Subgroups — #26373232

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255148	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245613	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245579	C₂	1	2	yes	maximal, complemented
4	#20245537	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245472	C₂	1	2	yes	maximal, complemented
6	#20245423	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245309	C₂	1	2	yes	maximal, complemented
8	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245093	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3341	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	2	yes	—
13	#15689794	—	2	1	no	—
14	C₂×#13093968	C₂²	1	1	yes	characteristic
15	C₂×#13093965	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
17	#15689500	C₂²	1	1	yes	characteristic
18	#15689359	C₂²	1	2	yes	—
19	#15689324	—	2	1	no	—
20	#15689270	—	2	2	no	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	#15688950	C₂²	1	1	yes	characteristic
23	#15688948	C₂²	1	2	yes	—
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#15688926	C₂²	1	2	yes	—
26	#15688926	C₂²	1	1	yes	characteristic
27	#15688924	C₂²	1	1	yes	characteristic
28	#15688923	C₂²	1	2	yes	—
29	#15688923	C₂²	1	2	yes	—
30	#15688923	C₂²	1	2	yes	—
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₇×#555	—	2	1	no	—
34	C₄₇×#500	C₂²	1	1	yes	characteristic
35	C₄₇×#482	—	2	2	no	—
36	C₄₇×#476	—	2	1	no	—
37	C₄₇×#466	C₂²	1	2	yes	—
38	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#427	C₂²	1	2	yes	—
41	C₂×#10625350	—	2	2	no	—
42	#13094046	—	2	4	no	—
43	#13094046	—	4	2	no	—
44	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
45	C₂×#10625325	—	2	2	no	—
46	C₂×D₃₇₆	—	2	1	no	—
47	C₂×D₃₇₆	C₂³	1	2	yes	—
48	#13094021	—	2	1	no	—
49	#13094020	—	4	1	no	—
50	#13093998	—	2	2	no	—
51	#13093997	—	4	2	no	—
52	C₄×Dic₉₄	—	2	1	no	—
53	#13093986	—	4	1	no	—
54	#13093968	—	2	2	no	—
55	#13093965	—	4	2	no	—
56	C₂×C₄₇⋊C₁₆	—	4	2	no	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13093958	C₂³	1	1	yes	characteristic
61	#13093949	—	2	1	no	—
62	#13093946	—	2	2	no	—
63	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	#124×C₄₇	—	4	2	no	—
67	#121×C₄₇	—	2	2	no	—
68	#121×C₄₇	—	4	1	no	—
69	C₄×D₄×C₄₇	—	2	1	no	—
70	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
71	#117×C₄₇	—	2	2	no	—
72	#117×C₄₇	—	4	1	no	—
73	#107×C₄₇	—	2	1	no	—
74	#107×C₄₇	C₂³	1	2	yes	—
75	#107×C₄₇	C₂³	1	1	yes	characteristic
76	#106×C₄₇	—	2	2	no	—
77	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
78	#102×C₄₇	—	2	2	no	—
79	#102×C₄₇	—	2	1	no	—
80	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	4	1	no	—
83	C₂×D₄×C₄₇	—	2	1	no	—
84	C₂×D₄×C₄₇	—	4	2	no	—
85	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	—	4	1	no	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₂²⋊D₄₇	—	4	2	no	—
89	C₂×C₂²⋊D₄₇	—	4	2	no	—
90	#10625350	—	4	4	no	—
91	#10625350	—	8	2	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	4	1	no	—
94	C₂×D₁₈₈	—	4	1	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×Dic₉₄	—	2	1	no	—
97	D₈×C₄₇	—	2	1	no	—
98	D₈×C₄₇	—	2	1	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	4	1	no	—
107	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₂²⋊C₄×C₄₇	—	4	1	no	—
111	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
112	C₄²×C₄₇	—	4	2	no	—
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	(C₂×D₄₇)⋊C₄	—	4	2	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	Dic₄₇⋊C₄	—	4	1	no	—
117	C₄×Dic₄₇	—	4	1	no	—
