	no	—
229	C₂×C₄²	—	94	4	no	—
230	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
234	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
235	#95	—	47	8	no	—
236	#95	—	47	8	no	—
237	#95	—	47	16	no	—
238	#95	—	47	8	no	—
239	#95	—	47	16	no	—
240	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
241	C₂×Q₈	—	4	8	no	—
242	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	47	4	no	—
245	C₂²×C₄	—	47	8	no	—
246	C₂²×C₄	—	47	4	no	—
247	C₂²×C₄	—	47	8	no	—
248	C₂²×C₄	—	94	4	no	—
249	C₂²×C₄	—	188	4	no	—
250	C₂²×C₄	—	188	4	no	—
251	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
252	C₂²×C₄	#10625354	1	4	yes	—
253	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
254	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
255	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
256	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
257	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
258	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
260	C₄⋊C₄	—	94	8	no	—
261	C₄⋊C₄	—	188	16	no	—
262	C₄⋊C₄	—	188	8	no	—
263	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
264	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
265	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
266	C₂²⋊C₄	—	94	8	no	—
267	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
268	C₄²	—	47	8	no	—
269	C₄²	—	47	8	no	—
270	C₄²	—	94	4	no	—
271	C₄²	—	94	4	no	—
272	C₄²	—	94	8	no	—
273	C₄²	—	188	16	no	—
274	C₄²	C₂²×Dic₄₇	1	8	yes	—
275	C₄²	C₂²×Dic₄₇	1	4	yes	—
276	C₄²	C₂²×Dic₄₇	1	4	yes	—
277	C₄²	C₂²×Dic₄₇	1	16	yes	—
278	C₂³	—	2	4	no	—
279	C₂³	C₂³×Dic₄₇	1	2	yes	—
280	C₂³	C₂×Q₈×D₄₇	1	2	yes	—
281	C₂³	C₂×#10625354	1	2	yes	—
282	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
283	Q₈	—	4	16	no	—
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	47	8	no	—
289	C₂×C₄	—	94	8	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	94	8	no	—
292	C₂×C₄	—	94	16	no	—
293	C₂×C₄	—	94	8	no	—
294	C₂×C₄	—	188	8	no	—
295	C₂×C₄	—	188	8	no	—
296	C₂×C₄	Q₈×Dic₄₇	1	16	yes	—
297	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
298	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
299	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
300	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
301	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
302	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
303	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
304	C₂×C₄	C₂×#10625354	1	2	yes	—
305	C₂×C₄	C₂×#10625354	1	2	yes	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	C₂×Q₈×Dic₄₇	1	4	yes	—
310	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
311	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
312	C₂²	C₂²×#10625354	1	1	yes	characteristic
313	C₂²	#15689959	1	4	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	2	4	no	—
316	C₄	—	2	16	no	—
317	C₄	—	94	8	no	—
318	C₄	—	188	4	no	—
319	C₄	—	188	8	no	—
320	C₄	C₂×Q₈×Dic₄₇	1	4	yes	—
321	C₄	C₂×Q₈×Dic₄₇	1	4	yes	—
322	C₂	—	2	4	no	—
323	C₂	C₂²×Q₈×Dic₄₇	1	1	yes	characteristic
324	C₂	#20256591	1	2	yes	—
325	C₂	#20256780	1	2	yes	—
326	C₂	#20256791	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#34074
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20246137	C₂	semidirect
2. #20246130	C₂	semidirect
5. #20245866	C₂	semidirect
32. C₄₇×#533	C₄	semidirect
47. C₂×C₄×Dic₄₇	Q₈	semidirect
62. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
63. #125×C₄₇	C₂×C₄	semidirect
65. C₄×Q₈×C₄₇	C₂×C₄	semidirect
173. #533	Dic₄₇	semidirect
202. C₄₇	#34074	semidirect
208. C₄⋊Q₈	C₂×Dic₄₇	semidirect
209. #125	C₂×Dic₄₇	semidirect
211. C₄×Q₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20246137	2
#20246130	2
#20246123	2
#20245872	2
#20245866	2
#20245795	2
#117×Dic₄₇	2
C₂×#15688969	2
C₂×C₄₇×#533	2
#34074	47