Subgroups — C₈₃×#9897 #35237721

Order 21248.

430 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1296	C₂	1	4	yes	maximal
5	C₂×#125×C₈₃	—	2	1	no	—
6	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	—	2	1	no	—
9	#389×C₈₃	—	2	2	no	—
10	#389×C₈₃	C₄	1	4	yes	—
11	#384×C₈₃	C₂²	1	2	yes	—
12	#382×C₈₃	—	2	1	no	—
13	C₄³×C₈₃	C₂²	1	1	yes	characteristic
14	#336×C₈₃	—	2	1	no	—
15	#336×C₈₃	—	2	2	no	—
16	C₂²×C₈×C₈₃	—	2	2	no	—
17	C₂²×C₈×C₈₃	—	2	2	no	—
18	#125×C₈₃	—	4	4	no	—
19	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	D₄	1	2	yes	—
26	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₈₃	—	2	2	no	—
29	C₄⋊C₈×C₈₃	—	2	4	no	—
30	C₄⋊C₈×C₈₃	Q₈	1	4	yes	—
31	C₄×C₈×C₈₃	—	4	2	no	—
32	#95×C₈₃	—	2	4	no	—
33	C₂²×C₄×C₈₃	—	2	4	no	—
34	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	—	2	4	no	—
40	C₂×C₈×C₈₃	—	4	2	no	—
41	C₂×C₈×C₈₃	—	4	2	no	—
42	C₂×C₈×C₈₃	—	4	2	no	—
43	C₂×C₈×C₈₃	—	4	4	no	—
44	C₄⋊C₄×C₈₃	—	4	4	no	—
45	C₄⋊C₄×C₈₃	—	4	8	no	—
46	C₄⋊C₄×C₈₃	—	4	4	no	—
47	C₄⋊C₄×C₈₃	—	4	4	no	—
48	C₄²×C₈₃	—	2	4	no	—
49	C₄²×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	—	4	2	no	—
51	C₄²×C₈₃	SD₁₆	1	2	yes	—
52	C₄²×C₈₃	Q₁₆	1	2	yes	—
53	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
54	C₄²×C₈₃	#41	1	2	yes	—
55	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	—	2	2	no	—
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	—	4	4	no	—
63	C₂×C₄×C₈₃	—	4	4	no	—
64	C₂×C₄×C₈₃	—	4	4	no	—
65	C₂×C₄×C₈₃	#103	1	4	yes	—
66	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#135	1	4	yes	—
73	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₈₃	—	4	4	no	—
75	C₈×C₈₃	—	8	2	no	—
76	C₂²×C₈₃	#391	1	1	yes	characteristic
77	C₂²×C₈₃	#413	1	2	yes	—
78	C₂²×C₈₃	C₂×#103	1	1	yes	characteristic
79	C₂²×C₈₃	#416	1	2	yes	—
80	C₂²×C₈₃	#418	1	1	yes	characteristic
81	C₄×C₈₃	—	2	4	no	—
82	C₄×C₈₃	—	2	2	no	—
83	C₄×C₈₃	—	4	2	no	—
84	C₄×C₈₃	—	8	4	no	—
85	C₄×C₈₃	#476	1	2	yes	—
86	C₄×C₈₃	#477	1	2	yes	—
87	C₄×C₈₃	#488	1	2	yes	—
88	C₄×C₈₃	#499	1	2	yes	—
89	#9897	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₈₃	#1542	1	2	yes	—
91	C₂×C₈₃	#1553	1	2	yes	—
92	C₂×C₈₃	#1987	1	1	yes	characteristic
93	C₂×C₈₃	#1991	1	1	yes	characteristic
94	C₂×C₈₃	#1998	1	1	yes	characteristic
95	#2859	C₂×C₈₃	1	1	yes	characteristic
96	#1831	C₂×C₈₃	1	1	yes	characteristic
97	C₄×C₄⋊C₈	C₂×C₈₃	1	1	yes	characteristic
98	#1296	C₂×C₈₃	1	4	yes	—
99	C₈₃	#9897	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×#125	—	2	1	no	—
101	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	—
103	C₂×C₄×C₈	—	2	1	no	—
104	#389	—	2	2	no	—
105	#389	C₄×C₈₃	1	4	yes	—
106	#384	C₂²×C₈₃	1	2	yes	—
107	#382	—	2	1	no	—
108	C₄³	C₂²×C₈₃	1	1	yes	characteristic
109	#336	—	2	2	no	—
110	#336	—	2	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	#125	—	4	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	4	yes	—
118	C₂×C₄²	C₂×C₄×C₈₃	1	2	yes	—
119	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
120	C₂×C₄²	D₄×C₈₃	1	2	yes	—
121	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	Q₈×C₈₃	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
130	C₂²×C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₈₃	1	2	yes	—
147	C₄²	Q₁₆×C₈₃	1	2	yes	—
148	C₄²	C₂×Q₈×C₈₃	1	2	yes	—
149	C₄²	#41×C₈₃	1	2	yes	—
150	C₂³	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	#103×C₈₃	1	4	yes	—
161	C₂×C₄	C₄×Q₈×C₈₃	1	2	yes	—
162	C₂×C₄	C₂²⋊Q₈×C₈₃	1	2	yes	—
163	C₂×C₄	#124×C₈₃	1	1	yes	characteristic
164	C₂×C₄	C₄⋊Q₈×C₈₃	1	1	yes	characteristic
165	C₂×C₄	C₂×SD₁₆×C₈₃	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₈₃	1	1	yes	characteristic
167	C₂×C₄	#135×C₈₃	1	4	yes	—
168	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	8	2	no	—
171	C₂²	#391×C₈₃	1	1	yes	characteristic
172	C₂²	#413×C₈₃	1	2	yes	—
173	C₂²	C₂×#103×C₈₃	1	1	yes	characteristic
174	C₂²	#416×C₈₃	1	2	yes	—
175	C₂²	#418×C₈₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#476×C₈₃	1	2	yes	—
181	C₄	#477×C₈₃	1	2	yes	—
182	C₄	#488×C₈₃	1	2	yes	—
183	C₄	#499×C₈₃	1	2	yes	—
184	C₂	C₈₃×#1542	1	2	yes	—
185	C₂	C₈₃×#1553	1	2	yes	—
186	C₂	C₈₃×#1987	1	1	yes	characteristic
187	C₂	C₈₃×#1991	1	1	yes	characteristic
188	C₂	C₈₃×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₈₃
Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ C₄³×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
89. #9897	C₈₃	direct (N × H)
99. C₈₃	#9897	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₃×#2859	2
C₈₃×#1831	2
C₄×C₄⋊C₈×C₈₃	2
C₈₃×#1296	2
#9897	83

Subgroups — C83×#9897 #35237721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#9897 #35237721