Subgroups — #35313253

Order 21248.

574 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127750	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127471	C₂	1	2	yes	maximal
3	C₂×#19127466	C₂	1	2	yes	maximal
4	C₂×C₈₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13050574	C₂²	1	2	yes	—
7	C₂²×#13050574	C₂²	1	2	yes	—
8	#19127750	—	2	2	no	—
9	#19127750	C₄	1	4	yes	complemented
10	C₂×#14581728	—	2	2	no	—
11	#19127471	—	2	4	no	—
12	#19127466	—	2	4	no	—
13	C₈₃⋊#338	—	2	2	no	—
14	C₂²×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
17	#336×C₈₃	—	2	2	no	—
18	C₂×#13050574	—	2	2	no	—
19	C₂×#13050574	—	2	2	no	—
20	C₂×#13050574	—	2	2	no	—
21	C₂×#13050574	—	2	2	no	—
22	C₂×#13050574	—	2	2	no	—
23	C₂×#13050574	—	2	2	no	—
24	C₂×#13050574	—	2	2	no	—
25	C₂×#13050574	—	2	2	no	—
26	C₂×#13050574	—	2	2	no	—
27	C₂×#13050574	—	2	2	no	—
28	C₂×#13050574	C₂×C₄	1	4	yes	—
29	C₂×#13050574	D₄	1	4	yes	—
30	C₂²×C₈₃⋊C₈	—	2	2	no	—
31	C₂²×C₈₃⋊C₈	—	2	2	no	—
32	#14581728	—	2	8	no	—
33	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₈₃	—	2	2	no	—
35	C₂×M₁₆×C₈₃	—	2	2	no	—
36	C₂×M₁₆×C₈₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
38	C₂²×C₈×C₈₃	—	2	1	no	—
39	C₂²×C₈×C₈₃	—	2	2	no	—
40	C₂²×C₈×C₈₃	—	2	2	no	—
41	C₂²×C₈×C₈₃	—	2	1	no	—
42	C₂²×C₈×C₈₃	D₄	1	2	yes	—
43	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
44	#117×C₈₃	—	2	2	no	—
45	#117×C₈₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₈₃	Q₈	1	2	yes	—
48	C₂×C₄²×C₈₃	—	2	1	no	—
49	#13050574	—	2	4	no	—
50	#13050574	—	2	4	no	—
51	#13050574	—	2	4	no	—
52	#13050574	—	2	4	no	—
53	#13050574	—	4	2	no	—
54	#13050574	—	4	2	no	—
55	#13050574	—	4	2	no	—
56	#13050574	—	4	2	no	—
57	C₂×C₈₃⋊C₈	—	2	4	no	—
58	C₂×C₈₃⋊C₈	—	2	4	no	—
59	C₂×C₈₃⋊C₈	—	4	2	no	—
60	C₂×C₈₃⋊C₈	—	4	2	no	—
61	C₂×C₈₃⋊C₈	—	4	2	no	—
62	C₂×C₈₃⋊C₈	—	4	2	no	—
63	C₂⁴×C₈₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₃	—	2	2	no	—
65	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	#41	1	2	yes	—
71	C₂²×C₄×C₈₃	#41	1	2	yes	—
72	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	#41	1	2	yes	—
75	M₁₆×C₈₃	—	2	4	no	—
76	M₁₆×C₈₃	—	2	4	no	—
77	C₂×C₈×C₈₃	—	2	2	no	—
78	C₂×C₈×C₈₃	—	2	2	no	—
79	C₂×C₈×C₈₃	—	2	4	no	—
80	C₂×C₈×C₈₃	—	2	2	no	—
81	C₂×C₈×C₈₃	—	4	1	no	—
82	C₂×C₈×C₈₃	—	4	2	no	—
83	C₂×C₈×C₈₃	—	4	1	no	—
84	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₈₃	—	4	2	no	—
86	C₂²⋊C₄×C₈₃	—	2	4	no	—
87	C₄²×C₈₃	—	4	1	no	—
88	C₄²×C₈₃	—	4	1	no	—
89	C₂³×C₈₃	—	4	2	no	—
90	C₂³×C₈₃	C₄×Q₈	1	1	yes	characteristic
91	C₂³×C₈₃	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂³×C₈₃	#125	1	2	yes	—
94	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₃	—	4	4	no	—
96	C₂×C₄×C₈₃	—	4	2	no	—
