Subgroups — #26373240

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	#20255812	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255147	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245612	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245581	C₂	1	2	yes	maximal, complemented
5	#20245540	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	#20245310	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245094	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094047	C₂²	1	2	yes	—
13	#15689838	—	2	1	no	—
14	#15689795	—	2	1	no	—
15	C₂×#13093965	C₂²	1	2	yes	—
16	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
17	#15689528	C₂²	1	1	yes	characteristic
18	#15689500	C₂²	1	1	yes	characteristic
19	#15689445	—	2	2	no	—
20	#15689413	—	2	1	no	—
21	#15689360	C₂²	1	2	yes	—
22	#15689356	C₂²	1	2	yes	—
23	#15689323	—	2	1	no	—
24	#15689271	—	2	2	no	—
25	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
26	#15689160	C₂²	1	2	yes	—
27	#15688948	C₂²	1	2	yes	—
28	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
29	#15688927	C₂²	1	2	yes	—
30	#15688925	C₂²	1	2	yes	—
31	#15688925	C₂²	1	2	yes	—
32	#15688923	C₂²	1	2	yes	—
33	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₄₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#367	C₂²	1	1	yes	characteristic
38	C₄₇×#358	C₂²	1	1	yes	characteristic
39	C₄₇×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625354	—	2	2	no	—
42	#13094047	—	2	4	no	—
43	#13094047	—	4	2	no	—
44	C₂×#10625336	—	2	2	no	—
45	C₂×D₃₇₆	—	2	1	no	—
46	C₂×D₃₇₆	C₂³	1	2	yes	—
47	C₂×C₈×D₄₇	C₂³	1	2	yes	—
48	#13094024	—	2	1	no	—
49	#13094021	—	2	1	no	—
50	#13094021	—	4	1	no	—
51	#13094010	—	4	2	no	—
52	#13093999	—	2	2	no	—
53	#13093999	—	4	2	no	—
54	#13093998	—	4	1	no	—
55	#13093995	—	2	2	no	—
56	C₄×D₁₈₈	—	4	1	no	—
57	C₄×D₁₈₈	D₄	1	1	yes	characteristic
58	#13093983	—	4	1	no	—
59	#13093982	—	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	#13093955	—	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×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₄₇	—	2	1	no	—
73	C₄×D₄×C₄₇	—	2	1	no	—
74	SD₃₂×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	#103×C₄₇	—	2	1	no	—
80	#102×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	2	1	no	—
83	C₂×D₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	4	1	no	—
85	C₂×C₂²⋊D₄₇	—	4	2	no	—
86	#10625354	—	4	4	no	—
87	#10625354	—	8	2	no	—
88	C₂×D₁₈₈	—	2	2	no	—
89	C₂×D₁₈₈	—	4	2	no	—
90	C₂×D₁₈₈	—	4	1	no	—
91	C₂×D₁₈₈	—	4	2	no	—
92	C₂×D₁₈₈	—	4	1	no	—
93	C₂×D₁₈₈	D₈	1	1	yes	characteristic
94	C₂×C₄×D₄₇	—	4	2	no	—
95	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
96	C₂×Dic₉₄	—	4	1	no	—
97	Q₁₆×C₄₇	—	2	1	no	—
98	Q₁₆×C₄₇	#41	1	2	yes	—
99	D₈×C₄₇	—	2	1	no	—
100	D₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	1	no	—
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₄⋊C₄×C₄₇	—	4	1	no	—
106	C₂²⋊C₄×C₄₇	—	4	1	no	—
107	C₄²×C₄₇	—	4	1	no	—
108	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂²⋊Dic₄₇	—	4	2	no	—
110	#10625336	—	4	4	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
117	C₄×Dic₄₇	—	4	2	no	—
118	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
119	D₃₇₆	—	2	2	no	—
120	D₃₇₆	—	2	2	no	—
121	D₃₇₆	—	8	1	no	—