118	Dic₁₈₈	—	2	1	no	—
119	Dic₁₈₈	#41	1	2	yes	—
120	D₃₇₆	—	2	2	no	—
121	D₃₇₆	—	2	2	no	—
122	D₃₇₆	—	8	1	no	—
123	#10625325	—	4	4	no	—
124	C₄₇⋊C₁₆	—	2	1	no	—
125	C₄₇⋊C₁₆	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₇	—	4	1	no	—
128	C₂³×C₄₇	C₂×D₈	1	2	yes	—
129	C₂²×D₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	16	1	no	—
131	D₄×C₄₇	—	2	2	no	—
132	D₄×C₄₇	—	4	1	no	—
133	D₄×C₄₇	—	4	1	no	—
134	D₄×C₄₇	—	16	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	8	2	no	—
140	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₄₇	—	8	4	no	—
145	C₂²⋊D₄₇	—	16	2	no	—
146	C₂×Dic₄₇	—	4	1	no	—
147	C₂×Dic₄₇	—	4	1	no	—
148	D₁₈₈	—	4	2	no	—
149	D₁₈₈	—	4	2	no	—
150	D₁₈₈	—	8	2	no	—
151	D₁₈₈	—	16	1	no	—
152	C₄×D₄₇	—	8	4	no	—
153	Dic₉₄	—	2	2	no	—
154	Dic₉₄	—	4	1	no	—
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	—	2	1	no	—
157	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
158	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
159	#33872	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	2	2	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	8	1	no	—
164	C₂²×C₄₇	—	8	1	no	—
165	C₂²×C₄₇	—	8	1	no	—
166	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₄₇	—	8	2	no	—
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	8	2	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	—	8	1	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#508	1	2	yes	—
175	C₄×C₄₇	#546	1	1	yes	characteristic
176	C₄×C₄₇	#577	1	1	yes	characteristic
177	Dic₄₇	—	4	2	no	—
178	Dic₄₇	—	8	1	no	—
179	#3341	D₄₇	1	1	yes	characteristic, complemented
180	#3287	—	47	1	no	—
181	#2240	—	47	1	no	—
182	#2234	—	47	2	no	—
183	#2227	—	47	1	no	—
184	#2215	—	47	2	no	—
185	#2206	—	47	1	no	—
186	#2186	—	47	2	no	—
187	C₄×SD₃₂	—	47	1	no	—
188	D₄×C₁₆	—	47	1	no	—
189	#2138	—	47	2	no	—
190	C₂×C₄₇	—	4	2	no	—
191	C₂×C₄₇	—	16	1	no	—
192	C₂×C₄₇	#3294	1	1	yes	characteristic
193	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
194	C₂×C₄₇	#3415	1	1	yes	characteristic
195	D₄₇	—	16	2	no	—
196	C₂×#135	—	47	2	no	—
197	#555	—	2	1	no	—
198	#550	—	94	1	no	—
199	C₂×SD₃₂	—	47	1	no	—
200	C₂×D₁₆	—	47	2	no	—
201	C₂²×C₁₆	—	47	2	no	—
202	#500	C₂×D₄₇	1	1	yes	characteristic
203	#493	—	47	1	no	—
204	#482	—	2	2	no	—
205	#476	—	2	1	no	—
206	#466	—	47	2	no	—
207	#466	C₂×D₄₇	1	2	yes	—
208	#460	—	94	1	no	—
209	#449	—	94	2	no	—
210	C₄×Q₁₆	—	47	1	no	—
211	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
212	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
213	#427	C₂×D₄₇	1	2	yes	—
214	#367	—	47	1	no	—
215	#366	—	47	2	no	—
216	C₄⋊C₁₆	—	47	1	no	—
217	#358	—	47	1	no	—
218	#358	—	47	2	no	—
219	#357	—	47	1	no	—
220	#357	—	47	2	no	—
221	#357	—	47	2	no	—
222	#357	—	47	2	no	—
223	C₂²⋊C₁₆	—	47	2	no	—
224	C₄×C₁₆	—	47	1	no	—
225	C₄₇	#33872	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	94	2	no	—
227	#135	—	94	4	no	—
228	#135	—	188	2	no	—
229	C₂×Q₁₆	—	47	1	no	—
230	C₂×SD₁₆	—	94	2	no	—
231	C₂×D₈	—	47	2	no	—
232	C₂×D₈	—	94	1	no	—
233	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
235	C₄⋊Q₈	—	2	1	no	—
236	#127	—	94	1	no	—
237	#127	—	188	1	no	—
238	#124	—	4	2	no	—
239	#121	—	2	2	no	—
240	#121	—	4	1	no	—
241	#121	—	94	2	no	—
242	#121	—	188	2	no	—
243	C₄×Q₈	—	94	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	188	1	no	—
246	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
247	#117	—	2	2	no	—
248	#117	—	4	1	no	—
249	SD₃₂	—	94	2	no	—
250	D₁₆	—	188	2	no	—
251	C₂×C₁₆	—	47	1	no	—
252	C₂×C₁₆	—	47	1	no	—
