Subgroups — #35319781

Order 21248.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909067	C₂	1	2	yes	maximal
2	C₄×#14581732	C₂	1	1	yes	maximal, characteristic
3	#24907342	C₂	1	1	yes	maximal, characteristic
4	#24905550	C₂	1	2	yes	maximal
5	C₂×#364×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	1	yes	characteristic
7	C₂×#14581732	C₂²	1	1	yes	characteristic
8	C₂×#14581728	C₄	1	1	yes	characteristic
9	#19127739	C₄	1	1	yes	characteristic
10	#19127656	C₂²	1	2	yes	—
11	#19127478	—	2	4	no	—
12	#19127477	C₄	1	2	yes	complemented
13	C₄×C₈₃⋊C₁₆	C₄	1	2	yes	complemented
14	#19127461	C₄	1	2	yes	—
15	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#364×C₈₃	C₄	1	2	yes	complemented
18	#364×C₈₃	C₄	1	2	yes	complemented
19	C₂×#13050574	C₂×C₄	1	2	yes	—
20	#14581751	C₂×C₄	1	2	yes	—
21	#14581732	—	2	1	no	—
22	#14581732	—	2	1	no	—
23	#14581732	D₄	1	2	yes	—
24	#14581732	Q₈	1	2	yes	—
25	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
27	#14581728	—	2	1	no	—
28	#14581728	—	2	1	no	—
29	#14581726	—	2	1	no	—
30	#14581724	—	2	1	no	—
31	#14581702	—	2	2	no	—
32	C₈×Dic₈₃	—	2	2	no	—
33	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₈₃	—	2	2	no	—
37	M₃₂×C₈₃	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₈₃	—	2	2	no	—
39	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	#13050574	—	2	2	no	—
44	#13050574	—	2	2	no	—
45	C₂×C₈₃⋊C₈	—	2	2	no	—
46	C₄⋊Dic₈₃	—	2	2	no	—
47	C₂²⋊Dic₈₃	—	2	2	no	—
48	C₄×Dic₈₃	—	2	2	no	—
49	C₈₃⋊C₁₆	—	2	2	no	—
50	C₈₃⋊C₁₆	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₈₃	—	2	1	no	—
52	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
64	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
65	C₄²×C₈₃	—	2	1	no	—
66	C₄²×C₈₃	C₄²	1	1	yes	characteristic
67	C₄²×C₈₃	C₄²	1	1	yes	characteristic
68	C₁₆×C₈₃	—	2	4	no	—
69	C₂³×C₈₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₈₃	—	2	1	no	—
71	C₂×C₄×C₈₃	—	2	1	no	—
72	C₂×C₄×C₈₃	—	4	2	no	—
73	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
79	C₂×Dic₈₃	—	4	2	no	—
80	C₈×C₈₃	#97	1	1	yes	characteristic
81	C₈×C₈₃	#97	1	1	yes	characteristic
82	C₈×C₈₃	#97	1	1	yes	characteristic
83	C₈×C₈₃	#97	1	1	yes	characteristic
84	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₈₃	#117	1	1	yes	characteristic
86	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
87	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
88	C₈₃⋊C₈	—	4	2	no	—
89	C₂²×C₈₃	—	4	1	no	—
90	C₂²×C₈₃	#405	1	1	yes	characteristic
91	C₂²×C₈₃	#436	1	1	yes	characteristic
92	C₂²×C₈₃	#446	1	1	yes	characteristic
93	C₄×C₈₃	—	2	2	no	—
94	C₄×C₈₃	—	4	1	no	—
95	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
97	C₄×C₈₃	#436	1	1	yes	characteristic
98	C₄×C₈₃	#446	1	1	yes	characteristic
99	Dic₈₃	—	8	2	no	—
100	#12667	—	83	1	no	maximal, Sylow
101	C₂×C₈₃	—	2	1	no	—
102	C₂×C₈₃	—	2	1	no	—
103	C₂×C₈₃	—	2	1	no	—
104	C₂×C₈₃	#1770	1	1	yes	characteristic
105	C₂×#364	D₈₃	1	1	yes	characteristic
106	#2110	—	83	2	no	—
107	C₄×M₃₂	—	83	1	no	—
108	#1772	—	83	1	no	—
109	#1378	—	83	2	no	—
110	C₈₃	#12667	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	83	1	no	—
112	C₂×M₃₂	—	83	1	no	—
113	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
114	C₂×#108	—	83	1	no	—
115	#427	—	83	1	no	—
116	#405	—	83	2	no	—
117	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
118	#364	Dic₈₃	1	2	yes	complemented
