Subgroups — #35303106

Order 21248.

298 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908236	C₂	1	2	yes	maximal
2	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
3	C₈×#13050574	C₂	1	1	yes	maximal, characteristic
4	#24905358	C₂	1	2	yes	maximal
5	C₈₃×#1567	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	C₂×#14581727	—	2	2	no	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₄×#13050574	C₂²	1	1	yes	characteristic
10	#19127465	—	2	2	no	—
11	#19127454	C₄	1	2	yes	—
12	#19127454	C₄	1	2	yes	—
13	#19127423	—	2	1	no	—
14	C₈×C₈₃⋊C₈	—	2	1	no	—
15	#423×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#335×C₈₃	—	2	1	no	—
18	#335×C₈₃	—	2	1	no	—
19	C₂×#13050574	D₄	1	1	yes	characteristic
20	C₂×#13050574	D₄	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	—	2	2	no	—
22	#14581727	—	4	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
25	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂²⋊(C₈₃⋊C₈)	Q₈	1	2	yes	—
27	#14581703	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₄×C₈₃⋊C₈	—	2	1	no	—
30	C₂×M₁₆×C₈₃	—	2	2	no	—
31	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₃	—	2	2	no	—
35	C₄⋊C₈×C₈₃	—	2	2	no	—
36	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
40	#13050574	—	2	1	no	—
41	#13050574	—	2	1	no	—
42	#13050574	D₈	1	2	yes	—
43	#13050574	Q₁₆	1	2	yes	—
44	C₂×C₈₃⋊C₈	—	2	1	no	—
45	C₂×C₈₃⋊C₈	—	2	1	no	—
46	C₂×C₈₃⋊C₈	—	2	4	no	—
47	C₂×C₈₃⋊C₈	—	2	2	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
53	M₁₆×C₈₃	—	4	2	no	—
54	M₁₆×C₈₃	—	4	2	no	—
55	C₂×C₈×C₈₃	—	2	2	no	—
56	C₂×C₈×C₈₃	—	4	1	no	—
57	C₂×C₈×C₈₃	—	4	2	no	—
58	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₃	#41	1	1	yes	characteristic
65	C₄²×C₈₃	#41	1	1	yes	characteristic
66	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	1	no	—
68	C₂×C₄×C₈₃	—	2	1	no	—
69	C₂×C₄×C₈₃	—	4	1	no	—
70	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
80	C₈×C₈₃	—	2	1	no	—
81	C₈×C₈₃	—	2	1	no	—
82	C₈×C₈₃	—	2	1	no	—
83	C₈×C₈₃	—	8	2	no	—
84	C₈×C₈₃	#108	1	2	yes	complemented
85	C₈₃⋊C₈	—	2	2	no	—
86	C₈₃⋊C₈	—	2	2	no	—
87	C₈₃⋊C₈	—	4	2	no	—
88	C₈₃⋊C₈	—	4	4	no	—
89	C₂²×C₈₃	—	2	1	no	—
90	C₂²×C₈₃	—	2	1	no	—
91	C₂²×C₈₃	#389	1	1	yes	characteristic
92	C₂²×C₈₃	#499	1	1	yes	characteristic
93	C₂²×C₈₃	#500	1	1	yes	characteristic
94	C₄×C₈₃	—	2	1	no	—
95	C₄×C₈₃	—	2	1	no	—
96	C₄×C₈₃	—	2	1	no	—
97	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
98	C₄×C₈₃	#430	1	1	yes	characteristic
99	C₄×C₈₃	C₄×D₈	1	1	yes	characteristic
100	C₄×C₈₃	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₈₃	#466	1	1	yes	characteristic
102	C₄×C₈₃	#467	1	1	yes	characteristic
103	#9624	—	83	1	no	maximal, Sylow
104	C₂×C₈₃	—	2	1	no	—
105	C₂×C₈₃	—	2	1	no	—
106	C₂×C₈₃	#1942	1	1	yes	characteristic
107	C₂×C₈₃	#1950	1	1	yes	characteristic
108	C₂×C₈₃	#1953	1	1	yes	characteristic
109	#1950	—	83	2	no	—
110	#1567	D₈₃	1	1	yes	characteristic
111	C₂×#335	—	83	1	no	—
112	C₈×M₁₆	—	83	1	no	—
113	#1301	—	83	2	no	—
114	C₈₃	#9624	1	1	yes	characteristic, complemented, Sylow
115	#431	—	83	1	no	—
116	C₂×#108	—	166	2	no	—
117	#423	C₂×D₈₃	1	2	yes	—
118	C₂×C₄⋊C₈	—	83	2	no	—
119	C₄×M₁₆	—	83	1	no	—
