Subgroups — #26373885

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 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	#20255737	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245670	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245578	C₂	1	2	yes	maximal, complemented
5	#20245572	C₂	1	2	yes	maximal, complemented
6	#20245535	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245525	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245488	C₂	1	2	yes	maximal, complemented
9	#20245221	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245173	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689838	—	2	1	no	—
13	#15689778	—	2	1	no	—
14	C₂×#13093963	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689445	—	2	2	no	—
18	#15689435	—	2	2	no	—
19	#15689413	—	2	1	no	—
20	#15689403	—	2	1	no	—
21	#15689356	C₂²	1	2	yes	—
22	#15689356	C₂²	1	2	yes	—
23	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
24	#15689160	C₂²	1	2	yes	—
25	#107×D₄₇	C₂²	1	2	yes	—
26	#15688951	C₂²	1	2	yes	—
27	#15688951	C₂²	1	2	yes	—
28	#15688950	C₂²	1	2	yes	—
29	#15688948	C₂²	1	2	yes	—
30	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
31	#15688924	C₂²	1	2	yes	—
32	#15688924	C₂²	1	2	yes	—
33	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#366	C₂²	1	1	yes	characteristic
38	C₄₇×#357	C₂²	1	1	yes	characteristic
39	C₄₇×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625350	—	4	1	no	—
42	C₂×C₈×D₄₇	C₂³	1	2	yes	—
43	#13094024	—	2	1	no	—
44	#13094024	—	2	1	no	—
45	#13094010	—	4	2	no	—
46	#13094006	—	4	2	no	—
47	#13094003	—	4	1	no	—
48	#13093998	—	4	1	no	—
49	#13093995	—	2	2	no	—
50	#13093995	—	2	2	no	—
51	C₄×D₁₈₈	D₄	1	1	yes	characteristic
52	#13093983	—	4	1	no	—
53	#13093982	—	2	2	no	—
54	C₄⋊C₄×D₄₇	—	2	2	no	—
55	#13093963	—	2	4	no	—
56	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
57	#13093958	—	2	1	no	—
58	#13093958	—	2	1	no	—
59	#13093958	C₂³	1	2	yes	—
60	#13093958	C₂³	1	2	yes	—
61	#13093957	—	2	2	no	—
62	#13093955	—	2	2	no	—
63	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
64	#13093945	—	2	2	no	—
65	#13093945	—	2	2	no	—
66	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
67	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₄₇	—	2	1	no	—
70	C₄×D₄×C₄₇	—	2	1	no	—
71	D₁₆×C₄₇	—	2	2	no	—
72	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	#107×C₄₇	C₂³	1	1	yes	characteristic
75	#107×C₄₇	C₂³	1	1	yes	characteristic
76	#102×C₄₇	—	2	1	no	—
77	#102×C₄₇	—	2	1	no	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₄₇	—	2	1	no	—
80	C₂×D₄×C₄₇	—	2	1	no	—
81	C₂²×C₄×C₄₇	—	4	1	no	—
82	C₂²×C₄×C₄₇	—	4	1	no	—
83	C₂×C₂²⋊D₄₇	—	4	2	no	—
84	C₂×C₂²⋊D₄₇	—	4	2	no	—
85	#10625350	—	8	4	no	—
86	C₂×D₁₈₈	D₈	1	1	yes	characteristic
87	C₂×C₄×D₄₇	—	4	2	no	—
88	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
89	C₂×Dic₉₄	—	4	1	no	—
90	C₂×Dic₉₄	—	4	1	no	—
91	D₈×C₄₇	—	2	1	no	—
92	D₈×C₄₇	—	2	1	no	—
93	D₈×C₄₇	—	2	1	no	—
94	D₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₄₇	—	2	1	no	—
98	C₄⋊C₄×C₄₇	—	2	1	no	—
99	C₂²⋊C₄×C₄₇	—	4	1	no	—
100	C₂²⋊C₄×C₄₇	—	4	1	no	—
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂²⋊Dic₄₇	—	4	2	no	—
103	C₂²⋊Dic₄₇	—	4	2	no	—
104	(C₂×D₄₇)⋊C₄	—	4	2	no	—
105	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	4	1	no	—
