Subgroups — #26373103

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255737	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245612	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245570	C₂	1	2	yes	maximal, complemented
5	#20245525	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245473	C₂	1	2	yes	maximal, complemented
7	#20245423	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245298	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₄₇	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₇	C₂²	1	2	yes	—
13	#15689778	—	2	1	no	—
14	D₄×D₁₈₈	—	2	1	no	—
15	C₂×#13093965	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₄₇	C₂²	1	2	yes	complemented
17	#15689528	C₂²	1	1	yes	characteristic, complemented
18	#15689500	C₂²	1	1	yes	characteristic
19	#15689435	—	2	2	no	—
20	#15689403	—	2	1	no	—
21	#15689360	C₂²	1	2	yes	—
22	#15689356	C₂²	1	2	yes	complemented
23	#15689318	—	2	1	no	—
24	#15689258	—	2	2	no	—
25	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
26	#107×D₄₇	C₂²	1	2	yes	complemented
27	#15688948	C₂²	1	2	yes	complemented
28	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
29	#15688925	C₂²	1	2	yes	complemented
30	#15688925	C₂²	1	2	yes	complemented
31	#15688924	C₂²	1	2	yes	—
32	#15688923	C₂²	1	2	yes	complemented
33	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
37	C₄₇×#366	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#357	C₂²	1	1	yes	characteristic
39	C₄₇×#357	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
41	C₂×#10625350	—	4	1	no	—
42	C₂²×D₁₈₈	—	4	1	no	—
43	C₂×D₄×D₄₇	—	2	2	no	—
44	C₂×D₃₇₆	—	2	1	no	—
45	C₂×D₃₇₆	C₂³	1	2	yes	—
46	C₂×#10625334	—	2	2	no	—
47	D₈×D₄₇	—	2	4	no	—
48	D₈×D₄₇	—	4	2	no	—
49	C₂×C₈×D₄₇	C₂³	1	2	yes	—
50	#13094024	—	2	1	no	—
51	#13094021	—	2	1	no	—
52	#13094006	—	4	2	no	—
53	#13094003	—	4	1	no	—
54	#13093999	—	2	2	no	—
55	#13093998	—	4	1	no	—
56	#13093995	—	2	2	no	—
57	#13093994	—	4	2	no	—
58	C₄×D₁₈₈	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₄₇	—	2	2	no	—
60	#13093965	—	4	2	no	—
61	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₄₇	—	4	2	no	—
63	#13093958	—	2	1	no	—
64	#13093958	C₂³	1	2	yes	—
65	#13093957	—	2	2	no	—
66	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
67	#13093945	—	2	2	no	—
68	#13093944	—	2	2	no	—
69	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
70	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₄₇	—	2	1	no	—
73	C₄×D₄×C₄₇	—	2	1	no	—
74	D₁₆×C₄₇	—	2	2	no	—
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
77	#107×C₄₇	C₂³	1	1	yes	characteristic
78	#107×C₄₇	C₂³	1	1	yes	characteristic
79	#102×C₄₇	—	2	1	no	—
80	#102×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂³×D₄₇	—	4	2	no	—
83	C₂×D₄×C₄₇	—	2	1	no	—
84	C₂×D₄×C₄₇	—	2	1	no	—
85	C₂²×C₄×C₄₇	—	4	1	no	—
86	C₂²×C₄×C₄₇	—	4	1	no	—
87	C₂×C₂²⋊D₄₇	—	4	2	no	—
88	C₂×C₂²⋊D₄₇	—	4	2	no	—
89	D₄×D₄₇	—	4	4	no	—
90	D₄×D₄₇	—	8	2	no	—
91	#10625350	—	8	4	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	4	2	no	—
