Subgroups — #26373083

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256538	C₂	1	1	yes	maximal, characteristic, complemented
2	#20256526	C₂	1	1	yes	maximal, characteristic, complemented
3	#20255176	C₂	1	2	yes	maximal, complemented
4	#20245649	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245514	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245509	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245457	C₂	1	2	yes	maximal, complemented
8	#20245360	C₂	1	2	yes	maximal, complemented
9	#20245353	C₂	1	2	yes	maximal, complemented
10	#20245226	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689937	C₂²	1	2	yes	complemented
13	#15689935	—	2	2	no	—
14	#15689925	C₂²	1	2	yes	—
15	#15689563	C₂²	1	2	yes	complemented
16	#15689562	—	2	2	no	—
17	#15689560	—	2	2	no	—
18	#15689558	C₂²	1	2	yes	—
19	#15689556	—	2	1	no	—
20	#15689552	—	2	1	no	—
21	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
22	Q₃₂×D₄₇	—	2	1	no	—
23	C₂×#13093969	C₂²	1	2	yes	complemented
24	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
25	D₁₆×D₄₇	—	2	1	no	—
26	#15689539	C₂²	1	1	yes	characteristic
27	#15689539	C₂²	1	1	yes	characteristic, complemented
28	#15689507	C₂²	1	2	yes	complemented
29	#15689399	C₂²	1	2	yes	—
30	#15689394	C₂²	1	2	yes	complemented
31	#15689386	—	2	2	no	—
32	#15689302	—	2	2	no	—
33	#15689297	—	2	2	no	—
34	#15689241	C₂²	1	1	yes	characteristic
35	#15688944	C₂²	1	1	yes	characteristic, complemented
36	#15688938	C₂²	1	2	yes	complemented
37	#15688935	C₂²	1	2	yes	complemented
38	#15688906	C₂²	1	1	yes	characteristic
39	#15688906	C₂²	1	1	yes	characteristic, complemented
40	C₄₇×#508	C₂²	1	1	yes	characteristic
41	C₄₇×#443	C₂²	1	2	yes	complemented
42	C₄₇×#368	C₂²	1	1	yes	characteristic, complemented
43	C₄₇×#359	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
45	#13094071	—	2	2	no	—
46	#13094070	—	2	2	no	—
47	#13094057	—	2	2	no	—
48	#13094057	—	4	2	no	—
49	#13094054	—	2	2	no	—
50	#13094053	—	4	2	no	—
51	#13094052	—	2	2	no	—
52	#13094052	—	4	2	no	—
53	#13094051	—	4	2	no	—
54	#13094049	—	2	2	no	—
55	#13094047	—	2	2	no	—
56	#13094046	—	4	2	no	—
57	#13094046	D₄	1	1	yes	characteristic
58	#13094043	—	2	2	no	—
59	C₂×Dic₁₈₈	C₂³	1	2	yes	—
60	Q₁₆×D₄₇	—	2	2	no	—
61	C₂×#10625325	—	2	2	no	—
62	C₂×#10625325	—	4	2	no	—
63	C₂×D₃₇₆	C₂³	1	2	yes	—
64	D₈×D₄₇	—	2	2	no	—
65	#13094030	C₂³	1	1	yes	characteristic
66	#13094030	C₂³	1	1	yes	characteristic
67	#13094012	—	2	2	no	—
68	Dic₃₇₆	—	2	1	no	—
69	Dic₃₇₆	—	2	1	no	—
70	#13093970	—	2	2	no	—
71	#13093969	—	4	2	no	—
72	#13093968	—	2	2	no	—
73	#13093967	—	2	2	no	—
74	D₇₅₂	—	2	1	no	—
