Subgroups — #26226673

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20244158	C₂	1	4	yes	maximal, complemented
4	#20244152	C₂	1	2	yes	maximal, complemented
5	#20244147	C₂	1	2	yes	maximal, complemented
6	#20243421	C₂	1	4	yes	maximal
7	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	—	2	1	no	—
9	C₂×#13093958	C₂²	1	4	yes	—
10	C₂×#13093956	C₂²	1	4	yes	—
11	#15688993	C₂²	1	2	yes	—
12	#15688991	—	2	1	no	—
13	#15688989	—	2	1	no	—
14	#15688989	C₂²	1	2	yes	—
15	#15688986	—	2	1	no	—
16	#15688983	—	2	1	no	—
17	#15688983	C₂²	1	2	yes	—
18	#15688983	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	—	2	1	no	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
24	#15688969	—	2	1	no	—
25	#15688886	C₂²	1	4	yes	—
26	#15688886	C₂²	1	4	yes	—
27	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#423	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
32	#13093983	—	4	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	C₂×C₄×Dic₄₇	—	2	1	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	#13093958	—	4	4	no	—
44	#13093956	—	4	4	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₄₇⋊#95	—	2	2	no	—
47	C₄₇⋊#95	—	2	2	no	—
48	C₄₇⋊#95	—	2	2	no	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	1	no	—
51	C₄₇⋊#95	—	2	1	no	—
52	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₄₇	—	4	4	no	—
56	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
66	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	—	2	2	no	—
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	4	no	—
85	M₁₆×C₄₇	—	4	8	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
89	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
95	C₂²⋊Dic₄₇	—	4	4	no	—
96	C₂²⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	4	4	no	—
104	C₄⋊Dic₄₇	—	4	4	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	C₄×Dic₄₇	—	4	2	no	—
107	C₄×Dic₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	2	1	no	—
110	C₂³×C₄₇	—	2	1	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₄₇	#137	1	2	yes	—
131	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
134	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	4	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₈×C₄₇	—	2	8	no	—
145	#21963	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	4	2	no	—
151	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₇	#537	1	2	yes	—
153	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
155	C₂²×C₄₇	#575	1	2	yes	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	#428	1	4	yes	—
161	C₄×C₄₇	#518	1	1	yes	characteristic
162	C₄×C₄₇	#519	1	1	yes	characteristic
163	C₄×C₄₇	#544	1	2	yes	—
164	Dic₄₇	—	8	4	no	—
165	Dic₄₇	—	8	2	no	—
166	Dic₄₇	—	8	2	no	—
167	C₂×#423	D₄₇	1	1	yes	characteristic
168	#2285	—	47	1	no	—
169	#2283	—	47	1	no	—
170	#1949	—	47	4	no	—
171	#1948	—	47	2	no	—
172	#1947	—	47	2	no	—
173	#1809	—	47	4	no	—
174	C₂×C₄₇	—	4	2	no	—
175	C₂×C₄₇	#2316	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#428	1	1	yes	characteristic
177	C₂×C₄₇	#2911	1	1	yes	characteristic
178	C₂×C₄₇	#3251	1	2	yes	—
179	C₂×C₄₇	#3252	1	2	yes	—
180	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
181	C₂×#117	—	94	1	no	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#107	—	47	4	no	—
184	C₂×#106	—	47	4	no	—
185	#423	Dic₄₇	1	8	yes	complemented
186	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
187	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
188	#384	—	47	2	no	—
189	#383	—	94	1	no	—
190	#382	—	47	2	no	—
191	#382	—	94	1	no	—
192	#381	—	94	1	no	—
193	#380	—	47	2	no	—
194	#380	—	47	2	no	—
195	#380	—	94	1	no	—
196	C₄×C₄⋊C₄	—	47	1	no	—
