Subgroups — #35559151

Order 21248.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917991	C₂	1	1	yes	maximal, characteristic, complemented
2	#24916380	C₂	1	1	yes	maximal, characteristic
3	#24909637	C₂	1	4	yes	maximal, complemented
4	#24909632	C₂	1	4	yes	maximal, complemented
5	#24909236	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127523	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909228	C₂	1	1	yes	maximal, characteristic, complemented
8	#24909199	C₂	1	1	yes	maximal, characteristic
9	C₂×#360×C₈₃	C₂	1	1	yes	maximal, characteristic
10	#19128468	—	2	1	no	—
11	#19128332	—	2	1	no	—
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	#19128108	—	2	2	no	—
14	#19128101	C₂²	1	2	yes	complemented
15	#19128098	C₂²	1	2	yes	complemented
16	#19127823	—	2	2	no	—
17	#19127818	C₂²	1	4	yes	—
18	C₂×#14581728	C₂²	1	1	yes	characteristic
19	#19127741	—	2	1	no	—
20	#19127739	C₂²	1	1	yes	characteristic
21	#19127718	—	2	1	no	—
22	#19127656	C₂²	1	1	yes	characteristic
23	#19127524	C₂²	1	2	yes	complemented
24	#19127524	C₂²	1	2	yes	complemented
25	#19127523	C₂²	1	4	yes	complemented
26	#19127521	C₂²	1	4	yes	complemented
27	#19127511	C₂²	1	2	yes	—
28	#19127511	C₂²	1	2	yes	—
29	C₂×#135×C₈₃	C₂²	1	1	yes	characteristic
30	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×#106×C₈₃	C₂²	1	1	yes	characteristic
32	#360×C₈₃	C₂²	1	4	yes	—
33	#14581799	—	2	4	no	—
34	#14581799	—	4	2	no	—
35	C₂×#13050574	—	2	1	no	—
36	C₂×#13050574	—	4	2	no	—
37	C₂×#13050574	D₄	1	1	yes	characteristic
38	#14581791	—	2	2	no	—
39	#14581791	—	4	1	no	—
40	#14581783	—	2	2	no	—
41	#14581783	—	4	1	no	—
42	#14581771	—	4	2	no	—
43	#14581751	—	2	1	no	—
44	#14581751	D₄	1	1	yes	characteristic
45	#14581732	—	2	1	no	—
46	#14581732	C₂³	1	2	yes	—
47	C₂×C₈₃⋊C₁₆	—	2	1	no	—
48	#14581728	—	2	1	no	—
49	#14581728	C₂³	1	2	yes	—
50	#14581726	—	2	1	no	—
51	#14581725	—	2	2	no	—
52	#14581724	C₂³	1	1	yes	characteristic
53	#14581724	C₂³	1	1	yes	characteristic
54	#14581723	—	2	2	no	—
55	#14581719	—	2	4	no	—
56	#14581702	D₄	1	2	yes	—
57	C₈×Dic₈₃	C₂³	1	2	yes	—
58	C₂×#41×C₈₃	—	2	1	no	—
59	#135×C₈₃	—	2	1	no	—
60	#135×C₈₃	—	2	1	no	—
61	#135×C₈₃	Q₈	1	2	yes	complemented
62	#135×C₈₃	C₂³	1	2	yes	—
63	C₂×Q₁₆×C₈₃	Q₈	1	1	yes	characteristic, complemented
64	C₂×SD₁₆×C₈₃	—	2	1	no	—
65	C₂×D₈×C₈₃	Q₈	1	1	yes	characteristic, complemented
66	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
68	M₃₂×C₈₃	—	2	1	no	—
69	M₃₂×C₈₃	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₃	—	2	1	no	—
71	#106×C₈₃	—	2	1	no	—
72	#106×C₈₃	C₂³	1	2	yes	—
73	C₂×Dic₁₆₆	—	4	2	no	—
74	#13050574	—	2	2	no	—
75	#13050574	—	4	1	no	—
76	#13050574	—	4	4	no	—
77	#13050574	SD₁₆	1	2	yes	—
78	#13050574	SD₁₆	1	2	yes	—
79	C₂×C₈₃⋊C₈	—	4	1	no	—
80	C₂×C₈₃⋊C₈	—	4	4	no	—
81	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	—	2	1	no	—
84	C₄⋊Dic₈₃	—	2	1	no	—
85	C₄⋊Dic₈₃	—	4	2	no	—
86	Dic₈₃⋊C₄	—	4	2	no	—
87	Dic₈₃⋊C₄	—	4	2	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	4	1	no	—
90	C₄×Dic₈₃	—	4	1	no	—
91	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
92	C₈₃⋊C₁₆	—	2	2	no	—
93	#41×C₈₃	—	2	2	no	—
94	#41×C₈₃	—	2	2	no	—
95	#41×C₈₃	—	4	1	no	—
96	#41×C₈₃	—	4	1	no	—
97	C₂×Q₈×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	—	2	1	no	—
99	C₂×D₄×C₈₃	—	4	1	no	—