119	#364	Dic₈₃	1	2	yes	complemented
120	#347	—	166	4	no	—
121	#346	—	83	2	no	—
122	C₄×C₁₆	—	83	2	no	—
123	#339	—	83	2	no	—
124	C₂×M₁₆	—	83	2	no	—
125	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
127	#117	—	83	2	no	—
128	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	83	2	no	—
131	M₃₂	—	83	2	no	—
132	M₃₂	—	166	1	no	—
133	M₃₂	—	166	1	no	—
134	M₃₂	C₂×Dic₈₃	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	83	2	no	—
137	C₂×C₁₆	—	83	2	no	—
138	#108	—	166	1	no	—
139	#108	—	166	1	no	—
140	#107	—	166	1	no	—
141	#106	—	166	1	no	—
142	#97	—	166	2	no	—
143	C₄×C₈	—	166	2	no	—
144	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
145	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
150	M₁₆	—	166	2	no	—
151	M₁₆	—	166	2	no	—
152	C₂×C₈	—	166	2	no	—
153	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
155	C₂×C₈	#13050573	1	1	yes	characteristic
156	C₂×C₈	#13050573	1	1	yes	characteristic
157	C₂×C₈	#13050573	1	1	yes	characteristic
158	C₂×C₈	#13050573	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
160	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
161	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
163	C₂×C₈	#13050589	1	1	yes	characteristic
164	C₂×C₈	#13050591	1	1	yes	characteristic
165	C₄⋊C₄	—	166	2	no	—
166	C₂²⋊C₄	—	166	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	166	2	no	—
169	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
170	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
171	C₁₆	—	2	4	no	—
172	C₁₆	—	83	4	no	—
173	C₁₆	—	166	2	no	—
174	C₂³	#14581800	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	332	2	no	—
179	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
180	C₂×C₄	#14581754	1	1	yes	characteristic
181	C₂×C₄	#14581757	1	1	yes	characteristic
182	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
183	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
184	C₂×C₄	#14581800	1	1	yes	characteristic
185	C₈	—	332	2	no	—
186	C₈	#14581702	1	1	yes	characteristic
187	C₈	#14581702	1	1	yes	characteristic
188	C₈	#14581702	1	1	yes	characteristic
189	C₈	#14581702	1	1	yes	characteristic
190	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
191	C₈	#14581750	1	1	yes	characteristic
192	C₈	D₄×Dic₈₃	1	1	yes	characteristic
193	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#19127659	1	1	yes	characteristic
196	C₂²	#19127778	1	1	yes	characteristic
197	C₂²	#19127828	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	664	2	no	—
201	C₄	C₄⋊C₄×Dic₈₃	1	1	yes	characteristic
202	C₄	C₂×#14581702	1	1	yes	characteristic
203	C₄	#19127778	1	1	yes	characteristic
204	C₄	#19127828	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#24907333	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14581750
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#364×C₈₃	C₂
socle	C₂×C₈₃	#1770
prime radical	C₂³×C₈₃	#131
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#364	D₈₃
hypocenter	C₈₃	#12667
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
12. #19127477	C₄	semidirect
13. C₄×C₈₃⋊C₁₆	C₄	semidirect
17. #364×C₈₃	C₄	semidirect
18. #364×C₈₃	C₄	semidirect
110. C₈₃	#12667	semidirect
118. #364	Dic₈₃	semidirect
119. #364	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24909067	2
C₄×#14581732	2
#24907342	2
#24905550	2
C₂×#364×C₈₃	2
#12667	83