97	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#117	1	2	yes	—
99	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₃	#121	1	2	yes	—
102	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
105	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
106	C₂×C₄×C₈₃	#123	1	2	yes	—
107	C₂×C₄×C₈₃	#123	1	2	yes	—
108	C₂×C₄×C₈₃	#123	1	2	yes	—
109	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₈₃	#125	1	2	yes	—
112	C₂×C₄×C₈₃	#126	1	2	yes	—
113	C₂×C₄×C₈₃	#126	1	2	yes	—
114	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
115	C₈×C₈₃	—	2	4	no	—
116	C₈×C₈₃	—	4	2	no	—
117	C₈₃⋊C₈	—	4	4	no	—
118	C₈₃⋊C₈	—	4	4	no	—
119	C₂²×C₈₃	—	2	2	no	—
120	C₂²×C₈₃	—	2	4	no	—
121	C₂²×C₈₃	—	4	1	no	—
122	C₂²×C₈₃	—	4	2	no	—
123	C₂²×C₈₃	—	4	1	no	—
124	C₂²×C₈₃	#387	1	2	yes	—
125	C₂²×C₈₃	#391	1	1	yes	characteristic
126	C₂²×C₈₃	#398	1	2	yes	—
127	C₂²×C₈₃	#399	1	1	yes	characteristic
128	C₂²×C₈₃	#400	1	1	yes	characteristic
129	C₄×C₈₃	—	8	2	no	—
130	C₄×C₈₃	#381	1	1	yes	characteristic
131	C₄×C₈₃	#387	1	2	yes	—
132	C₄×C₈₃	#390	1	1	yes	characteristic
133	C₄×C₈₃	#397	1	2	yes	—
134	C₄×C₈₃	#399	1	1	yes	characteristic
135	C₄×C₈₃	#400	1	1	yes	characteristic
136	#11368	—	83	1	no	maximal, Sylow
137	C₂×C₈₃	—	2	1	no	—
138	C₂×C₈₃	—	2	2	no	—
139	C₂×C₈₃	—	2	1	no	—
140	C₂×C₈₃	—	2	2	no	—
141	C₂×C₈₃	#1445	1	1	yes	characteristic
142	C₂×C₈₃	#2103	1	2	yes	—
143	C₂×#430	—	83	1	no	—
144	C₂×#344	—	83	2	no	—
145	C₂×#341	—	83	2	no	—
146	C₂×#338	—	83	1	no	—
147	#1729	D₈₃	1	1	yes	characteristic
148	C₈₃	#11368	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	83	2	no	—
150	C₂²×M₁₆	—	83	2	no	—
151	C₂²×M₁₆	C₂×D₈₃	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
153	C₂×#117	C₂×D₈₃	1	1	yes	characteristic
154	#430	—	83	4	no	—
155	#430	—	166	2	no	—
156	C₂×#108	—	166	2	no	—
157	#344	—	166	4	no	—
158	#341	—	166	4	no	—
159	#338	—	166	2	no	—
160	#336	—	2	2	no	—
161	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	83	4	no	—
165	C₂×M₁₆	—	83	4	no	—
166	C₂×M₁₆	—	166	2	no	—
167	C₂×M₁₆	—	166	2	no	—
168	C₂×M₁₆	—	166	2	no	—
169	C₂×M₁₆	—	166	2	no	—
170	C₂×M₁₆	—	166	2	no	—
171	C₂×M₁₆	—	166	2	no	—
172	C₂×M₁₆	—	166	2	no	—
173	C₂×M₁₆	—	166	2	no	—
174	C₂×M₁₆	—	166	2	no	—
175	C₂×M₁₆	—	166	2	no	—
176	C₂×M₁₆	C₄×D₈₃	1	2	yes	—
177	C₂×M₁₆	D₃₃₂	1	2	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	166	2	no	—
183	C₂²×C₈	—	166	2	no	—
184	C₂²×C₈	Dic₁₆₆	1	2	yes	—
185	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	Dic₁₆₆	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	#108	—	166	8	no	—
192	C₂⁴	C₂×Dic₁₆₆	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
195	C₂²×C₄	D₄×D₈₃	1	2	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
