Subgroups — #26373088

Order 12032.

636 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256540	C₂	1	2	yes	maximal, complemented
2	#20255193	C₂	1	1	yes	maximal, characteristic, complemented
3	#20255192	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245640	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245512	C₂	1	2	yes	maximal, complemented
6	#20245470	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245369	C₂	1	2	yes	maximal, complemented
8	#20245363	C₂	1	2	yes	maximal, complemented
9	#20245267	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245224	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2214	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689940	—	2	1	no	—
13	#15689937	C₂²	1	2	yes	—
14	#15689936	—	2	1	no	—
15	#15689936	C₂²	1	2	yes	—
16	#15689566	C₂²	1	2	yes	—
17	#15689565	—	2	2	no	—
18	#15689562	—	2	2	no	—
19	#15689554	—	2	2	no	—
20	#15689553	C₂²	1	1	yes	characteristic
21	C₂×#13093970	C₂²	1	2	yes	—
22	C₂×#13093970	C₂²	1	2	yes	—
23	Q₃₂×D₄₇	—	2	2	no	—
24	#15689541	C₂²	1	2	yes	—
25	#15689503	C₂²	1	1	yes	characteristic
26	#15689399	C₂²	1	2	yes	—
27	#15689398	—	2	1	no	—
28	#15689395	—	2	1	no	—
29	#15689395	C₂²	1	2	yes	—
30	#15689307	—	2	2	no	—
31	#15689305	—	2	2	no	—
32	#15689241	C₂²	1	1	yes	characteristic
33	#15689241	C₂²	1	1	yes	characteristic
34	#15688952	C₂²	1	1	yes	characteristic
35	#15688940	C₂²	1	2	yes	—
36	#15688930	C₂²	1	1	yes	characteristic
37	#15688929	C₂²	1	1	yes	characteristic
38	#15688908	C₂²	1	2	yes	—
39	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#510	C₂²	1	2	yes	—
41	C₄₇×#494	C₂²	1	1	yes	characteristic
42	C₄₇×#443	C₂²	1	1	yes	characteristic
43	C₄₇×#364	C₂²	1	1	yes	characteristic
44	C₄₇×#362	C₂²	1	2	yes	—
45	#13094072	—	2	2	no	—
46	#13094071	—	2	2	no	—
47	#13094057	—	2	2	no	—
48	#13094057	—	4	2	no	—
49	#13094056	—	4	2	no	—
50	#13094056	—	4	2	no	—
51	#13094055	—	2	2	no	—
52	#13094055	—	2	2	no	—
53	#13094054	—	2	2	no	—
54	#13094053	—	4	2	no	—
55	#13094047	—	2	2	no	—
56	#13094047	—	4	1	no	—
57	#13094046	—	4	1	no	—
58	#13094046	D₄	1	1	yes	characteristic
59	#13094046	C₂³	1	1	yes	characteristic
60	#13094043	—	2	2	no	—
61	C₂×Dic₁₈₈	—	2	1	no	—
62	C₂×Dic₁₈₈	—	4	1	no	—
63	C₂×Dic₁₈₈	C₂³	1	2	yes	—
64	Q₁₆×D₄₇	—	2	2	no	—
65	Q₁₆×D₄₇	—	2	2	no	—
66	Q₁₆×D₄₇	—	4	1	no	—
67	#13094030	—	2	1	no	—
68	#13094030	C₂³	1	2	yes	—
69	#13094024	—	2	1	no	—
70	Dic₃₇₆	—	2	2	no	—
71	#13093970	—	2	2	no	—
72	#13093970	—	2	2	no	—
73	#13093970	—	4	2	no	—
