Subgroups — #26106889

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093986	C₂	1	1	yes	maximal, characteristic
2	#20243243	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688852	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688850	C₂	1	1	yes	maximal, characteristic
6	#20241229	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×#13093986	C₄	1	2	yes	—
10	C₂×C₄²×D₄₇	—	2	1	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13093934	—	2	2	no	—
13	C₂×#13093934	—	2	1	no	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
16	C₄×(C₂×D₄₇)⋊C₄	—	2	1	no	—
17	#15688970	—	2	2	no	—
18	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688877	—	2	1	no	—
20	#15688852	C₄	1	4	yes	complemented
21	#15688850	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	—	2	1	no	—
27	C₂²×C₄×D₄₇	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₄×D₁₈₈	—	2	1	no	—
31	C₄×D₁₈₈	—	2	1	no	—
32	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
33	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
34	#13093986	—	2	4	no	—
35	#13093986	—	2	4	no	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
38	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
42	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
44	C₄²×D₄₇	—	4	2	no	—
45	C₄²×D₄₇	—	4	2	no	—
46	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	complemented
48	#13093934	—	2	8	no	—
49	#13093934	—	4	2	no	—
50	C₄×C₄₇⋊C₈	—	2	1	no	—
51	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	complemented
52	C₄₇⋊#95	—	2	2	no	—
53	C₂²×C₈×C₄₇	—	2	1	no	—
54	C₂²×C₈×C₄₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	—	2	1	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	—	2	1	no	—
59	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	complemented
60	C₄×C₈×C₄₇	—	2	1	no	—
61	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	complemented
62	#95×C₄₇	—	2	1	no	—
63	C₂³×D₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂×D₁₈₈	—	2	1	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	C₄²	1	2	yes	—
74	C₂×D₁₈₈	M₁₆	1	4	yes	—
75	C₂×C₄×D₄₇	—	2	8	no	—
76	C₂×C₄×D₄₇	—	2	2	no	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	—	4	4	no	—
79	C₂×C₄×D₄₇	—	4	1	no	—
80	C₂×C₄×D₄₇	—	4	1	no	—
81	C₂×C₄×D₄₇	—	4	1	no	—
82	C₂×C₄×D₄₇	—	4	1	no	—
83	C₂×C₄×D₄₇	—	4	4	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	4	1	no	—
88	C₂×C₈×C₄₇	—	4	1	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	M₁₆	1	4	yes	—
91	C₄²×C₄₇	—	4	1	no	—
92	C₄²×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₇	#41	1	1	yes	characteristic
96	C₄²×C₄₇	#41	1	1	yes	characteristic
97	(C₂×D₄₇)⋊C₄	—	2	8	no	—
98	(C₂×D₄₇)⋊C₄	—	2	2	no	—
99	(C₂×D₄₇)⋊C₄	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	1	no	—
101	C₄⋊Dic₄₇	C₄²	1	2	yes	—
102	C₄×Dic₄₇	—	2	2	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	—	2	1	no	—
106	C₄×Dic₄₇	—	2	1	no	—
107	C₄×Dic₄₇	—	2	1	no	—
108	C₄×Dic₄₇	—	2	4	no	—
109	C₄×Dic₄₇	—	4	4	no	—
110	C₂×C₄₇⋊C₈	—	2	4	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	4	1	no	—
114	C₂×C₄₇⋊C₈	—	4	1	no	—
115	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×D₄₇	—	2	2	no	—
117	C₂²×D₄₇	—	2	4	no	—
118	C₂²×D₄₇	—	4	1	no	—
119	C₂²×D₄₇	—	4	1	no	—
120	C₂²×D₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
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	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	4	1	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	#104	1	4	yes	—
