Subgroups — #26370589

Order 12032.

748 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689558	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093965	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252241	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245440	C₂	1	4	yes	maximal, complemented
5	#20245420	C₂	1	4	yes	maximal, complemented
6	#20245147	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245104	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245095	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	#15689597	—	2	1	no	—
12	#15689558	—	2	4	no	—
13	C₂×#13093967	C₂²	1	2	yes	complemented
14	C₂×#13093965	—	2	4	no	—
15	C₂×#13093965	C₂²	1	4	yes	—
16	C₂×#13093965	C₂²	1	2	yes	complemented
17	C₂×#13093964	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	#15689236	—	2	2	no	—
20	C₄×D₃₇₆	C₂²	1	4	yes	complemented
21	#15689161	C₂²	1	1	yes	characteristic
22	#15689129	—	2	1	no	—
23	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
24	#15688927	C₂²	1	2	yes	complemented
25	#15688924	C₂²	1	2	yes	complemented
26	#15688924	C₂²	1	4	yes	complemented
27	C₂×AGL(1,ℤ/8)×C₄₇	—	2	1	no	—
28	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#535	—	2	1	no	—
31	C₄₇×#496	—	2	2	no	—
32	C₄₇×#495	C₂²	1	2	yes	complemented
33	C₄₇×#493	C₂²	1	2	yes	complemented
34	C₄₇×#405	C₂²	1	1	yes	characteristic
35	C₂²×D₁₈₈	—	2	1	no	—
36	C₂×D₃₇₆	—	2	1	no	—
37	C₂×D₃₇₆	—	2	2	no	—
38	C₂×D₃₇₆	D₄	1	4	yes	complemented
39	C₂×D₃₇₆	C₂³	1	2	yes	—
40	C₄×D₁₈₈	—	2	4	no	—
41	#13093986	—	4	2	no	—
42	#13093983	—	2	1	no	—
43	C₂²⋊C₄×D₄₇	—	4	2	no	—
44	#13093967	—	2	4	no	—
45	#13093965	—	2	8	no	—
46	#13093965	—	2	4	no	—
47	#13093964	—	2	2	no	—
48	C₂×C₄₇⋊C₁₆	—	2	2	no	—
49	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	#13093958	C₂³	1	2	yes	—
52	#13093956	—	2	1	no	—
53	#13093946	—	2	4	no	—
54	C₂×#41×C₄₇	—	2	1	no	—
55	C₂²×D₄×C₄₇	—	2	1	no	—
56	AGL(1,ℤ/8)×C₄₇	—	4	4	no	—
57	#135×C₄₇	—	2	2	no	—
58	#135×C₄₇	—	2	2	no	—
59	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₄₇	—	2	2	no	—
61	C₂×SD₁₆×C₄₇	—	2	1	no	—
62	C₂×D₈×C₄₇	—	2	2	no	—
63	C₂×D₈×C₄₇	—	2	2	no	—
64	C₂×D₈×C₄₇	—	2	1	no	—
65	C₂×D₈×C₄₇	—	2	2	no	—
66	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₄₇	C₂³	1	2	yes	—
68	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	#127×C₄₇	—	2	2	no	—
71	#124×C₄₇	—	2	2	no	—
72	#121×C₄₇	—	4	2	no	—
73	#120×C₄₇	—	4	2	no	—
74	#117×C₄₇	D₄	1	1	yes	characteristic
75	#97×C₄₇	D₄	1	2	yes	—
76	C₄×C₈×C₄₇	C₂³	1	2	yes	—
77	C₂⁴×C₄₇	—	4	1	no	—
78	C₂³×D₄₇	—	4	1	no	—
79	#41×C₄₇	—	4	2	no	—
80	#41×C₄₇	—	4	2	no	—
81	C₂×Q₈×C₄₇	—	2	1	no	—
82	C₂×D₄×C₄₇	—	2	2	no	—
83	C₂×D₄×C₄₇	—	2	1	no	—
84	C₂×D₄×C₄₇	—	4	1	no	—