253	C₂×C₁₆	—	47	2	no	—
254	C₂×C₁₆	—	188	2	no	—
255	#107	—	2	1	no	—
256	#107	—	47	1	no	—
257	#107	C₂²×D₄₇	1	2	yes	—
258	#107	C₂²×D₄₇	1	1	yes	characteristic
259	#106	—	2	2	no	—
260	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
261	#103	—	94	1	no	—
262	#102	—	2	1	no	—
263	#102	—	2	2	no	—
264	#102	—	94	2	no	—
265	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
266	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
267	#41	—	376	2	no	—
268	#41	—	188	4	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	94	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂×D₄	—	188	2	no	—
275	C₂×D₄	—	188	2	no	—
276	C₂×D₄	—	188	1	no	—
277	C₂×D₄	—	188	1	no	—
278	C₂×D₄	#10625334	1	1	yes	characteristic
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	188	2	no	—
281	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
282	Q₁₆	—	47	2	no	—
283	Q₁₆	—	94	1	no	—
284	SD₁₆	—	188	4	no	—
285	D₈	—	2	1	no	—
286	D₈	—	2	1	no	—
287	D₈	—	94	2	no	—
288	D₈	—	94	2	no	—
289	D₈	—	376	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
294	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	94	1	no	—
299	C₄⋊C₄	—	188	1	no	—
300	C₄⋊C₄	#10625334	1	1	yes	characteristic
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	188	2	no	—
305	C₂²⋊C₄	#10625334	1	2	yes	—
306	C₄²	—	4	2	no	—
307	C₄²	—	188	1	no	—
308	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₁₆	—	94	1	no	—
310	C₁₆	—	94	1	no	—
311	C₁₆	—	94	2	no	—
312	C₂³	—	4	1	no	—
313	C₂³	—	188	2	no	—
314	C₂³	C₂×#10625334	1	2	yes	—
315	Q₈	—	16	1	no	—
316	Q₈	—	94	2	no	—
317	Q₈	—	188	1	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	1	no	—
321	D₄	—	16	2	no	—
322	D₄	—	376	4	no	—
323	D₄	—	376	2	no	—
324	D₄	—	188	2	no	—
325	D₄	—	188	2	no	—
326	D₄	—	752	1	no	—
327	D₄	—	752	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	376	4	no	—
334	C₂×C₄	—	188	1	no	—
335	C₂×C₄	—	188	1	no	—
336	C₂×C₄	C₂×#10625334	1	2	yes	—
337	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	#41×D₄₇	1	1	yes	characteristic
343	C₈	#13094070	1	1	yes	characteristic
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	8	1	no	—
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	376	2	no	—
352	C₂²	—	376	2	no	—
353	C₂²	C₂²×#10625334	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	376	1	no	—
359	C₄	—	188	2	no	—
360	C₄	#15689553	1	2	yes	—
361	C₄	#15689761	1	1	yes	characteristic
362	C₄	#15689931	1	1	yes	characteristic
363	C₂	—	4	2	no	—
364	C₂	—	16	1	no	—
365	C₂	—	752	2	no	—
366	C₂	#20255195	1	1	yes	characteristic
367	C₂	C₂×#15689553	1	1	yes	characteristic
368	C₂	#20256527	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3341	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3341	D₄₇
hypocenter	C₄₇	#33872
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3341

Complements

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

N	H ≅ G/N	type
1. #20255148	C₂	semidirect
2. #20245613	C₂	semidirect
3. #20245579	C₂	semidirect
4. #20245537	C₂	semidirect
5. #20245472	C₂	semidirect
6. #20245423	C₂	semidirect
7. #20245309	C₂	semidirect
8. C₄×#13093968	C₂	semidirect
9. D₄×C₄₇⋊C₁₆	C₂	semidirect
10. #20245093	C₂	semidirect
11. C₄₇×#3341	C₂	semidirect
179. #3341	D₄₇	semidirect
225. C₄₇	#33872	semidirect

Maximal subgroups

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

subgroup	index
#20255148	2
#20245613	2
#20245579	2
#20245537	2
#20245472	2
#20245423	2
#20245309	2
C₄×#13093968	2
D₄×C₄₇⋊C₁₆	2
#20245093	2
C₄₇×#3341	2
#33872	47