Subgroups — #35302949

Order 21248.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908179	C₂	1	2	yes	maximal
2	C₄×#14581726	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127422	C₂	1	1	yes	maximal, characteristic
4	#24905302	C₂	1	2	yes	maximal
5	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	C₄	1	1	yes	characteristic
7	C₂×#14581726	C₄	1	1	yes	characteristic
8	C₂×Dic₈₃⋊C₈	—	2	2	no	—
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₈₃	C₂²	1	2	yes	—
12	#19127464	C₄	1	2	yes	—
13	#19127455	—	2	2	no	—
14	#19127422	—	2	2	no	—
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#334×C₈₃	—	2	2	no	—
18	C₂²×C₈₃⋊C₈	—	2	1	no	—
19	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₈₃	—	2	2	no	—
23	#14581726	—	2	2	no	—
24	#14581726	C₈	1	4	yes	complemented
25	Dic₈₃⋊C₈	—	4	4	no	—
26	C₄×C₈₃⋊C₈	—	2	1	no	—
27	C₄×C₈₃⋊C₈	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₄×C₈₃⋊C₈	Q₈	1	2	yes	—
30	C₈₃⋊#95	—	2	2	no	—
31	C₂²×C₈×C₈₃	—	2	2	no	—
32	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₈₃	—	2	2	no	—
36	C₄⋊C₈×C₈₃	—	2	2	no	—
37	C₄×C₈×C₈₃	D₄	1	2	yes	—
38	C₄×C₈×C₈₃	Q₈	1	2	yes	—
39	C₂²×Dic₈₃	—	2	2	no	—
40	C₂²×Dic₈₃	—	2	2	no	—
41	C₂×C₈₃⋊C₈	—	2	2	no	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	2	2	no	—
45	C₂×C₈₃⋊C₈	—	4	1	no	—
46	C₂×C₈₃⋊C₈	—	4	1	no	—
47	C₄⋊Dic₈₃	—	2	2	no	—
48	C₄⋊Dic₈₃	—	2	2	no	—
49	C₄⋊Dic₈₃	—	2	2	no	—
50	C₄⋊Dic₈₃	C₂×C₈	1	4	yes	—
51	C₄×Dic₈₃	—	4	2	no	—
52	C₄×Dic₈₃	—	4	2	no	—
53	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	—	2	1	no	—
58	C₂×C₈×C₈₃	—	4	2	no	—
59	C₂×C₈×C₈₃	—	4	2	no	—
60	C₂×C₈×C₈₃	—	4	2	no	—
61	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
65	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₈₃	#41	1	1	yes	characteristic
69	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	—	2	1	no	—
71	C₂×C₄×C₈₃	—	2	1	no	—
72	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₈₃	#137	1	2	yes	—
84	C₂×Dic₈₃	—	2	4	no	—
85	C₂×Dic₈₃	—	4	2	no	—
86	C₂×Dic₈₃	—	4	2	no	—
87	C₂×Dic₈₃	—	4	2	no	—
88	C₂×Dic₈₃	—	4	2	no	—
89	C₂×Dic₈₃	—	4	2	no	—
90	C₈×C₈₃	—	2	2	no	—
91	C₈×C₈₃	—	8	2	no	—
92	C₈×C₈₃	C₄⋊C₈	1	4	yes	complemented
93	C₈₃⋊C₈	—	4	2	no	—
94	C₈₃⋊C₈	—	4	2	no	—
95	C₂²×C₈₃	#389	1	1	yes	characteristic
96	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
97	C₂²×C₈₃	#436	1	1	yes	characteristic
98	C₂²×C₈₃	#441	1	1	yes	characteristic
99	C₂²×C₈₃	#442	1	1	yes	characteristic
100	C₂²×C₈₃	#444	1	1	yes	characteristic
101	C₂²×C₈₃	#444	1	1	yes	characteristic
102	C₄×C₈₃	—	2	1	no	—
103	C₄×C₈₃	—	2	1	no	—
104	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₈₃	#424	1	1	yes	characteristic
106	C₄×C₈₃	C₈×Q₈	1	1	yes	characteristic
107	C₄×C₈₃	#446	1	1	yes	characteristic
108	C₄×C₈₃	#468	1	2	yes	—
109	C₄×C₈₃	#501	1	2	yes	—
110	Dic₈₃	—	4	4	no	—
111	Dic₈₃	—	8	2	no	—
112	#9595	—	83	1	no	maximal, Sylow
113	C₂×C₈₃	#1579	1	1	yes	characteristic
114	C₂×C₈₃	#1580	1	1	yes	characteristic
115	C₂×C₈₃	#1593	1	1	yes	characteristic
116	C₂×C₈₃	#1594	1	1	yes	characteristic
117	C₂×C₈₃	#1939	1	1	yes	characteristic
118	C₂×C₈₃	#1943	1	1	yes	characteristic
119	C₂×C₈₃	#1946	1	1	yes	characteristic
120	#1939	—	83	2	no	—
121	C₄×#107	—	83	1	no	—
122	C₂×#334	D₈₃	1	1	yes	characteristic
123	C₂×#322	—	83	1	no	—
124	#1279	—	83	2	no	—
125	C₈₃	#9595	1	1	yes	characteristic, complemented, Sylow
126	C₂×#107	—	83	1	no	—
127	C₂×#107	—	83	1	no	—
128	C₂×C₄⋊C₈	—	166	2	no	—
129	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
130	C₂×C₄×C₈	—	83	1	no	—
131	C₂×C₄×C₈	—	83	1	no	—
132	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	83	2	no	—
134	#340	—	83	2	no	—
135	#336	—	166	2	no	—
