Subgroups — #2868×D₈₃ #35610499

Order 21248.

3774 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×D₈₃	C₂	1	6	yes	maximal
2	#24915861	C₂	1	1	yes	maximal, characteristic, complemented
3	#24915860	C₂	1	8	yes	maximal, complemented
4	#24915859	C₂	1	6	yes	maximal, complemented
5	#384×D₈₃	C₂	1	8	yes	maximal
6	C₄³×D₈₃	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₈₃	C₂²	1	4	yes	—
9	C₂×#14581792	C₂²	1	6	yes	—
10	C₂×#14581791	C₂²	1	24	yes	—
11	C₂×#14581790	C₂²	1	12	yes	—
12	C₄⋊Q₈×D₈₃	—	2	48	no	—
13	C₂×C₄⋊C₄×D₈₃	C₂²	1	24	yes	—
14	C₂×C₄²×D₈₃	C₂²	1	3	yes	—
15	C₂×C₄²×D₈₃	C₂²	1	4	yes	—
16	#19127571	C₂²	1	8	yes	—
17	#19127570	C₂²	1	48	yes	—
18	C₄²×Dic₈₃	C₂²	1	3	yes	—
19	C₄²×Dic₈₃	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₈₃	C₂²	1	6	yes	—
21	#384×C₈₃	C₂²	1	8	yes	—
22	C₄³×C₈₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₆₆	C₂³	1	4	yes	—
24	C₂²×Dic₁₆₆	C₂³	1	8	yes	—
25	C₂×Q₈×D₈₃	—	2	48	no	—
26	C₂²×C₄×D₈₃	C₂³	1	8	yes	—
27	C₂²×C₄×D₈₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×D₈₃	C₂³	1	6	yes	—
29	#14581792	—	2	24	no	—
30	#14581791	—	2	96	no	—
31	#14581790	—	2	48	no	—
32	C₂×C₄⋊Dic₈₃	C₂³	1	24	yes	—
33	C₂×Dic₈₃⋊C₄	C₂³	1	48	yes	—
34	C₄⋊C₄×D₈₃	—	2	96	no	—
35	C₂×C₄×Dic₈₃	C₂³	1	3	yes	—
36	C₂×C₄×Dic₈₃	C₂³	1	6	yes	—
37	C₂×C₄×Dic₈₃	C₂³	1	12	yes	—
38	C₄²×D₈₃	D₄	1	24	yes	—
39	C₄²×D₈₃	Q₈	1	32	yes	complemented
40	C₂²×Q₈×C₈₃	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₈₃	—	2	24	no	—
42	C₂×C₄⋊C₄×C₈₃	C₂³	1	24	yes	—
43	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
44	C₂×C₄²×C₈₃	C₂³	1	4	yes	—
45	C₂³×D₈₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₆₆	—	2	24	no	—
47	C₂×Dic₁₆₆	—	2	48	no	—
48	Q₈×D₈₃	—	4	64	no	—
49	C₂²×Dic₈₃	C₂⁴	1	8	yes	—
50	C₂²×Dic₈₃	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₈₃	C₂⁴	1	6	yes	—
52	C₂×C₄×D₈₃	—	2	48	no	—
53	C₂×C₄×D₈₃	C₂×D₄	1	12	yes	—
54	C₂×C₄×D₈₃	C₂×D₄	1	24	yes	—
55	C₂×C₄×D₈₃	C₂×Q₈	1	48	yes	—
56	C₄⋊Dic₈₃	—	2	48	no	—
57	Dic₈₃⋊C₄	—	2	96	no	—
58	C₄×Dic₈₃	C₂×D₄	1	12	yes	—
59	C₄×Dic₈₃	C₂×D₄	1	24	yes	—
60	C₄×Dic₈₃	C₂×Q₈	1	48	yes	complemented
61	C₂×Q₈×C₈₃	—	2	24	no	—
62	C₂²×C₄×C₈₃	C₂⁴	1	8	yes	—
63	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₈₃	C₂⁴	1	6	yes	—
65	C₄⋊C₄×C₈₃	—	2	48	no	—
66	C₄²×C₈₃	C₂×D₄	1	12	yes	—
67	C₄²×C₈₃	C₂×Q₈	1	16	yes	complemented
