Subgroups — #35322144

Order 21248.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127486	C₂	1	1	yes	maximal, characteristic
2	#24909078	C₂	1	1	yes	maximal, characteristic
3	#24909062	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907300	C₂	1	1	yes	maximal, characteristic
5	#24905553	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905541	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#364×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581798	C₂²	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14581720	C₄	1	1	yes	characteristic
12	#19127698	C₄	1	1	yes	characteristic
13	#19127656	C₂²	1	1	yes	characteristic
14	#19127486	—	2	2	no	—
15	(C₂×D₈₃)⋊C₁₆	—	2	1	no	—
16	(C₂×D₈₃)⋊C₁₆	—	2	1	no	—
17	#19127476	—	2	1	no	—
18	C₁₆×Dic₈₃	—	2	1	no	—
19	#19127466	C₄	1	1	yes	characteristic
20	#19127461	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#364×C₈₃	—	2	2	no	—
25	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
26	#14581798	—	2	1	no	—
27	#14581798	—	2	1	no	—
28	#14581798	—	2	1	no	—
29	#14581798	—	2	1	no	—
30	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
31	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
32	C₂×#13050573	—	2	1	no	—
33	C₂×C₈×D₈₃	—	2	1	no	—
34	#14581751	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈₃⋊C₁₆	—	2	1	no	—
36	C₂×C₈₃⋊C₁₆	—	2	1	no	—
37	C₂×C₈₃⋊C₁₆	—	2	1	no	—
38	C₂×C₈₃⋊C₁₆	—	2	1	no	—
39	C₂×C₈₃⋊C₁₆	—	2	1	no	—
40	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
42	#14581720	—	2	2	no	—
43	#14581717	C₈	1	1	yes	characteristic
44	#14581717	C₈	1	1	yes	characteristic
45	#14581714	C₈	1	1	yes	characteristic
46	#14581714	C₈	1	1	yes	characteristic
47	#14581702	—	2	1	no	—
48	C₈×Dic₈₃	—	2	1	no	—
49	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₈₃	—	2	1	no	—
53	M₃₂×C₈₃	—	2	1	no	—
54	M₃₂×C₈₃	—	2	1	no	—
55	M₃₂×C₈₃	D₄	1	2	yes	—
56	C₂×C₁₆×C₈₃	—	2	1	no	—
57	C₂×C₁₆×C₈₃	—	2	1	no	—
58	C₄×C₈×C₈₃	—	2	1	no	—
59	C₄×C₈×C₈₃	—	2	1	no	—
60	#13050590	—	2	1	no	—
61	#13050590	—	2	1	no	—
62	#13050590	—	2	1	no	—
63	#13050590	—	2	1	no	—
64	C₂×Dic₁₆₆	C₂×C₈	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂×C₈	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	#13050574	—	2	1	no	—
69	#13050574	—	2	1	no	—
70	#13050574	—	2	1	no	—
71	#13050574	M₁₆	1	2	yes	—
72	#13050573	—	4	1	no	—
73	#13050573	—	4	1	no	—
74	C₂×C₈₃⋊C₈	—	2	1	no	—
75	C₂×C₈₃⋊C₈	—	2	1	no	—
76	C₈×D₈₃	—	4	1	no	—
77	C₈×D₈₃	—	4	1	no	—
78	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
80	C₂²⋊Dic₈₃	—	2	1	no	—
81	C₄×Dic₈₃	—	2	1	no	—
82	C₈₃⋊C₁₆	—	2	1	no	—
83	C₈₃⋊C₁₆	—	2	1	no	—
84	C₈₃⋊C₁₆	—	4	1	no	—
85	C₂²×C₄×C₈₃	—	2	1	no	—
86	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₃	—	2	1	no	—
88	C₂×C₈×C₈₃	—	2	1	no	—
89	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
97	C₄²×C₈₃	—	2	1	no	—
98	C₄²×C₈₃	—	2	1	no	—
99	C₁₆×C₈₃	—	2	2	no	—
100	C₁₆×C₈₃	—	4	1	no	—
101	C₂³×C₈₃	#131	1	1	yes	characteristic
102	C₂²×D₈₃	—	2	1	no	—
103	C₂×C₄×C₈₃	—	2	2	no	—