120	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
121	#341	—	166	2	no	—
122	#335	—	2	1	no	—
123	#335	—	2	1	no	—
124	#335	—	83	2	no	—
125	#335	—	83	2	no	—
126	#322	—	166	1	no	—
127	C₈²	—	166	1	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	83	1	no	—
130	C₂×M₁₆	—	83	1	no	—
131	C₂²×C₈	—	166	2	no	—
132	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
133	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
134	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
135	#108	—	332	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	83	4	no	—
139	C₄⋊C₈	—	166	2	no	—
140	C₄⋊C₈	—	166	1	no	—
141	C₂²⋊C₈	—	83	2	no	—
142	#97	—	166	1	no	—
143	C₄×C₈	—	166	1	no	—
144	C₄×C₈	—	166	1	no	—
145	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
146	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
150	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	83	2	no	—
155	M₁₆	—	83	2	no	—
156	M₁₆	—	166	1	no	—
157	M₁₆	—	166	1	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	332	2	no	—
162	C₂×C₈	—	332	2	no	—
163	C₂×C₈	—	166	2	no	—
164	C₂×C₈	—	166	4	no	—
165	C₂×C₈	—	166	1	no	—
166	C₂×C₈	—	166	1	no	—
167	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
168	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
169	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
170	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
173	C₄²	#13050589	1	1	yes	characteristic
174	C₄²	#13050591	1	1	yes	characteristic
175	C₂³	#14581791	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
180	C₂×C₄	#14581754	1	1	yes	characteristic
181	C₂×C₄	#14581757	1	1	yes	characteristic
182	C₂×C₄	#14581763	1	1	yes	characteristic
183	C₂×C₄	#14581768	1	1	yes	characteristic
184	C₂×C₄	#14581783	1	1	yes	characteristic
185	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
186	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
187	C₂×C₄	#14581811	1	1	yes	characteristic
188	C₂×C₄	#14581815	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	8	2	no	—
193	C₈	—	332	2	no	—
194	C₈	—	332	4	no	—
195	C₈	—	166	2	no	—
196	C₈	—	166	2	no	—
197	C₈	#14581727	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#19127597	1	1	yes	characteristic
201	C₂²	#19128102	1	1	yes	characteristic
202	C₂²	#19128105	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#14581727	1	1	yes	characteristic
207	C₄	#19127749	1	1	yes	characteristic
208	C₄	#19127783	1	1	yes	characteristic
209	C₄	#19127795	1	1	yes	characteristic
210	C₄	#19127934	1	1	yes	characteristic
211	C₄	#19127939	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#24908194	1	1	yes	characteristic
215	C₂	#24908236	1	1	yes	characteristic
216	C₂	#24908251	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581763
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1567	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1567	D₈₃
hypocenter	C₈₃	#9624
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
84. C₈×C₈₃	#108	semidirect
114. C₈₃	#9624	semidirect
197. C₈	#14581727	semidirect

Maximal subgroups

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

subgroup	index
#24908236	2
C₂×#19127454	2
C₈×#13050574	2
#24905358	2
C₈₃×#1567	2
#9624	83