Subgroups — #26372982

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689553	C₂	1	2	yes	maximal, complemented
2	#20255148	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245645	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245486	C₂	1	2	yes	maximal, complemented
5	#20245435	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245423	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245326	C₂	1	2	yes	maximal, complemented
8	#20245311	C₂	1	2	yes	maximal, complemented
9	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	2	yes	—
13	#15689794	—	2	1	no	—
14	#15689553	—	2	8	no	—
15	C₂×#13093970	C₂²	1	2	yes	—
16	C₂×#13093968	C₂²	1	1	yes	characteristic
17	C₂×#13093967	C₂²	1	2	yes	—
18	C₂×#13093967	C₂²	1	2	yes	—
19	C₂×#13093965	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
21	#15689500	C₂²	1	1	yes	characteristic
22	#15689359	C₂²	1	2	yes	—
23	#15689324	—	2	1	no	—
24	#15689270	—	2	2	no	—
25	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
26	#15688950	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#15688927	C₂²	1	2	yes	—
29	#15688926	C₂²	1	1	yes	characteristic
30	#15688924	C₂²	1	1	yes	characteristic
31	#15688923	C₂²	1	2	yes	—
32	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂×#135×C₄₇	C₂²	1	2	yes	—
35	C₄₇×#547	—	2	1	no	—
36	C₄₇×#494	C₂²	1	1	yes	characteristic
37	C₄₇×#467	C₂²	1	2	yes	—
38	C₄₇×#461	—	2	1	no	—
39	C₄₇×#452	—	2	2	no	—
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×#10625350	—	2	2	no	—
43	#13094046	—	2	4	no	—
44	#13094046	—	4	2	no	—
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625325	—	2	2	no	—
47	C₂×D₃₇₆	—	2	1	no	—
48	C₂×D₃₇₆	C₂³	1	2	yes	—
49	#13094021	—	2	1	no	—
50	#13094020	—	4	1	no	—
51	#13093998	—	2	2	no	—
52	#13093997	—	4	2	no	—
53	C₄×Dic₉₄	—	2	1	no	—
54	#13093986	—	4	1	no	—
55	#13093970	—	2	4	no	—
56	#13093968	D₄	1	4	yes	complemented
57	#13093967	—	2	4	no	—
58	#13093967	—	4	2	no	—
59	#13093965	—	2	4	no	—
60	C₂×C₄₇⋊C₁₆	—	2	4	no	—
61	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093949	—	2	1	no	—
66	#13093946	—	2	2	no	—
67	C₂×#41×C₄₇	—	2	2	no	—
68	#135×C₄₇	—	2	4	no	—
69	#135×C₄₇	—	4	2	no	—
70	C₂×Q₁₆×C₄₇	—	2	1	no	—
71	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₄₇	—	2	2	no	—
73	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₇	—	2	1	no	—
76	#123×C₄₇	—	4	2	no	—
77	C₂²⋊Q₈×C₄₇	—	2	2	no	—
78	#121×C₄₇	—	4	1	no	—
79	C₄×D₄×C₄₇	—	2	1	no	—
80	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
82	#107×C₄₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
84	#103×C₄₇	—	2	2	no	—
85	#102×C₄₇	—	2	1	no	—
86	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	#41×C₄₇	—	4	4	no	—
89	#41×C₄₇	—	8	2	no	—
90	C₂×Q₈×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	1	no	—
92	C₂×D₄×C₄₇	—	4	2	no	—
93	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	—	4	1	no	—
