Subgroups — #26370961

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252241	C₂	1	1	yes	maximal, characteristic
2	#20245542	C₂	1	2	yes	maximal, complemented
3	#20245542	C₂	1	2	yes	maximal, complemented
4	#20245532	C₂	1	2	yes	maximal, complemented
5	#20245522	C₂	1	2	yes	maximal, complemented
6	#20245174	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245147	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245103	C₂	1	1	yes	maximal, characteristic
10	C₂×#15688923	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
13	#15689597	—	2	1	no	—
14	C₂×#13093964	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#15689236	—	2	2	no	—
17	C₄×D₃₇₆	C₂²	1	2	yes	—
18	C₄×D₃₇₆	C₂²	1	2	yes	—
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689129	—	2	1	no	—
21	#15688950	C₂²	1	1	yes	characteristic, complemented
22	#15688950	C₂²	1	2	yes	complemented
23	#15688950	C₂²	1	1	yes	characteristic, complemented
24	#15688948	C₂²	1	1	yes	characteristic, complemented
25	#15688948	C₂²	1	2	yes	complemented
26	#15688948	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
28	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
29	#15688923	C₂²	1	2	yes	—
30	#15688923	C₂²	1	2	yes	—
31	#15688923	C₂²	1	2	yes	—
32	#15688923	C₂²	1	2	yes	—
33	C₄₇×#544	—	2	1	no	—
34	C₄₇×#501	—	2	2	no	—
35	C₄₇×#500	C₂²	1	2	yes	complemented
36	C₄₇×#499	C₂²	1	2	yes	complemented
37	C₄₇×#428	—	2	1	no	—
38	C₄₇×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#405	C₂²	1	1	yes	characteristic
41	C₂²×D₁₈₈	—	2	1	no	—
42	C₂×D₃₇₆	—	2	1	no	—
43	C₂×D₃₇₆	—	2	2	no	—
44	C₂×D₃₇₆	Q₈	1	4	yes	complemented
45	C₂×D₃₇₆	C₂³	1	2	yes	—
46	C₄×D₁₈₈	—	2	2	no	—
47	C₄×D₁₈₈	—	2	2	no	—
48	#13093986	—	4	1	no	—
49	#13093986	—	4	1	no	—
50	#13093983	—	2	1	no	—
51	C₂²⋊C₄×D₄₇	—	4	2	no	—
52	#13093964	—	2	2	no	—
53	C₂×C₄₇⋊C₁₆	—	2	2	no	—
54	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
57	#13093958	C₂³	1	1	yes	characteristic
58	#13093958	C₂³	1	1	yes	characteristic
59	#13093956	—	2	1	no	—
60	#13093946	—	2	2	no	—
61	#13093946	—	2	2	no	—
62	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₄₇	—	2	2	no	—
65	C₄⋊Q₈×C₄₇	—	4	1	no	—
66	#125×C₄₇	—	2	2	no	—
67	#125×C₄₇	—	4	1	no	—
68	C₂²⋊Q₈×C₄₇	—	4	2	no	—
69	#117×C₄₇	—	2	1	no	—
70	#117×C₄₇	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
72	#107×C₄₇	—	2	2	no	—
73	#107×C₄₇	—	2	1	no	—
74	#107×C₄₇	C₂³	1	1	yes	characteristic
75	#107×C₄₇	C₂³	1	2	yes	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	#106×C₄₇	—	2	2	no	—
78	#106×C₄₇	—	2	1	no	—
79	#97×C₄₇	D₄	1	2	yes	—
80	C₄×C₈×C₄₇	C₂³	1	2	yes	—
81	C₂³×D₄₇	—	4	1	no	—
82	C₂×Q₈×C₄₇	—	4	2	no	—
83	C₂²×C₄×C₄₇	—	4	1	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×D₁₈₈	—	2	4	no	—
86	C₂×D₁₈₈	—	2	2	no	—
87	C₂×D₁₈₈	—	4	1	no	—
88	C₂×D₁₈₈	—	4	2	no	—
89	C₂×C₄×D₄₇	—	4	2	no	—
90	C₂×C₄×D₄₇	—	4	2	no	—
91	M₁₆×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₄₇	#41	1	2	yes	—
96	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₄₇	—	2	1	no	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	1	no	—
100	C₄⋊C₄×C₄₇	—	4	1	no	—
101	C₄⋊C₄×C₄₇	—	4	2	no	—
102	C₄⋊C₄×C₄₇	—	4	2	no	—
103	C₄⋊C₄×C₄₇	—	4	2	no	—
104	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₄₇	—	4	1	no	—
107	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
108	C₄²×C₄₇	—	4	1	no	—
109	C₄²×C₄₇	C₂×D₄	1	2	yes	—