199	C₂²×C₄	#13050589	1	2	yes	—
200	C₂²×C₄	#13050589	1	2	yes	—
201	C₂²×C₄	#13050590	1	1	yes	characteristic
202	C₂²×C₄	#13050590	1	1	yes	characteristic
203	C₂²×C₄	#13050591	1	2	yes	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	332	2	no	—
207	M₁₆	—	332	2	no	—
208	M₁₆	—	332	2	no	—
209	M₁₆	—	332	2	no	—
210	M₁₆	—	166	4	no	—
211	M₁₆	—	166	4	no	—
212	M₁₆	—	166	4	no	—
213	M₁₆	—	166	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	332	2	no	—
222	C₂×C₈	—	332	2	no	—
223	C₂×C₈	—	332	2	no	—
224	C₂×C₈	—	332	2	no	—
225	C₂×C₈	—	166	4	no	—
226	C₂×C₈	—	166	4	no	—
227	C₂×C₈	(C₂×D₈₃)⋊C₄	1	4	yes	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	1	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₄×Dic₁₆₆	1	1	yes	characteristic
234	C₂³	#14581769	1	2	yes	—
235	C₂³	#14581771	1	1	yes	characteristic
236	C₂³	#14581783	1	2	yes	—
237	C₂³	#14581792	1	1	yes	characteristic
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	#14581748	1	2	yes	—
242	C₂×C₄	#14581754	1	2	yes	—
243	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
244	C₂×C₄	#14581767	1	2	yes	—
245	C₂×C₄	#14581769	1	2	yes	—
246	C₂×C₄	#14581771	1	1	yes	characteristic
247	C₂×C₄	#14581773	1	1	yes	characteristic
248	C₂×C₄	#14581773	1	1	yes	characteristic
249	C₂×C₄	#14581774	1	2	yes	—
250	C₂×C₄	#14581774	1	2	yes	—
251	C₂×C₄	#14581776	1	2	yes	—
252	C₂×C₄	#14581780	1	1	yes	characteristic
253	C₂×C₄	#14581781	1	1	yes	characteristic
254	C₂×C₄	#14581783	1	2	yes	—
255	C₂×C₄	#14581787	1	2	yes	—
256	C₂×C₄	#14581787	1	2	yes	—
257	C₂×C₄	#14581788	1	1	yes	characteristic
258	C₈	—	2	4	no	—
259	C₈	—	4	2	no	—
260	C₈	—	332	4	no	—
261	C₈	—	332	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	#19127584	1	2	yes	—
268	C₂²	#19127607	1	1	yes	characteristic
269	C₂²	#19127633	1	2	yes	—
270	C₂²	#19127637	1	1	yes	characteristic
271	C₂²	#19127640	1	1	yes	characteristic
272	C₄	—	8	2	no	—
273	C₄	#19127562	1	1	yes	characteristic
274	C₄	#19127584	1	2	yes	—
275	C₄	#19127602	1	1	yes	characteristic
276	C₄	#19127628	1	2	yes	—
277	C₄	#19127637	1	1	yes	characteristic
278	C₄	#19127640	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	2	no	—
283	C₂	#24905732	1	1	yes	characteristic
284	C₂	#24909043	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14581781
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₃
Fitting	C₈₃×#1729	C₂
socle	C₂²×C₈₃	#400
prime radical	C₂⁴×C₈₃	C₂×Q₈
cosocle	C₂³×C₄×C₈₃	C₂³
hypercenter	#1729	D₈₃
hypocenter	C₈₃	#11368
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
9. #19127750	C₄	semidirect
148. C₈₃	#11368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127750	2
C₂×#19127471	2
C₂×#19127466	2
C₂×C₈₃⋊#338	2
C₈₃×#1729	2
#11368	83