Subgroups — #26373205

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255819	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245615	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245593	C₂	1	2	yes	maximal, complemented
4	#20245525	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245477	C₂	1	2	yes	maximal, complemented
6	#20245435	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245328	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	#20245095	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3283	C₂	1	1	yes	maximal, characteristic
12	#15689843	—	2	1	no	—
13	C₂×#13093970	C₂²	1	2	yes	—
14	C₂×#13093968	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689465	—	2	2	no	—
18	#15689409	—	2	1	no	—
19	#15689364	C₂²	1	2	yes	—
20	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
21	#15689161	C₂²	1	2	yes	—
22	#15688950	C₂²	1	1	yes	characteristic
23	#15688948	C₂²	1	2	yes	—
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#15688927	C₂²	1	2	yes	—
26	#15688927	C₂²	1	2	yes	—
27	#15688927	C₂²	1	2	yes	—
28	#15688926	C₂²	1	1	yes	characteristic
29	#15688924	C₂²	1	2	yes	—
30	#15688924	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂×#135×C₄₇	C₂²	1	2	yes	—
34	C₄₇×#547	—	2	1	no	—
35	C₄₇×#494	C₂²	1	1	yes	characteristic
36	C₄₇×#467	C₂²	1	2	yes	—
37	C₄₇×#461	—	2	1	no	—
38	C₄₇×#452	—	2	2	no	—
39	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
42	#13094024	—	2	1	no	—
43	#13094017	—	4	2	no	—
44	#13094011	—	4	1	no	—
45	#13094003	—	2	2	no	—
46	C₄×Dic₉₄	—	2	1	no	—
47	#13093989	—	4	1	no	—
48	#13093983	—	2	2	no	—
49	#13093970	—	4	2	no	—
50	#13093968	—	2	2	no	—
51	C₂×C₄₇⋊C₁₆	—	4	2	no	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	#13093958	—	2	1	no	—
56	#13093958	C₂³	1	1	yes	characteristic
57	#13093958	C₂³	1	2	yes	—
58	#13093956	—	2	2	no	—
59	#13093949	—	2	1	no	—
60	#13093949	—	2	2	no	—
61	C₂×#41×C₄₇	—	2	2	no	—
62	#135×C₄₇	—	2	4	no	—
63	#135×C₄₇	—	4	2	no	—
64	C₂×Q₁₆×C₄₇	—	2	1	no	—
65	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
66	C₂×SD₁₆×C₄₇	—	2	2	no	—
67	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	#123×C₄₇	—	4	2	no	—
71	C₂²⋊Q₈×C₄₇	—	2	2	no	—
72	#121×C₄₇	—	4	1	no	—
73	C₄×D₄×C₄₇	—	2	1	no	—
74	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
78	#103×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	#41×C₄₇	—	4	4	no	—
83	#41×C₄₇	—	8	2	no	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	1	no	—
86	C₂×D₄×C₄₇	—	4	2	no	—
87	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	—	4	1	no	—
89	C₂²×C₄×C₄₇	—	4	2	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×Dic₉₄	—	2	1	no	—
92	C₂×Dic₉₄	—	4	1	no	—
93	Q₁₆×C₄₇	—	2	2	no	—
94	Q₁₆×C₄₇	—	2	2	no	—
95	Q₁₆×C₄₇	—	8	1	no	—
96	SD₁₆×C₄₇	—	4	4	no	—
97	D₈×C₄₇	—	2	1	no	—
98	D₈×C₄₇	#41	1	2	yes	—
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	—	4	1	no	—
106	C₄⋊C₄×C₄₇	—	4	2	no	—
107	C₄⋊C₄×C₄₇	—	4	1	no	—
108	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₄₇	—	4	2	no	—
114	C₂²⋊Dic₄₇	—	4	2	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	4	1	no	—
118	Dic₄₇⋊C₄	—	4	2	no	—
119	Dic₄₇⋊C₄	—	4	1	no	—
120	C₄×Dic₄₇	—	4	2	no	—
121	C₄×Dic₄₇	—	4	1	no	—
122	Dic₁₈₈	—	2	1	no	—
123	Dic₁₈₈	—	2	1	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	Q₈×C₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	4	2	no	—
131	D₄×C₄₇	—	2	2	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	4	1	no	—
134	D₄×C₄₇	—	8	4	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	4	no	—
140	C₂×C₄×C₄₇	—	8	2	no	—
141	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₄₇	—	4	1	no	—
146	C₂×Dic₄₇	—	4	1	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	Dic₉₄	—	4	1	no	—
149	Dic₉₄	—	4	1	no	—
150	Dic₉₄	—	8	2	no	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
154	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
155	#33869	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	—	2	2	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	4	2	no	—
160	C₂²×C₄₇	—	8	1	no	—
161	C₂²×C₄₇	—	8	1	no	—
162	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	8	1	no	—
166	C₄×C₄₇	—	8	2	no	—
167	C₄×C₄₇	#508	1	2	yes	—
168	C₄×C₄₇	#546	1	1	yes	characteristic
169	C₄×C₄₇	#577	1	1	yes	characteristic
170	Dic₄₇	—	8	2	no	—
171	Dic₄₇	—	8	1	no	—
172	Dic₄₇	—	8	1	no	—
173	#3342	—	47	1	no	—
174	#3283	D₄₇	1	1	yes	characteristic
175	#2241	—	47	1	no	—
176	#2237	—	47	2	no	—
177	#2224	—	47	1	no	—
178	#2216	—	47	2	no	—
179	#2209	—	47	1	no	—
180	#2189	—	47	2	no	—
181	C₄×SD₃₂	—	47	1	no	—
182	D₄×C₁₆	—	47	1	no	—
183	#2138	—	47	2	no	—
184	C₂×C₄₇	—	4	2	no	—
