Subgroups — #26216702

Order 12032.

1190 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689162	C₂	1	4	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20245793	C₂	1	1	yes	maximal, characteristic
4	#384×D₄₇	C₂	1	1	yes	maximal, characteristic
5	#20243553	C₂	1	4	yes	maximal, complemented
6	#20243548	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243547	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243544	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	2	yes	—
11	C₂×#13093982	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×D₄₇	—	2	2	no	—
13	C₂×C₄⋊C₄×D₄₇	C₂²	1	2	yes	—
14	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
15	#15689162	—	2	16	no	—
16	C₂×#13093958	C₂²	1	4	yes	—
17	C₂×#13093957	C₂²	1	4	yes	—
18	C₂×#13093955	C₂²	1	4	yes	—
19	C₄×#10625324	C₄	1	8	yes	complemented
20	C₂×#13093934	C₂²	1	2	yes	—
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	#15688993	C₂²	1	1	yes	characteristic
23	#15688993	C₂²	1	1	yes	characteristic
24	#15688992	—	2	2	no	—
25	#15688992	C₂²	1	2	yes	—
26	#15688988	C₂²	1	2	yes	—
27	#15688980	—	2	2	no	—
28	C₄⋊C₄×Dic₄₇	—	2	2	no	—
29	C₂×#106×C₄₇	C₂²	1	4	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#384	C₂²	1	1	yes	characteristic
32	C₄₇×#384	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₄₇	—	2	2	no	—
34	C₂²×C₄×D₄₇	D₄	1	2	yes	—
35	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
36	C₂×#10625324	C₂×C₄	1	8	yes	—
37	C₂×#10625324	D₄	1	8	yes	—
38	C₂×#10625324	Q₈	1	8	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	#13093982	—	2	8	no	—
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
45	C₂×Dic₄₇⋊C₄	—	2	2	no	—
46	C₂×Dic₄₇⋊C₄	—	2	4	no	—
47	C₄⋊C₄×D₄₇	—	2	8	no	—
48	C₄⋊C₄×D₄₇	—	4	8	no	—
49	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
50	C₂×C₄×Dic₄₇	—	2	2	no	—
51	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₄²×D₄₇	C₂×C₄	1	4	yes	—
54	C₄²×D₄₇	Q₈	1	4	yes	—
55	#13093958	—	2	8	no	—
56	#13093957	—	2	8	no	—
57	#13093955	—	2	8	no	—
58	#13093934	C₂×C₄	1	8	yes	complemented
59	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
60	C₄₇⋊#95	—	2	4	no	—
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	#106×C₄₇	—	2	8	no	—
68	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
69	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×C₄×D₄₇	—	2	4	no	—
80	C₂×C₄×D₄₇	—	4	8	no	—
81	C₂×C₄×D₄₇	—	4	4	no	—
82	C₂×C₄×D₄₇	C₄⋊C₄	1	8	yes	—
83	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
85	C₂×C₄×D₄₇	C₂×Q₈	1	4	yes	—
86	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	4	4	no	—
93	C₄⋊C₄×C₄₇	—	4	4	no	—
94	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
96	(C₂×D₄₇)⋊C₄	—	4	4	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	4	4	no	—
100	C₄⋊Dic₄₇	—	4	4	no	—
101	Dic₄₇⋊C₄	—	4	8	no	—
102	Dic₄₇⋊C₄	—	4	4	no	—
103	C₄×Dic₄₇	—	4	4	no	—
104	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
105	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
106	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
109	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
110	#10625324	C₄⋊C₄	1	32	yes	complemented
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂²×D₄₇	—	2	2	no	—
113	C₂²×D₄₇	—	2	2	no	—
114	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂²×D₄₇	#127	1	2	yes	—