94	C₂×D₁₈₈	—	4	1	no	—
95	C₂×D₁₈₈	—	4	1	no	—
96	C₂×D₁₈₈	—	8	4	no	—
97	C₂×D₁₈₈	D₈	1	1	yes	characteristic
98	C₂×C₄×D₄₇	—	4	2	no	—
99	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
100	C₂×Dic₉₄	—	4	1	no	—
101	D₈×C₄₇	—	2	1	no	—
102	D₈×C₄₇	—	2	1	no	—
103	D₈×C₄₇	—	2	1	no	—
104	D₈×C₄₇	#41	1	2	yes	complemented
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₄₇	—	2	1	no	—
108	C₄⋊C₄×C₄₇	—	2	1	no	—
109	C₂²⋊C₄×C₄₇	—	4	1	no	—
110	C₂²⋊C₄×C₄₇	—	4	1	no	—
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	4	2	no	—
113	#10625334	—	4	4	no	—
114	(C₂×D₄₇)⋊C₄	—	4	2	no	—
115	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	4	1	no	—
118	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
119	Dic₄₇⋊C₄	—	4	2	no	—
120	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
121	D₃₇₆	—	2	2	no	—
122	D₃₇₆	—	2	2	no	—
123	D₃₇₆	—	8	1	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₈×D₄₇	—	2	2	no	—
126	C₁₆×C₄₇	—	2	1	no	—
127	C₁₆×C₄₇	—	2	1	no	—
128	C₄₇⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₇	—	4	1	no	—
130	C₂³×C₄₇	—	4	1	no	—
131	C₂²×D₄₇	—	4	2	no	—
132	C₂²×D₄₇	—	8	4	no	—
133	C₂²×D₄₇	—	8	4	no	—
134	C₂²×D₄₇	—	16	2	no	—
135	C₂²×D₄₇	C₂×D₈	1	2	yes	—
136	D₄×C₄₇	—	2	2	no	—
137	D₄×C₄₇	—	4	1	no	—
138	D₄×C₄₇	—	4	1	no	—
139	D₄×C₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	—	4	1	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	—	8	2	no	—
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₄₇	—	8	4	no	—
148	C₂²⋊D₄₇	—	8	4	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
151	D₁₈₈	—	4	2	no	—
152	D₁₈₈	—	4	2	no	—
153	D₁₈₈	—	8	4	no	—
154	D₁₈₈	—	8	2	no	—
155	D₁₈₈	—	16	1	no	—
156	D₁₈₈	D₁₆	1	4	yes	complemented
157	C₄×D₄₇	—	2	2	no	—
158	C₄×D₄₇	—	2	2	no	—
159	C₄×D₄₇	—	8	4	no	—
160	Dic₉₄	—	8	2	no	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
163	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
164	C₄₇⋊C₈	—	2	2	no	—
165	#33861	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	—	8	1	no	—
168	C₂²×C₄₇	—	8	1	no	—
169	C₂²×C₄₇	—	8	1	no	—
170	C₂²×C₄₇	—	8	1	no	—
171	C₂²×C₄₇	—	8	1	no	—
172	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
173	C₂×D₄₇	—	2	2	no	—
174	C₂×D₄₇	—	2	2	no	—
175	C₂×D₄₇	—	8	2	no	—
176	C₂×D₄₇	—	8	2	no	—
177	C₂×D₄₇	—	8	2	no	—
178	C₂×D₄₇	—	16	8	no	—
179	C₂×D₄₇	C₂×D₁₆	1	4	yes	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	8	1	no	—
182	C₄×C₄₇	—	8	1	no	—
183	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
184	C₄×C₄₇	#546	1	1	yes	characteristic
185	C₄×C₄₇	#577	1	1	yes	characteristic
186	Dic₄₇	—	2	2	no	—
187	Dic₄₇	—	8	2	no	—
188	#3334	—	47	1	no	—
189	D₄×D₈	—	47	1	no	—
190	#2240	—	47	1	no	—
191	#2232	—	47	2	no	—
192	#2224	—	47	1	no	—
193	#2215	—	47	2	no	—
194	#2206	—	47	1	no	—
195	#2184	—	47	2	no	—
196	C₄×D₁₆	D₄₇	1	1	yes	characteristic, complemented
197	D₄×C₁₆	—	47	1	no	—
198	C₂×#357	—	47	2	no	—
199	C₂×C₄₇	—	8	2	no	—
200	C₂×C₄₇	—	16	1	no	—
201	C₂×C₄₇	#3294	1	1	yes	characteristic
202	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