75	D₇₅₂	—	2	1	no	—
76	#13093965	—	2	2	no	—
77	#13093964	C₂³	1	1	yes	characteristic
78	#13093964	C₂³	1	1	yes	characteristic
79	#13093963	—	2	1	no	—
80	#13093963	—	2	1	no	—
81	C₁₆×D₄₇	—	2	1	no	—
82	C₁₆×D₄₇	—	2	1	no	—
83	#13093960	D₄	1	1	yes	characteristic
84	#13093952	D₄	1	1	yes	characteristic
85	#13093952	D₄	1	1	yes	characteristic
86	#13093942	—	2	2	no	—
87	#13093940	—	2	2	no	—
88	#135×C₄₇	C₂³	1	2	yes	—
89	#131×C₄₇	—	2	2	no	—
90	Q₃₂×C₄₇	D₄	1	1	yes	characteristic
91	SD₃₂×C₄₇	D₄	1	2	yes	—
92	D₁₆×C₄₇	D₄	1	1	yes	characteristic
93	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
95	#108×C₄₇	C₂³	1	2	yes	—
96	#104×C₄₇	—	2	2	no	—
97	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
98	#41×C₄₇	—	2	2	no	—
99	#10625354	—	4	2	no	—
100	Q₈×D₄₇	—	4	2	no	—
101	#10625352	—	4	2	no	—
102	#10625352	—	8	2	no	—
103	D₄×D₄₇	—	4	2	no	—
104	D₄×D₄₇	—	8	2	no	—
105	#10625350	—	4	2	no	—
106	#10625350	—	4	2	no	—
107	#10625350	C₂×D₄	1	1	yes	characteristic
108	#10625350	C₂×D₄	1	1	yes	characteristic
109	C₂×D₁₈₈	—	2	2	no	—
110	C₂×D₁₈₈	—	4	2	no	—
111	C₂×Dic₉₄	—	2	2	no	—
112	C₂×Dic₉₄	—	4	2	no	—
113	Q₁₆×C₄₇	C₂×D₄	1	2	yes	—
114	SD₁₆×C₄₇	—	2	2	no	—
115	D₈×C₄₇	C₂×D₄	1	2	yes	—
116	M₁₆×C₄₇	—	2	2	no	—
117	M₁₆×C₄₇	—	4	2	no	—
118	C₂×C₈×C₄₇	—	4	2	no	—
119	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
121	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
122	#10625337	—	4	2	no	—
123	#10625336	—	4	2	no	—
124	#10625335	—	4	2	no	—
125	#10625334	—	4	2	no	—
126	(C₂×D₄₇)⋊C₄	—	8	2	no	—
127	#10625329	C₂×D₄	1	1	yes	characteristic
128	#10625329	C₂×D₄	1	1	yes	characteristic
129	Dic₁₈₈	—	2	2	no	—
130	Dic₁₈₈	—	2	2	no	—
131	Dic₁₈₈	—	4	2	no	—
132	Dic₁₈₈	D₈	1	1	yes	characteristic
133	D₃₇₆	—	2	2	no	—
134	D₃₇₆	—	2	2	no	—
135	D₃₇₆	—	4	2	no	—
136	D₃₇₆	D₈	1	1	yes	characteristic
137	#10625325	—	4	2	no	—
138	#10625325	—	4	2	no	—
139	#10625325	—	4	2	no	—
140	#10625325	—	4	2	no	—
141	#10625325	D₈	1	1	yes	characteristic
142	#10625325	D₈	1	1	yes	characteristic
143	#10625324	—	2	1	no	—
144	#10625324	—	2	1	no	—
145	C₈×D₄₇	—	2	1	no	—
146	C₈×D₄₇	—	2	1	no	—
147	C₁₆×C₄₇	—	2	1	no	—
148	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
150	C₄₇⋊C₁₆	—	2	1	no	—
151	C₄₇⋊C₁₆	—	2	1	no	—
152	C₂²×D₄₇	—	4	2	no	—
153	C₂²×D₄₇	—	8	2	no	—
154	Q₈×C₄₇	—	2	2	no	—
155	D₄×C₄₇	—	2	2	no	—
156	D₄×C₄₇	—	4	2	no	—
157	C₂×C₄×C₄₇	—	2	1	no	—
158	C₂×C₄×C₄₇	—	4	2	no	—
159	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₄₇	—	2	1	no	—
161	C₂²⋊D₄₇	—	2	1	no	—
162	C₂²⋊D₄₇	—	8	2	no	—
163	C₂²⋊D₄₇	—	8	2	no	—
164	C₂×Dic₄₇	—	4	2	no	—
165	C₂×Dic₄₇	—	8	2	no	—
166	D₁₈₈	—	4	2	no	—
167	D₁₈₈	—	4	2	no	—
