Subgroups — #24907156

Order 10624.

480 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13050574	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₈₃	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	C₂³×Dic₈₃	C₄	1	2	yes	—
6	C₂×#13050574	—	2	2	no	—
7	C₂×#13050574	—	2	2	no	—
8	C₂×#13050574	C₄	1	4	yes	complemented
9	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₈₃	—	2	2	no	—
11	C₂×C₄×Dic₈₃	—	2	2	no	—
12	C₂×C₄×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	—	2	4	no	—
20	C₂²×Dic₈₃	—	2	8	no	—
21	C₂²×Dic₈₃	C₂×C₄	1	4	yes	—
22	#13050574	—	2	8	no	—
23	C₂×C₈₃⋊C₈	—	2	2	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	complemented
26	C₄×Dic₈₃	—	2	4	no	—
27	C₄×Dic₈₃	—	2	2	no	—
28	C₄×Dic₈₃	—	2	2	no	—
29	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₈₃	D₄	1	2	yes	—
34	C₂²×C₄×C₈₃	D₄	1	2	yes	—
35	C₂²×C₄×C₈₃	D₄	1	2	yes	—
36	C₂²×C₄×C₈₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
38	M₁₆×C₈₃	—	2	8	no	—
39	C₂×C₈×C₈₃	—	2	2	no	—
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	complemented
42	C₂³×C₈₃	—	2	4	no	—
43	C₂³×C₈₃	C₄²	1	2	yes	—
44	C₂³×C₈₃	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₈₃	C₄⋊C₄	1	2	yes	—
46	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₄²	1	2	yes	—
48	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂²×C₄	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×D₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₈₃	—	2	16	no	—
61	C₂×Dic₈₃	—	2	4	no	—
62	C₂×Dic₈₃	—	2	4	no	—
63	C₂×Dic₈₃	M₁₆	1	8	yes	complemented
64	C₈×C₈₃	—	2	4	no	—
65	C₈₃⋊C₈	—	2	4	no	—
66	C₂²×C₈₃	—	2	4	no	—
67	C₂²×C₈₃	—	2	4	no	—
68	C₂²×C₈₃	—	2	4	no	—
69	C₂²×C₈₃	#95	1	4	yes	—
70	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
71	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₈₃	C₂×M₁₆	1	4	yes	—
74	C₄×C₈₃	#95	1	4	yes	—
75	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Dic₈₃	—	2	8	no	—
80	C₂×C₈₃	—	2	4	no	—
81	C₂×C₈₃	C₂×#95	1	1	yes	characteristic
82	C₂×C₈₃	C₄×M₁₆	1	2	yes	—
83	C₂×C₈₃	#407	1	2	yes	—
84	C₂×C₈₃	#423	1	2	yes	—
85	#1728	—	83	1	no	maximal, Sylow
86	C₈₃	#1728	1	1	yes	characteristic, complemented, Sylow
87	C₂²×M₁₆	—	83	1	no	—
88	C₂²×M₁₆	D₈₃	1	1	yes	characteristic
89	C₂²×C₄²	—	83	1	no	—
90	#336	—	83	4	no	—
91	C₂³×C₄	—	83	2	no	—
92	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	83	4	no	—
96	C₂×M₁₆	—	166	2	no	—
97	C₂×M₁₆	—	166	2	no	—
98	C₂×M₁₆	Dic₈₃	1	4	yes	complemented
99	C₂²×C₈	—	83	2	no	—
100	C₂²×C₈	C₂×D₈₃	1	2	yes	—
101	C₂×C₄²	—	83	2	no	—
102	C₂×C₄²	—	83	2	no	—
103	C₂×C₄²	—	166	2	no	—
104	C₂×C₄²	—	166	2	no	—
105	C₂⁴	C₄×D₈₃	1	1	yes	characteristic
106	C₂²×C₄	—	83	4	no	—
107	C₂²×C₄	—	166	8	no	—
108	C₂²×C₄	—	166	4	no	—
109	C₂²×C₄	Dic₁₆₆	1	2	yes	—
110	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₈₃	1	2	yes	—
112	C₂²×C₄	D₃₃₂	1	2	yes	—
113	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
114	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
117	M₁₆	—	2	8	no	—
118	M₁₆	—	166	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	83	4	no	—
122	C₂×C₈	—	166	2	no	—
123	C₂×C₈	—	166	2	no	—
124	C₂×C₈	C₂×Dic₈₃	1	4	yes	complemented
125	C₄²	—	166	4	no	—
126	C₄²	—	166	2	no	—
127	C₄²	—	166	2	no	—
128	C₂³	—	2	4	no	—
129	C₂³	C₄×Dic₈₃	1	2	yes	—
130	C₂³	(C₂×D₈₃)⋊C₄	1	2	yes	—
131	C₂³	Dic₈₃⋊C₄	1	2	yes	—
132	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
133	C₂×C₄	—	83	8	no	—
134	C₂×C₄	—	166	16	no	—
135	C₂×C₄	—	166	4	no	—
136	C₂×C₄	—	166	4	no	—
137	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
138	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₈₃)⋊C₄	1	4	yes	—
140	C₂×C₄	C₂²⋊Dic₈₃	1	4	yes	—
141	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
142	C₂×C₄	Dic₈₃⋊C₄	1	4	yes	—
143	C₂×C₄	C₄⋊Dic₈₃	1	4	yes	—
144	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
148	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	166	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₈₃⋊#95	1	4	yes	—
156	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
157	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
158	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
159	C₂²	M₁₆×D₈₃	1	4	yes	—
160	C₄	—	166	8	no	—
161	C₄	C₈₃⋊#95	1	4	yes	—
162	C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
164	C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
165	C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
166	C₂	—	2	4	no	—
167	C₂	C₂×C₈₃⋊#95	1	1	yes	characteristic
168	C₂	M₁₆×Dic₈₃	1	2	yes	—
169	C₂	#19127664	1	2	yes	—
170	C₂	#19127724	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	#1728
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
8. C₂×#13050574	C₄	semidirect
17. C₂×M₁₆×C₈₃	C₄	semidirect
25. C₂×C₈₃⋊C₈	C₂×C₄	semidirect
41. C₂×C₈×C₈₃	C₂×C₄	semidirect
63. C₂×Dic₈₃	M₁₆	semidirect
86. C₈₃	#1728	semidirect
98. C₂×M₁₆	Dic₈₃	semidirect
124. C₂×C₈	C₂×Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13050574	2
C₂²×C₄×Dic₈₃	2
#19127455	2
C₂²×M₁₆×C₈₃	2
#1728	83