110	C₂²⋊Dic₄₇	—	4	1	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄×Dic₄₇	—	4	1	no	—
116	D₃₇₆	—	2	4	no	—
117	D₃₇₆	—	2	2	no	—
118	D₃₇₆	C₂×Q₈	1	4	yes	—
119	C₄₇⋊C₁₆	—	2	1	no	—
120	C₄₇⋊C₁₆	—	2	1	no	—
121	C₄₇⋊C₁₆	—	2	2	no	—
122	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₄₇	—	4	2	no	—
124	C₂²×D₄₇	—	4	4	no	—
125	C₂²×D₄₇	—	16	1	no	—
126	C₂²×D₄₇	—	16	1	no	—
127	Q₈×C₄₇	—	16	2	no	—
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	16	1	no	—
132	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#137	1	2	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	4	1	no	—
140	D₁₈₈	—	2	4	no	—
141	D₁₈₈	—	4	2	no	—
142	D₁₈₈	—	4	4	no	—
143	C₄×D₄₇	—	16	2	no	—
144	C₄×D₄₇	—	16	2	no	—
145	C₈×C₄₇	—	2	2	no	—
146	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
149	#33631	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	#474	1	2	yes	—
153	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₄₇	—	4	4	no	—
155	C₂×D₄₇	—	8	2	no	—
156	C₂×D₄₇	—	8	2	no	—
157	C₂×D₄₇	—	16	4	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	—	8	1	no	—
162	C₄×C₄₇	—	8	1	no	—
163	C₄×C₄₇	#474	1	2	yes	—
164	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
166	Dic₄₇	—	8	1	no	—
167	Dic₄₇	—	8	1	no	—
168	#3164	D₄₇	1	1	yes	characteristic, complemented
169	#2948	—	47	1	no	—
170	#2228	—	47	2	no	—
171	#2228	—	47	2	no	—
172	#2226	—	47	2	no	—
173	#2224	—	47	2	no	—
174	#2159	—	47	1	no	—
175	#2158	—	47	1	no	—
176	#2151	—	47	1	no	—
177	#2140	—	47	1	no	—
178	C₂×#357	—	47	1	no	—
179	C₂×C₄₇	—	2	2	no	—
180	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
181	C₂×C₄₇	#3415	1	1	yes	characteristic
182	C₂×C₄₇	#3416	1	1	yes	characteristic
183	D₄₇	—	8	4	no	—
184	C₂²×D₈	—	47	1	no	—
185	#544	—	2	1	no	—
186	#518	—	94	1	no	—
187	C₂×M₃₂	—	47	1	no	—
188	C₂²×C₁₆	—	47	1	no	—
189	#501	—	2	2	no	—
190	#500	C₂×D₄₇	1	2	yes	complemented
191	#499	C₂×D₄₇	1	2	yes	complemented
192	#442	—	94	2	no	—
193	C₄×D₈	—	47	2	no	—
194	C₄×D₈	—	47	2	no	—
195	#428	—	2	1	no	—
196	#427	—	47	1	no	—
197	#427	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
199	#418	—	94	1	no	—
200	#405	C₂×D₄₇	1	1	yes	characteristic
201	#367	—	47	1	no	—
202	#367	—	47	1	no	—
203	#367	—	47	2	no	—
204	#366	—	47	1	no	—
205	#366	—	47	1	no	—
206	#366	—	47	2	no	—
207	C₄⋊C₁₆	—	47	2	no	—
208	C₄⋊C₁₆	—	47	2	no	—
209	#357	—	47	2	no	—
210	#357	—	47	2	no	—
211	#357	—	47	2	no	—
212	#357	—	47	2	no	—
213	C₄₇	#33631	1	1	yes	characteristic, complemented, Sylow
214	C₂²×D₄	—	94	1	no	—
215	C₂×D₈	—	47	4	no	—
216	C₂×D₈	—	47	2	no	—
217	C₂×D₈	—	94	2	no	—
218	C₂×D₈	—	94	1	no	—
219	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	1	no	—
225	C₂²⋊Q₈	—	4	2	no	—
226	C₄×D₄	—	94	2	no	—
227	C₄×D₄	—	94	2	no	—
228	C₄×D₄	—	188	1	no	—
229	C₄×D₄	—	188	1	no	—
230	#117	—	2	1	no	—
231	#117	—	94	1	no	—
232	#117	C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	188	2	no	—
235	M₃₂	—	94	2	no	—
236	C₂×C₁₆	—	47	1	no	—
237	C₂×C₁₆	—	47	2	no	—
238	C₂×C₁₆	—	47	1	no	—
239	C₂×C₁₆	—	94	2	no	—
240	#107	—	2	1	no	—
241	#107	—	2	2	no	—
242	#107	—	47	1	no	—
243	#107	—	47	1	no	—
244	#107	C₂²×D₄₇	1	2	yes	—
245	#107	C₂²×D₄₇	1	1	yes	characteristic
246	#107	C₂²×D₄₇	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	2	no	—
249	#106	—	94	1	no	—
250	#102	—	94	2	no	—
251	#102	—	94	2	no	—
252	#97	C₂²⋊D₄₇	1	2	yes	—
253	C₄×C₈	C₂²×D₄₇	1	2	yes	—
