Subgroups — #26376759

Order 12032.

2094 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246488	C₂	1	8	yes	maximal, complemented
2	#20246265	C₂	1	8	yes	maximal
3	#20246130	C₂	1	4	yes	maximal, complemented
4	#20246006	C₂	1	2	yes	maximal
5	#125×Dic₄₇	C₂	1	4	yes	maximal, complemented
6	#20245795	C₂	1	1	yes	maximal, characteristic
7	#20245745	C₂	1	2	yes	maximal
8	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
9	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	#15689022	C₂²	1	8	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	#15688993	C₂²	1	2	yes	—
19	#15688989	C₂²	1	2	yes	—
20	#15688983	C₂²	1	4	yes	—
21	#15688983	C₂²	1	8	yes	—
22	C₄⋊C₄×Dic₄₇	C₂²	1	8	yes	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
24	C₄⋊C₄×Dic₄₇	C₂²	1	8	yes	—
25	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
26	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
27	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
28	C₄×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
29	#15688969	C₂²	1	2	yes	—
30	C₄²×Dic₄₇	C₂²	1	2	yes	—
31	C₂×#125×C₄₇	C₄	1	16	yes	complemented
32	C₂×#125×C₄₇	C₂²	1	2	yes	—
33	C₂×#125×C₄₇	C₂²	1	2	yes	—
34	C₂×#125×C₄₇	C₂²	1	4	yes	—
35	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
37	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
38	#13093983	—	2	8	no	—
39	#13093983	—	2	4	no	—
40	#13093983	Q₈	1	8	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
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×C₄⋊Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₂²⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₂²⋊Dic₄₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊Dic₄₇	C₂³	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
54	C₄₇⋊#95	C₂³	1	8	yes	—
55	C₄₇⋊#95	C₂³	1	16	yes	—
56	C₄₇⋊#95	C₂³	1	8	yes	—
57	C₄₇⋊#95	C₂³	1	16	yes	—
58	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	#125×C₄₇	C₂×C₄	1	16	yes	—
62	#125×C₄₇	C₂×C₄	1	16	yes	—
63	#125×C₄₇	C₂×C₄	1	32	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
66	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	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×C₄²×C₄₇	C₂×C₄	1	8	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
74	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	—	4	4	no	—
76	C₂²×C₄×C₄₇	—	4	8	no	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₄₇	#41	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
88	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
89	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
90	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
91	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
92	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
93	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
94	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
97	C₂²⋊Dic₄₇	—	2	4	no	—
98	C₂²⋊Dic₄₇	—	2	8	no	—
99	C₂²⋊Dic₄₇	C₂×Q₈	1	8	yes	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	8	no	—
102	C₄⋊Dic₄₇	—	4	8	no	—
103	C₄⋊Dic₄₇	—	4	16	no	—
104	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	—
105	C₄×Dic₄₇	—	2	8	no	—
106	C₄×Dic₄₇	—	2	4	no	—
107	C₄×Dic₄₇	—	4	16	no	—
108	C₄×Dic₄₇	—	4	8	no	—
109	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
110	C₂³×C₄₇	—	2	4	no	—
111	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
112	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
113	C₂³×C₄₇	C₂×#41	1	2	yes	—
114	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	16	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	4	8	no	—
119	C₂×C₄×C₄₇	—	4	16	no	—
120	C₂×C₄×C₄₇	C₄×Q₈	1	16	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	8	no	—
134	C₂×Dic₄₇	—	2	8	no	—
135	C₂×Dic₄₇	—	2	8	no	—
136	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
137	#34217	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
142	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
145	C₂²×C₄₇	#585	1	4	yes	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	2	16	no	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
150	C₄×C₄₇	C₂×C₄×Q₈	1	4	yes	—
151	Dic₄₇	—	2	8	no	—
152	Dic₄₇	—	4	8	no	—
153	Dic₄₇	—	4	4	no	—
154	C₂²×#125	D₄₇	1	1	yes	characteristic
155	#2370	—	47	8	no	—
156	#2345	—	47	8	no	—
157	#2330	—	47	4	no	—
158	#2316	—	47	2	no	—
159	C₄×#125	—	47	4	no	—
160	#2285	—	47	1	no	—
161	#2274	—	47	2	no	—
162	C₄×#117	—	47	1	no	—
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
165	C₂×C₄₇	#3428	1	2	yes	—
166	C₂×C₄₇	#3504	1	2	yes	—
167	C₂×C₄₇	#3512	1	2	yes	—
168	C₂×#125	Dic₄₇	1	16	yes	complemented
169	C₂×#125	C₂×D₄₇	1	2	yes	—
170	C₂×#125	C₂×D₄₇	1	2	yes	—
171	C₂×#125	C₂×D₄₇	1	4	yes	—
172	C₂×#117	—	47	1	no	—
173	C₂×#117	—	47	1	no	—
174	C₂×#117	—	47	2	no	—