95	C₂²×C₄×C₄₇	—	4	2	no	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₂²⋊D₄₇	—	4	2	no	—
98	C₂×C₂²⋊D₄₇	—	4	2	no	—
99	#10625350	—	4	4	no	—
100	#10625350	—	8	2	no	—
101	C₂×D₁₈₈	—	2	2	no	—
102	C₂×D₁₈₈	—	4	1	no	—
103	C₂×D₁₈₈	—	4	1	no	—
104	C₂×C₄×D₄₇	—	4	2	no	—
105	C₂×Dic₉₄	—	2	1	no	—
106	Q₁₆×C₄₇	—	2	2	no	—
107	Q₁₆×C₄₇	—	2	2	no	—
108	Q₁₆×C₄₇	—	8	1	no	—
109	SD₁₆×C₄₇	—	4	4	no	—
110	D₈×C₄₇	—	2	1	no	—
111	D₈×C₄₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₄₇	—	2	1	no	—
118	C₄⋊C₄×C₄₇	—	4	1	no	—
119	C₄⋊C₄×C₄₇	—	4	2	no	—
120	C₄⋊C₄×C₄₇	—	4	1	no	—
121	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
122	C₂²⋊C₄×C₄₇	—	4	2	no	—
123	C₂²⋊C₄×C₄₇	—	4	1	no	—
124	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
125	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
126	(C₂×D₄₇)⋊C₄	—	4	2	no	—
127	C₄⋊Dic₄₇	—	2	1	no	—
128	Dic₄₇⋊C₄	—	4	1	no	—
129	C₄×Dic₄₇	—	4	1	no	—
130	Dic₁₈₈	—	2	1	no	—
131	Dic₁₈₈	C₂×D₄	1	2	yes	—
132	D₃₇₆	—	2	2	no	—
133	D₃₇₆	—	2	2	no	—
134	D₃₇₆	—	8	1	no	—
135	#10625325	—	4	4	no	—
136	C₄₇⋊C₁₆	—	2	2	no	—
137	C₄₇⋊C₁₆	—	2	1	no	—
138	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
139	C₂³×C₄₇	—	4	1	no	—
140	C₂³×C₄₇	C₂×D₈	1	2	yes	—
141	C₂²×D₄₇	—	4	2	no	—
142	Q₈×C₄₇	—	4	2	no	—
143	Q₈×C₄₇	—	4	2	no	—
144	D₄×C₄₇	—	2	2	no	—
145	D₄×C₄₇	—	2	2	no	—
146	D₄×C₄₇	—	4	1	no	—
147	D₄×C₄₇	—	8	4	no	—
148	C₂×C₄×C₄₇	—	2	2	no	—
149	C₂×C₄×C₄₇	—	2	2	no	—
150	C₂×C₄×C₄₇	—	4	1	no	—
151	C₂×C₄×C₄₇	—	4	2	no	—
152	C₂×C₄×C₄₇	—	8	4	no	—
153	C₂×C₄×C₄₇	—	8	2	no	—
154	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
158	C₂²⋊D₄₇	—	8	4	no	—
159	C₂²⋊D₄₇	—	16	2	no	—
160	C₂×Dic₄₇	—	4	1	no	—
161	C₂×Dic₄₇	—	4	1	no	—
162	D₁₈₈	—	4	2	no	—
163	D₁₈₈	—	4	2	no	—
164	D₁₈₈	—	8	2	no	—
165	D₁₈₈	—	16	1	no	—
166	C₄×D₄₇	—	8	4	no	—
167	Dic₉₄	—	2	2	no	—
168	Dic₉₄	—	4	1	no	—
169	C₈×C₄₇	—	2	2	no	—
170	C₈×C₄₇	—	2	1	no	—
171	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
173	#33850	—	47	1	no	maximal, Sylow
174	C₂²×C₄₇	—	2	2	no	—
175	C₂²×C₄₇	—	2	2	no	—
176	C₂²×C₄₇	—	2	2	no	—
177	C₂²×C₄₇	—	4	2	no	—
178	C₂²×C₄₇	—	8	1	no	—
179	C₂²×C₄₇	—	8	1	no	—
180	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₄₇	—	8	2	no	—
182	C₂×D₄₇	—	8	2	no	—
183	C₂×D₄₇	—	8	2	no	—
184	C₄×C₄₇	—	2	1	no	—
185	C₄×C₄₇	—	2	2	no	—
186	C₄×C₄₇	—	8	1	no	—
187	C₄×C₄₇	—	8	2	no	—
188	C₄×C₄₇	#508	1	2	yes	complemented
189	C₄×C₄₇	D₄²	1	1	yes	characteristic
190	C₄×C₄₇	#576	1	1	yes	characteristic
191	Dic₄₇	—	4	2	no	—
192	Dic₄₇	—	8	1	no	—
193	C₂×#508	—	47	2	no	—
194	#3287	—	47	1	no	—
195	#3283	D₄₇	1	1	yes	characteristic, complemented
196	#2248	—	47	1	no	—
197	#2217	—	47	2	no	—
198	#2209	—	47	1	no	—
199	#2206	—	47	1	no	—
200	#2189	—	47	2	no	—
201	#2186	—	47	2	no	—
202	C₄×SD₃₂	—	47	1	no	—
203	D₄×C₁₆	—	47	1	no	—
204	C₂×C₄₇	—	4	2	no	—
205	C₂×C₄₇	—	8	2	no	—
206	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