104	C₂×C₄×C₈₃	—	4	1	no	—
105	C₂×C₄×C₈₃	—	4	1	no	—
106	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
112	C₂²⋊D₈₃	—	4	1	no	—
113	C₂²⋊D₈₃	—	4	1	no	—
114	C₂×Dic₈₃	—	2	1	no	—
115	C₂×Dic₈₃	—	2	1	no	—
116	C₂×Dic₈₃	—	2	1	no	—
117	D₃₃₂	—	2	1	no	—
118	D₃₃₂	—	2	1	no	—
119	C₄×D₈₃	—	4	1	no	—
120	C₄×D₈₃	—	4	1	no	—
121	Dic₁₆₆	—	2	1	no	—
122	Dic₁₆₆	—	2	1	no	—
123	C₈×C₈₃	—	2	1	no	—
124	C₈×C₈₃	—	2	1	no	—
125	C₈×C₈₃	#121	1	1	yes	characteristic
126	C₈×C₈₃	#123	1	1	yes	characteristic
127	C₈×C₈₃	#124	1	1	yes	characteristic
128	C₈×C₈₃	#126	1	1	yes	characteristic
129	C₈₃⋊C₈	—	2	2	no	—
130	C₈₃⋊C₈	—	4	1	no	—
131	C₂²×C₈₃	—	2	1	no	—
132	C₂²×C₈₃	—	2	1	no	—
133	C₂²×C₈₃	#433	1	1	yes	characteristic
134	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
135	C₂²×C₈₃	#436	1	1	yes	characteristic
136	C₂×D₈₃	—	4	1	no	—
137	C₂×D₈₃	—	4	1	no	—
138	C₂×D₈₃	—	4	1	no	—
139	C₄×C₈₃	—	4	2	no	—
140	C₄×C₈₃	#388	1	1	yes	characteristic
141	C₄×C₈₃	#431	1	1	yes	characteristic
142	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
143	C₄×C₈₃	#435	1	1	yes	characteristic
144	Dic₈₃	—	4	1	no	—
145	Dic₈₃	—	4	1	no	—
146	Dic₈₃	—	4	1	no	—
147	#13122	—	83	1	no	maximal, Sylow
148	C₂×C₈₃	—	2	1	no	—
149	C₂×C₈₃	—	2	1	no	—
150	C₂×C₈₃	#1923	1	1	yes	characteristic
151	C₂×C₈₃	#2170	1	1	yes	characteristic
152	C₂×C₈₃	#2171	1	1	yes	characteristic
153	D₈₃	—	8	1	no	—
154	C₂×#364	D₈₃	1	1	yes	characteristic, complemented
155	C₂×#350	—	83	1	no	—
156	#2113	—	83	1	no	—
157	#2108	—	83	1	no	—
158	#1764	—	83	1	no	—
159	#1379	—	83	1	no	—
160	#1374	—	83	1	no	—
161	C₈₃	#13122	1	1	yes	characteristic, complemented, Sylow
162	C₂×#131	—	83	1	no	—
163	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
165	C₂²×C₁₆	—	83	1	no	—
166	C₂²×C₁₆	—	83	1	no	—
167	C₂×#104	—	83	1	no	—
168	#416	—	83	1	no	—
169	#405	—	83	1	no	—
170	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
171	#364	—	2	2	no	—
172	#350	—	166	2	no	—
173	C₂²⋊C₁₆	—	166	1	no	—
174	C₂²⋊C₁₆	—	166	1	no	—
175	#346	—	166	1	no	—
176	C₄×C₁₆	—	166	1	no	—
177	#341	—	83	1	no	—
178	#339	—	83	1	no	—
179	C₂×#41	—	83	1	no	—
180	#131	—	166	1	no	—
181	#131	—	166	1	no	—
182	#131	—	166	1	no	—
183	#131	—	166	1	no	—
184	C₂×M₁₆	—	83	1	no	—
185	C₂×M₁₆	—	83	1	no	—
186	C₂×M₁₆	—	166	1	no	—
187	C₂²×C₈	—	166	1	no	—
188	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
190	#117	—	83	1	no	—
191	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
192	M₃₂	—	2	1	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	M₃₂	C₂²⋊D₈₃	1	2	yes	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	83	1	no	—
199	C₂×C₁₆	—	83	1	no	—
200	C₂×C₁₆	—	166	1	no	—
201	C₂×C₁₆	—	166	1	no	—
202	C₂×C₁₆	—	166	1	no	—
203	C₂×C₁₆	—	166	1	no	—
204	C₂×C₁₆	—	166	1	no	—
205	#104	—	166	2	no	—
206	#103	—	83	1	no	—
207	#103	—	83	1	no	—
208	#102	—	83	1	no	—
209	#102	—	83	1	no	—
210	#97	—	166	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	166	1	no	—
214	#41	—	166	1	no	—
215	#41	—	166	1	no	—
216	#41	—	166	1	no	—
