Subgroups — #26242851

Order 12032.

756 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253841	C₂	1	1	yes	maximal, characteristic
2	#20243878	C₂	1	2	yes	maximal, complemented
3	#20243872	C₂	1	2	yes	maximal, complemented
4	#20243868	C₂	1	2	yes	maximal, complemented
5	#20243866	C₂	1	2	yes	maximal, complemented
6	#20243785	C₂	1	2	yes	maximal, complemented
7	#20243772	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243534	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1884	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094049	C₂²	1	2	yes	—
11	C₂×#13094049	C₂²	1	2	yes	—
12	#15689890	C₂²	1	1	yes	characteristic
13	C₂×#13094021	C₂²	1	1	yes	characteristic
14	#15689597	C₂²	1	2	yes	—
15	#15689513	C₄	1	2	yes	complemented
16	#15689513	C₄	1	2	yes	complemented
17	#15689499	C₄	1	2	yes	complemented
18	#15689497	C₄	1	2	yes	complemented
19	#15689140	C₂²	1	2	yes	—
20	#15689129	C₂²	1	2	yes	—
21	#15689127	C₂²	1	2	yes	—
22	C₂×#13093940	C₂²	1	2	yes	—
23	#15689097	C₂²	1	1	yes	characteristic
24	C₄×#10625329	C₂²	1	1	yes	characteristic
25	#15688895	C₂²	1	2	yes	—
26	#15688893	C₂²	1	2	yes	—
27	#15688883	C₂²	1	2	yes	—
28	C₄₇⋊#338	C₂²	1	2	yes	—
29	#15688879	C₂²	1	2	yes	—
30	C₄₇×#568	C₂²	1	1	yes	characteristic
31	C₄₇×#532	C₂²	1	1	yes	characteristic
32	C₄₇×#421	C₂²	1	2	yes	—
33	C₄₇×#410	C₂²	1	1	yes	characteristic
34	C₄₇×#339	C₂²	1	2	yes	—
35	C₂²×D₁₈₈	—	4	1	no	—
36	C₂²×D₁₈₈	C₂³	1	2	yes	—
37	#13094049	—	4	2	no	—
38	#13094049	—	4	2	no	—
39	#13094049	—	4	2	no	—
40	#13094049	—	4	2	no	—
41	C₂×#10625336	C₂×C₄	1	4	yes	complemented
42	C₂×#10625336	C₂×C₄	1	4	yes	complemented
43	C₂×#10625334	C₂×C₄	1	4	yes	complemented
44	C₂×#10625334	C₂×C₄	1	4	yes	complemented
45	#13094031	—	4	2	no	—
46	C₂×#10625329	—	4	1	no	—
47	C₂×#10625329	C₂³	1	2	yes	—
48	C₂×#10625329	C₂³	1	2	yes	—
49	#13094021	—	4	1	no	—
50	#13094021	C₂×C₄	1	2	yes	—
51	#13094021	C₂×C₄	1	2	yes	—
52	C₄×D₁₈₈	—	4	2	no	—
53	#13093986	—	4	2	no	—
54	#13093983	C₂³	1	2	yes	—
55	C₂²⋊C₄×D₄₇	—	4	2	no	—
56	#13093951	—	4	2	no	—
57	#13093946	—	4	2	no	—
58	#13093944	—	4	2	no	—
59	#13093940	—	4	2	no	—
60	#13093939	—	4	1	no	—
61	#13093935	C₂×C₄	1	2	yes	complemented
62	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	complemented
63	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
65	#131×C₄₇	—	4	2	no	—
66	C₂×M₁₆×C₄₇	—	4	1	no	—
67	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
69	#127×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
70	#124×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
71	#124×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
72	#121×C₄₇	—	4	1	no	—
73	C₄×D₄×C₄₇	—	4	1	no	—
74	#117×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#104×C₄₇	—	4	2	no	—
77	#99×C₄₇	—	4	1	no	—
78	C₂³×D₄₇	—	2	2	no	—
79	#41×C₄₇	—	2	2	no	—
80	#41×C₄₇	—	2	2	no	—
81	#41×C₄₇	—	2	2	no	—
82	#41×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Q₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×D₄×C₄₇	—	2	1	no	—
86	C₂×D₄×C₄₇	—	2	1	no	—
87	C₂×D₄×C₄₇	—	4	1	no	—