136	#334	—	2	2	no	—
137	#322	—	166	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	83	2	no	—
140	C₂²×C₈	—	166	1	no	—
141	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
142	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	83	2	no	—
144	C₂×C₄⋊C₄	—	83	2	no	—
145	C₂×C₄²	—	166	2	no	—
146	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
147	#107	—	83	4	no	—
148	#107	—	166	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	332	4	no	—
152	C₄×C₈	—	83	2	no	—
153	C₄×C₈	—	166	1	no	—
154	C₄×C₈	—	166	1	no	—
155	C₄×C₈	—	166	1	no	—
156	C₄×C₈	Dic₁₆₆	1	2	yes	—
157	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
158	#95	—	166	2	no	—
159	C₂²×C₄	—	166	2	no	—
160	C₂²×C₄	—	166	2	no	—
161	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
162	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
163	C₂²×C₄	#13050591	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	332	1	no	—
170	C₂×C₈	—	332	1	no	—
171	C₂×C₈	—	166	2	no	—
172	C₂×C₈	—	166	2	no	—
173	C₂×C₈	—	166	2	no	—
174	C₂×C₈	—	166	2	no	—
175	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
176	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
177	C₂×C₈	C₈×D₈₃	1	2	yes	—
178	C₂×C₈	#13050573	1	2	yes	—
179	C₄⋊C₄	—	83	4	no	—
180	C₄⋊C₄	—	166	2	no	—
181	C₄⋊C₄	—	166	2	no	—
182	C₄⋊C₄	—	166	2	no	—
183	C₄²	—	332	2	no	—
184	C₄²	—	332	2	no	—
185	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
186	C₄²	Q₈×D₈₃	1	1	yes	characteristic
187	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
188	C₄²	#13050589	1	1	yes	characteristic
189	C₂³	#14581754	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	332	2	no	—
193	C₂×C₄	—	332	2	no	—
194	C₂×C₄	—	332	2	no	—
195	C₂×C₄	—	332	2	no	—
196	C₂×C₄	—	332	2	no	—
197	C₂×C₄	—	166	4	no	—
198	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#14581757	1	1	yes	characteristic
200	C₂×C₄	#14581763	1	1	yes	characteristic
201	C₂×C₄	#14581768	1	1	yes	characteristic
202	C₂×C₄	#14581783	1	1	yes	characteristic
203	C₂×C₄	#14581791	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
205	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
206	C₂×C₄	#14581800	1	1	yes	characteristic
207	C₂×C₄	#14581800	1	1	yes	characteristic
208	C₂×C₄	#14581819	1	2	yes	—
209	C₂×C₄	#14581821	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	—	8	2	no	—
212	C₈	—	332	2	no	—
213	C₈	—	332	2	no	—
214	C₈	Dic₈₃⋊C₈	1	4	yes	complemented
215	C₂²	#19127597	1	1	yes	characteristic
216	C₂²	#19127766	1	1	yes	characteristic
217	C₂²	#19127778	1	1	yes	characteristic
218	C₂²	#19127807	1	1	yes	characteristic
219	C₂²	#19127812	1	1	yes	characteristic
220	C₂²	#19127822	1	1	yes	characteristic
221	C₂²	#19127822	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	332	4	no	—
225	C₄	—	664	2	no	—
226	C₄	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
227	C₄	#19127729	1	1	yes	characteristic
228	C₄	#19127825	1	1	yes	characteristic
229	C₄	#19127828	1	1	yes	characteristic
230	C₄	#19127944	1	2	yes	—
231	C₄	#19128108	1	2	yes	—
232	C₂	#24906281	1	1	yes	characteristic
233	C₂	#24906286	1	1	yes	characteristic
234	C₂	#24906371	1	1	yes	characteristic
235	C₂	#24906376	1	1	yes	characteristic
236	C₂	#24908179	1	1	yes	characteristic
237	C₂	#24908199	1	1	yes	characteristic
238	C₂	#24908214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	#9595
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
24. #14581726	C₈	semidirect
92. C₈×C₈₃	C₄⋊C₈	semidirect
125. C₈₃	#9595	semidirect
214. C₈	Dic₈₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#24908179	2
C₄×#14581726	2
C₂×#19127422	2
#24905302	2
C₂×#334×C₈₃	2
#9595	83