175	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
176	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	#389	—	47	8	no	—
179	#389	—	47	8	no	—
180	#389	—	47	16	no	—
181	#387	—	47	16	no	—
182	#387	—	47	16	no	—
183	#384	—	47	2	no	—
184	#382	—	47	2	no	—
185	#380	—	47	8	no	—
186	#380	—	47	4	no	—
187	C₄×C₄⋊C₄	—	47	8	no	—
188	C₄×C₄⋊C₄	—	47	16	no	—
189	C₄×C₄⋊C₄	—	47	8	no	—
190	C₄×C₄⋊C₄	—	47	4	no	—
191	C₄×C₄⋊C₄	—	47	4	no	—
192	C₄×C₂²⋊C₄	—	47	4	no	—
193	C₄×C₂²⋊C₄	—	47	8	no	—
194	#376	—	47	2	no	—
195	C₄³	—	47	2	no	—
196	C₄₇	#34217	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
198	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
199	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	#125	C₂×Dic₄₇	1	16	yes	—
201	#125	C₂×Dic₄₇	1	16	yes	—
202	#125	C₂×Dic₄₇	1	32	yes	—
203	#117	—	47	8	no	—
204	#117	—	94	4	no	—
205	#117	—	94	8	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	47	4	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	—	47	8	no	—
210	C₂×C₄⋊C₄	—	47	16	no	—
211	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
212	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
213	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
214	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
215	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
216	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
217	C₂×C₂²⋊C₄	—	47	2	no	—
218	C₂×C₂²⋊C₄	—	47	2	no	—
219	C₂×C₂²⋊C₄	—	47	4	no	—
220	C₂×C₄²	—	47	2	no	—
221	C₂×C₄²	—	47	2	no	—
222	C₂×C₄²	—	47	4	no	—
223	C₂×C₄²	—	47	8	no	—
224	C₂×C₄²	—	47	16	no	—
225	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
226	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
229	#95	—	47	8	no	—
230	#95	—	47	16	no	—
231	#95	—	47	8	no	—
232	#95	—	47	16	no	—
233	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	8	no	—
236	C₂²×C₄	—	47	4	no	—
237	C₂²×C₄	—	47	8	no	—
238	C₂²×C₄	—	47	4	no	—
239	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
240	C₂²×C₄	#10625354	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
245	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
249	C₄⋊C₄	—	47	8	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	94	8	no	—
252	C₄⋊C₄	—	188	16	no	—
253	C₄⋊C₄	—	188	8	no	—
254	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
255	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
256	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
257	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
258	C₂²⋊C₄	—	47	8	no	—
259	C₂²⋊C₄	—	94	4	no	—
260	C₂²⋊C₄	—	94	8	no	—
261	C₄²	—	47	8	no	—
262	C₄²	—	94	8	no	—
263	C₄²	—	94	4	no	—
264	C₄²	—	188	8	no	—
265	C₄²	—	188	16	no	—
266	C₄²	C₂²×Dic₄₇	1	4	yes	—
267	C₄²	C₂²×Dic₄₇	1	4	yes	—
268	C₄²	C₂²×Dic₄₇	1	8	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	C₂³×Dic₄₇	1	2	yes	—
271	C₂³	C₂×Q₈×D₄₇	1	2	yes	—
272	C₂³	C₂×#10625354	1	2	yes	—
273	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	47	8	no	—
280	C₂×C₄	—	94	8	no	—
281	C₂×C₄	—	94	8	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	—	94	8	no	—
285	C₂×C₄	—	94	8	no	—
286	C₂×C₄	Q₈×Dic₄₇	1	16	yes	—
287	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
288	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
289	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
290	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
291	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
292	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
293	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
294	C₂×C₄	C₂×#10625354	1	2	yes	—
295	C₂×C₄	C₂×#10625354	1	2	yes	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	C₂×Q₈×Dic₄₇	1	4	yes	—
300	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
301	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
302	C₂²	C₂²×#10625354	1	1	yes	characteristic
303	C₂²	#15689959	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	16	no	—
306	C₄	—	2	4	no	—
307	C₄	—	94	8	no	—
308	C₄	—	188	8	no	—
309	C₄	—	188	4	no	—
310	C₄	C₂×Q₈×Dic₄₇	1	4	yes	—
311	C₄	C₂×Q₈×Dic₄₇	1	4	yes	—
312	C₂	—	2	4	no	—
313	C₂	C₂²×Q₈×Dic₄₇	1	1	yes	characteristic
314	C₂	#20256591	1	2	yes	—
315	C₂	#20257212	1	2	yes	—
316	C₂	#20257312	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂²×#125×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#34217
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. #20246488	C₂	semidirect
3. #20246130	C₂	semidirect
5. #125×Dic₄₇	C₂	semidirect
31. C₂×#125×C₄₇	C₄	semidirect
168. C₂×#125	Dic₄₇	semidirect
196. C₄₇	#34217	semidirect

Maximal subgroups

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

subgroup	index
#20246488	2
#20246265	2
#20246130	2
#20246006	2
#125×Dic₄₇	2
#20245795	2
#20245745	2
C₄×#13093983	2
C₂²×#125×C₄₇	2
#34217	47