68	C₂³×C₈₃	C₂⁵	1	1	yes	characteristic
69	C₂²×D₈₃	C₂²×D₄	1	6	yes	—
70	C₂²×D₈₃	C₂²×Q₈	1	8	yes	—
71	Q₈×C₈₃	—	4	16	no	—
72	C₂×C₄×C₈₃	—	2	24	no	—
73	C₂×C₄×C₈₃	C₂²×D₄	1	6	yes	—
74	C₂×C₄×C₈₃	C₂²×D₄	1	12	yes	—
75	C₂×C₄×C₈₃	C₂²×Q₈	1	24	yes	—
76	C₂×Dic₈₃	—	2	24	no	—
77	C₂×Dic₈₃	C₂²×D₄	1	6	yes	—
78	C₂×Dic₈₃	C₂²×D₄	1	12	yes	—
79	C₂×Dic₈₃	C₂²×Q₈	1	24	yes	—
80	C₄×D₈₃	—	4	32	no	—
81	C₄×D₈₃	#127	1	16	yes	—
82	C₄×D₈₃	C₄⋊Q₈	1	96	yes	complemented
83	Dic₁₆₆	—	4	16	no	—
84	Dic₁₆₆	—	4	32	no	—
85	C₂²×C₈₃	C₂³×D₄	1	3	yes	—
86	C₂²×C₈₃	C₂³×Q₈	1	4	yes	—
87	C₂×D₈₃	C₂×#127	1	4	yes	—
88	C₂×D₈₃	C₂×C₄⋊Q₈	1	24	yes	—
89	C₄×C₈₃	—	4	8	no	—
90	C₄×C₈₃	C₂×#127	1	4	yes	—
91	C₄×C₈₃	C₂×C₄⋊Q₈	1	24	yes	complemented
92	Dic₈₃	—	4	8	no	—
93	Dic₈₃	C₂×#127	1	4	yes	—
94	Dic₈₃	C₂×C₄⋊Q₈	1	24	yes	complemented
95	C₂×#2868	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	C₂²×#127	1	1	yes	characteristic
97	C₂×C₈₃	C₂²×C₄⋊Q₈	1	6	yes	—
98	D₈₃	#2868	1	8	yes	direct product factor
99	C₂²×C₄⋊Q₈	—	83	6	no	—
100	#2868	—	83	8	no	—
101	#2868	—	83	1	no	—
102	#2868	—	83	6	no	—
103	#2868	D₈₃	1	1	yes	characteristic, direct product factor
104	C₂×#384	—	83	8	no	—
105	C₂×C₄³	—	83	1	no	—
106	C₈₃	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
107	C₂³×Q₈	—	83	4	no	—
108	C₂×C₄⋊Q₈	—	83	12	no	—
109	C₂×C₄⋊Q₈	—	83	24	no	—
110	C₂×C₄⋊Q₈	—	83	6	no	—
111	C₂×C₄⋊Q₈	—	166	48	no	—
112	C₂×C₄⋊Q₈	C₂×D₈₃	1	6	yes	—
113	C₂²×C₄⋊C₄	—	83	24	no	—
114	C₂²×C₄²	—	83	3	no	—
115	C₂²×C₄²	—	83	4	no	—
116	#384	—	83	48	no	—
117	#384	—	83	8	no	—
118	#384	C₂×D₈₃	1	8	yes	—
119	C₄³	—	83	4	no	—
120	C₄³	—	83	3	no	—
121	C₄³	C₂×D₈₃	1	1	yes	characteristic
122	C₂²×Q₈	—	83	8	no	—
123	C₂²×Q₈	—	83	4	no	—
124	C₂²×Q₈	—	166	48	no	—
125	C₂²×Q₈	C₂²×D₈₃	1	4	yes	—
126	C₂³×C₄	—	83	6	no	—
127	C₂³×C₄	—	83	1	no	—
128	C₂³×C₄	—	83	8	no	—
129	C₄⋊Q₈	—	2	24	no	—
130	C₄⋊Q₈	—	166	96	no	—
131	C₄⋊Q₈	—	166	48	no	—
132	C₄⋊Q₈	—	166	24	no	—
133	C₂×C₄⋊C₄	—	83	48	no	—
134	C₂×C₄⋊C₄	—	83	24	no	—
135	C₂×C₄⋊C₄	—	166	96	no	—
136	C₂×C₄⋊C₄	C₂²×D₈₃	1	24	yes	—
137	C₂×C₄²	—	83	3	no	—
138	C₂×C₄²	—	83	6	no	—
139	C₂×C₄²	—	83	12	no	—
140	C₂×C₄²	—	83	24	no	—
141	C₂×C₄²	—	83	32	no	—
142	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
143	C₂×C₄²	C₂²×D₈₃	1	4	yes	—
144	C₂⁴	—	83	1	no	—
145	C₂×Q₈	—	2	24	no	—
146	C₂×Q₈	—	332	64	no	—
147	C₂×Q₈	—	166	48	no	—