116	C₂²×D₄₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
123	C₂×C₄×C₄₇	#137	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×Dic₄₇	#127	1	2	yes	—
137	C₂×Dic₄₇	C₄⋊Q₈	1	2	yes	—
138	C₄×D₄₇	—	2	16	no	—
139	C₄×D₄₇	—	8	8	no	—
140	C₄×D₄₇	C₂×C₄⋊C₄	1	8	yes	—
141	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
142	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	complemented
143	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	complemented
144	#20800	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₄₇	C₂×#137	1	2	yes	—
150	C₂×D₄₇	—	2	2	no	—
151	C₂×D₄₇	—	2	2	no	—
152	C₂×D₄₇	—	2	4	no	—
153	C₂×D₄₇	—	2	4	no	—
154	C₂×D₄₇	#384	1	4	yes	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	#428	1	8	yes	complemented
158	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
159	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
160	Dic₄₇	—	2	4	no	—
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	—	8	2	no	—
163	Dic₄₇	#384	1	4	yes	—
164	C₂×#428	—	47	4	no	—
165	C₂×C₄×M₁₆	—	47	1	no	—
166	#2285	—	47	1	no	—
167	C₂×#384	—	47	1	no	—
168	#1847	—	47	4	no	—
169	#1845	—	47	1	no	—
170	#1845	—	47	1	no	—
171	#1844	—	47	1	no	—
172	#1844	D₄₇	1	1	yes	characteristic, complemented
173	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#428	1	2	yes	—
175	C₂×C₄₇	#3151	1	1	yes	characteristic
176	C₂×C₄₇	#3152	1	1	yes	characteristic
177	C₂×C₄₇	#3165	1	2	yes	—
178	D₄₇	—	2	4	no	—
179	C₂²×M₁₆	—	47	2	no	—
180	C₂×#117	—	47	2	no	—
181	C₂²×C₄⋊C₄	—	47	2	no	—
182	C₂²×C₄⋊C₄	—	94	2	no	—
183	C₂²×C₄²	—	47	1	no	—
184	#428	—	94	16	no	—
185	C₂×#107	—	47	4	no	—
186	C₂×#107	—	47	4	no	—
187	C₂×#106	—	47	4	no	—
188	C₂×#106	C₂×D₄₇	1	4	yes	—
189	C₄×M₁₆	—	47	8	no	—
190	C₂×#97	—	47	2	no	—
191	C₂×C₄×C₈	—	47	1	no	—
192	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
193	#384	—	47	2	no	—
194	#384	—	47	1	no	—
195	#384	—	47	1	no	—
196	#384	—	94	2	no	—
197	#384	C₂×D₄₇	1	1	yes	characteristic
198	#384	C₂×D₄₇	1	1	yes	characteristic
199	#382	—	47	2	no	—
200	#380	—	94	2	no	—
201	C₄×C₄⋊C₄	—	94	2	no	—
202	C₄₇	#20800	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	47	1	no	—
204	C₂³×C₄	—	47	2	no	—
205	C₂³×C₄	—	94	2	no	—
206	C₂×M₁₆	—	47	8	no	—
207	C₂×M₁₆	—	47	8	no	—
208	C₂×M₁₆	—	47	8	no	—
209	C₂²×C₈	—	47	2	no	—
210	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
211	#117	—	94	8	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₄⋊C₄	—	47	2	no	—
216	C₂×C₄⋊C₄	—	94	8	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄⋊C₄	—	94	4	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	94	2	no	—
221	C₂×C₄⋊C₄	—	188	8	no	—
222	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
224	C₂×C₂²⋊C₄	—	94	2	no	—
225	C₂×C₄²	—	47	2	no	—
226	C₂×C₄²	—	47	1	no	—
227	C₂×C₄²	—	47	4	no	—
228	C₂×C₄²	—	47	4	no	—
229	C₂×C₄²	—	94	2	no	—
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	#107	—	94	8	no	—
232	#107	—	94	8	no	—
233	#106	—	2	8	no	—
234	#106	—	94	8	no	—
235	#97	—	47	8	no	—
236	C₄×C₈	—	47	4	no	—
237	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
238	#95	—	94	4	no	—
239	C₂⁴	—	47	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	47	1	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	—	47	4	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	4	no	—
