Subgroups — #24907161

Order 10624.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581751	C₂	1	1	yes	maximal, characteristic
3	#19127455	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₈	—	2	2	no	—
6	C₂²×C₈₃⋊C₈	—	2	2	no	—
7	C₂²×C₈₃⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#14581751	—	2	2	no	—
10	#14581751	—	2	2	no	—
11	C₂×C₄⋊Dic₈₃	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₈₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₈₃	—	2	2	no	—
16	C₂×M₁₆×C₈₃	—	2	2	no	—
17	C₂×M₁₆×C₈₃	C₄	1	4	yes	complemented
18	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
19	C₂²×Dic₈₃	C₂×C₄	1	4	yes	—
20	C₂×C₈₃⋊C₈	—	2	8	no	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₈₃⋊C₈	—	2	2	no	—
23	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₈₃	—	2	4	no	—
25	C₂²⋊Dic₈₃	—	2	4	no	—
26	C₄×Dic₈₃	—	2	2	no	—
27	C₄×Dic₈₃	—	2	2	no	—
28	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₈₃	D₄	1	2	yes	—
33	C₂²×C₄×C₈₃	D₄	1	2	yes	—
34	C₂²×C₄×C₈₃	D₄	1	2	yes	—
35	C₂²×C₄×C₈₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₈₃	—	2	8	no	—
38	C₂×C₈×C₈₃	—	2	2	no	—
39	C₂×C₈×C₈₃	—	2	2	no	—
40	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
41	C₂³×C₈₃	—	2	4	no	—
42	C₂³×C₈₃	C₄²	1	2	yes	—
43	C₂³×C₈₃	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₈₃	C₄⋊C₄	1	2	yes	—
45	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₄²	1	2	yes	—
47	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×Dic₈₃	—	2	4	no	—
60	C₂×Dic₈₃	—	2	8	no	—
61	C₈×C₈₃	—	2	4	no	—
62	C₈₃⋊C₈	—	2	4	no	—
63	C₂²×C₈₃	—	2	4	no	—
64	C₂²×C₈₃	—	2	4	no	—
65	C₂²×C₈₃	—	2	4	no	—
66	C₂²×C₈₃	#95	1	4	yes	—
67	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₈₃	#131	1	4	yes	—
71	C₄×C₈₃	#95	1	4	yes	—
72	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	Dic₈₃	—	4	4	no	—
77	C₂×C₈₃	—	2	4	no	—
78	C₂×C₈₃	C₂×#95	1	1	yes	characteristic
79	C₂×C₈₃	#405	1	2	yes	—
80	C₂×C₈₃	#408	1	2	yes	—
81	C₂×C₈₃	#424	1	2	yes	—
82	#1729	—	83	1	no	maximal, Sylow
83	C₈₃	#1729	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	D₈₃	1	1	yes	characteristic
85	C₂³×C₈	—	83	1	no	—
86	C₂×#117	—	83	1	no	—
87	#336	—	83	4	no	—
88	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	Dic₈₃	1	4	yes	complemented
92	C₂²×C₈	—	83	4	no	—
93	C₂²×C₈	—	83	2	no	—
94	C₂²×C₈	—	166	2	no	—
95	C₂²×C₈	—	166	2	no	—
96	C₂²×C₈	C₂×D₈₃	1	2	yes	—
97	#117	—	166	2	no	—
98	#117	—	166	2	no	—
99	C₂×C₄⋊C₄	—	83	2	no	—
100	C₂×C₂²⋊C₄	—	83	2	no	—
101	C₂×C₄²	—	83	2	no	—
102	C₂⁴	C₄×D₈₃	1	1	yes	characteristic
103	C₂²×C₄	—	83	4	no	—
104	C₂²×C₄	Dic₁₆₆	1	2	yes	—
105	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₈₃	1	2	yes	—
107	C₂²×C₄	D₃₃₂	1	2	yes	—
108	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
109	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
112	M₁₆	—	2	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	83	4	no	—
116	C₂×C₈	—	166	8	no	—
117	C₂×C₈	—	166	2	no	—
118	C₂×C₈	—	166	2	no	—
119	C₂×C₈	C₂×Dic₈₃	1	4	yes	—
120	C₄⋊C₄	—	166	4	no	—
121	C₂²⋊C₄	—	166	4	no	—
122	C₄²	—	166	2	no	—
123	C₄²	—	166	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₈₃	1	2	yes	—
126	C₂³	(C₂×D₈₃)⋊C₄	1	2	yes	—
127	C₂³	Dic₈₃⋊C₄	1	2	yes	—
128	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₂×C₄	—	166	8	no	—
130	C₂×C₄	—	166	4	no	—
131	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
132	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
133	C₂×C₄	(C₂×D₈₃)⋊C₄	1	4	yes	—
134	C₂×C₄	C₂²⋊Dic₈₃	1	4	yes	—
135	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
136	C₂×C₄	Dic₈₃⋊C₄	1	4	yes	—
137	C₂×C₄	C₄⋊Dic₈₃	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
142	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	166	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	C₈₃⋊#95	1	4	yes	—
150	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
151	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
152	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
153	C₂²	#14581800	1	4	yes	—
154	C₄	—	332	4	no	—
155	C₄	C₈₃⋊#95	1	4	yes	—
156	C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
158	C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
159	C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×C₈₃⋊#95	1	1	yes	characteristic
162	C₂	#19127659	1	2	yes	—
163	C₂	#19127669	1	2	yes	—
164	C₂	#19127729	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄²
Fitting	C₂²×M₁₆×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×C₄
prime radical	C₂⁴×C₈₃	C₂×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂²×M₁₆	D₈₃
hypocenter	C₈₃	#1729
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #19127455	C₂	semidirect
7. C₂²×C₈₃⋊C₈	C₄	semidirect
17. C₂×M₁₆×C₈₃	C₄	semidirect
83. C₈₃	#1729	semidirect
91. C₂×M₁₆	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈₃⋊C₈	2
C₂×#14581751	2
#19127455	2
C₂²×M₁₆×C₈₃	2
#1729	83