Subgroups — #35314331

Order 21248.

360 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909266	C₂	1	1	yes	maximal, characteristic, complemented
2	#24909262	C₂	1	1	yes	maximal, characteristic, complemented
3	#24905421	C₂	1	4	yes	maximal, complemented
4	C₈₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
6	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₈₃⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₈₃⋊C₁₆)	—	2	1	no	—
9	#19127477	C₂²	1	2	yes	—
10	C₄×C₈₃⋊C₁₆	—	2	1	no	—
11	C₂×#131×C₈₃	—	2	1	no	—
12	#532×C₈₃	C₄	1	1	yes	characteristic
13	#436×C₈₃	—	2	1	no	—
14	#436×C₈₃	C₂²	1	2	yes	—
15	C₈×D₄×C₈₃	—	2	1	no	—
16	C₈×D₄×C₈₃	C₄	1	2	yes	—
17	#423×C₈₃	C₄	1	1	yes	characteristic
18	#408×C₈₃	—	2	1	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₈₃⋊C₁₆	—	2	2	no	—
21	C₂×C₈₃⋊C₁₆	—	2	2	no	—
22	C₂×#41×C₈₃	—	2	1	no	—
23	#131×C₈₃	—	4	4	no	—
24	C₂×M₁₆×C₈₃	—	2	1	no	—
25	C₂×M₁₆×C₈₃	—	2	2	no	—
26	C₂²×C₈×C₈₃	—	2	2	no	—
27	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₈₃	C₈	1	1	yes	characteristic
30	#127×C₈₃	C₈	1	1	yes	characteristic
31	#124×C₈₃	—	2	1	no	—
32	#121×C₈₃	—	2	1	no	—
33	#121×C₈₃	C₈	1	2	yes	—
34	C₄×D₄×C₈₃	—	2	1	no	—
35	C₄×D₄×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₈₃	—	2	1	no	—
38	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
39	C₂²⋊C₈×C₈₃	—	2	2	no	—
40	C₂²⋊C₈×C₈₃	—	2	2	no	—
41	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₈₃⋊C₁₆	—	4	2	no	—
46	C₈₃⋊C₁₆	—	4	2	no	—
47	#41×C₈₃	—	4	2	no	—
48	#41×C₈₃	—	4	2	no	—
49	C₂×Q₈×C₈₃	—	2	1	no	—
50	C₂×D₄×C₈₃	—	2	2	no	—
51	C₂×D₄×C₈₃	—	2	1	no	—
52	C₂×D₄×C₈₃	—	2	1	no	—
53	C₂×D₄×C₈₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₈₃	—	2	2	no	—
55	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
58	M₁₆×C₈₃	—	4	2	no	—
59	M₁₆×C₈₃	—	4	4	no	—
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	—	2	2	no	—
62	C₂×C₈×C₈₃	—	4	1	no	—
63	C₂×C₈×C₈₃	—	4	4	no	—
64	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₈₃	—	2	1	no	—
69	C₄⋊C₄×C₈₃	C₂×C₈	1	2	yes	—
70	C₂²⋊C₄×C₈₃	—	2	2	no	—
71	C₂²⋊C₄×C₈₃	—	2	2	no	—
72	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
75	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₈₃	—	2	2	no	—
77	C₂³×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
78	Q₈×C₈₃	—	4	2	no	—
79	D₄×C₈₃	—	4	2	no	—
80	D₄×C₈₃	—	4	2	no	—
81	D₄×C₈₃	—	4	4	no	—
82	D₄×C₈₃	—	4	2	no	—
83	C₂×C₄×C₈₃	—	2	2	no	—
84	C₂×C₄×C₈₃	—	2	1	no	—
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	—	4	1	no	—
88	C₂×C₄×C₈₃	—	4	2	no	—
89	C₂×C₄×C₈₃	—	4	2	no	—
90	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₈₃	—	2	1	no	—
97	C₈×C₈₃	—	2	1	no	—
98	C₈×C₈₃	—	2	1	no	—
99	C₈×C₈₃	—	4	2	no	—
100	C₈×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
101	C₂²×C₈₃	—	2	1	no	—
102	C₂²×C₈₃	—	2	1	no	—
103	C₂²×C₈₃	—	2	1	no	—
104	C₂²×C₈₃	—	4	2	no	—
105	C₂²×C₈₃	—	4	2	no	—
106	C₂²×C₈₃	—	4	2	no	—
107	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₈₃	—	2	1	no	—
109	C₄×C₈₃	—	2	1	no	—
110	C₄×C₈₃	—	2	1	no	—
111	C₄×C₈₃	—	2	1	no	—