247	C₂²×C₄	—	47	2	no	—
248	C₂²×C₄	—	47	8	no	—
249	C₂²×C₄	—	94	4	no	—
250	C₂²×C₄	—	94	4	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	—	188	4	no	—
254	C₂²×C₄	—	188	8	no	—
255	C₂²×C₄	D₄×D₄₇	1	2	yes	—
256	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
257	M₁₆	—	47	32	no	—
258	C₂×C₈	—	47	4	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	47	4	no	—
261	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
262	C₂×C₈	D₄×D₄₇	1	4	yes	—
263	C₂×C₈	Q₈×D₄₇	1	4	yes	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₄⋊C₄	—	94	4	no	—
270	C₄⋊C₄	—	188	4	no	—
271	C₄⋊C₄	—	188	8	no	—
272	C₄⋊C₄	—	188	4	no	—
273	C₄⋊C₄	—	188	4	no	—
274	C₂²⋊C₄	—	188	4	no	—
275	C₄²	—	47	8	no	—
276	C₄²	—	47	2	no	—
277	C₄²	—	47	2	no	—
278	C₄²	—	188	4	no	—
279	C₄²	C₂×C₄×D₄₇	1	2	yes	—
280	C₄²	Q₈×D₄₇	1	2	yes	—
281	C₂³	—	47	2	no	—
282	C₂³	—	47	2	no	—
283	C₂³	—	47	2	no	—
284	C₂³	—	94	2	no	—
285	C₂³	—	94	2	no	—
286	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	47	2	no	—
292	C₂×C₄	—	47	2	no	—
293	C₂×C₄	—	47	2	no	—
294	C₂×C₄	—	47	8	no	—
295	C₂×C₄	—	47	8	no	—
296	C₂×C₄	—	47	4	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	16	no	—
300	C₂×C₄	—	376	8	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
306	C₂×C₄	#13094051	1	4	yes	—
307	C₂×C₄	#13094053	1	4	yes	—
308	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
310	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
311	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
312	C₈	—	47	8	no	—
313	C₈	C₄⋊C₄×D₄₇	1	8	yes	complemented
314	C₂²	—	47	4	no	—
315	C₂²	—	94	4	no	—
316	C₂²	—	94	2	no	—
317	C₂²	—	94	2	no	—
318	C₂²	—	94	4	no	—
319	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
320	C₂²	#127×D₄₇	1	1	yes	characteristic
321	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
322	C₂²	C₂×#13094051	1	2	yes	—
323	C₂²	C₂×#13094053	1	2	yes	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	47	4	no	—
327	C₄	—	94	4	no	—
328	C₄	—	376	2	no	—
329	C₄	—	376	2	no	—
330	C₄	#15689162	1	8	yes	complemented
331	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
332	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
333	C₂	—	94	4	no	—
334	C₂	#384×D₄₇	1	1	yes	characteristic
335	C₂	C₂×#15689162	1	2	yes	—
336	C₂	#20253835	1	1	yes	characteristic
337	C₂	#20253842	1	1	yes	characteristic
338	C₂	#20253929	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#20800
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
5. #20243553	C₂	semidirect
6. #20243548	C₂	semidirect
7. #20243547	C₂	semidirect
8. #20243544	C₂	semidirect
9. C₄₇×#1844	C₂	semidirect
19. C₄×#10625324	C₄	semidirect
58. #13093934	C₂×C₄	semidirect
59. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
68. C₄×C₈×C₄₇	C₂×C₄	semidirect
110. #10625324	C₄⋊C₄	semidirect
142. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
143. C₄₇⋊C₈	C₂×C₄⋊C₄	semidirect
157. C₄×C₄₇	#428	semidirect
172. #1844	D₄₇	semidirect
202. C₄₇	#20800	semidirect
237. C₄×C₈	C₄×D₄₇	semidirect
313. C₈	C₄⋊C₄×D₄₇	semidirect
330. C₄	#15689162	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689162	2
C₂×C₄×#10625324	2
#20245793	2
#384×D₄₇	2
#20243553	2
#20243548	2
#20243547	2
#20243544	2
C₄₇×#1844	2
#20800	47