100	C₂²×C₄×C₈₃	—	4	1	no	—
101	C₂²×C₄×C₈₃	—	4	1	no	—
102	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₈₃	—	2	1	no	—
104	Q₁₆×C₈₃	C₂×Q₈	1	2	yes	—
105	SD₁₆×C₈₃	—	2	2	no	—
106	SD₁₆×C₈₃	—	2	2	no	—
107	D₈×C₈₃	—	2	1	no	—
108	D₈×C₈₃	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₃	—	2	2	no	—
116	C₄⋊C₄×C₈₃	—	4	1	no	—
117	C₁₆×C₈₃	—	2	2	no	—
118	C₂³×C₈₃	—	4	1	no	—
119	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₈₃	—	2	2	no	—
121	Q₈×C₈₃	—	4	1	no	—
122	D₄×C₈₃	—	2	2	no	—
123	D₄×C₈₃	—	4	2	no	—
124	D₄×C₈₃	—	4	2	no	—
125	D₄×C₈₃	—	4	1	no	—
126	C₂×C₄×C₈₃	—	4	1	no	—
127	C₂×C₄×C₈₃	—	4	2	no	—
128	C₂×C₄×C₈₃	—	4	2	no	—
129	C₂×C₄×C₈₃	—	4	2	no	—
130	C₂×C₄×C₈₃	—	16	1	no	—
131	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₈₃	—	2	2	no	—
138	C₂×Dic₈₃	—	4	1	no	—
139	C₂×Dic₈₃	—	4	1	no	—
140	Dic₁₆₆	—	16	2	no	—
141	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
144	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
145	C₈₃⋊C₈	—	4	2	no	—
146	C₈₃⋊C₈	C₂×SD₁₆	1	4	yes	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	—	4	2	no	—
149	C₂²×C₈₃	—	16	1	no	—
150	C₂²×C₈₃	#476	1	1	yes	characteristic
151	C₂²×C₈₃	#476	1	1	yes	characteristic
152	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₈₃	—	4	2	no	—
154	C₄×C₈₃	—	8	2	no	—
155	C₄×C₈₃	#476	1	1	yes	characteristic
156	C₄×C₈₃	#476	1	1	yes	characteristic
157	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₈₃	—	4	2	no	—
160	Dic₈₃	—	8	1	no	—
161	Dic₈₃	—	8	1	no	—
162	#33638	—	83	1	no	maximal, Sylow
163	C₂×C₈₃	—	2	1	no	—
164	C₂×C₈₃	—	2	1	no	—
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	—	8	2	no	—
167	C₂×C₈₃	C₂×#476	1	1	yes	characteristic
168	#3163	—	83	1	no	—
169	#2957	—	83	1	no	—
170	#2231	—	83	4	no	—
171	#2230	—	83	4	no	—
172	#2155	—	83	1	no	—
173	C₂×#365	—	83	1	no	—
174	#2153	—	83	1	no	—
175	#2145	—	83	1	no	—
176	C₂×#360	D₈₃	1	1	yes	characteristic
177	C₈₃	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	C₂×D₈₃	1	1	yes	characteristic
179	#568	—	166	1	no	—
180	#544	—	166	1	no	—
181	C₂×M₃₂	—	83	1	no	—
182	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
183	#501	—	166	2	no	—
184	#499	—	83	2	no	—
185	#498	—	83	2	no	—
186	#444	—	166	2	no	—
187	#443	—	83	4	no	—
188	C₂×#108	—	83	1	no	—
189	#428	—	166	1	no	—
190	#427	—	83	1	no	—
191	C₂×#106	C₂×D₈₃	1	1	yes	characteristic
192	#421	—	166	1	no	—
193	#405	—	83	1	no	—
194	#365	—	83	2	no	—
195	#365	—	83	2	no	—
196	#365	—	83	4	no	—
197	#364	—	83	4	no	—
198	#360	—	83	2	no	—
199	#360	—	83	2	no	—
200	#360	C₂×D₈₃	1	4	yes	—
201	C₂×#41	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	Dic₁₆₆	1	2	yes	complemented
205	#135	C₂²×D₈₃	1	2	yes	—
206	C₂×Q₁₆	Dic₁₆₆	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	Dic₁₆₆	1	1	yes	characteristic, complemented
209	#131	—	332	2	no	—
210	#131	—	166	4	no	—
211	C₂×M₁₆	—	332	2	no	—
212	C₂×M₁₆	—	83	1	no	—
213	C₂×M₁₆	—	166	1	no	—
214	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	332	1	no	—
216	C₄⋊Q₈	—	166	2	no	—
217	#125	—	332	1	no	—
218	#125	—	166	2	no	—
219	C₂²⋊Q₈	—	332	2	no	—
220	#117	—	83	1	no	—
221	#117	—	166	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	83	2	no	—
225	M₃₂	—	166	1	no	—
226	M₃₂	C₂²×D₈₃	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	166	1	no	—
229	#108	—	83	2	no	—