88	C₂×D₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×D₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	—	2	1	no	—
91	C₂²×C₄×C₄₇	—	2	1	no	—
92	C₂²×C₄×C₄₇	—	2	1	no	—
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂×D₁₈₈	—	2	2	no	—
95	C₂×D₁₈₈	—	2	2	no	—
96	C₂×D₁₈₈	—	2	2	no	—
97	C₂×D₁₈₈	—	2	2	no	—
98	C₂×D₁₈₈	—	2	2	no	—
99	C₂×D₁₈₈	—	4	2	no	—
100	C₂×D₁₈₈	—	4	2	no	—
101	C₂×D₁₈₈	—	8	4	no	—
102	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
103	C₂×C₄×D₄₇	—	4	4	no	—
104	M₁₆×C₄₇	—	2	2	no	—
105	M₁₆×C₄₇	—	2	2	no	—
106	M₁₆×C₄₇	—	4	2	no	—
107	C₂×C₈×C₄₇	—	2	2	no	—
108	C₂×C₈×C₄₇	—	4	2	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	4	1	no	—
111	C₂²⋊C₄×C₄₇	—	2	2	no	—
112	C₂²⋊C₄×C₄₇	—	4	1	no	—
113	C₂²⋊C₄×C₄₇	—	4	1	no	—
114	C₄²×C₄₇	—	2	1	no	—
115	C₄²×C₄₇	—	2	2	no	—
116	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊Dic₄₇	—	2	2	no	—
119	#10625336	—	4	4	no	—
120	#10625336	—	4	4	no	—
121	#10625334	—	4	4	no	—
122	#10625334	—	4	4	no	—
123	(C₂×D₄₇)⋊C₄	—	4	4	no	—
124	C₄⋊Dic₄₇	—	2	2	no	—
125	C₄×Dic₄₇	—	2	2	no	—
126	#10625329	—	2	2	no	—
127	#10625329	—	2	2	no	—
128	#10625329	—	2	2	no	—
129	#10625329	—	2	2	no	—
130	#10625329	—	4	2	no	—
131	C₂×C₄₇⋊C₈	—	2	2	no	—
132	C₂×C₄₇⋊C₈	—	4	2	no	—
133	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
134	C₂³×C₄₇	—	2	1	no	—
135	C₂³×C₄₇	—	2	1	no	—
136	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×D₄₇	—	2	4	no	—
138	C₂²×D₄₇	—	4	2	no	—
139	C₂²×D₄₇	—	4	2	no	—
140	C₂²×D₄₇	—	8	4	no	—
141	Q₈×C₄₇	—	2	2	no	—
142	Q₈×C₄₇	—	2	2	no	—
143	D₄×C₄₇	—	2	2	no	—
144	D₄×C₄₇	—	2	2	no	—
145	D₄×C₄₇	—	4	2	no	—
146	D₄×C₄₇	—	4	2	no	—
147	D₄×C₄₇	—	8	2	no	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	—	2	1	no	—
150	C₂×C₄×C₄₇	—	2	1	no	—
151	C₂×C₄×C₄₇	—	2	1	no	—
152	C₂×C₄×C₄₇	—	4	2	no	—
153	C₂×C₄×C₄₇	—	4	1	no	—
154	C₂×C₄×C₄₇	—	4	1	no	—
155	C₂×C₄×C₄₇	—	4	1	no	—
156	C₂×C₄×C₄₇	—	4	1	no	—
157	C₂×C₄×C₄₇	—	4	2	no	—
158	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
160	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
162	C₂×Dic₄₇	—	4	2	no	—
163	D₁₈₈	—	2	4	no	—
164	D₁₈₈	—	2	4	no	—
165	D₁₈₈	—	4	2	no	—
166	D₁₈₈	—	4	2	no	—
167	D₁₈₈	—	8	4	no	—
168	D₁₈₈	—	8	4	no	—
169	C₄×D₄₇	—	16	4	no	—
170	C₈×C₄₇	—	4	2	no	—
171	C₄₇⋊C₈	—	2	4	no	—
172	C₄₇⋊C₈	—	4	2	no	—
173	#23361	—	47	1	no	maximal, Sylow
174	C₂²×C₄₇	—	4	1	no	—
175	C₂²×C₄₇	—	4	1	no	—
176	C₂²×C₄₇	—	4	1	no	—
177	C₂²×C₄₇	—	8	1	no	—
178	C₂²×C₄₇	—	8	1	no	—
179	C₂²×C₄₇	#518	1	1	yes	characteristic
180	C₂²×C₄₇	#560	1	2	yes	—
181	C₂×D₄₇	—	4	4	no	—
182	C₂×D₄₇	—	8	4	no	—
183	C₂×D₄₇	—	8	4	no	—
184	C₂×D₄₇	—	16	2	no	—
185	C₄×C₄₇	—	2	2	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	—	4	1	no	—
188	C₄×C₄₇	—	4	1	no	—
189	C₄×C₄₇	—	8	2	no	—
190	C₄×C₄₇	#518	1	2	yes	—
191	C₄×C₄₇	#560	1	1	yes	characteristic
192	C₄×C₄₇	#560	1	1	yes	characteristic
193	Dic₄₇	—	8	2	no	—
194	#3151	—	47	1	no	—
195	#1904	—	47	2	no	—
196	#1904	—	47	2	no	—
197	#1903	—	47	2	no	—
198	#1903	—	47	2	no	—
199	#1887	—	47	2	no	—
200	#1884	—	47	1	no	—
201	#1884	D₄₇	1	1	yes	characteristic, complemented
202	#1841	—	47	2	no	—
203	C₂×C₄₇	—	2	2	no	—
