Subgroups — #26224748

Order 12032.

1230 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689224	C₂	1	4	yes	maximal, complemented
2	C₄⋊Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#127×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20244271	C₂	1	4	yes	maximal, complemented
5	#20244195	C₂	1	1	yes	maximal, characteristic
6	#20244191	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₇×#492	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×Dic₄₇	C₂²	1	2	yes	—
11	C₂×D₄×Dic₄₇	—	2	2	no	—
12	C₂×D₄×Dic₄₇	C₂²	1	2	yes	—
13	#15689224	—	2	16	no	—
14	C₂×#13093958	C₂²	1	4	yes	—
15	C₂×#13093948	C₂²	1	4	yes	—
16	C₂×#13093945	C₂²	1	4	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15689030	C₂²	1	2	yes	—
21	#15689026	—	2	2	no	—
22	#15689026	C₂²	1	2	yes	—
23	#15689022	—	2	2	no	—
24	#15688993	C₂²	1	1	yes	characteristic
25	#15688993	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×Dic₄₇	—	2	2	no	—
27	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂²×SD₁₆×C₄₇	C₂²	1	4	yes	—
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#492	C₄	1	8	yes	complemented
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂³×Dic₄₇	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	Q₈×Dic₄₇	—	2	8	no	—
36	D₄×Dic₄₇	—	2	8	no	—
37	D₄×Dic₄₇	—	4	8	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
44	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
47	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	#13093958	—	2	8	no	—
51	#13093948	—	2	8	no	—
52	#13093945	—	2	8	no	—
53	#13093934	—	2	4	no	—
54	#13093934	D₄	1	8	yes	complemented
55	C₄×C₄₇⋊C₈	—	2	2	no	—
56	C₄₇⋊#95	—	2	4	no	—
57	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
58	C₂²×D₄×C₄₇	—	2	2	no	—
59	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
60	C₂×SD₁₆×C₄₇	—	2	8	no	—
61	C₂×SD₁₆×C₄₇	—	2	8	no	—
62	C₂×SD₁₆×C₄₇	C₂×C₄	1	16	yes	complemented
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₇	—	2	2	no	—
65	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	complemented
66	#127×C₄₇	—	2	2	no	—
67	#127×C₄₇	C₂×C₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
71	C₂⁴×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	2	4	no	—
74	C₂×Q₈×C₄₇	—	2	4	no	—
75	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂×D₄×C₄₇	—	2	8	no	—
77	C₂×D₄×C₄₇	—	2	4	no	—
78	C₂×D₄×C₄₇	—	2	4	no	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂×D₄×C₄₇	—	4	4	no	—
81	C₂×D₄×C₄₇	—	4	4	no	—
82	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₇	—	2	2	no	—
87	C₂²×Dic₄₇	—	2	2	no	—
88	C₂²×Dic₄₇	—	4	8	no	—
89	C₂²×Dic₄₇	—	4	4	no	—
90	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
91	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
93	SD₁₆×C₄₇	—	2	32	no	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₄₇	#41	1	4	yes	—
97	C₄⋊C₄×C₄₇	—	2	8	no	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	#41	1	2	yes	—
100	C₂²⋊Dic₄₇	—	4	8	no	—
101	C₂²⋊Dic₄₇	—	4	4	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	—	4	4	no	—
105	C₄⋊Dic₄₇	—	4	4	no	—
106	C₄⋊Dic₄₇	—	4	4	no	—
107	C₄×Dic₄₇	—	2	4	no	—
108	C₄×Dic₄₇	—	2	4	no	—
109	C₄×Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	—	4	4	no	—
111	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
115	C₂³×C₄₇	—	2	4	no	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	—	4	4	no	—
119	C₂³×C₄₇	—	4	4	no	—
120	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₄₇	—	2	8	no	—
122	Q₈×C₄₇	—	4	4	no	—
123	D₄×C₄₇	—	2	8	no	—
124	D₄×C₄₇	—	4	4	no	—
125	D₄×C₄₇	—	4	16	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
131	C₂×C₄×C₄₇	#137	1	4	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	4	no	—
144	C₂×Dic₄₇	—	4	4	no	—
145	C₂×Dic₄₇	—	8	8	no	—
146	C₂×Dic₄₇	#127	1	2	yes	—
147	C₂×Dic₄₇	#127	1	2	yes	—
148	C₈×C₄₇	C₄×D₄	1	8	yes	complemented
149	C₄₇⋊C₈	—	2	4	no	—
150	#21692	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	4	4	no	—
152	C₂²×C₄₇	—	4	8	no	—
153	C₂²×C₄₇	—	8	2	no	—
154	C₂²×C₄₇	—	8	2	no	—
155	C₂²×C₄₇	—	8	4	no	—
156	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
158	C₂²×C₄₇	#532	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₂²×C₄₇	C₂×#137	1	2	yes	—
161	C₄×C₄₇	—	4	4	no	—
162	C₄×C₄₇	#440	1	8	yes	complemented
163	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
164	C₄×C₄₇	#532	1	2	yes	—
165	Dic₄₇	—	2	2	no	—
166	Dic₄₇	—	2	2	no	—
167	Dic₄₇	—	4	2	no	—
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	—	8	2	no	—
170	C₂×#492	D₄₇	1	1	yes	characteristic
171	C₂×#440	—	47	4	no	—
172	C₄×C₄⋊Q₈	—	47	1	no	—
173	C₄×#127	—	47	1	no	—
