Subgroups — C₈₃×#2563 #24904248

Order 10624.

564 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
3	#393×C₈₃	C₂	1	4	yes	maximal, complemented
4	#390×C₈₃	C₂	1	2	yes	maximal
5	#382×C₈₃	C₂	1	2	yes	maximal, complemented
6	#380×C₈₃	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₈₃	C₂²	1	2	yes	—
9	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₈₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₈₃	—	2	4	no	—
12	C₄×D₄×C₈₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₈₃	C₂²	1	8	yes	—
19	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
21	#95×C₈₃	C₂²	1	2	yes	—
22	#95×C₈₃	C₂²	1	8	yes	—
23	C₂⁴×C₈₃	C₂³	1	2	yes	—
24	C₂×Q₈×C₈₃	—	2	4	no	—
25	C₂×Q₈×C₈₃	—	4	4	no	—
26	C₂×D₄×C₈₃	—	2	2	no	—
27	C₂×D₄×C₈₃	—	2	2	no	—
28	C₂×D₄×C₈₃	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₈₃	—	2	4	no	—
30	C₂²×C₄×C₈₃	—	2	8	no	—
31	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₈₃	—	2	2	no	—
38	C₄⋊C₄×C₈₃	—	2	4	no	—
39	C₄⋊C₄×C₈₃	—	2	4	no	—
40	C₄⋊C₄×C₈₃	—	2	4	no	—
41	C₂²⋊C₄×C₈₃	—	2	4	no	—
42	C₂²⋊C₄×C₈₃	—	4	8	no	—
43	C₄²×C₈₃	—	2	4	no	—
44	C₄²×C₈₃	D₄	1	4	yes	—
45	C₂³×C₈₃	—	2	2	no	—
46	C₂³×C₈₃	—	2	4	no	—
47	C₂³×C₈₃	—	2	2	no	—
48	C₂³×C₈₃	—	2	4	no	—
49	C₂³×C₈₃	C₂×Q₈	1	4	yes	—
50	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
51	Q₈×C₈₃	—	4	8	no	—
52	D₄×C₈₃	—	2	8	no	—
53	C₂×C₄×C₈₃	—	2	2	no	—
54	C₂×C₄×C₈₃	—	2	2	no	—
55	C₂×C₄×C₈₃	—	2	4	no	—
56	C₂×C₄×C₈₃	—	2	4	no	—
57	C₂×C₄×C₈₃	—	2	4	no	—
58	C₂×C₄×C₈₃	—	2	4	no	—
59	C₂×C₄×C₈₃	—	2	4	no	—
60	C₂×C₄×C₈₃	—	2	8	no	—
61	C₂×C₄×C₈₃	—	4	8	no	—
62	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₈₃	#41	1	2	yes	—
67	C₂×C₄×C₈₃	#41	1	4	yes	—
68	C₂²×C₈₃	—	2	4	no	—
69	C₂²×C₈₃	—	2	8	no	—
70	C₂²×C₈₃	—	4	2	no	—
71	C₂²×C₈₃	—	4	2	no	—
72	C₂²×C₈₃	—	4	4	no	—
73	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
76	C₂²×C₈₃	C₂×#41	1	2	yes	—
77	C₂²×C₈₃	H(5,𝔽₂)	1	2	yes	—
78	C₄×C₈₃	—	2	2	no	—
79	C₄×C₈₃	—	4	2	no	—
80	C₄×C₈₃	—	4	4	no	—
81	C₄×C₈₃	—	4	4	no	—
82	C₄×C₈₃	C₂²⋊Q₈	1	4	yes	—
83	C₄×C₈₃	#124	1	4	yes	—
84	C₂×C₈₃	—	4	4	no	—
85	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₈₃	C₂×#124	1	1	yes	characteristic
87	C₂×C₈₃	#535	1	1	yes	characteristic
88	C₂×C₈₃	#554	1	2	yes	—
89	C₂×C₈₃	#555	1	2	yes	—
90	#2563	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₈₃	#2563	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×C₄⋊Q₈	C₂×C₈₃	1	1	yes	characteristic, complemented
93	C₂×C₄×D₄	C₂×C₈₃	1	1	yes	characteristic
94	#393	C₂×C₈₃	1	4	yes	complemented
95	#390	C₂×C₈₃	1	2	yes	—