204	C₂×C₄₇	—	2	1	no	—
205	C₂×C₄₇	—	8	1	no	—
206	C₂×C₄₇	—	8	1	no	—
207	C₂×C₄₇	#2380	1	1	yes	characteristic
208	D₄₇	—	8	4	no	—
209	C₂×AGL(1,ℤ/8)	—	47	2	no	—
210	C₂×AGL(1,ℤ/8)	—	47	2	no	—
211	#568	—	47	1	no	—
212	#568	C₂×D₄₇	1	1	yes	characteristic
213	#532	C₂×D₄₇	1	1	yes	characteristic
214	C₂×#127	—	47	1	no	—
215	#518	—	47	2	no	—
216	#496	—	47	2	no	—
217	#496	—	47	2	no	—
218	#493	—	47	2	no	—
219	#492	—	47	2	no	—
220	#421	—	47	2	no	—
221	#421	C₂×D₄₇	1	2	yes	—
222	#418	—	47	2	no	—
223	#418	—	47	2	no	—
224	C₂×#100	—	47	2	no	—
225	#410	—	47	1	no	—
226	#410	C₂×D₄₇	1	1	yes	characteristic
227	C₄×M₁₆	—	47	1	no	—
228	#344	—	47	2	no	—
229	#344	—	47	2	no	—
230	#339	—	47	2	no	—
231	#339	C₂×D₄₇	1	2	yes	—
232	#338	—	47	2	no	—
233	H(3,ℤ/4)	—	47	2	no	—
234	C₄₇	#23361	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
236	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
237	C₂²×D₄	—	47	2	no	—
238	C₂²×D₄	—	188	1	no	—
239	AGL(1,ℤ/8)	—	188	2	no	—
240	AGL(1,ℤ/8)	—	188	2	no	—
241	AGL(1,ℤ/8)	—	188	2	no	—
242	AGL(1,ℤ/8)	—	188	2	no	—
243	C₂×SD₁₆	—	47	4	no	—
244	C₂×SD₁₆	—	47	4	no	—
245	C₂×D₈	—	47	4	no	—
246	C₂×D₈	—	47	4	no	—
247	#131	—	4	2	no	—
248	#131	—	188	2	no	—
249	C₂×M₁₆	—	4	1	no	—
250	C₂×M₁₆	—	47	2	no	—
251	C₂×M₁₆	—	47	2	no	—
252	C₂×M₁₆	—	188	1	no	—
253	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
254	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic, complemented
255	#127	—	47	2	no	—
256	#127	—	47	2	no	—
257	#127	—	188	1	no	—
258	#127	C₄×D₄₇	1	1	yes	characteristic, complemented
259	#124	C₄×D₄₇	1	1	yes	characteristic, complemented
260	#124	C₄×D₄₇	1	1	yes	characteristic, complemented
261	#121	—	4	1	no	—
262	C₄×D₄	—	4	1	no	—
263	C₄×D₄	—	188	2	no	—
264	C₄×D₄	—	188	2	no	—
265	#117	—	47	2	no	—
266	#117	C₂²×D₄₇	1	2	yes	—
267	C₂×C₂²⋊C₄	—	188	2	no	—
268	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
269	#104	—	4	2	no	—
270	#104	—	188	2	no	—
271	#102	—	188	2	no	—
272	#102	—	188	2	no	—
273	#100	—	188	2	no	—
274	#99	—	4	1	no	—
275	#99	—	188	1	no	—
276	#97	—	47	2	no	—
277	C₄×C₈	—	47	2	no	—
278	C₂⁴	—	94	2	no	—
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	#41	—	2	2	no	—
282	#41	—	2	2	no	—
283	C₂×Q₈	C₂×C₄×D₄₇	1	1	yes	characteristic
284	C₂×Q₈	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	47	4	no	—
289	C₂×D₄	—	94	2	no	—
290	C₂×D₄	—	94	2	no	—
291	C₂×D₄	—	94	2	no	—
292	C₂×D₄	—	94	2	no	—
293	C₂×D₄	—	94	2	no	—
294	C₂×D₄	—	376	4	no	—
295	C₂×D₄	—	188	2	no	—
296	C₂×D₄	—	188	2	no	—
297	C₂×D₄	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₂×D₄	C₂×C₄×D₄₇	1	1	yes	characteristic
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	188	4	no	—
303	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
304	SD₁₆	—	188	4	no	—
305	SD₁₆	—	188	4	no	—
306	D₈	—	188	4	no	—
307	D₈	—	188	4	no	—
308	M₁₆	—	2	2	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	94	2	no	—
312	M₁₆	—	94	2	no	—
313	M₁₆	—	94	2	no	—
314	M₁₆	—	94	2	no	—
315	M₁₆	—	188	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	4	2	no	—
318	C₂×C₈	—	47	4	no	—
319	C₂×C₈	—	94	2	no	—