203	C₂×C₄₇	#3416	1	1	yes	characteristic
204	D₄₇	—	2	4	no	—
205	D₄₇	—	16	2	no	—
206	C₂²×D₈	—	47	2	no	—
207	#547	—	94	1	no	—
208	D₄²	—	94	1	no	—
209	C₂×D₁₆	—	47	2	no	—
210	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
211	C₂²×C₁₆	—	47	2	no	—
212	#500	—	47	1	no	—
213	#493	—	47	1	no	—
214	#480	—	94	2	no	—
215	#474	—	94	1	no	—
216	#466	—	47	2	no	—
217	#466	—	47	2	no	—
218	#459	—	94	1	no	—
219	#447	—	94	2	no	—
220	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
221	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic, complemented
222	C₈×D₄	—	47	1	no	—
223	C₂×#107	—	47	2	no	—
224	#366	—	47	2	no	—
225	#366	C₂×D₄₇	1	1	yes	characteristic, complemented
226	C₄⋊C₁₆	—	47	1	no	—
227	#357	—	47	2	no	—
228	#357	—	47	2	no	—
229	#357	—	47	2	no	—
230	#357	—	47	2	no	—
231	#357	C₂×D₄₇	1	1	yes	characteristic
232	#357	C₂×D₄₇	1	1	yes	characteristic, complemented
233	C₂²⋊C₁₆	—	47	2	no	—
234	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
235	C₄₇	#33861	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	188	1	no	—
237	C₂²×D₄	—	94	2	no	—
238	C₂²×D₄	—	188	1	no	—
239	C₂×D₈	—	47	2	no	—
240	C₂×D₈	—	94	2	no	—
241	C₂×D₈	—	94	4	no	—
242	C₂×D₈	—	94	1	no	—
243	C₂×D₈	—	188	2	no	—
244	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
245	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₂²×C₈	—	47	2	no	—
247	C₄⋊Q₈	—	94	1	no	—
248	#127	—	94	1	no	—
249	#123	—	188	2	no	—
250	C₂²⋊Q₈	—	188	1	no	—
251	#121	—	94	2	no	—
252	#121	—	94	2	no	—
253	#121	—	188	1	no	—
254	#120	—	188	2	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	47	1	no	—
258	C₂×C₄⋊C₄	—	94	2	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	188	2	no	—
261	C₂×C₁₆	—	47	2	no	—
262	C₂×C₁₆	—	188	2	no	—
263	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
265	#107	—	47	2	no	—
266	#107	—	94	2	no	—
267	#107	—	94	1	no	—
268	#107	C₂²×D₄₇	1	1	yes	characteristic
269	#107	C₂²×D₄₇	1	1	yes	characteristic
270	C₄⋊C₈	—	47	1	no	—
271	#102	—	2	1	no	—
272	#102	—	2	1	no	—
273	#102	—	94	2	no	—
274	#102	—	94	2	no	—
275	C₂²⋊C₈	—	47	2	no	—
276	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
277	C₂⁴	—	188	2	no	—
278	#41	—	376	4	no	—
279	C₂×Q₈	—	188	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	47	1	no	—
283	C₂×D₄	—	94	2	no	—
284	C₂×D₄	—	376	4	no	—
285	C₂×D₄	—	376	2	no	—
286	C₂×D₄	—	188	2	no	—
287	C₂×D₄	—	188	2	no	—
288	C₂×D₄	—	188	4	no	—
289	C₂×D₄	—	188	2	no	—
290	C₂×D₄	—	188	1	no	—
291	C₂×D₄	—	188	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	47	2	no	—
295	C₂²×C₄	—	188	2	no	—
296	D₈	—	2	1	no	—
297	D₈	—	2	1	no	—
298	D₈	—	2	1	no	—
299	D₈	—	94	2	no	—
300	D₈	—	94	2	no	—
301	D₈	—	376	1	no	—
302	D₈	—	188	4	no	—
303	D₈	#10625350	1	2	yes	complemented
304	C₂×C₈	—	47	2	no	—
305	C₂×C₈	—	94	2	no	—
306	C₂×C₈	—	94	2	no	—
307	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	47	1	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	—	188	2	no	—