254	C₂⁴	—	188	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	94	4	no	—
257	C₂×D₄	—	94	2	no	—
258	C₂×D₄	—	188	1	no	—
259	C₂×D₄	—	188	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	188	2	no	—
262	C₂²×C₄	—	188	2	no	—
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
264	D₈	—	47	4	no	—
265	D₈	—	94	4	no	—
266	D₈	—	94	2	no	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	Q₈×D₄₇	1	2	yes	—
269	C₂×C₈	#10625354	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
271	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	94	1	no	—
281	C₄⋊C₄	—	94	1	no	—
282	C₄⋊C₄	#10625334	1	1	yes	characteristic
283	C₄⋊C₄	#10625334	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	188	1	no	—
286	C₂²⋊C₄	—	188	2	no	—
287	C₂²⋊C₄	—	188	2	no	—
288	C₂²⋊C₄	#10625334	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	188	1	no	—
291	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
292	C₁₆	—	94	1	no	—
293	C₁₆	—	94	1	no	—
294	C₁₆	—	94	2	no	—
295	C₂³	—	188	2	no	—
296	C₂³	—	188	4	no	—
297	C₂³	—	752	1	no	—
298	C₂³	—	752	1	no	—
299	C₂³	C₂×#10625334	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	94	4	no	—
302	D₄	—	188	4	no	—
303	D₄	—	188	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	188	1	no	—
309	C₂×C₄	—	188	1	no	—
310	C₂×C₄	—	752	2	no	—
311	C₂×C₄	—	752	2	no	—
312	C₂×C₄	#13094000	1	2	yes	—
313	C₂×C₄	C₂×#10625334	1	2	yes	—
314	C₂×C₄	C₂×#10625334	1	2	yes	—
315	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
316	C₂×C₄	#13094055	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#13094000	1	2	yes	—
320	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
321	C₈	C₂×#10625354	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	376	2	no	—
325	C₂²	—	376	2	no	—
326	C₂²	—	188	4	no	—
327	C₂²	—	752	4	no	—
328	C₂²	#15689400	1	2	yes	—
329	C₂²	C₂²×#10625334	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	376	1	no	—
336	C₄	—	376	1	no	—
337	C₄	#15689400	1	2	yes	—
338	C₄	C₂×#13094000	1	1	yes	characteristic
339	C₄	C₂×#13094055	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	376	4	no	—
342	C₂	C₂×#15689400	1	1	yes	characteristic
343	C₂	#20256527	1	1	yes	characteristic
344	C₂	#20256532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3164	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	#3164	D₄₇
hypocenter	C₄₇	#33631
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #20245542	C₂	semidirect
3. #20245542	C₂	semidirect
4. #20245532	C₂	semidirect
5. #20245522	C₂	semidirect
6. #20245174	C₂	semidirect
7. #20245170	C₂	semidirect
8. #20245147	C₂	semidirect
11. C₄₇×#3164	C₂	semidirect
21. #15688950	C₂²	semidirect
22. #15688950	C₂²	semidirect
23. #15688950	C₂²	semidirect
24. #15688948	C₂²	semidirect
25. #15688948	C₂²	semidirect
26. #15688948	C₂²	semidirect
27. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
28. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
35. C₄₇×#500	C₂²	semidirect
36. C₄₇×#499	C₂²	semidirect
44. C₂×D₃₇₆	Q₈	semidirect
168. #3164	D₄₇	semidirect
190. #500	C₂×D₄₇	semidirect
191. #499	C₂×D₄₇	semidirect
213. C₄₇	#33631	semidirect

Maximal subgroups

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

subgroup	index
#20252241	2
#20245542	2
#20245532	2
#20245522	2
#20245174	2
#20245170	2
#20245147	2
#20245103	2
C₂×#15688923	2
C₄₇×#3164	2
#33631	47