174	#1973	—	47	4	no	—
175	#1957	—	47	1	no	—
176	#1956	—	47	1	no	—
177	#1845	—	47	1	no	—
178	C₄×#97	—	47	1	no	—
179	C₂×C₄₇	—	8	4	no	—
180	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#440	1	2	yes	—
182	C₂×C₄₇	#3151	1	1	yes	characteristic
183	C₂×C₄₇	#3152	1	1	yes	characteristic
184	C₂×C₄₇	#3182	1	2	yes	—
185	C₂²×SD₁₆	C₂×D₄₇	1	4	yes	—
186	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
188	C₂×C₄×Q₈	—	47	2	no	—
189	C₂×C₄×D₄	—	47	2	no	—
190	C₂×C₄×D₄	—	94	2	no	—
191	#492	Dic₄₇	1	8	yes	complemented
192	#440	—	94	16	no	—
193	C₂×#107	—	47	4	no	—
194	C₂×#103	—	47	4	no	—
195	C₂×#102	—	47	4	no	—
196	C₂×#97	—	47	2	no	—
197	C₂×#97	—	47	2	no	—
198	C₂×C₄×C₈	—	47	1	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
200	#391	—	47	2	no	—
201	#390	—	47	2	no	—
202	#390	—	94	2	no	—
203	#389	—	94	2	no	—
204	#384	—	47	1	no	—
205	#384	—	47	1	no	—
206	C₄×C₄⋊C₄	—	94	2	no	—
207	C₄³	—	47	1	no	—
208	C₄₇	#21692	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
212	C₂³×C₄	—	94	2	no	—
213	C₂×SD₁₆	—	2	8	no	—
214	C₂×SD₁₆	—	2	8	no	—
215	C₂×SD₁₆	C₂×Dic₄₇	1	16	yes	complemented
216	C₂²×C₈	—	47	2	no	—
217	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	complemented
220	#127	—	2	2	no	—
221	#127	C₂×Dic₄₇	1	4	yes	—
222	C₄×Q₈	—	94	8	no	—
223	C₄×D₄	—	94	8	no	—
224	C₄×D₄	—	188	8	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	47	2	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	—	94	2	no	—
229	C₂×C₄⋊C₄	—	94	2	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₂²⋊C₄	—	94	2	no	—
232	C₂×C₂²⋊C₄	—	94	4	no	—
233	C₂×C₄²	—	47	1	no	—
234	C₂×C₄²	—	47	2	no	—
235	C₂×C₄²	—	47	2	no	—
236	C₂×C₄²	—	47	1	no	—
237	C₂×C₄²	—	94	2	no	—
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	#107	—	94	8	no	—
240	#103	—	94	8	no	—
241	#102	—	94	8	no	—
242	#97	—	47	8	no	—
243	#97	—	94	4	no	—
244	C₄×C₈	—	94	2	no	—
245	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
246	#95	—	94	4	no	—
247	C₂⁴	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	4	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	47	1	no	—
261	C₂²×C₄	—	47	1	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	—	94	2	no	—
264	C₂²×C₄	—	94	2	no	—
265	C₂²×C₄	—	188	4	no	—
266	C₂²×C₄	—	188	8	no	—
267	C₂²×C₄	D₄×D₄₇	1	2	yes	—
268	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	SD₁₆	—	2	32	no	—
270	C₂×C₈	—	47	4	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	D₄×D₄₇	1	4	yes	—
274	C₂×C₈	#10625352	1	4	yes	—
275	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
276	C₄⋊C₄	—	2	8	no	—
277	C₄⋊C₄	—	94	4	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	188	4	no	—
280	C₄⋊C₄	—	188	4	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₂²⋊C₄	—	188	4	no	—
283	C₂²⋊C₄	—	188	8	no	—
284	C₄²	—	47	4	no	—
285	C₄²	—	94	4	no	—
286	C₄²	—	94	4	no	—
287	C₄²	—	94	2	no	—
288	C₄²	—	188	4	no	—
289	C₄²	#10625352	1	2	yes	—
290	C₄²	C₂²×Dic₄₇	1	2	yes	—
291	C₂³	—	2	4	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
297	Q₈	—	2	8	no	—
298	Q₈	—	4	4	no	—
299	D₄	—	2	8	no	—
300	D₄	—	4	16	no	—
301	D₄	—	4	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	47	2	no	—
306	C₂×C₄	—	47	2	no	—
307	C₂×C₄	—	94	2	no	—
308	C₂×C₄	—	94	2	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	376	8	no	—
313	C₂×C₄	—	188	4	no	—
314	C₂×C₄	—	188	4	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
318	C₂×C₄	#13094051	1	4	yes	—
319	C₂×C₄	#13094053	1	4	yes	—
320	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
322	C₂×C₄	C₂×#10625352	1	2	yes	—
323	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
324	C₈	—	94	4	no	—
325	C₈	D₄×Dic₄₇	1	8	yes	complemented
326	C₂²	—	4	8	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	4	no	—
331	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
332	C₂²	#127×D₄₇	1	1	yes	characteristic
333	C₂²	#15689671	1	1	yes	characteristic
334	C₂²	C₂×#13094051	1	2	yes	—
335	C₂²	C₂×#13094053	1	2	yes	—
336	C₄	—	4	4	no	—
337	C₄	—	94	2	no	—
338	C₄	—	94	2	no	—
339	C₄	—	376	2	no	—
340	C₄	—	376	2	no	—
341	C₄	—	188	2	no	—
342	C₄	#15689224	1	8	yes	complemented
343	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
344	C₄	#15689671	1	2	yes	—
345	C₂	—	8	4	no	—
346	C₂	#127×Dic₄₇	1	1	yes	characteristic
347	C₂	C₂×#15689224	1	2	yes	—
348	C₂	#20253835	1	1	yes	characteristic
349	C₂	#20253842	1	1	yes	characteristic
350	C₂	#20254104	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#492	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#492	D₄₇
hypocenter	C₄₇	#21692
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#492