314	C₄⋊C₄	—	188	1	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	47	2	no	—
318	C₂²⋊C₄	—	188	2	no	—
319	C₂²⋊C₄	—	188	2	no	—
320	C₄²	D₄×D₄₇	1	1	yes	characteristic
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	94	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	47	2	no	—
327	C₂³	—	376	4	no	—
328	C₂³	—	376	4	no	—
329	C₂³	—	188	2	no	—
330	C₂³	—	752	2	no	—
331	Q₈	—	376	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	47	4	no	—
337	D₄	—	376	4	no	—
338	D₄	—	376	4	no	—
339	D₄	—	376	4	no	—
340	D₄	—	376	2	no	—
341	D₄	—	188	2	no	—
342	D₄	—	188	2	no	—
343	D₄	—	752	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	47	2	no	—
349	C₂×C₄	—	94	2	no	—
350	C₂×C₄	—	94	2	no	—
351	C₂×C₄	—	376	4	no	—
352	C₂×C₄	—	188	2	no	—
353	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
354	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	94	2	no	—
358	C₈	C₂×#10625350	1	1	yes	characteristic
359	C₈	#13094070	1	1	yes	characteristic
360	C₂²	—	4	2	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	47	4	no	—
367	C₂²	—	94	2	no	—
368	C₂²	—	94	2	no	—
369	C₂²	—	376	2	no	—
370	C₂²	—	376	2	no	—
371	C₂²	—	376	2	no	—
372	C₂²	—	752	8	no	—
373	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	94	2	no	—
378	C₄	—	376	2	no	—
379	C₄	D₁₆×D₄₇	1	2	yes	—
380	C₄	#15689769	1	1	yes	characteristic
381	C₄	#15689930	1	1	yes	characteristic
382	C₂	—	8	2	no	—
383	C₂	—	16	1	no	—
384	C₂	—	94	4	no	—
385	C₂	—	752	2	no	—
386	C₂	#20255203	1	1	yes	characteristic
387	C₂	C₂×D₁₆×D₄₇	1	1	yes	characteristic
388	C₂	#20256531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄×D₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₄×D₁₆	D₄₇
hypocenter	C₄₇	#33861
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20255737	C₂	semidirect
2. D₈×D₁₈₈	C₂	semidirect
3. #20245612	C₂	semidirect
4. #20245570	C₂	semidirect
5. #20245525	C₂	semidirect
6. #20245473	C₂	semidirect
7. #20245423	C₂	semidirect
8. #20245298	C₂	semidirect
9. C₁₆×D₁₈₈	C₂	semidirect
10. #357×D₄₇	C₂	semidirect
11. C₄×D₁₆×C₄₇	C₂	semidirect
15. C₂×#13093965	C₂²	semidirect
16. C₂×C₁₆×D₄₇	C₂²	semidirect
17. #15689528	C₂²	semidirect
22. #15689356	C₂²	semidirect
26. #107×D₄₇	C₂²	semidirect
27. #15688948	C₂²	semidirect
28. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
29. #15688925	C₂²	semidirect
30. #15688925	C₂²	semidirect
32. #15688923	C₂²	semidirect
33. (C₂×D₄₇)⋊C₁₆	C₂²	semidirect
36. C₄×D₈×C₄₇	C₂²	semidirect
37. C₄₇×#366	C₂²	semidirect
39. C₄₇×#357	C₂²	semidirect
40. C₄×C₁₆×C₄₇	C₂²	semidirect
104. D₈×C₄₇	#41	semidirect
156. D₁₈₈	D₁₆	semidirect
196. C₄×D₁₆	D₄₇	semidirect
221. C₄×D₈	C₂×D₄₇	semidirect
225. #366	C₂×D₄₇	semidirect
232. #357	C₂×D₄₇	semidirect
234. C₄×C₁₆	C₂×D₄₇	semidirect
235. C₄₇	#33861	semidirect
303. D₈	#10625350	semidirect

Maximal subgroups

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

subgroup	index
#20255737	2
D₈×D₁₈₈	2
#20245612	2
#20245570	2
#20245525	2
#20245473	2
#20245423	2
#20245298	2
C₁₆×D₁₈₈	2
#357×D₄₇	2
C₄×D₁₆×C₄₇	2
#33861	47