122	C₈×D₄₇	—	2	2	no	—
123	C₈×D₄₇	—	2	2	no	—
124	C₁₆×C₄₇	—	2	1	no	—
125	C₁₆×C₄₇	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₇	—	4	1	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	C₂²×D₄₇	C₂×D₈	1	2	yes	—
130	Q₈×C₄₇	—	2	2	no	—
131	Q₈×C₄₇	—	4	1	no	—
132	D₄×C₄₇	—	4	1	no	—
133	D₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	—	8	2	no	—
138	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₄₇	—	16	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
144	D₁₈₈	—	2	2	no	—
145	D₁₈₈	—	4	2	no	—
146	D₁₈₈	—	4	2	no	—
147	D₁₈₈	—	8	4	no	—
148	D₁₈₈	—	8	2	no	—
149	D₁₈₈	—	16	1	no	—
150	D₁₈₈	—	16	2	no	—
151	C₄×D₄₇	—	2	2	no	—
152	C₄×D₄₇	—	2	2	no	—
153	C₄×D₄₇	—	8	4	no	—
154	Dic₉₄	—	16	1	no	—
155	C₈×C₄₇	—	2	1	no	—
156	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
157	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
158	C₄₇⋊C₈	—	2	2	no	—
159	#33872	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	8	1	no	—
161	C₂²×C₄₇	—	8	1	no	—
162	C₂²×C₄₇	—	8	1	no	—
163	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₄₇	—	2	2	no	—
165	C₂×D₄₇	—	2	2	no	—
166	C₂×D₄₇	—	2	2	no	—
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	—	4	2	no	—
172	C₄×C₄₇	—	8	1	no	—
173	C₄×C₄₇	—	8	1	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	—	2	2	no	—
178	Dic₄₇	—	8	2	no	—
179	#3341	—	47	1	no	—
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	D₄₇	1	1	yes	characteristic, complemented
188	D₄×C₁₆	—	47	1	no	—
189	#2138	—	47	2	no	—
190	C₂×C₄₇	—	16	1	no	—
191	C₂×C₄₇	#3294	1	1	yes	characteristic
192	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
193	C₂×C₄₇	#3415	1	1	yes	characteristic
194	D₄₇	—	4	2	no	—
195	D₄₇	—	16	2	no	—
196	C₂×#135	—	47	2	no	—
197	#555	—	94	1	no	—
198	#550	—	94	1	no	—
199	C₂×SD₃₂	C₂×D₄₇	1	1	yes	characteristic
200	C₂×D₁₆	—	47	2	no	—
201	C₂²×C₁₆	—	47	2	no	—
202	#500	—	47	1	no	—
203	#493	—	47	1	no	—
204	#482	—	94	2	no	—
205	#476	—	94	1	no	—
206	#466	—	47	2	no	—
207	#466	—	47	2	no	—
208	#460	—	94	1	no	—
209	#449	—	94	2	no	—
210	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
211	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
212	C₈×D₄	—	47	1	no	—
213	#427	—	47	2	no	—
214	#367	C₂×D₄₇	1	1	yes	characteristic
215	#366	—	47	2	no	—
216	C₄⋊C₁₆	—	47	1	no	—
217	#358	—	47	2	no	—
218	#358	C₂×D₄₇	1	1	yes	characteristic
219	#357	—	47	2	no	—
220	#357	—	47	2	no	—
221	#357	—	47	2	no	—
222	#357	C₂×D₄₇	1	1	yes	characteristic
223	C₂²⋊C₁₆	—	47	2	no	—
224	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
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	C₂²×D₄₇	1	1	yes	characteristic
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	—	47	2	no	—
235	C₄⋊Q₈	—	94	1	no	—
236	#127	—	94	1	no	—
237	#127	—	188	1	no	—
238	#124	—	188	2	no	—
239	#121	—	94	2	no	—
240	#121	—	94	2	no	—
241	#121	—	188	2	no	—
242	#121	—	188	1	no	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	47	1	no	—
246	C₄×D₄	—	188	1	no	—
247	#117	—	94	2	no	—
248	#117	—	188	1	no	—
249	SD₃₂	—	2	2	no	—
250	D₁₆	—	188	2	no	—
251	C₂×C₁₆	—	47	2	no	—
252	C₂×C₁₆	—	188	2	no	—
253	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
254	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
255	#107	—	47	2	no	—
256	#107	—	94	1	no	—