109	C₄⋊Dic₄₇	—	4	1	no	—
110	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
111	Dic₄₇⋊C₄	—	4	2	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
114	C₈×D₄₇	—	2	2	no	—
115	C₈×D₄₇	—	2	2	no	—
116	C₁₆×C₄₇	—	2	1	no	—
117	C₁₆×C₄₇	#41	1	2	yes	complemented
118	C₄₇⋊C₁₆	—	2	2	no	—
119	C₂³×C₄₇	—	4	1	no	—
120	C₂³×C₄₇	—	4	1	no	—
121	C₂²×D₄₇	C₂×D₈	1	2	yes	—
122	D₄×C₄₇	—	4	1	no	—
123	D₄×C₄₇	—	4	1	no	—
124	D₄×C₄₇	—	4	1	no	—
125	D₄×C₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	4	1	no	—
127	C₂×C₄×C₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	—	8	2	no	—
130	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₄₇	—	8	4	no	—
134	C₂²⋊D₄₇	—	16	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
138	D₁₈₈	—	2	2	no	—
139	C₄×D₄₇	—	2	2	no	—
140	C₄×D₄₇	—	2	2	no	—
141	C₄×D₄₇	—	8	4	no	—
142	Dic₉₄	—	8	2	no	—
143	Dic₉₄	—	16	1	no	—
144	C₈×C₄₇	—	2	1	no	—
145	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄₇⋊C₈	—	2	2	no	—
148	#33921	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	8	1	no	—
150	C₂²×C₄₇	—	8	1	no	—
151	C₂²×C₄₇	—	8	1	no	—
152	C₂²×C₄₇	—	8	1	no	—
153	C₂²×C₄₇	—	8	1	no	—
154	C₂²×C₄₇	—	8	1	no	—
155	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₇	—	2	2	no	—
157	C₂×D₄₇	—	2	2	no	—
158	C₂×D₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	8	1	no	—
161	C₄×C₄₇	—	8	1	no	—
162	C₄×C₄₇	#509	1	2	yes	—
163	C₄×C₄₇	#547	1	1	yes	characteristic
164	C₄×C₄₇	#578	1	1	yes	characteristic
165	Dic₄₇	—	2	2	no	—
166	Dic₄₇	—	8	2	no	—
167	Dic₄₇	—	8	2	no	—
168	#3341	—	47	1	no	—
169	#3334	—	47	1	no	—
170	#2254	—	47	1	no	—
171	#2234	—	47	2	no	—
172	#2233	—	47	2	no	—
173	#2226	—	47	1	no	—
174	#2224	—	47	1	no	—
175	#2218	—	47	2	no	—
176	C₄×D₁₆	D₄₇	1	1	yes	characteristic, complemented
177	#2169	—	47	1	no	—
178	#2159	—	47	2	no	—
179	C₂×C₄₇	—	16	1	no	—
180	C₂×C₄₇	—	16	1	no	—
181	C₂×C₄₇	#3334	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
183	C₂×C₄₇	#3416	1	1	yes	characteristic
184	D₄₇	—	4	2	no	—
185	#555	—	94	1	no	—
186	#547	—	94	1	no	—
187	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
188	C₂×M₃₂	—	47	2	no	—
189	#500	—	47	1	no	—
190	#500	—	47	1	no	—
191	#482	—	94	2	no	—
192	#480	—	94	2	no	—
193	#476	—	94	1	no	—
194	#474	—	94	1	no	—
195	#466	—	47	2	no	—
196	#466	—	47	2	no	—
197	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
198	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
199	C₈×D₄	—	47	1	no	—
200	#427	—	47	2	no	—
201	C₂×#107	—	47	2	no	—
202	#367	—	47	2	no	—
203	#367	—	47	2	no	—
204	#367	—	47	2	no	—
205	#366	—	47	2	no	—
206	#366	C₂×D₄₇	1	1	yes	characteristic
207	C₄⋊C₁₆	—	47	1	no	—
208	#357	—	47	2	no	—
209	#357	—	47	2	no	—
210	#357	C₂×D₄₇	1	1	yes	characteristic
211	#357	C₂×D₄₇	1	1	yes	characteristic
212	C₂²⋊C₁₆	—	47	2	no	—
213	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
214	C₄₇	#33921	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	188	1	no	—
216	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₈	—	47	2	no	—
219	C₄⋊Q₈	—	94	1	no	—
220	C₄⋊Q₈	—	94	1	no	—
221	#124	—	188	2	no	—
222	#123	—	188	2	no	—
223	C₂²⋊Q₈	—	188	1	no	—
224	#121	—	94	2	no	—
225	#121	—	94	2	no	—
226	#121	—	188	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	47	1	no	—
230	#117	—	94	2	no	—
231	#117	—	188	1	no	—