74	#13093970	D₄	1	1	yes	characteristic
75	#13093969	—	2	2	no	—
76	#13093968	D₄	1	1	yes	characteristic
77	#13093967	D₄	1	1	yes	characteristic
78	#13093965	D₄	1	1	yes	characteristic
79	#13093963	—	2	2	no	—
80	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
82	C₁₆×D₄₇	—	2	2	no	—
83	#13093960	D₄	1	1	yes	characteristic
84	#13093960	C₂³	1	1	yes	characteristic
85	#13093952	—	2	1	no	—
86	#13093952	D₄	1	2	yes	—
87	#13093943	—	2	2	no	—
88	#13093942	—	2	2	no	—
89	#135×C₄₇	C₂³	1	1	yes	characteristic
90	C₂×Q₁₆×C₄₇	—	2	1	no	—
91	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
92	#131×C₄₇	—	2	1	no	—
93	C₄⋊Q₈×C₄₇	—	2	1	no	—
94	Q₃₂×C₄₇	—	2	2	no	—
95	SD₃₂×C₄₇	—	2	2	no	—
96	M₃₂×C₄₇	C₂³	1	2	yes	—
97	#108×C₄₇	C₂³	1	1	yes	characteristic
98	#104×C₄₇	—	2	1	no	—
99	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
100	#41×C₄₇	—	2	1	no	—
101	C₂×Q₈×C₄₇	—	2	2	no	—
102	#10625354	—	4	2	no	—
103	#10625354	—	4	2	no	—
104	Q₈×D₄₇	—	4	2	no	—
105	Q₈×D₄₇	—	4	2	no	—
106	Q₈×D₄₇	—	4	2	no	—
107	#10625352	—	4	2	no	—
108	#10625352	—	8	2	no	—
109	#10625350	—	2	1	no	—
110	#10625350	—	4	2	no	—
111	#10625350	—	8	2	no	—
112	#10625350	C₂×D₄	1	2	yes	—
113	C₂×Dic₉₄	—	2	2	no	—
114	C₂×Dic₉₄	—	4	2	no	—
115	Q₁₆×C₄₇	—	2	2	no	—
116	Q₁₆×C₄₇	—	2	2	no	—
117	Q₁₆×C₄₇	—	4	1	no	—
118	Q₁₆×C₄₇	—	4	1	no	—
119	Q₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
120	SD₁₆×C₄₇	—	2	1	no	—
121	D₈×C₄₇	C₂×D₄	1	1	yes	characteristic
122	M₁₆×C₄₇	—	2	1	no	—
123	M₁₆×C₄₇	—	4	1	no	—
124	C₂×C₈×C₄₇	—	4	1	no	—
125	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₄₇	—	8	1	no	—
128	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
129	#10625337	—	4	2	no	—
130	#10625337	—	4	2	no	—
131	#10625337	—	4	2	no	—
132	#10625336	—	4	2	no	—
133	#10625336	—	4	2	no	—
134	#10625335	—	4	2	no	—
135	C₄⋊Dic₄₇	—	8	1	no	—
136	#10625329	—	2	1	no	—
137	#10625329	C₂×D₄	1	2	yes	—
138	Dic₁₈₈	—	2	2	no	—
139	Dic₁₈₈	—	2	2	no	—
140	Dic₁₈₈	—	4	2	no	—
141	Dic₁₈₈	—	4	1	no	—
142	Dic₁₈₈	—	4	1	no	—
143	Dic₁₈₈	D₈	1	1	yes	characteristic
144	Dic₁₈₈	C₂×D₄	1	1	yes	characteristic
145	D₃₇₆	D₈	1	1	yes	characteristic
146	D₃₇₆	C₂×D₄	1	1	yes	characteristic
147	#10625325	—	2	1	no	—
148	#10625325	—	4	2	no	—
149	#10625325	D₈	1	2	yes	—
150	#10625324	—	2	2	no	—
151	#10625324	—	4	1	no	—
152	C₈×D₄₇	—	2	2	no	—
153	C₈×D₄₇	—	4	1	no	—
154	C₁₆×C₄₇	—	2	2	no	—
155	C₄₇⋊C₁₆	—	2	1	no	—
156	C₄₇⋊C₁₆	C₂×D₄	1	1	yes	characteristic
157	C₄₇⋊C₁₆	C₂×D₄	1	1	yes	characteristic
158	Q₈×C₄₇	—	2	1	no	—