230	#108	—	166	1	no	—
231	#107	—	166	2	no	—
232	#107	—	166	1	no	—
233	#106	—	2	1	no	—
234	#106	—	83	1	no	—
235	#106	—	83	1	no	—
236	#106	—	166	2	no	—
237	#106	C₂²×D₈₃	1	2	yes	—
238	#104	—	166	4	no	—
239	#97	—	83	2	no	—
240	C₄×C₈	—	83	2	no	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	332	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Dic₁₆₆	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Dic₁₆₆	1	2	yes	—
258	M₁₆	—	332	1	no	—
259	M₁₆	—	332	4	no	—
260	M₁₆	—	83	2	no	—
261	M₁₆	—	83	2	no	—
262	M₁₆	—	166	2	no	—
263	C₂×C₈	—	332	1	no	—
264	C₂×C₈	—	332	4	no	—
265	C₂×C₈	—	83	2	no	—
266	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
267	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
268	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
269	C₂×C₈	#13050589	1	1	yes	characteristic
270	C₂×C₈	#13050589	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	332	2	no	—
275	C₄⋊C₄	—	332	2	no	—
276	C₄⋊C₄	—	332	2	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	—	166	1	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	332	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₄²	—	332	1	no	—
283	C₄²	—	83	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	166	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#14581816	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	1328	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	332	1	no	—
301	C₂×C₄	—	332	1	no	—
302	C₂×C₄	—	166	2	no	—
303	C₂×C₄	#14581769	1	1	yes	characteristic
304	C₂×C₄	#14581769	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
307	C₂×C₄	#14581816	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
309	C₈	—	332	2	no	—
310	C₈	—	83	4	no	—
311	C₈	#14581769	1	1	yes	characteristic
312	C₈	#14581769	1	1	yes	characteristic
313	C₈	C₂²×Dic₁₆₆	1	1	yes	characteristic
314	C₈	C₂×#13050589	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#19127989	1	1	yes	characteristic
319	C₂²	#19127989	1	1	yes	characteristic
320	C₂²	C₂×#14581816	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	332	2	no	—
324	C₄	—	664	1	no	—
325	C₄	—	664	1	no	—
326	C₄	#19127989	1	1	yes	characteristic
327	C₄	#19127989	1	1	yes	characteristic
328	C₄	C₂×#14581769	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#19127989	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19127989
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#360×C₈₃	C₂
socle	C₂×C₈₃	C₂×#476
prime radical	C₂×#135×C₈₃	C₂²
hypercenter	C₂×#360	D₈₃
hypocenter	C₈₃	#33638
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. #24917991	C₂	semidirect
3. #24909637	C₂	semidirect
4. #24909632	C₂	semidirect
5. #24909236	C₂	semidirect
6. C₂×#19127523	C₂	semidirect
7. #24909228	C₂	semidirect
14. #19128101	C₂²	semidirect
15. #19128098	C₂²	semidirect
23. #19127524	C₂²	semidirect
24. #19127524	C₂²	semidirect
25. #19127523	C₂²	semidirect
26. #19127521	C₂²	semidirect
61. #135×C₈₃	Q₈	semidirect
63. C₂×Q₁₆×C₈₃	Q₈	semidirect
65. C₂×D₈×C₈₃	Q₈	semidirect
177. C₈₃	#33638	semidirect
204. #135	Dic₁₆₆	semidirect
206. C₂×Q₁₆	Dic₁₆₆	semidirect
208. C₂×D₈	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24917991	2
#24916380	2
#24909637	2
#24909632	2
#24909236	2
C₂×#19127523	2
#24909228	2
#24909199	2
C₂×#360×C₈₃	2
#33638	83