232	C₂×C₄⋊C₄	—	94	2	no	—
233	D₁₆	—	2	2	no	—
234	M₃₂	—	94	4	no	—
235	C₂×C₁₆	—	47	2	no	—
236	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
238	#107	—	47	2	no	—
239	#107	—	47	2	no	—
240	#107	—	94	2	no	—
241	#107	—	94	1	no	—
242	#107	—	94	1	no	—
243	#107	C₂²×D₄₇	1	1	yes	characteristic
244	#107	C₂²×D₄₇	1	1	yes	characteristic
245	#106	—	94	2	no	—
246	C₄⋊C₈	—	47	1	no	—
247	#102	—	2	1	no	—
248	#102	—	2	1	no	—
249	#102	—	94	2	no	—
250	#102	—	94	2	no	—
251	C₂²⋊C₈	—	47	2	no	—
252	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
253	#41	—	376	4	no	—
254	C₂×Q₈	—	188	1	no	—
255	C₂×Q₈	—	188	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	47	1	no	—
259	C₂×D₄	—	188	2	no	—
260	C₂×D₄	—	188	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	47	2	no	—
264	C₂²×C₄	—	188	2	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	1	no	—
269	C₂×C₈	—	47	2	no	—
270	C₂×C₈	—	94	2	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	47	1	no	—
277	C₄⋊C₄	—	94	2	no	—
278	C₄⋊C₄	—	94	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	—	188	1	no	—
281	C₄⋊C₄	—	188	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	47	2	no	—
285	C₂²⋊C₄	—	188	2	no	—
286	C₂²⋊C₄	—	188	2	no	—
287	C₂²⋊C₄	—	188	2	no	—
288	C₄²	—	188	2	no	—
289	C₄²	D₄×D₄₇	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	94	2	no	—
292	C₁₆	#10625350	1	2	yes	complemented
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	47	2	no	—
296	Q₈	—	376	2	no	—
297	Q₈	—	752	1	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	94	2	no	—
303	D₄	—	376	4	no	—
304	D₄	—	752	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	94	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	376	4	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
316	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	94	2	no	—
320	C₈	C₂×#10625350	1	1	yes	characteristic
321	C₈	#13094071	1	1	yes	characteristic
322	C₂²	—	8	1	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	94	2	no	—
329	C₂²	—	94	2	no	—
330	C₂²	—	94	2	no	—
331	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	94	2	no	—
336	C₄	—	376	2	no	—
337	C₄	—	376	2	no	—
338	C₄	#15689557	1	2	yes	—
339	C₄	#15689778	1	1	yes	characteristic
340	C₄	#15689937	1	1	yes	characteristic
341	C₂	—	16	1	no	—
342	C₂	—	16	1	no	—
343	C₂	—	188	2	no	—
344	C₂	#20255737	1	1	yes	characteristic
345	C₂	C₂×#15689557	1	1	yes	characteristic
346	C₂	#20256531	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×D₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₄×D₁₆	D₄₇
hypocenter	C₄₇	#33921
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20255812	C₂	semidirect
2. #20255737	C₂	semidirect
3. #20245670	C₂	semidirect
4. #20245578	C₂	semidirect
5. #20245572	C₂	semidirect
6. #20245535	C₂	semidirect
7. #20245525	C₂	semidirect
8. #20245488	C₂	semidirect
9. #20245221	C₂	semidirect
10. #20245173	C₂	semidirect
11. C₄×D₁₆×C₄₇	C₂	semidirect
117. C₁₆×C₄₇	#41	semidirect
176. C₄×D₁₆	D₄₇	semidirect
214. C₄₇	#33921	semidirect
292. C₁₆	#10625350	semidirect

Maximal subgroups

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

subgroup	index
#20255812	2
#20255737	2
#20245670	2
#20245578	2
#20245572	2
#20245535	2
#20245525	2
#20245488	2
#20245221	2
#20245173	2
C₄×D₁₆×C₄₇	2
#33921	47