197	C₄×C₄⋊C₄	—	47	1	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₄⋊C₄	—	94	1	no	—
200	C₄×C₂²⋊C₄	—	94	1	no	—
201	#376	—	94	1	no	—
202	#340	—	47	4	no	—
203	#340	—	47	4	no	—
204	C₄₇	#21963	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
206	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×M₁₆	—	4	4	no	—
209	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
210	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
211	#117	—	188	4	no	—
212	C₂×C₄⋊C₄	—	47	2	no	—
213	C₂×C₄⋊C₄	—	47	2	no	—
214	C₂×C₄⋊C₄	—	47	4	no	—
215	C₂×C₄⋊C₄	—	94	2	no	—
216	C₂×C₄⋊C₄	—	94	2	no	—
217	C₂×C₂²⋊C₄	—	94	2	no	—
218	C₂×C₂²⋊C₄	—	94	2	no	—
219	C₂×C₄²	—	94	2	no	—
220	C₂×C₄²	—	94	1	no	—
221	C₂×C₄²	—	94	1	no	—
222	C₂×C₄²	Dic₉₄	1	4	yes	—
223	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
224	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
226	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
228	#107	—	188	4	no	—
229	#106	—	188	4	no	—
230	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
231	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
232	#95	—	94	2	no	—
233	#95	—	94	2	no	—
234	#95	—	94	2	no	—
235	#95	—	94	2	no	—
236	#95	—	94	1	no	—
237	#95	—	94	1	no	—
238	#95	—	94	2	no	—
239	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	94	2	no	—
247	C₂²×C₄	—	94	4	no	—
248	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
249	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
250	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
251	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
257	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
258	M₁₆	—	4	8	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
262	C₄⋊C₄	—	94	2	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	—	94	2	no	—
265	C₄⋊C₄	—	94	2	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	4	no	—
268	C₄⋊C₄	—	188	4	no	—
269	C₄⋊C₄	—	188	4	no	—
270	C₂²⋊C₄	—	188	4	no	—
271	C₂²⋊C₄	—	188	4	no	—
272	C₄²	—	188	4	no	—
273	C₄²	—	188	2	no	—
274	C₄²	—	188	2	no	—
275	C₄²	C₄⋊Dic₄₇	1	4	yes	—
276	C₄²	C₄⋊Dic₄₇	1	4	yes	—
277	C₄²	C₂×Dic₉₄	1	2	yes	—
278	C₄²	C₂×Dic₉₄	1	2	yes	—
279	C₄²	C₂²×Dic₄₇	1	2	yes	—
280	C₄²	C₂²×Dic₄₇	1	2	yes	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
286	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	—	188	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
308	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
309	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
311	C₂×C₄	#13094052	1	2	yes	—
312	C₂×C₄	#13094057	1	2	yes	—
313	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
315	C₂×C₄	#13094069	1	2	yes	—
316	C₂×C₄	#13094072	1	2	yes	—
317	C₈	—	2	8	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
324	C₂²	#15689704	1	2	yes	—
325	C₂²	C₂×#13094052	1	1	yes	characteristic
326	C₂²	C₂×#13094057	1	1	yes	characteristic
327	C₂²	#15689923	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	376	4	no	—
333	C₄	—	376	2	no	—
334	C₄	—	376	2	no	—
335	C₄	#15689165	1	4	yes	—
336	C₄	#15689598	1	1	yes	characteristic
337	C₄	#15689601	1	1	yes	characteristic
338	C₄	#15689754	1	2	yes	—
339	C₂	—	4	2	no	—
340	C₂	#20246006	1	1	yes	characteristic
341	C₂	C₂×#15689165	1	1	yes	characteristic
342	C₂	#20251978	1	1	yes	characteristic
343	C₂	#20254751	1	2	yes	—
344	C₂	#20254762	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#21963
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20244158	C₂	semidirect
4. #20244152	C₂	semidirect
5. #20244147	C₂	semidirect
29. C₄₇×#423	C₄	semidirect
185. #423	Dic₄₇	semidirect
204. C₄₇	#21963	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
#20245786	2
#20244158	2
#20244152	2
#20244147	2
#20243421	2
C₂×C₄₇×#423	2
#21963	47