159	Q₈×C₄₇	—	4	2	no	—
160	Q₈×C₄₇	—	4	2	no	—
161	D₄×C₄₇	—	2	1	no	—
162	D₄×C₄₇	—	4	1	no	—
163	C₂×C₄×C₄₇	—	2	1	no	—
164	C₂×C₄×C₄₇	—	4	1	no	—
165	C₂×C₄×C₄₇	—	4	2	no	—
166	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
167	C₂²⋊D₄₇	—	2	2	no	—
168	C₂²⋊D₄₇	—	4	1	no	—
169	C₂²⋊D₄₇	—	8	2	no	—
170	C₂×Dic₄₇	—	4	2	no	—
171	C₂×Dic₄₇	—	8	2	no	—
172	D₁₈₈	—	2	1	no	—
173	D₁₈₈	—	8	2	no	—
174	D₁₈₈	C₂×D₈	1	2	yes	—
175	C₄×D₄₇	—	2	2	no	—
176	C₄×D₄₇	—	4	1	no	—
177	C₄×D₄₇	—	8	2	no	—
178	C₄×D₄₇	—	8	2	no	—
179	C₄×D₄₇	—	8	2	no	—
180	Dic₉₄	—	2	1	no	—
181	Dic₉₄	—	4	2	no	—
182	Dic₉₄	—	4	2	no	—
183	Dic₉₄	—	8	2	no	—
184	Dic₉₄	—	8	2	no	—
185	Dic₉₄	C₂×D₈	1	2	yes	—
186	C₈×C₄₇	—	4	1	no	—
187	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
188	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
189	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
190	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
191	C₄₇⋊C₈	—	2	2	no	—
192	C₄₇⋊C₈	—	4	1	no	—
193	#33860	—	47	1	no	maximal, Sylow
194	C₂²×C₄₇	—	4	1	no	—
195	C₂²×C₄₇	#576	1	1	yes	characteristic
196	C₂×D₄₇	—	2	2	no	—
197	C₂×D₄₇	—	4	1	no	—
198	C₄×C₄₇	—	2	2	no	—
199	C₄×C₄₇	—	4	1	no	—
200	C₄×C₄₇	—	8	2	no	—
201	C₄×C₄₇	D₄²	1	1	yes	characteristic
202	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
203	Dic₄₇	—	2	2	no	—
204	Dic₄₇	—	4	1	no	—
205	Dic₄₇	—	8	2	no	—
206	#3417	—	47	2	no	—
207	#3293	—	47	1	no	—
208	#3293	—	47	1	no	—
209	#2247	—	47	1	no	—
210	#2222	—	47	2	no	—
211	#2214	—	47	1	no	—
212	#2214	D₄₇	1	1	yes	characteristic, complemented
213	#2195	—	47	2	no	—
214	#2195	—	47	2	no	—
215	#2178	—	47	1	no	—
216	#2170	—	47	1	no	—
217	C₂×C₄₇	—	2	1	no	—
218	C₂×C₄₇	—	8	1	no	—
219	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
220	D₄₇	—	4	2	no	—
221	D₄₇	—	8	1	no	—
222	#578	—	47	2	no	—
223	#578	—	47	2	no	—
224	#578	—	94	1	no	—
225	#578	—	94	1	no	—
226	#510	—	47	2	no	—
227	#510	—	94	2	no	—
228	#510	—	94	2	no	—
229	#510	C₂×D₄₇	1	2	yes	—
230	#508	—	47	1	no	—
231	#508	—	94	2	no	—
232	C₂×Q₃₂	—	47	2	no	—
233	C₂×Q₃₂	—	47	2	no	—
234	C₂×Q₃₂	—	94	2	no	—
235	#504	—	47	2	no	—
236	#494	—	47	1	no	—
237	#494	C₂×D₄₇	1	1	yes	characteristic
238	#473	—	47	2	no	—
239	#473	—	47	2	no	—
240	#473	—	94	1	no	—
241	#473	—	94	1	no	—
242	#456	—	94	2	no	—
243	#456	—	94	2	no	—
244	#443	—	47	1	no	—
245	#443	—	47	1	no	—
246	#443	C₂×D₄₇	1	1	yes	characteristic
247	#368	—	47	1	no	—
248	#364	C₂×D₄₇	1	1	yes	characteristic
249	#362	—	47	2	no	—
250	#362	C₂×D₄₇	1	2	yes	—
251	#359	—	47	1	no	—
252	#359	—	47	1	no	—