217	#41	—	166	1	no	—
218	C₂×Q₈	—	83	1	no	—
219	C₂×D₄	—	83	1	no	—
220	C₂×D₄	—	166	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	166	1	no	—
223	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
224	M₁₆	—	332	1	no	—
225	M₁₆	—	332	1	no	—
226	M₁₆	—	83	2	no	—
227	M₁₆	—	166	1	no	—
228	M₁₆	—	166	1	no	—
229	M₁₆	—	166	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	332	1	no	—
233	C₂×C₈	—	332	1	no	—
234	C₂×C₈	—	166	1	no	—
235	C₂×C₈	—	166	1	no	—
236	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
237	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
238	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
240	C₂×C₈	#13050589	1	1	yes	characteristic
241	C₂×C₈	#13050590	1	1	yes	characteristic
242	C₂×C₈	#13050591	1	1	yes	characteristic
243	C₂×C₈	#13050591	1	1	yes	characteristic
244	C₄⋊C₄	—	83	1	no	—
245	C₄⋊C₄	—	83	1	no	—
246	C₂²⋊C₄	—	166	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	166	1	no	—
250	C₁₆	—	2	2	no	—
251	C₁₆	—	4	1	no	—
252	C₁₆	—	332	1	no	—
253	C₁₆	—	166	1	no	—
254	C₁₆	—	166	1	no	—
255	C₂³	—	166	1	no	—
256	C₂³	#14581800	1	1	yes	characteristic
257	Q₈	—	166	1	no	—
258	Q₈	—	166	1	no	—
259	D₄	—	332	1	no	—
260	D₄	—	332	1	no	—
261	D₄	—	166	1	no	—
262	D₄	—	166	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	332	1	no	—
267	C₂×C₄	—	332	1	no	—
268	C₂×C₄	—	166	1	no	—
269	C₂×C₄	—	166	1	no	—
270	C₂×C₄	—	166	1	no	—
271	C₂×C₄	#14581750	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
273	C₂×C₄	#14581754	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
276	C₂×C₄	#14581798	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	332	1	no	—
280	C₈	—	166	2	no	—
281	C₈	#14581764	1	1	yes	characteristic
282	C₈	#14581777	1	1	yes	characteristic
283	C₈	#14581781	1	1	yes	characteristic
284	C₈	#14581787	1	1	yes	characteristic
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	332	1	no	—
288	C₂²	—	332	1	no	—
289	C₂²	—	332	1	no	—
290	C₂²	#19127761	1	1	yes	characteristic
291	C₂²	#19127766	1	1	yes	characteristic
292	C₂²	#19127777	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	332	1	no	—
295	C₄	—	332	1	no	—
296	C₄	—	332	1	no	—
297	C₄	#19127591	1	1	yes	characteristic
298	C₄	#19127753	1	1	yes	characteristic
299	C₄	C₈×C₂²⋊D₈₃	1	1	yes	characteristic
300	C₄	#19127771	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	664	1	no	—
304	C₂	#24908068	1	1	yes	characteristic
305	C₂	#24909299	1	1	yes	characteristic
306	C₂	#24909304	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#364×C₈₃	C₂
socle	C₂²×C₈₃	C₈×D₄
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#364	D₈₃
hypocenter	C₈₃	#13122
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #24909062	C₂	semidirect
5. #24905553	C₂	semidirect
6. #24905541	C₂	semidirect
7. C₂×#364×C₈₃	C₂	semidirect
154. C₂×#364	D₈₃	semidirect
161. C₈₃	#13122	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127486	2
#24909078	2
#24909062	2
#24907300	2
#24905553	2
#24905541	2
C₂×#364×C₈₃	2
#13122	83