320	C₂×C₈	—	188	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	94	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₂²⋊C₄	—	4	1	no	—
326	C₂²⋊C₄	—	4	1	no	—
327	C₂²⋊C₄	—	94	2	no	—
328	C₂²⋊C₄	—	188	4	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	94	2	no	—
332	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
333	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
334	C₂³	—	2	1	no	—
335	C₂³	—	2	1	no	—
336	C₂³	—	94	4	no	—
337	C₂³	—	376	4	no	—
338	C₂³	—	188	2	no	—
339	C₂³	—	188	2	no	—
340	C₂³	#13094069	1	1	yes	characteristic
341	Q₈	—	2	2	no	—
342	Q₈	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	4	2	no	—
346	D₄	—	4	2	no	—
347	D₄	—	8	2	no	—
348	D₄	—	94	4	no	—
349	D₄	—	94	4	no	—
350	D₄	—	376	4	no	—
351	D₄	—	376	4	no	—
352	D₄	—	188	2	no	—
353	D₄	—	188	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	188	2	no	—
365	C₂×C₄	—	752	4	no	—
366	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
367	C₂×C₄	#13094069	1	2	yes	—
368	C₂×C₄	#13094069	1	1	yes	characteristic
369	C₂×C₄	#13094070	1	2	yes	—
370	C₈	—	4	2	no	—
371	C₈	—	94	4	no	—
372	C₈	—	188	2	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	376	4	no	—
379	C₂²	—	376	4	no	—
380	C₂²	—	188	4	no	—
381	C₂²	—	752	2	no	—
382	C₂²	#15689596	1	1	yes	characteristic
383	C₂²	#15689861	1	2	yes	—
384	C₄	—	2	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	8	2	no	—
389	C₄	—	376	2	no	—
390	C₄	#15689595	1	2	yes	—
391	C₄	#15689862	1	1	yes	characteristic
392	C₄	#15689863	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	2	no	—
395	C₂	—	8	1	no	—
396	C₂	—	8	1	no	—
397	C₂	—	376	4	no	—
398	C₂	#20246563	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20246563
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1884	C₂
socle	C₂×C₄₇	#2380
prime radical	#20253841	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1884	D₄₇
hypocenter	C₄₇	#23361
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1884

Complements

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

N	H ≅ G/N	type
2. #20243878	C₂	semidirect
3. #20243872	C₂	semidirect
4. #20243868	C₂	semidirect
5. #20243866	C₂	semidirect
6. #20243785	C₂	semidirect
7. #20243772	C₂	semidirect
8. #20243534	C₂	semidirect
9. C₄₇×#1884	C₂	semidirect
15. #15689513	C₄	semidirect
16. #15689513	C₄	semidirect
17. #15689499	C₄	semidirect
18. #15689497	C₄	semidirect
41. C₂×#10625336	C₂×C₄	semidirect
42. C₂×#10625336	C₂×C₄	semidirect
43. C₂×#10625334	C₂×C₄	semidirect
44. C₂×#10625334	C₂×C₄	semidirect
61. #13093935	C₂×C₄	semidirect
62. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
68. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
69. #127×C₄₇	C₂×C₄	semidirect
70. #124×C₄₇	C₂×C₄	semidirect
71. #124×C₄₇	C₂×C₄	semidirect
201. #1884	D₄₇	semidirect
234. C₄₇	#23361	semidirect
254. C₄⋊Q₈	C₄×D₄₇	semidirect
258. #127	C₄×D₄₇	semidirect
259. #124	C₄×D₄₇	semidirect
260. #124	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20253841	2
#20243878	2
#20243872	2
#20243868	2
#20243866	2
#20243785	2
#20243772	2
#20243534	2
C₄₇×#1884	2
#23361	47