96	#382	C₂×C₈₃	1	2	yes	complemented
97	#380	C₂×C₈₃	1	4	yes	complemented
98	C₄×C₄⋊C₄	C₂×C₈₃	1	1	yes	characteristic, complemented
99	C₂²×Q₈	C₂²×C₈₃	1	2	yes	—
100	C₂²×D₄	C₂²×C₈₃	1	1	yes	characteristic
101	C₂³×C₄	C₂²×C₈₃	1	2	yes	—
102	C₄⋊Q₈	—	2	4	no	—
103	C₄×D₄	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₈₃	1	8	yes	—
110	C₂×C₄²	C₂²×C₈₃	1	2	yes	—
111	C₂×C₄²	C₂²×C₈₃	1	1	yes	characteristic
112	#95	C₂²×C₈₃	1	2	yes	—
113	#95	C₂²×C₈₃	1	8	yes	—
114	C₂⁴	C₂³×C₈₃	1	2	yes	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	Q₈×C₈₃	1	4	yes	complemented
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₈₃	1	4	yes	—
124	C₂²×C₄	C₂³×C₈₃	1	4	yes	—
125	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	4	8	no	—
134	C₄²	—	2	4	no	—
135	C₄²	D₄×C₈₃	1	4	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	4	no	—
140	C₂³	C₂×Q₈×C₈₃	1	4	yes	—
141	C₂³	C₂⁴×C₈₃	1	1	yes	characteristic
142	Q₈	—	4	8	no	—
143	D₄	—	2	8	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
154	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
156	C₂×C₄	C₂×Q₈×C₈₃	1	2	yes	—
157	C₂×C₄	#41×C₈₃	1	2	yes	—
158	C₂×C₄	#41×C₈₃	1	4	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	8	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂²×D₄×C₈₃	1	1	yes	characteristic
165	C₂²	C₂²×Q₈×C₈₃	1	1	yes	characteristic
166	C₂²	C₂×#41×C₈₃	1	1	yes	characteristic
167	C₂²	C₂×#41×C₈₃	1	2	yes	—
168	C₂²	H(5,𝔽₂)×C₈₃	1	2	yes	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	4	4	no	—
173	C₄	C₂²⋊Q₈×C₈₃	1	4	yes	—
174	C₄	#124×C₈₃	1	4	yes	—
175	C₂	—	4	4	no	—
176	C₂	C₂×C₂²⋊Q₈×C₈₃	1	1	yes	characteristic
177	C₂	C₂×#124×C₈₃	1	1	yes	characteristic
178	C₂	#535×C₈₃	1	1	yes	characteristic
179	C₂	#554×C₈₃	1	2	yes	—
180	C₂	#555×C₈₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₈₃
prime radical	C₂²×D₄×C₈₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₈₃	C₂	semidirect
3. #393×C₈₃	C₂	semidirect
5. #382×C₈₃	C₂	semidirect
6. #380×C₈₃	C₂	semidirect
7. C₄×C₄⋊C₄×C₈₃	C₂	semidirect
28. C₂×D₄×C₈₃	Q₈	semidirect
90. #2563	C₈₃	direct (N × H)
91. C₈₃	#2563	direct (N × H)
92. C₂×C₄⋊Q₈	C₂×C₈₃	semidirect
94. #393	C₂×C₈₃	semidirect
96. #382	C₂×C₈₃	semidirect
97. #380	C₂×C₈₃	semidirect
98. C₄×C₄⋊C₄	C₂×C₈₃	semidirect
119. C₂×D₄	Q₈×C₈₃	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×C₈₃	2
C₂×C₄×D₄×C₈₃	2
#393×C₈₃	2
#390×C₈₃	2
#382×C₈₃	2
#380×C₈₃	2
C₄×C₄⋊C₄×C₈₃	2
#2563	83

Subgroups — C83×#2563 #24904248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#2563 #24904248