Complements

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

N	H ≅ G/N	type
1. C₂×#15689224	C₂	semidirect
2. C₄⋊Q₈×Dic₄₇	C₂	semidirect
4. #20244271	C₂	semidirect
6. #20244191	C₂	semidirect
7. #20243548	C₂	semidirect
8. C₄×#13093934	C₂	semidirect
31. C₄₇×#492	C₄	semidirect
54. #13093934	D₄	semidirect
62. C₂×SD₁₆×C₄₇	C₂×C₄	semidirect
65. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
70. C₄×C₈×C₄₇	C₂×C₄	semidirect
148. C₈×C₄₇	C₄×D₄	semidirect
162. C₄×C₄₇	#440	semidirect
191. #492	Dic₄₇	semidirect
208. C₄₇	#21692	semidirect
215. C₂×SD₁₆	C₂×Dic₄₇	semidirect
219. C₄⋊Q₈	C₂×Dic₄₇	semidirect
245. C₄×C₈	C₂×Dic₄₇	semidirect
325. C₈	D₄×Dic₄₇	semidirect
342. C₄	#15689224	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689224	2
C₄⋊Q₈×Dic₄₇	2
#127×Dic₄₇	2
#20244271	2
#20244195	2
#20244191	2
#20243548	2
C₄×#13093934	2
C₂×C₄₇×#492	2
#21692	47