253	#350	—	47	2	no	—
254	C₄×C₁₆	—	47	1	no	—
255	C₄₇	#33860	1	1	yes	characteristic, complemented, Sylow
256	2₋¹⁺⁴	—	94	2	no	—
257	2₋¹⁺⁴	—	94	2	no	—
258	#137	—	94	2	no	—
259	#137	—	94	2	no	—
260	#137	—	94	2	no	—
261	#137	—	94	2	no	—
262	#137	—	188	2	no	—
263	#137	—	188	2	no	—
264	#137	—	188	2	no	—
265	#137	—	188	2	no	—
266	#135	—	47	1	no	—
267	#135	—	47	1	no	—
268	#135	—	94	2	no	—
269	#135	—	94	2	no	—
270	#135	—	188	1	no	—
271	#135	—	188	1	no	—
272	#135	C₂²×D₄₇	1	1	yes	characteristic
273	C₂×Q₁₆	—	2	1	no	—
274	C₂×Q₁₆	—	47	2	no	—
275	C₂×Q₁₆	—	94	2	no	—
276	C₂×Q₁₆	—	94	2	no	—
277	C₂×Q₁₆	—	94	1	no	—
278	C₂×Q₁₆	—	188	1	no	—
279	C₂×Q₁₆	—	188	1	no	—
280	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
281	#131	—	2	1	no	—
282	#131	—	47	2	no	—
283	#131	—	94	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	94	1	no	—
286	Q₃₂	—	2	2	no	—
287	Q₃₂	—	47	1	no	—
288	Q₃₂	—	94	2	no	—
289	Q₃₂	—	94	2	no	—
290	Q₃₂	—	94	2	no	—
291	Q₃₂	—	188	2	no	—
292	SD₃₂	—	2	2	no	—
293	SD₃₂	—	47	1	no	—
294	SD₃₂	—	47	1	no	—
295	SD₃₂	—	94	2	no	—
296	D₁₆	—	47	1	no	—
297	M₃₂	—	94	2	no	—
298	M₃₂	C₂²×D₄₇	1	2	yes	—
299	C₂×C₁₆	—	47	1	no	—
300	C₂×C₁₆	—	47	1	no	—
301	C₂×C₁₆	—	94	2	no	—
302	#108	—	47	1	no	—
303	#108	—	47	1	no	—
304	#108	C₂²×D₄₇	1	1	yes	characteristic
305	#104	—	2	1	no	—
306	#104	—	47	2	no	—
307	#104	—	94	1	no	—
308	#101	—	94	2	no	—
309	#101	—	94	2	no	—
310	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
311	#41	—	2	1	no	—
312	#41	—	47	2	no	—
313	#41	—	94	1	no	—
314	#41	—	376	2	no	—
315	#41	—	376	2	no	—
316	#41	—	188	2	no	—
317	#41	—	188	2	no	—
318	#41	—	188	2	no	—
319	#41	—	188	2	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	94	2	no	—
322	C₂×Q₈	—	188	2	no	—
323	C₂×Q₈	—	188	2	no	—
324	C₂×Q₈	—	188	2	no	—
325	C₂×Q₈	—	188	2	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	1	no	—
329	Q₁₆	—	4	1	no	—
330	Q₁₆	—	47	1	no	—
331	Q₁₆	—	47	1	no	—
332	Q₁₆	—	94	2	no	—
333	Q₁₆	—	94	2	no	—
334	Q₁₆	—	188	2	no	—
335	Q₁₆	—	188	2	no	—
336	Q₁₆	—	188	2	no	—
337	Q₁₆	—	188	2	no	—
338	Q₁₆	—	188	1	no	—
339	Q₁₆	—	188	1	no	—
340	Q₁₆	D₄×D₄₇	1	1	yes	characteristic
341	SD₁₆	—	2	1	no	—
342	SD₁₆	—	47	2	no	—
343	SD₁₆	—	94	1	no	—
344	SD₁₆	—	188	2	no	—
345	SD₁₆	—	188	2	no	—
346	SD₁₆	—	188	2	no	—
347	SD₁₆	—	188	2	no	—
348	D₈	—	47	1	no	—
349	D₈	—	47	1	no	—
350	D₈	D₄×D₄₇	1	1	yes	characteristic
351	M₁₆	—	2	1	no	—
352	M₁₆	—	4	1	no	—
353	M₁₆	—	47	2	no	—
354	M₁₆	—	94	2	no	—
355	M₁₆	—	94	1	no	—
356	M₁₆	—	188	1	no	—