207	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
208	C₂×C₄₇	#3416	1	1	yes	characteristic
209	D₄₇	—	16	2	no	—
210	C₂×#135	—	47	2	no	—
211	C₂×#135	C₂×D₄₇	1	2	yes	—
212	#550	—	94	1	no	—
213	#547	—	2	1	no	—
214	#508	—	94	8	no	—
215	C₂×Q₃₂	—	47	2	no	—
216	C₂×SD₃₂	—	47	1	no	—
217	C₂×SD₃₂	—	47	2	no	—
218	C₂×SD₃₂	—	47	2	no	—
219	C₂×D₁₆	—	47	2	no	—
220	C₂²×C₁₆	—	47	2	no	—
221	#494	C₂×D₄₇	1	1	yes	characteristic
222	#493	—	47	1	no	—
223	#467	C₂×D₄₇	1	2	yes	—
224	#466	—	47	2	no	—
225	#461	—	2	1	no	—
226	#460	—	94	1	no	—
227	#452	—	2	2	no	—
228	#449	—	94	2	no	—
229	C₄×Q₁₆	—	47	1	no	—
230	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
231	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
232	#367	—	47	1	no	—
233	C₄⋊C₁₆	—	47	1	no	—
234	#358	—	47	2	no	—
235	#358	—	47	1	no	—
236	#357	—	47	1	no	—
237	#357	—	47	2	no	—
238	C₂²⋊C₁₆	—	47	2	no	—
239	C₄×C₁₆	—	47	1	no	—
240	C₄₇	#33850	1	1	yes	characteristic, complemented, Sylow
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	94	2	no	—
243	#135	—	2	4	no	—
244	#135	—	4	2	no	—
245	#135	—	94	4	no	—
246	#135	—	188	2	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	47	1	no	—
249	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
250	C₂×SD₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	94	2	no	—
252	C₂×D₈	—	47	2	no	—
253	C₂×D₈	—	94	1	no	—
254	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
255	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
256	C₄⋊Q₈	—	2	1	no	—
257	#127	—	94	1	no	—
258	#127	—	188	1	no	—
259	#123	—	4	2	no	—
260	C₂²⋊Q₈	—	2	2	no	—
261	#121	—	4	1	no	—
262	#121	—	94	2	no	—
263	#121	—	188	2	no	—
264	C₄×Q₈	—	94	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	—	188	1	no	—
267	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂×C₄⋊C₄	—	4	1	no	—
269	Q₃₂	—	94	4	no	—
270	SD₃₂	—	47	4	no	—
271	SD₃₂	—	94	4	no	—
272	SD₃₂	—	188	2	no	—
273	D₁₆	—	94	4	no	—
274	C₂×C₁₆	—	47	1	no	—
275	C₂×C₁₆	—	47	1	no	—
276	C₂×C₁₆	—	47	2	no	—
277	C₂×C₁₆	—	94	4	no	—
278	#107	—	47	1	no	—
279	#107	C₂²×D₄₇	1	1	yes	characteristic
280	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
281	#103	—	2	2	no	—
282	#103	—	94	1	no	—
283	#102	—	2	1	no	—
284	#102	—	94	2	no	—
285	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	#41	—	4	4	no	—
288	#41	—	8	2	no	—
289	#41	—	376	2	no	—
290	#41	—	188	4	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	94	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	4	2	no	—
295	C₂×D₄	—	94	2	no	—
296	C₂×D₄	—	188	2	no	—
297	C₂×D₄	—	188	2	no	—
298	C₂×D₄	—	188	1	no	—
299	C₂×D₄	—	188	1	no	—
300	C₂×D₄	#10625334	1	1	yes	characteristic
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	188	2	no	—
304	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	2	2	no	—
307	Q₁₆	—	8	1	no	—
308	Q₁₆	—	47	2	no	—
309	Q₁₆	—	94	1	no	—
310	SD₁₆	—	4	4	no	—
311	SD₁₆	—	188	4	no	—