168	D₁₈₈	—	8	2	no	—
169	D₁₈₈	C₂×D₈	1	1	yes	characteristic
170	D₁₈₈	C₂×D₈	1	1	yes	characteristic
171	C₄×D₄₇	—	2	1	no	—
172	C₄×D₄₇	—	2	1	no	—
173	C₄×D₄₇	—	8	2	no	—
174	C₄×D₄₇	—	8	2	no	—
175	Dic₉₄	—	4	2	no	—
176	Dic₉₄	—	4	2	no	—
177	Dic₉₄	—	8	2	no	—
178	Dic₉₄	C₂×D₈	1	1	yes	characteristic
179	Dic₉₄	C₂×D₈	1	1	yes	characteristic
180	C₈×C₄₇	—	4	2	no	—
181	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
182	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
183	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
184	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
185	C₄₇⋊C₈	—	2	1	no	—
186	C₄₇⋊C₈	—	2	1	no	—
187	#33859	—	47	1	no	maximal, Sylow
188	C₂²×C₄₇	—	4	2	no	—
189	C₂²×C₄₇	#576	1	1	yes	characteristic
190	C₂×D₄₇	—	2	1	no	—
191	C₂×D₄₇	—	2	1	no	—
192	C₂×D₄₇	—	8	2	no	—
193	C₂×D₄₇	—	16	2	no	—
194	C₂×D₄₇	—	32	2	no	—
195	C₄×C₄₇	—	4	1	no	—
196	C₄×C₄₇	—	4	2	no	—
197	C₄×C₄₇	D₄²	1	1	yes	characteristic
198	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
199	Dic₄₇	—	2	1	no	—
200	Dic₄₇	—	2	1	no	—
201	Dic₄₇	—	8	2	no	—
202	#3417	—	47	1	no	—
203	#3415	—	47	1	no	—
204	#3290	—	47	2	no	—
205	#2249	—	47	1	no	—
206	#2222	—	47	1	no	—
207	#2221	—	47	1	no	—
208	#2212	—	47	2	no	—
209	#2194	—	47	2	no	—
210	#2193	—	47	2	no	—
211	#2178	D₄₇	1	1	yes	characteristic, complemented
212	#2170	—	47	1	no	—
213	C₂×C₄₇	—	2	1	no	—
214	C₂×C₄₇	—	8	2	no	—
215	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
216	D₄₇	—	4	1	no	—
217	D₄₇	—	4	1	no	—
218	D₄₇	—	16	2	no	—
219	#578	—	47	2	no	—
220	#577	—	94	2	no	—
221	#576	—	47	2	no	—
222	#510	—	47	2	no	—
223	#510	—	94	2	no	—
224	#509	—	47	2	no	—
225	#509	—	94	2	no	—
226	#508	—	94	1	no	—
227	#508	—	94	1	no	—
228	#508	C₂×D₄₇	1	1	yes	characteristic
229	C₂×Q₃₂	—	47	1	no	—
230	C₂×Q₃₂	—	94	1	no	—
231	C₂×SD₃₂	—	47	2	no	—
232	C₂×D₁₆	—	47	1	no	—
233	C₂×D₁₆	—	94	1	no	—
234	#504	—	47	1	no	—
235	#504	—	47	1	no	—
236	#495	—	47	2	no	—
237	#473	—	47	2	no	—
238	#472	—	47	2	no	—
239	#471	—	94	2	no	—
240	#455	—	94	2	no	—
241	#454	—	94	2	no	—
242	#443	—	47	1	no	—
243	#443	C₂×D₄₇	1	2	yes	complemented
244	#368	C₂×D₄₇	1	1	yes	characteristic, complemented
245	#364	—	47	1	no	—
246	#362	—	47	2	no	—
247	#361	—	47	2	no	—
248	#359	C₂×D₄₇	1	2	yes	complemented
249	#350	—	47	1	no	—
250	#350	—	47	1	no	—
251	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
252	C₄₇	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	94	2	no	—
254	H(5,𝔽₂)	—	94	2	no	—
255	#137	—	94	2	no	—
256	#137	—	94	2	no	—
257	#137	—	188	2	no	—
258	#137	—	188	2	no	—
259	AGL(1,ℤ/8)	—	94	2	no	—
260	AGL(1,ℤ/8)	—	94	2	no	—
261	AGL(1,ℤ/8)	—	188	2	no	—