85	C₂×D₄×C₄₇	—	4	2	no	—
86	C₂×D₄×C₄₇	—	4	1	no	—
87	C₂×D₄×C₄₇	—	4	4	no	—
88	C₂×D₄×C₄₇	—	4	2	no	—
89	C₂²×C₄×C₄₇	—	4	1	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×D₁₈₈	—	2	4	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	4	1	no	—
94	C₂×D₁₈₈	—	4	2	no	—
95	C₂×C₄×D₄₇	—	4	4	no	—
96	Q₁₆×C₄₇	—	2	1	no	—
97	Q₁₆×C₄₇	—	2	1	no	—
98	SD₁₆×C₄₇	—	4	4	no	—
99	SD₁₆×C₄₇	—	4	2	no	—
100	D₈×C₄₇	—	2	2	no	—
101	D₈×C₄₇	—	2	1	no	—
102	D₈×C₄₇	—	2	2	no	—
103	D₈×C₄₇	—	2	1	no	—
104	D₈×C₄₇	—	4	4	no	—
105	D₈×C₄₇	—	4	2	no	—
106	M₁₆×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₇	#41	1	2	yes	—
111	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
113	C₂²⋊C₄×C₄₇	—	4	4	no	—
114	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
115	C₄²×C₄₇	C₂×D₄	1	2	yes	—
116	C₂²⋊Dic₄₇	—	4	1	no	—
117	(C₂×D₄₇)⋊C₄	—	4	4	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄×Dic₄₇	—	4	1	no	—
120	D₃₇₆	—	2	4	no	—
121	D₃₇₆	—	2	2	no	—
122	D₃₇₆	C₂×D₄	1	4	yes	—
123	C₄₇⋊C₁₆	—	2	2	no	—
124	C₄₇⋊C₁₆	—	2	2	no	—
125	C₂³×C₄₇	—	4	2	no	—
126	C₂³×C₄₇	—	4	1	no	—
127	C₂³×C₄₇	—	8	2	no	—
128	C₂³×C₄₇	—	16	2	no	—
129	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₄₇	—	4	2	no	—
131	C₂²×D₄₇	—	4	4	no	—
132	C₂²×D₄₇	—	16	2	no	—
133	Q₈×C₄₇	—	4	1	no	—
134	Q₈×C₄₇	—	4	1	no	—
135	D₄×C₄₇	—	4	2	no	—
136	D₄×C₄₇	—	4	1	no	—
137	D₄×C₄₇	—	4	1	no	—
138	D₄×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	8	2	no	—
140	D₄×C₄₇	—	8	2	no	—
141	D₄×C₄₇	—	16	4	no	—
142	C₂×C₄×C₄₇	—	4	1	no	—
143	C₂×C₄×C₄₇	—	8	2	no	—
144	C₂×C₄×C₄₇	#121	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₄₇	—	4	2	no	—
150	D₁₈₈	—	2	4	no	—
151	D₁₈₈	—	4	2	no	—
152	D₁₈₈	—	4	4	no	—
153	C₄×D₄₇	—	16	4	no	—
154	C₈×C₄₇	—	2	2	no	—
155	C₈×C₄₇	#121	1	2	yes	—
156	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
158	#33585	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	—	8	4	no	—
162	C₂²×C₄₇	—	8	1	no	—
163	C₂²×C₄₇	—	8	2	no	—
164	C₂²×C₄₇	—	16	4	no	—
165	C₂²×C₄₇	#459	1	2	yes	—
166	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₇	—	4	4	no	—
168	C₂×D₄₇	—	8	4	no	—
169	C₂×D₄₇	—	16	4	no	—
170	C₄×C₄₇	—	2	4	no	—
171	C₄×C₄₇	—	8	1	no	—
172	C₄×C₄₇	#459	1	2	yes	—
173	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
174	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	Dic₄₇	—	8	2	no	—
176	C₂×#509	—	47	1	no	—
177	C₂²×D₁₆	—	47	1	no	—
178	#3155	D₄₇	1	1	yes	characteristic, complemented
179	#2948	—	47	1	no	—
180	#2210	—	47	4	no	—
181	#2206	—	47	4	no	—
182	#2151	—	47	1	no	—
183	#2140	—	47	1	no	—
184	#2138	—	47	1	no	—
185	C₂×C₄₇	—	2	2	no	—
186	C₂×C₄₇	—	16	1	no	—
187	C₂×C₄₇	—	16	2	no	—
188	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