312	D₈	—	2	1	no	—
313	D₈	—	94	2	no	—
314	D₈	—	94	2	no	—
315	D₈	—	376	1	no	—
316	D₈	D₄×D₄₇	1	2	yes	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
321	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	94	1	no	—
327	C₄⋊C₄	—	188	1	no	—
328	C₄⋊C₄	#10625334	1	1	yes	characteristic
329	C₂²⋊C₄	—	4	2	no	—
330	C₂²⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	188	2	no	—
332	C₂²⋊C₄	#10625334	1	2	yes	—
333	C₄²	—	188	1	no	—
334	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
335	C₁₆	—	47	2	no	—
336	C₁₆	—	94	2	no	—
337	C₁₆	—	94	1	no	—
338	C₂³	—	4	1	no	—
339	C₂³	—	188	2	no	—
340	C₂³	C₂×#10625334	1	2	yes	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	94	2	no	—
344	Q₈	—	188	1	no	—
345	D₄	—	2	2	no	—
346	D₄	—	2	2	no	—
347	D₄	—	4	1	no	—
348	D₄	—	8	4	no	—
349	D₄	—	376	4	no	—
350	D₄	—	376	2	no	—
351	D₄	—	188	2	no	—
352	D₄	—	188	2	no	—
353	D₄	—	752	1	no	—
354	D₄	—	752	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	8	4	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	376	4	no	—
362	C₂×C₄	—	188	1	no	—
363	C₂×C₄	—	188	1	no	—
364	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
365	C₂×C₄	C₂×#10625334	1	2	yes	—
366	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
367	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
371	C₈	#13094069	1	1	yes	characteristic
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	376	2	no	—
379	C₂²	—	376	2	no	—
380	C₂²	—	376	2	no	—
381	C₂²	C₂²×#10625334	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	2	no	—
384	C₄	—	8	2	no	—
385	C₄	—	8	1	no	—
386	C₄	—	376	1	no	—
387	C₄	—	188	2	no	—
388	C₄	#15689553	1	2	yes	complemented
389	C₄	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
390	C₄	#15689926	1	1	yes	characteristic
391	C₂	—	4	2	no	—
392	C₂	—	8	2	no	—
393	C₂	—	752	2	no	—
394	C₂	D₄×#10625334	1	1	yes	characteristic
395	C₂	C₂×#15689553	1	1	yes	characteristic
396	C₂	#20256532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3283	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3283	D₄₇
hypocenter	C₄₇	#33850
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3283

Complements

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

N	H ≅ G/N	type
1. C₂×#15689553	C₂	semidirect
2. #20255148	C₂	semidirect
3. #20245645	C₂	semidirect
4. #20245486	C₂	semidirect
5. #20245435	C₂	semidirect
6. #20245423	C₂	semidirect
7. #20245326	C₂	semidirect
8. #20245311	C₂	semidirect
9. C₄×#13093968	C₂	semidirect
10. D₄×C₄₇⋊C₁₆	C₂	semidirect
11. C₄₇×#3283	C₂	semidirect
56. #13093968	D₄	semidirect
188. C₄×C₄₇	#508	semidirect
195. #3283	D₄₇	semidirect
240. C₄₇	#33850	semidirect
388. C₄	#15689553	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689553	2
#20255148	2
#20245645	2
#20245486	2
#20245435	2
#20245423	2
#20245326	2
#20245311	2
C₄×#13093968	2
D₄×C₄₇⋊C₁₆	2
C₄₇×#3283	2
#33850	47