185	C₂×C₄₇	—	8	2	no	—
186	C₂×C₄₇	#3294	1	1	yes	characteristic
187	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
188	C₂×C₄₇	#3417	1	1	yes	characteristic
189	C₂×#135	C₂×D₄₇	1	2	yes	—
190	#556	—	94	1	no	—
191	#547	—	2	1	no	—
192	C₂×Q₃₂	—	47	2	no	—
193	C₂×SD₃₂	—	47	1	no	—
194	C₂²×C₁₆	—	47	2	no	—
195	#500	—	47	1	no	—
196	#494	C₂×D₄₇	1	1	yes	characteristic
197	#485	—	94	2	no	—
198	#475	—	94	1	no	—
199	#467	—	47	2	no	—
200	#467	C₂×D₄₇	1	2	yes	—
201	#461	—	2	1	no	—
202	#452	—	2	2	no	—
203	C₄×Q₁₆	—	47	1	no	—
204	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
205	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
206	#427	—	47	2	no	—
207	#367	—	47	1	no	—
208	#366	—	47	2	no	—
209	C₄⋊C₁₆	—	47	1	no	—
210	#358	—	47	2	no	—
211	#358	—	47	2	no	—
212	#358	—	47	2	no	—
213	#358	—	47	1	no	—
214	#357	—	47	1	no	—
215	#357	—	47	2	no	—
216	C₂²⋊C₁₆	—	47	2	no	—
217	C₄×C₁₆	—	47	1	no	—
218	C₄₇	#33869	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	2	2	no	—
220	#135	—	2	4	no	—
221	#135	—	4	2	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	—	47	1	no	—
224	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	94	1	no	—
230	#126	—	188	2	no	—
231	#124	—	188	1	no	—
232	#123	—	4	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	94	2	no	—
235	#121	—	4	1	no	—
236	C₄×Q₈	—	94	1	no	—
237	C₄×Q₈	—	188	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
240	#117	—	94	2	no	—
241	C₂×C₄⋊C₄	—	4	1	no	—
242	Q₃₂	—	188	2	no	—
243	SD₃₂	—	94	2	no	—
244	C₂×C₁₆	—	47	1	no	—
245	C₂×C₁₆	—	47	1	no	—
246	C₂×C₁₆	—	47	2	no	—
247	C₂×C₁₆	—	188	2	no	—
248	#107	—	47	2	no	—
249	#107	—	47	1	no	—
250	#107	—	94	1	no	—
251	#107	C₂²×D₄₇	1	1	yes	characteristic
252	#106	—	94	2	no	—
253	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
254	#103	—	2	2	no	—
255	#103	—	94	1	no	—
256	#103	—	94	2	no	—
257	#102	—	2	1	no	—
258	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
259	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
260	#41	—	4	4	no	—
261	#41	—	8	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	94	1	no	—
264	C₂×Q₈	—	188	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	#10625334	1	1	yes	characteristic
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	8	1	no	—
274	Q₁₆	—	94	1	no	—
275	Q₁₆	—	94	1	no	—
276	SD₁₆	—	4	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#10625352	1	2	yes	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
283	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	94	1	no	—
289	C₄⋊C₄	—	94	2	no	—
290	C₄⋊C₄	—	188	1	no	—
291	C₄⋊C₄	—	188	2	no	—
292	C₄⋊C₄	—	188	1	no	—
293	C₄⋊C₄	#10625334	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	188	2	no	—
297	C₂²⋊C₄	—	188	2	no	—
298	C₂²⋊C₄	#10625334	1	2	yes	—
299	C₄²	—	188	2	no	—
300	C₄²	—	188	1	no	—
301	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
302	C₁₆	—	94	1	no	—
303	C₁₆	—	94	1	no	—
304	C₁₆	—	94	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₂×#10625334	1	2	yes	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	376	2	no	—
310	Q₈	—	188	1	no	—
311	Q₈	—	188	1	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	188	1	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	—	188	1	no	—
325	C₂×C₄	C₂×#10625334	1	2	yes	—
326	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
327	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	C₂×#10625352	1	1	yes	characteristic
332	C₈	#13094069	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×#10625334	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	376	1	no	—
345	C₄	—	376	1	no	—
346	C₄	—	376	2	no	—
347	C₄	#15689553	1	2	yes	—
348	C₄	#15689767	1	1	yes	characteristic
349	C₄	#15689934	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	8	2	no	—
352	C₂	#20255201	1	1	yes	characteristic
353	C₂	C₂×#15689553	1	1	yes	characteristic
354	C₂	#20256539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3283	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3283	D₄₇
hypocenter	C₄₇	#33869
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. #20255819	C₂	semidirect
2. #20245615	C₂	semidirect
3. #20245593	C₂	semidirect
4. #20245525	C₂	semidirect
5. #20245477	C₂	semidirect
6. #20245435	C₂	semidirect
7. #20245328	C₂	semidirect
8. C₄×#13093968	C₂	semidirect
9. D₄×C₄₇⋊C₁₆	C₂	semidirect
10. #20245095	C₂	semidirect
218. C₄₇	#33869	semidirect

Maximal subgroups

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

subgroup	index
#20255819	2
#20245615	2
#20245593	2
#20245525	2
#20245477	2
#20245435	2
#20245328	2
C₄×#13093968	2
D₄×C₄₇⋊C₁₆	2
#20245095	2
C₄₇×#3283	2
#33869	47