189	C₂×C₄₇	#3415	1	2	yes	—
190	D₄₇	—	8	4	no	—
191	C₂×AGL(1,ℤ/8)	—	2	1	no	—
192	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
193	C₂²×D₈	—	47	1	no	—
194	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
195	#535	—	2	1	no	—
196	#518	—	94	1	no	—
197	#509	—	94	4	no	—
198	C₂×SD₃₂	—	47	2	no	—
199	C₂×D₁₆	—	47	4	no	—
200	C₂×D₁₆	—	47	2	no	—
201	C₂×D₁₆	—	94	4	no	—
202	C₂×M₃₂	—	47	1	no	—
203	C₂²×C₁₆	—	47	1	no	—
204	#496	—	2	2	no	—
205	#495	C₂×D₄₇	1	2	yes	complemented
206	#493	C₂×D₄₇	1	2	yes	complemented
207	#442	—	94	2	no	—
208	C₄×D₈	—	47	4	no	—
209	#427	—	47	1	no	—
210	#418	—	94	1	no	—
211	#405	C₂×D₄₇	1	1	yes	characteristic
212	C₄⋊C₁₆	—	47	4	no	—
213	#358	—	47	2	no	—
214	#357	—	47	2	no	—
215	#357	—	47	4	no	—
216	C₄₇	#33585	1	1	yes	characteristic, complemented, Sylow
217	C₂×#41	—	2	1	no	—
218	C₂²×D₄	—	2	1	no	—
219	C₂²×D₄	—	94	1	no	—
220	AGL(1,ℤ/8)	—	4	4	no	—
221	#135	—	2	2	no	—
222	#135	—	2	2	no	—
223	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
224	C₂×SD₁₆	—	2	1	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	—	2	2	no	—
227	C₂×D₈	—	2	2	no	—
228	C₂×D₈	—	2	1	no	—
229	C₂×D₈	—	2	2	no	—
230	C₂×D₈	—	47	4	no	—
231	C₂×D₈	—	47	2	no	—
232	C₂×D₈	—	94	2	no	—
233	C₂×D₈	—	94	1	no	—
234	C₂×D₈	C₂²×D₄₇	1	2	yes	—
235	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
238	#127	—	2	2	no	—
239	#124	—	2	2	no	—
240	#121	—	4	2	no	—
241	#120	—	4	2	no	—
242	C₄×D₄	—	94	4	no	—
243	C₄×D₄	—	188	2	no	—
244	#117	—	94	1	no	—
245	#117	C₂²⋊D₄₇	1	1	yes	characteristic
246	C₂×C₂²⋊C₄	—	188	2	no	—
247	SD₃₂	—	94	4	no	—
248	D₁₆	—	94	8	no	—
249	D₁₆	—	94	4	no	—
250	M₃₂	—	94	2	no	—
251	C₂×C₁₆	—	47	2	no	—
252	C₂×C₁₆	—	47	2	no	—
253	C₂×C₁₆	—	94	2	no	—
254	#107	—	47	2	no	—
255	#106	—	94	1	no	—
256	#102	—	94	4	no	—
257	#97	C₂²⋊D₄₇	1	2	yes	—
258	C₄×C₈	C₂²×D₄₇	1	2	yes	—
259	C₂⁴	—	4	1	no	—
260	C₂⁴	—	188	1	no	—
261	#41	—	4	2	no	—
262	#41	—	4	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	4	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	94	4	no	—
272	C₂×D₄	—	94	2	no	—
273	C₂×D₄	—	188	1	no	—
274	C₂×D₄	—	188	2	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	188	4	no	—
277	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
278	Q₁₆	—	2	1	no	—
279	Q₁₆	—	2	1	no	—
280	SD₁₆	—	4	4	no	—
281	SD₁₆	—	4	2	no	—
282	D₈	—	2	2	no	—
283	D₈	—	2	1	no	—
284	D₈	—	2	2	no	—
285	D₈	—	2	1	no	—
286	D₈	—	4	4	no	—
287	D₈	—	4	2	no	—
288	D₈	—	47	4	no	—
289	D₈	—	94	4	no	—
290	D₈	—	94	2	no	—
291	M₁₆	—	2	2	no	—
292	C₂×C₈	D₄×D₄₇	1	2	yes	—
293	C₂×C₈	#10625352	1	2	yes	—
294	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
295	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	#10625334	1	2	yes	—