262	AGL(1,ℤ/8)	—	188	2	no	—
263	#135	—	47	1	no	—
264	#135	—	94	2	no	—
265	#135	—	94	2	no	—
266	#135	—	188	2	no	—
267	#135	C₂²×D₄₇	1	2	yes	—
268	C₂×Q₁₆	—	47	2	no	—
269	C₂×Q₁₆	—	94	2	no	—
270	C₂×SD₁₆	—	94	2	no	—
271	C₂×SD₁₆	—	188	2	no	—
272	C₂×D₈	—	47	2	no	—
273	C₂×D₈	—	94	2	no	—
274	#131	—	2	2	no	—
275	#131	—	47	1	no	—
276	#131	—	47	1	no	—
277	#124	—	94	2	no	—
278	Q₃₂	—	94	2	no	—
279	Q₃₂	—	94	1	no	—
280	Q₃₂	—	94	1	no	—
281	Q₃₂	D₁₈₈	1	1	yes	characteristic
282	SD₃₂	—	94	2	no	—
283	SD₃₂	—	94	2	no	—
284	SD₃₂	—	188	2	no	—
285	SD₃₂	D₁₈₈	1	2	yes	—
286	D₁₆	—	94	2	no	—
287	D₁₆	—	94	1	no	—
288	D₁₆	—	94	1	no	—
289	D₁₆	D₁₈₈	1	1	yes	characteristic
290	M₃₂	—	47	1	no	—
291	M₃₂	—	47	1	no	—
292	M₃₂	—	94	1	no	—
293	M₃₂	—	94	1	no	—
294	C₂×C₁₆	—	94	1	no	—
295	C₂×C₁₆	—	94	1	no	—
296	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
297	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
298	#108	—	47	1	no	—
299	#108	C₂²×D₄₇	1	2	yes	—
300	#104	—	2	2	no	—
301	#104	—	47	1	no	—
302	#104	—	47	1	no	—
303	#101	—	94	2	no	—
304	#100	—	94	2	no	—
305	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
306	#41	—	2	2	no	—
307	#41	—	47	1	no	—
308	#41	—	47	1	no	—
309	#41	—	376	2	no	—
310	#41	—	188	2	no	—
311	#41	—	188	2	no	—
312	#41	—	188	2	no	—
313	#41	—	188	2	no	—
314	C₂×Q₈	—	94	2	no	—
315	C₂×Q₈	—	188	2	no	—
316	C₂×Q₈	—	188	2	no	—
317	C₂×D₄	—	94	2	no	—
318	C₂×D₄	—	376	2	no	—
319	C₂×D₄	—	188	2	no	—
320	C₂×D₄	—	188	2	no	—
321	Q₁₆	—	47	1	no	—
322	Q₁₆	—	94	2	no	—
323	Q₁₆	—	94	2	no	—
324	Q₁₆	—	188	2	no	—
325	Q₁₆	—	188	2	no	—
326	Q₁₆	C₂×D₁₈₈	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	47	1	no	—
329	SD₁₆	—	47	1	no	—
330	SD₁₆	—	188	2	no	—
331	SD₁₆	—	188	2	no	—
332	SD₁₆	—	188	2	no	—
333	SD₁₆	—	188	2	no	—
334	SD₁₆	—	188	2	no	—
335	SD₁₆	—	188	2	no	—
336	D₈	—	47	1	no	—
337	D₈	—	94	2	no	—
338	D₈	—	94	2	no	—
339	D₈	—	188	2	no	—
340	D₈	—	188	2	no	—
341	D₈	C₂×D₁₈₈	1	2	yes	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	4	2	no	—
344	M₁₆	—	47	1	no	—
345	M₁₆	—	47	1	no	—
346	M₁₆	—	94	1	no	—
347	M₁₆	—	94	1	no	—
348	C₂×C₈	—	4	2	no	—
349	C₂×C₈	—	94	1	no	—
350	C₂×C₈	—	94	1	no	—
351	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
352	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
353	C₂²⋊C₄	—	376	2	no	—
354	C₄²	D₄×D₄₇	1	1	yes	characteristic
355	C₁₆	—	2	1	no	—
356	C₁₆	—	94	1	no	—
357	C₁₆	—	94	1	no	—
358	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
359	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
360	C₂³	—	376	2	no	—
361	C₂³	—	188	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	47	1	no	—
364	Q₈	—	47	1	no	—
365	Q₈	—	376	2	no	—