257	#107	C₂²×D₄₇	1	1	yes	characteristic
258	#107	C₂²×D₄₇	1	1	yes	characteristic
259	#106	—	94	2	no	—
260	C₄⋊C₈	—	47	1	no	—
261	#103	—	2	1	no	—
262	#102	—	2	1	no	—
263	#102	—	94	2	no	—
264	#102	—	94	2	no	—
265	C₂²⋊C₈	—	47	2	no	—
266	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
267	#41	—	376	2	no	—
268	#41	—	188	4	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	188	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	47	1	no	—
273	C₂×D₄	—	94	2	no	—
274	C₂×D₄	—	188	2	no	—
275	C₂×D₄	—	188	2	no	—
276	C₂×D₄	—	188	2	no	—
277	C₂×D₄	—	188	1	no	—
278	C₂×D₄	—	188	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	47	2	no	—
281	C₂²×C₄	—	188	2	no	—
282	Q₁₆	—	2	1	no	—
283	Q₁₆	#10625350	1	2	yes	—
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	—	47	2	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
294	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	47	1	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	188	1	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	47	2	no	—
303	C₂²⋊C₄	—	188	2	no	—
304	C₂²⋊C₄	—	188	2	no	—
305	C₂²⋊C₄	—	188	2	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	188	2	no	—
308	C₄²	D₄×D₄₇	1	1	yes	characteristic
309	C₁₆	—	2	1	no	—
310	C₁₆	—	2	1	no	—
311	C₁₆	—	94	2	no	—
312	C₂³	—	4	1	no	—
313	C₂³	—	47	2	no	—
314	C₂³	—	188	2	no	—
315	Q₈	—	2	2	no	—
316	Q₈	—	4	1	no	—
317	Q₈	—	752	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	94	2	no	—
321	D₄	—	376	4	no	—
322	D₄	—	376	2	no	—
323	D₄	—	188	2	no	—
324	D₄	—	188	2	no	—
325	D₄	—	752	1	no	—
326	D₄	—	752	2	no	—
327	D₄	—	752	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	47	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	376	4	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
338	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	94	2	no	—
342	C₈	C₂×#10625350	1	1	yes	characteristic
343	C₈	#13094070	1	1	yes	characteristic
344	C₂²	—	8	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	94	2	no	—
348	C₂²	—	94	2	no	—
349	C₂²	—	94	2	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	376	2	no	—
352	C₂²	—	376	2	no	—
353	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	8	1	no	—
357	C₄	—	8	1	no	—
358	C₄	—	94	2	no	—
359	C₄	—	376	2	no	—
360	C₄	#15689554	1	2	yes	—
361	C₄	#15689769	1	1	yes	characteristic
362	C₄	#15689930	1	1	yes	characteristic
363	C₂	—	16	1	no	—
364	C₂	—	188	2	no	—
365	C₂	—	752	2	no	—
366	C₂	#20255203	1	1	yes	characteristic
367	C₂	C₂×#15689554	1	1	yes	characteristic
368	C₂	#20256526	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×SD₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₄×SD₃₂	D₄₇
hypocenter	C₄₇	#33872
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20255812	C₂	semidirect
2. #20255147	C₂	semidirect
3. #20245612	C₂	semidirect
4. #20245581	C₂	semidirect
5. #20245540	C₂	semidirect
6. #20245473	C₂	semidirect
7. #20245423	C₂	semidirect
8. #20245310	C₂	semidirect
9. C₁₆×D₁₈₈	C₂	semidirect
10. #20245094	C₂	semidirect
11. C₄×SD₃₂×C₄₇	C₂	semidirect
187. C₄×SD₃₂	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
#20255812	2
#20255147	2
#20245612	2
#20245581	2
#20245540	2
#20245473	2
#20245423	2
#20245310	2
C₁₆×D₁₈₈	2
#20245094	2
C₄×SD₃₂×C₄₇	2
#33872	47