112	C₄×C₈₃	—	4	2	no	—
113	C₄×C₈₃	#350	1	2	yes	—
114	C₄×C₈₃	#418	1	1	yes	characteristic
115	C₄×C₈₃	#421	1	1	yes	characteristic
116	#11633	—	83	1	no	maximal, Sylow
117	C₂×C₈₃	—	4	1	no	—
118	C₂×C₈₃	—	8	2	no	—
119	C₂×C₈₃	#1478	1	1	yes	characteristic
120	C₂×C₈₃	#2115	1	1	yes	characteristic
121	C₂×C₈₃	C₂×#350	1	1	yes	characteristic
122	#2931	D₈₃	1	1	yes	characteristic
123	#2163	—	83	1	no	—
124	#2162	—	83	1	no	—
125	#1330	—	83	4	no	—
126	C₈₃	#11633	1	1	yes	characteristic, complemented, Sylow
127	C₂×#131	—	2	1	no	—
128	#532	Dic₈₃	1	1	yes	characteristic
129	#436	—	2	1	no	—
130	#436	C₂×D₈₃	1	2	yes	—
131	C₈×D₄	—	2	1	no	—
132	C₈×D₄	Dic₈₃	1	2	yes	—
133	#423	Dic₈₃	1	1	yes	characteristic
134	#408	—	2	1	no	—
135	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
136	C₄⋊C₁₆	—	83	2	no	—
137	C₄⋊C₁₆	—	166	1	no	—
138	C₂²⋊C₁₆	—	166	1	no	—
139	C₂²⋊C₁₆	—	166	1	no	—
140	#346	—	83	2	no	—
141	C₄×C₁₆	—	166	1	no	—
142	C₂×#41	—	2	1	no	—
143	#131	—	4	4	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
149	C₄⋊Q₈	C₈₃⋊C₈	1	1	yes	characteristic
150	#127	C₈₃⋊C₈	1	1	yes	characteristic
151	#124	—	2	1	no	—
152	#121	—	2	1	no	—
153	#121	C₈₃⋊C₈	1	2	yes	—
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	C₂×Dic₈₃	1	2	yes	—
156	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
157	C₂×C₁₆	—	166	2	no	—
158	C₂×C₁₆	—	166	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	C₂×Dic₈₃	1	2	yes	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
167	#41	—	4	2	no	—
168	#41	—	4	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₂×C₈₃⋊C₈	1	2	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
177	C₂²×C₄	#13050574	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	C₂×C₈₃⋊C₈	1	2	yes	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
193	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
194	C₄²	#13050574	1	1	yes	characteristic
195	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
196	C₁₆	—	332	2	no	—
197	C₁₆	—	332	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
200	Q₈	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
217	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	#14581818	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	#19127485	1	2	yes	—
236	C₄	#19127706	1	1	yes	characteristic
237	C₄	#19127718	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	#24905854	1	1	yes	characteristic
241	C₂	#24909084	1	1	yes	characteristic
242	C₂	C₂×#19127485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#2931	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₈
prime radical	#532×C₈₃	C₄
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#2931	D₈₃
hypocenter	C₈₃	#11633
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #24909266	C₂	semidirect
2. #24909262	C₂	semidirect
3. #24905421	C₂	semidirect
126. C₈₃	#11633	semidirect

Maximal subgroups

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

subgroup	index
#24909266	2
#24909262	2
#24905421	2
C₈₃×#2931	2
#11633	83