299	C₂²⋊C₄	—	4	4	no	—
300	C₂²⋊C₄	—	188	1	no	—
301	C₂²⋊C₄	—	188	4	no	—
302	C₂²⋊C₄	#10625334	1	2	yes	—
303	C₄²	—	188	1	no	—
304	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
305	C₁₆	—	94	2	no	—
306	C₁₆	—	94	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	—	8	2	no	—
310	C₂³	—	16	2	no	—
311	C₂³	—	188	2	no	—
312	C₂³	—	188	4	no	—
313	C₂³	—	752	2	no	—
314	C₂³	C₂×#10625334	1	1	yes	characteristic
315	Q₈	—	4	1	no	—
316	Q₈	—	4	1	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	2	no	—
322	D₄	—	8	2	no	—
323	D₄	—	16	4	no	—
324	D₄	—	94	4	no	—
325	D₄	—	188	4	no	—
326	D₄	—	188	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	752	4	no	—
331	C₂×C₄	#13093995	1	2	yes	—
332	C₂×C₄	C₂×#10625334	1	4	yes	—
333	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
334	C₂×C₄	#13094050	1	2	yes	—
335	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	#13093995	1	2	yes	—
338	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
339	C₈	C₂×#10625352	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	16	4	no	—
346	C₂²	—	376	4	no	—
347	C₂²	—	188	4	no	—
348	C₂²	—	752	4	no	—
349	C₂²	#15689315	1	2	yes	—
350	C₂²	C₂²×#10625334	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	8	1	no	—
353	C₄	—	376	2	no	—
354	C₄	#15689315	1	2	yes	—
355	C₄	C₂×#13093995	1	1	yes	characteristic
356	C₄	C₂×#13094050	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	16	2	no	—
359	C₂	—	16	1	no	—
360	C₂	—	376	4	no	—
361	C₂	C₂×#15689315	1	1	yes	characteristic
362	C₂	#20256527	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3155	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3155	D₄₇
hypocenter	C₄₇	#33585
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3155

Complements

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

N	H ≅ G/N	type
1. C₂×#15689558	C₂	semidirect
2. C₂²×#13093965	C₂	semidirect
3. #20252241	C₂	semidirect
4. #20245440	C₂	semidirect
5. #20245420	C₂	semidirect
6. #20245147	C₂	semidirect
7. #20245104	C₂	semidirect
8. #20245095	C₂	semidirect
9. C₄₇×#3155	C₂	semidirect
13. C₂×#13093967	C₂²	semidirect
16. C₂×#13093965	C₂²	semidirect
20. C₄×D₃₇₆	C₂²	semidirect
23. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
24. #15688927	C₂²	semidirect
25. #15688924	C₂²	semidirect
26. #15688924	C₂²	semidirect
32. C₄₇×#495	C₂²	semidirect
33. C₄₇×#493	C₂²	semidirect
38. C₂×D₃₇₆	D₄	semidirect
178. #3155	D₄₇	semidirect
205. #495	C₂×D₄₇	semidirect
206. #493	C₂×D₄₇	semidirect
216. C₄₇	#33585	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689558	2
C₂²×#13093965	2
#20252241	2
#20245440	2
#20245420	2
#20245147	2
#20245104	2
#20245095	2
C₄₇×#3155	2
#33585	47