146	D₁₈₈	—	2	2	no	—
147	D₁₈₈	—	2	4	no	—
148	D₁₈₈	#97	1	4	yes	complemented
149	C₄×D₄₇	—	4	16	no	—
150	C₄×D₄₇	—	4	2	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₄×D₄₇	—	4	2	no	—
153	C₄×D₄₇	—	4	2	no	—
154	C₈×C₄₇	—	2	4	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₄₇⋊C₈	—	2	4	no	—
157	C₄₇⋊C₈	—	4	2	no	—
158	#8172	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄₇	#407	1	2	yes	—
161	C₂²×C₄₇	#417	1	1	yes	characteristic
162	C₂²×C₄₇	#418	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#104	1	2	yes	—
164	C₂×D₄₇	—	2	4	no	—
165	C₂×D₄₇	—	4	4	no	—
166	C₂×D₄₇	—	4	4	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₂×D₄₇	—	4	2	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	4	4	no	—
172	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
173	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
174	C₄×C₄₇	#432	1	2	yes	—
175	C₄×C₄₇	#440	1	1	yes	characteristic
176	C₄×C₄₇	#440	1	1	yes	characteristic
177	C₄×C₄₇	#442	1	1	yes	characteristic
178	C₄×C₄₇	#442	1	1	yes	characteristic
179	Dic₄₇	—	2	4	no	—
180	Dic₄₇	—	4	2	no	—
181	Dic₄₇	—	4	2	no	—
182	C₄²×D₄	—	47	1	no	—
183	#1767	—	47	1	no	—
184	C₄×#97	—	47	1	no	—
185	C₂×D₄⋊C₈	—	47	1	no	—
186	C₂×D₄⋊C₈	—	47	1	no	—
187	#1279	—	47	1	no	—
188	#1279	D₄₇	1	1	yes	characteristic, complemented
189	C₂×C₄₇	#1489	1	2	yes	—
190	C₂×C₄₇	#1490	1	2	yes	—
191	C₂×C₄₇	#1749	1	1	yes	characteristic
192	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
193	C₂×C₄₇	#1762	1	1	yes	characteristic
194	D₄₇	—	4	4	no	—
195	C₂×C₄×D₄	—	47	2	no	—
196	C₂×C₄×D₄	—	47	1	no	—
197	C₂²×C₄²	—	94	1	no	—
198	C₂×C₄⋊C₈	—	47	1	no	—
199	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
200	C₂×#97	—	94	2	no	—
201	C₂×#97	—	94	1	no	—
202	C₂×C₄×C₈	—	47	1	no	—
203	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
204	C₄×C₄⋊C₄	—	47	2	no	—
205	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
206	C₄×C₂²⋊C₄	—	94	2	no	—
207	C₄×C₂²⋊C₄	—	94	1	no	—
208	C₄³	—	47	1	no	—
209	#336	—	94	1	no	—
210	D₄⋊C₈	—	47	4	no	—
211	D₄⋊C₈	—	47	4	no	—
212	C₄₇	#8172	1	1	yes	characteristic, complemented, Sylow
213	C₂²×D₄	—	47	1	no	—
214	C₂³×C₄	—	94	2	no	—
215	C₂³×C₄	—	94	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	94	1	no	—
219	C₂²×C₈	—	94	1	no	—
220	C₄×D₄	—	47	2	no	—
221	C₄×D₄	—	47	2	no	—
222	C₄×D₄	—	94	4	no	—
223	C₄×D₄	—	94	4	no	—
224	C₄×D₄	—	94	1	no	—
225	C₄×D₄	—	94	1	no	—
226	C₂×C₄⋊C₄	—	47	1	no	—
227	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
228	C₂×C₂²⋊C₄	—	94	2	no	—
229	C₂×C₂²⋊C₄	—	94	1	no	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	—	47	2	no	—
232	C₂×C₄²	—	47	2	no	—
233	C₂×C₄²	—	47	1	no	—
234	C₂×C₄²	—	47	1	no	—
235	C₂×C₄²	—	94	2	no	—
236	C₂×C₄²	—	188	2	no	—
237	C₂×C₄²	—	188	2	no	—
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	47	2	no	—
241	C₄⋊C₈	—	94	1	no	—
242	C₄⋊C₈	C₄×D₄₇	1	2	yes	complemented
243	#97	—	94	8	no	—
244	#97	—	188	2	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	—	47	2	no	—
247	C₄×C₈	—	94	1	no	—
248	C₄×C₈	C₄×D₄₇	1	2	yes	complemented
249	#95	—	2	1	no	—
250	#95	—	94	2	no	—
251	C₂⁴	—	94	1	no	—
252	C₂×D₄	—	47	2	no	—
253	C₂×D₄	—	47	4	no	—
254	C₂×D₄	—	94	1	no	—
255	C₂×D₄	—	94	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	2	no	—
259	C₂²×C₄	—	94	8	no	—
260	C₂²×C₄	—	94	1	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	—	188	4	no	—
264	C₂²×C₄	—	188	4	no	—
265	C₂²×C₄	—	188	1	no	—
266	C₂²×C₄	—	188	1	no	—
267	C₂²×C₄	—	188	1	no	—
268	C₂²×C₄	—	188	1	no	—
