Subgroups — #35322969

Order 21248.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127522	C₂	1	1	yes	maximal, characteristic
2	#24907342	C₂	1	1	yes	maximal, characteristic
3	#24905570	C₂	1	2	yes	maximal
4	#24905561	C₂	1	2	yes	maximal
5	C₂×#499×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	C₂×#14581728	C₂²	1	1	yes	characteristic
8	#19127739	C₂²	1	1	yes	characteristic
9	#19127656	—	2	1	no	—
10	#19127522	—	2	2	no	—
11	#19127461	—	2	1	no	—
12	C₂×#125×C₈₃	—	2	1	no	—
13	#499×C₈₃	—	2	2	no	—
14	#499×C₈₃	C₄	1	4	yes	complemented
15	C₂×#107×C₈₃	—	2	1	no	—
16	C₂×#106×C₈₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050574	—	2	1	no	—
19	#14581751	—	2	1	no	—
20	#14581732	—	2	2	no	—
21	#14581732	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	2	no	—
23	#14581728	—	2	1	no	—
24	#14581728	Q₈	1	2	yes	—
25	#14581726	Q₈	1	2	yes	—
26	#14581724	—	2	1	no	—
27	#14581702	—	4	1	no	—
28	C₈×Dic₈₃	—	4	1	no	—
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	#125×C₈₃	—	2	4	no	—
32	#125×C₈₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₃	—	4	2	no	—
35	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
36	#107×C₈₃	—	2	4	no	—
37	#106×C₈₃	—	2	2	no	—
38	#106×C₈₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₃	D₄	1	2	yes	—
41	#13050574	—	2	2	no	—
42	#13050574	—	4	1	no	—
43	C₂×C₈₃⋊C₈	—	4	1	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	C₂²⋊Dic₈₃	—	4	1	no	—
46	C₄×Dic₈₃	—	4	1	no	—
47	C₈₃⋊C₁₆	—	4	2	no	—
48	C₂²×C₄×C₈₃	—	2	1	no	—
49	C₂²×C₄×C₈₃	—	4	2	no	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
55	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₈₃	—	2	4	no	—
62	C₄⋊C₄×C₈₃	—	4	8	no	—
63	C₄⋊C₄×C₈₃	—	4	2	no	—
64	C₄²×C₈₃	—	2	1	no	—
65	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
66	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	2	no	—
68	C₂×C₄×C₈₃	—	4	2	no	—
69	C₂×C₄×C₈₃	—	4	4	no	—
70	C₂×C₄×C₈₃	—	8	4	no	—
71	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×Dic₈₃	—	4	2	no	—
78	C₈×C₈₃	#103	1	2	yes	—
79	C₈×C₈₃	#103	1	2	yes	—
80	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₈₃	#124	1	1	yes	characteristic
83	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₈₃⋊C₈	—	4	2	no	—
85	C₂²×C₈₃	—	2	2	no	—
86	C₂²×C₈₃	#418	1	1	yes	characteristic
87	C₂²×C₈₃	#476	1	1	yes	characteristic
88	C₂²×C₈₃	#477	1	1	yes	characteristic
89	C₄×C₈₃	—	4	2	no	—
90	C₄×C₈₃	—	8	4	no	—
91	C₄×C₈₃	#391	1	1	yes	characteristic
92	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
93	C₄×C₈₃	#476	1	1	yes	characteristic
94	C₄×C₈₃	#477	1	1	yes	characteristic
95	Dic₈₃	—	8	2	no	—
96	#13273	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	—	2	1	no	—
98	C₂×C₈₃	—	2	1	no	—
99	C₂×C₈₃	#1987	1	1	yes	characteristic
100	C₂×C₈₃	#2231	1	2	yes	—
101	C₂×#499	D₈₃	1	1	yes	characteristic
102	C₂×#364	—	83	1	no	—
103	#1772	—	83	1	no	—
104	#1386	—	83	2	no	—
105	#1383	—	83	2	no	—
106	C₈₃	#13273	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	2	1	no	—
108	C₂×M₃₂	—	83	2	no	—
109	#499	—	2	2	no	—
110	#499	Dic₈₃	1	4	yes	complemented
111	C₂×#108	—	83	1	no	—
112	#427	—	83	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	C₂×D₈₃	1	2	yes	—
115	#405	—	166	1	no	—
116	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
117	#364	—	166	2	no	—
118	#339	—	166	1	no	—
119	C₂×M₁₆	—	166	1	no	—
120	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
122	#125	—	2	4	no	—
123	#125	—	4	2	no	—
124	#117	—	166	1	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	4	2	no	—
127	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
128	M₃₂	—	166	2	no	—
129	M₃₂	—	166	2	no	—
130	C₂×C₁₆	—	166	2	no	—
131	#108	—	83	2	no	—
132	#108	—	166	1	no	—
133	#107	—	2	4	no	—
134	#107	—	83	2	no	—
135	#106	—	2	2	no	—
136	#106	—	166	1	no	—
137	#106	C₂×Dic₈₃	1	4	yes	—
138	#97	—	332	1	no	—
139	C₄×C₈	—	332	1	no	—
140	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
145	M₁₆	—	332	1	no	—
146	M₁₆	—	166	2	no	—
147	C₂×C₈	—	332	1	no	—
148	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
149	C₂×C₈	#13050577	1	1	yes	characteristic
150	C₂×C₈	#13050577	1	1	yes	characteristic
151	C₂×C₈	#13050580	1	2	yes	—
152	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
156	C₂×C₈	#13050591	1	1	yes	characteristic
157	C₂×C₈	#13050591	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	166	2	no	—
162	C₂²⋊C₄	—	332	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	332	1	no	—
165	C₄²	C₂²⋊Dic₈₃	1	2	yes	—
166	C₁₆	—	332	2	no	—
167	C₂³	#14581818	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	—	332	2	no	—
173	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
174	C₂×C₄	#14581768	1	1	yes	characteristic
175	C₂×C₄	#14581768	1	1	yes	characteristic
176	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
177	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
178	C₂×C₄	#14581818	1	1	yes	characteristic
179	C₈	—	332	2	no	—
180	C₈	#14581716	1	2	yes	—
181	C₈	#14581716	1	2	yes	—
182	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
183	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
184	C₈	#14581781	1	1	yes	characteristic
185	C₈	#14581790	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#19127706	1	1	yes	characteristic
188	C₂²	#19127988	1	1	yes	characteristic
189	C₂²	#19127993	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	4	no	—
192	C₄	—	664	2	no	—
193	C₄	#19127608	1	1	yes	characteristic
194	C₄	C₂×#14581716	1	1	yes	characteristic
195	C₄	#19127988	1	1	yes	characteristic
196	C₄	#19127993	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#24908412	1	1	yes	characteristic
200	C₂	#24909636	1	2	yes	—

	subgroup	G/H
center	C₂²	#19127988
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#499×C₈₃	C₂
socle	C₂²×C₈₃	#476
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#499	D₈₃
hypocenter	C₈₃	#13273
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
14. #499×C₈₃	C₄	semidirect
106. C₈₃	#13273	semidirect
110. #499	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127522	2
#24907342	2
#24905570	2
#24905561	2
C₂×#499×C₈₃	2
#13273	83