366	Q₈	—	188	2	no	—
367	Q₈	—	188	2	no	—
368	D₄	—	2	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	47	1	no	—
371	D₄	—	47	1	no	—
372	D₄	—	94	1	no	—
373	D₄	—	94	1	no	—
374	D₄	—	376	2	no	—
375	D₄	—	376	2	no	—
376	D₄	—	376	2	no	—
377	D₄	—	188	2	no	—
378	D₄	—	188	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	94	1	no	—
382	C₂×C₄	—	94	1	no	—
383	C₂×C₄	—	376	2	no	—
384	C₂×C₄	—	376	2	no	—
385	C₂×C₄	—	376	2	no	—
386	C₂×C₄	—	188	2	no	—
387	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
388	C₈	—	4	2	no	—
389	C₈	—	94	1	no	—
390	C₈	—	94	1	no	—
391	C₈	D₈×D₄₇	1	1	yes	characteristic
392	C₈	D₈×D₄₇	1	1	yes	characteristic
393	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
394	C₈	#13094070	1	1	yes	characteristic
395	C₂²	—	4	2	no	—
396	C₂²	—	94	1	no	—
397	C₂²	—	94	1	no	—
398	C₂²	—	376	2	no	—
399	C₂²	—	1504	2	no	—
400	C₂²	—	752	2	no	—
401	C₂²	#15689925	1	1	yes	characteristic
402	C₄	—	4	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	94	1	no	—
405	C₄	—	94	1	no	—
406	C₄	—	376	2	no	—
407	C₄	D₄×D₁₈₈	1	1	yes	characteristic
408	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	8	2	no	—
411	C₂	—	188	1	no	—
412	C₂	—	188	1	no	—
413	C₂	—	752	2	no	—
414	C₂	D₈×D₁₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₈₈
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2178	C₂
socle	C₂×C₄₇	D₄×D₈
hypercenter	#2178	D₄₇
hypocenter	C₄₇	#33859
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
1. #20256538	C₂	semidirect
2. #20256526	C₂	semidirect
3. #20255176	C₂	semidirect
4. #20245649	C₂	semidirect
5. #20245514	C₂	semidirect
6. #20245509	C₂	semidirect
7. #20245457	C₂	semidirect
8. #20245360	C₂	semidirect
9. #20245353	C₂	semidirect
10. #20245226	C₂	semidirect
11. C₄₇×#2178	C₂	semidirect
12. #15689937	C₂²	semidirect
15. #15689563	C₂²	semidirect
21. C₂×Dic₃₇₆	C₂²	semidirect
23. C₂×#13093969	C₂²	semidirect
27. #15689539	C₂²	semidirect
28. #15689507	C₂²	semidirect
30. #15689394	C₂²	semidirect
35. #15688944	C₂²	semidirect
36. #15688938	C₂²	semidirect
37. #15688935	C₂²	semidirect
39. #15688906	C₂²	semidirect
41. C₄₇×#443	C₂²	semidirect
42. C₄₇×#368	C₂²	semidirect
43. C₄₇×#359	C₂²	semidirect
44. C₄×C₁₆×C₄₇	C₂²	semidirect
211. #2178	D₄₇	semidirect
243. #443	C₂×D₄₇	semidirect
244. #368	C₂×D₄₇	semidirect
248. #359	C₂×D₄₇	semidirect
251. C₄×C₁₆	C₂×D₄₇	semidirect
252. C₄₇	#33859	semidirect

Maximal subgroups

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

subgroup	index
#20256538	2
#20256526	2
#20255176	2
#20245649	2
#20245514	2
#20245509	2
#20245457	2
#20245360	2
#20245353	2
#20245226	2
C₄₇×#2178	2
#33859	47