357	C₂×C₈	—	4	1	no	—
358	C₂×C₈	—	94	2	no	—
359	C₂×C₈	—	188	1	no	—
360	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
361	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
362	C₄⋊C₄	—	8	1	no	—
363	C₄⋊C₄	—	376	1	no	—
364	C₄²	D₄×D₄₇	1	1	yes	characteristic
365	C₁₆	—	2	2	no	—
366	C₁₆	—	47	1	no	—
367	C₁₆	—	47	1	no	—
368	C₁₆	—	94	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	47	2	no	—
373	Q₈	—	94	1	no	—
374	Q₈	—	376	2	no	—
375	Q₈	—	376	2	no	—
376	Q₈	—	188	2	no	—
377	Q₈	—	188	2	no	—
378	D₄	—	2	1	no	—
379	D₄	—	4	1	no	—
380	D₄	—	47	2	no	—
381	D₄	—	94	2	no	—
382	D₄	—	94	1	no	—
383	D₄	—	376	2	no	—
384	D₄	—	376	2	no	—
385	D₄	—	188	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	94	2	no	—
390	C₂×C₄	—	376	2	no	—
391	C₂×C₄	—	376	2	no	—
392	C₂×C₄	—	376	2	no	—
393	C₂×C₄	—	376	2	no	—
394	C₂×C₄	—	188	2	no	—
395	C₂×C₄	—	188	1	no	—
396	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
397	C₈	—	4	1	no	—
398	C₈	—	94	2	no	—
399	C₈	—	188	1	no	—
400	C₈	D₈×D₄₇	1	1	yes	characteristic
401	C₈	D₈×D₄₇	1	1	yes	characteristic
402	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
403	C₈	#13094070	1	1	yes	characteristic
404	C₂²	—	4	1	no	—
405	C₂²	—	94	2	no	—
406	C₂²	—	188	1	no	—
407	C₂²	#15689927	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	4	1	no	—
410	C₄	—	8	2	no	—
411	C₄	—	94	2	no	—
412	C₄	—	376	2	no	—
413	C₄	—	188	1	no	—
414	C₄	#15689757	1	1	yes	characteristic
415	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	8	1	no	—
418	C₂	—	376	1	no	—
419	C₂	—	188	2	no	—
420	C₂	#20255060	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20255060
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2214	C₂
socle	C₂×C₄₇	D₄×D₈
hypercenter	#2214	D₄₇
hypocenter	C₄₇	#33860
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2214

Complements

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

N	H ≅ G/N	type
1. #20256540	C₂	semidirect
2. #20255193	C₂	semidirect
3. #20255192	C₂	semidirect
4. #20245640	C₂	semidirect
5. #20245512	C₂	semidirect
6. #20245470	C₂	semidirect
7. #20245369	C₂	semidirect
8. #20245363	C₂	semidirect
9. #20245267	C₂	semidirect
10. #20245224	C₂	semidirect
11. C₄₇×#2214	C₂	semidirect
212. #2214	D₄₇	semidirect
255. C₄₇	#33860	semidirect

Maximal subgroups

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

subgroup	index
#20256540	2
#20255193	2
#20255192	2
#20245640	2
#20245512	2
#20245470	2
#20245369	2
#20245363	2
#20245267	2
#20245224	2
C₄₇×#2214	2
#33860	47