148	C₂×Q₈	—	166	24	no	—
149	C₂²×C₄	—	83	6	no	—
150	C₂²×C₄	—	83	1	no	—
151	C₂²×C₄	—	83	24	no	—
152	C₂²×C₄	—	83	12	no	—
153	C₂²×C₄	—	83	48	no	—
154	C₂²×C₄	—	83	8	no	—
155	C₂²×C₄	—	166	48	no	—
156	C₂²×C₄	C₂³×D₈₃	1	8	yes	—
157	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×D₈₃	1	6	yes	—
159	C₄⋊C₄	—	2	48	no	—
160	C₄⋊C₄	—	166	96	no	—
161	C₄⋊C₄	—	166	48	no	—
162	C₄²	—	83	12	no	—
163	C₄²	—	83	24	no	—
164	C₄²	—	83	48	no	—
165	C₄²	D₄×D₈₃	1	12	yes	—
166	C₄²	Q₈×D₈₃	1	16	yes	complemented
167	C₂³	—	83	6	no	—
168	C₂³	—	83	8	no	—
169	C₂³	C₂⁴×D₈₃	1	1	yes	characteristic
170	Q₈	—	4	16	no	—
171	Q₈	—	332	32	no	—
172	Q₈	—	332	16	no	—
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	332	32	no	—
175	C₂×C₄	—	83	12	no	—
176	C₂×C₄	—	83	6	no	—
177	C₂×C₄	—	83	24	no	—
178	C₂×C₄	—	83	16	no	—
179	C₂×C₄	—	83	96	no	—
180	C₂×C₄	—	166	24	no	—
181	C₂×C₄	C₂×D₄×D₈₃	1	6	yes	—
182	C₂×C₄	C₂×D₄×D₈₃	1	12	yes	—
183	C₂×C₄	C₂×Q₈×D₈₃	1	24	yes	—
184	C₂²	—	83	4	no	—
185	C₂²	—	83	24	no	—
186	C₂²	C₂²×D₄×D₈₃	1	3	yes	—
187	C₂²	C₂²×Q₈×D₈₃	1	4	yes	—
188	C₄	—	4	8	no	—
189	C₄	—	332	8	no	—
190	C₄	—	83	4	no	—
191	C₄	—	83	24	no	—
192	C₄	#127×D₈₃	1	4	yes	—
193	C₄	C₄⋊Q₈×D₈₃	1	24	yes	complemented
194	C₂	—	83	8	no	—
195	C₂	C₂×#127×D₈₃	1	1	yes	characteristic
196	C₂	C₂×C₄⋊Q₈×D₈₃	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₈₃
commutator, socle, cosocle	C₂³×C₈₃	C₂⁵
Fitting	C₈₃×#2868	C₂
prime radical	C₂³×D₈₃	C₂⁴
hypercenter	#2868	D₈₃
hypocenter	C₈₃	C₂×#2868
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ #2868

Complements

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

N	H ≅ G/N	type
2. #24915861	C₂	semidirect
3. #24915860	C₂	semidirect
4. #24915859	C₂	semidirect
7. C₈₃×#2868	C₂	semidirect
39. C₄²×D₈₃	Q₈	semidirect
60. C₄×Dic₈₃	C₂×Q₈	semidirect
67. C₄²×C₈₃	C₂×Q₈	semidirect
82. C₄×D₈₃	C₄⋊Q₈	semidirect
91. C₄×C₈₃	C₂×C₄⋊Q₈	semidirect
94. Dic₈₃	C₂×C₄⋊Q₈	semidirect
98. D₈₃	#2868	direct (N × H)
103. #2868	D₈₃	direct (N × H)
106. C₈₃	C₂×#2868	semidirect
166. C₄²	Q₈×D₈₃	semidirect
193. C₄	C₄⋊Q₈×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×D₈₃	2
#24915861	2
#24915860	2
#24915859	2
#384×D₈₃	2
C₄³×D₈₃	2
C₈₃×#2868	2
C₂×#2868	83

Subgroups — #2868×D83 #35610499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2868×D₈₃ #35610499