269	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	94	4	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	188	1	no	—
281	C₂×C₈	—	188	1	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	47	2	no	—
284	C₄⋊C₄	—	94	1	no	—
285	C₄⋊C₄	#10625324	1	4	yes	—
286	C₂²⋊C₄	—	94	8	no	—
287	C₂²⋊C₄	—	94	2	no	—
288	C₂²⋊C₄	—	94	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	94	4	no	—
292	C₄²	—	94	2	no	—
293	C₄²	—	94	2	no	—
294	C₄²	—	94	1	no	—
295	C₄²	—	94	1	no	—
296	C₄²	—	94	1	no	—
297	C₄²	—	94	1	no	—
298	C₄²	—	188	4	no	—
299	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
301	C₄²	#10625352	1	1	yes	characteristic
302	C₄²	#10625354	1	1	yes	characteristic
303	C₂³	—	94	2	no	—
304	C₂³	—	94	4	no	—
305	C₂³	—	188	1	no	—
306	C₂³	—	188	1	no	—
307	C₂³	—	188	2	no	—
308	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
309	D₄	—	47	4	no	—
310	D₄	—	94	2	no	—
311	D₄	—	94	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	47	4	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	—	94	2	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	188	16	no	—
324	C₂×C₄	—	188	1	no	—
325	C₂×C₄	—	188	1	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
331	C₂×C₄	#13093982	1	1	yes	characteristic
332	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
334	C₂×C₄	#13093986	1	1	yes	characteristic
335	C₂×C₄	#13093986	1	1	yes	characteristic
336	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
337	C₂×C₄	C₂×#10625324	1	2	yes	—
338	C₂×C₄	#13094048	1	1	yes	characteristic
339	C₂×C₄	#13094051	1	1	yes	characteristic
340	C₂×C₄	#13094051	1	1	yes	characteristic
341	C₂×C₄	#13094056	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	4	2	no	—
344	C₈	—	94	4	no	—
345	C₈	—	188	2	no	—
346	C₂²	—	94	4	no	—
347	C₂²	—	188	2	no	—
348	C₂²	—	188	4	no	—
349	C₂²	—	188	4	no	—
350	C₂²	—	188	2	no	—
351	C₂²	#15688970	1	1	yes	characteristic
352	C₂²	#15689085	1	2	yes	—
353	C₂²	#15689123	1	1	yes	characteristic
354	C₂²	#15689126	1	1	yes	characteristic
355	C₂²	#104×D₄₇	1	2	yes	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	4	no	—
359	C₄	—	94	4	no	—
360	C₄	—	188	2	no	—
361	C₄	—	188	2	no	—
362	C₄	#97×D₄₇	1	1	yes	characteristic
363	C₄	C₄×#10625324	1	1	yes	characteristic
364	C₄	#15689179	1	2	yes	—
365	C₄	#15689222	1	1	yes	characteristic
366	C₄	#15689226	1	1	yes	characteristic
367	C₄	#15689232	1	1	yes	characteristic
368	C₄	#15689234	1	1	yes	characteristic
369	C₂	—	188	4	no	—
370	C₂	#20241826	1	2	yes	—
371	C₂	#20241829	1	2	yes	—
372	C₂	#20243149	1	1	yes	characteristic
373	C₂	#20243192	1	1	yes	characteristic
374	C₂	#20243212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1279	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1279	D₄₇
hypocenter	C₄₇	#8172
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
2. #20243243	C₂	semidirect
3. C₄×#13093934	C₂	semidirect
6. #20241229	C₂	semidirect
7. C₄₇×#1279	C₂	semidirect
20. #15688852	C₄	semidirect
21. #15688850	C₄	semidirect
47. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
51. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
59. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
61. C₄×C₈×C₄₇	C₂×C₄	semidirect
148. D₁₈₈	#97	semidirect
188. #1279	D₄₇	semidirect
212. C₄₇	#8172	semidirect
242. C₄⋊C₈	C₄×D₄₇	semidirect
248. C₄×C₈	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093986	2
#20243243	2
C₄×#13093934	2
C₂×#15688852	2
C₂×#15688850	2
#20241229	2
C₄₇×#1279	2
#8172	47