Subgroups — #35214229

Order 20864.

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×#14579683	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#23781419	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₆₃	C₄	1	2	yes	—
6	C₂×#14579683	—	2	2	no	—
7	C₂×#14579683	—	2	2	no	—
8	C₂×#14579683	C₄	1	4	yes	complemented
9	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₆₃	—	2	2	no	—
11	C₂×C₄×Dic₁₆₃	—	2	2	no	—
12	C₂×C₄×Dic₁₆₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₆₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₆₃	—	2	2	no	—
16	C₂×M₁₆×C₁₆₃	—	2	2	no	—
17	C₂×M₁₆×C₁₆₃	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
19	C₂²×Dic₁₆₃	—	2	8	no	—
20	C₂²×Dic₁₆₃	—	2	4	no	—
21	C₂²×Dic₁₆₃	C₂×C₄	1	4	yes	—
22	#14579683	—	2	8	no	—
23	C₂×C₁₆₃⋊C₈	—	2	2	no	—
24	C₂×C₁₆₃⋊C₈	—	2	2	no	—
25	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	complemented
26	C₄×Dic₁₆₃	—	2	4	no	—
27	C₄×Dic₁₆₃	—	2	2	no	—
28	C₄×Dic₁₆₃	—	2	2	no	—
29	C₂⁴×C₁₆₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
35	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
36	C₂²×C₄×C₁₆₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₆₃	—	2	8	no	—
39	C₂×C₈×C₁₆₃	—	2	2	no	—
40	C₂×C₈×C₁₆₃	—	2	2	no	—
41	C₂×C₈×C₁₆₃	C₂×C₄	1	4	yes	complemented
42	C₂×Dic₁₆₃	—	2	16	no	—
43	C₂×Dic₁₆₃	—	2	4	no	—
44	C₂×Dic₁₆₃	—	2	4	no	—
45	C₂×Dic₁₆₃	M₁₆	1	8	yes	complemented
46	C₁₆₃⋊C₈	—	2	4	no	—
47	C₂³×C₁₆₃	—	2	4	no	—
48	C₂³×C₁₆₃	C₄²	1	2	yes	—
49	C₂³×C₁₆₃	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₆₃	C₄⋊C₄	1	2	yes	—
51	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₆₃	C₄²	1	2	yes	—
53	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₆₃	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₁₆₃	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
65	C₈×C₁₆₃	—	2	4	no	—
66	C₂²×C₁₆₃	—	2	4	no	—
67	C₂²×C₁₆₃	—	2	4	no	—
68	C₂²×C₁₆₃	—	2	4	no	—
69	C₂²×C₁₆₃	#95	1	4	yes	—
70	C₂²×C₁₆₃	C₂×C₄²	1	1	yes	characteristic
71	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₆₃	C₂×M₁₆	1	4	yes	—
74	C₄×C₁₆₃	#95	1	4	yes	—
75	C₄×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Dic₁₆₃	—	2	8	no	—
80	C₂×C₁₆₃	—	2	4	no	—
81	C₂×C₁₆₃	C₂×#95	1	1	yes	characteristic
82	C₂×C₁₆₃	C₄×M₁₆	1	2	yes	—
83	C₂×C₁₆₃	#407	1	2	yes	—
84	C₂×C₁₆₃	#423	1	2	yes	—
85	C₁₆₃	#1728	1	1	yes	characteristic, complemented, Sylow
86	#1728	—	163	1	no	maximal, Sylow
87	C₂²×M₁₆	—	163	1	no	—
88	C₂²×M₁₆	D₁₆₃	1	1	yes	characteristic
89	C₂²×C₄²	—	163	1	no	—
90	#336	—	163	4	no	—
91	C₂³×C₄	—	163	2	no	—
92	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	326	2	no	—
96	C₂×M₁₆	—	326	2	no	—
97	C₂×M₁₆	—	163	4	no	—
98	C₂×M₁₆	Dic₁₆₃	1	4	yes	complemented
99	C₂²×C₈	—	163	2	no	—
100	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
101	C₂×C₄²	—	326	2	no	—
102	C₂×C₄²	—	326	2	no	—
103	C₂×C₄²	—	163	2	no	—
104	C₂×C₄²	—	163	2	no	—
105	C₂⁴	C₄×D₁₆₃	1	1	yes	characteristic
106	C₂²×C₄	—	326	4	no	—
107	C₂²×C₄	—	326	8	no	—
108	C₂²×C₄	—	163	4	no	—
109	C₂²×C₄	C₄×D₁₆₃	1	1	yes	characteristic
110	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
112	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
113	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
114	C₂²×C₄	D₆₅₂	1	2	yes	—
115	C₂²×C₄	Dic₃₂₆	1	2	yes	—
116	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
117	M₁₆	—	2	8	no	—
118	M₁₆	—	326	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	326	2	no	—
122	C₂×C₈	—	326	2	no	—
123	C₂×C₈	—	163	4	no	—
124	C₂×C₈	C₂×Dic₁₆₃	1	4	yes	complemented
125	C₄²	—	326	4	no	—
126	C₄²	—	326	2	no	—
127	C₄²	—	326	2	no	—
128	C₂³	—	2	4	no	—
129	C₂³	C₄×Dic₁₆₃	1	2	yes	—
130	C₂³	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
131	C₂³	Dic₁₆₃⋊C₄	1	2	yes	—
132	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
133	C₂×C₄	—	326	16	no	—
134	C₂×C₄	—	326	4	no	—
135	C₂×C₄	—	326	4	no	—
136	C₂×C₄	—	163	8	no	—
137	C₂×C₄	C₄×Dic₁₆₃	1	2	yes	—
138	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	4	yes	—
140	C₂×C₄	C₂²⋊Dic₁₆₃	1	4	yes	—
141	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
142	C₂×C₄	Dic₁₆₃⋊C₄	1	4	yes	—
143	C₂×C₄	C₄⋊Dic₁₆₃	1	4	yes	—
144	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
148	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	326	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₁₆₃⋊#95	1	4	yes	—
156	C₂²	C₂×C₄×Dic₁₆₃	1	1	yes	characteristic
157	C₂²	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
158	C₂²	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
159	C₂²	M₁₆×D₁₆₃	1	4	yes	—
160	C₄	—	326	8	no	—
161	C₄	C₁₆₃⋊#95	1	4	yes	—
162	C₄	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
164	C₄	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
165	C₄	C₂×C₄⋊Dic₁₆₃	1	1	yes	characteristic
166	C₂	—	2	4	no	—
167	C₂	C₂×C₁₆₃⋊#95	1	1	yes	characteristic
168	C₂	M₁₆×Dic₁₆₃	1	2	yes	—
169	C₂	#23781628	1	2	yes	—
170	C₂	#23781688	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator	C₂²×C₁₆₃	C₂×C₄²
Fitting	C₂²×M₁₆×C₁₆₃	C₂
socle	C₂³×C₁₆₃	C₂²×C₄
prime radical	C₂⁴×C₁₆₃	C₂×C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂²×M₁₆	D₁₆₃
hypocenter	C₁₆₃	#1728
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23781419	C₂	semidirect
8. C₂×#14579683	C₄	semidirect
17. C₂×M₁₆×C₁₆₃	C₄	semidirect
25. C₂×C₁₆₃⋊C₈	C₂×C₄	semidirect
41. C₂×C₈×C₁₆₃	C₂×C₄	semidirect
45. C₂×Dic₁₆₃	M₁₆	semidirect
85. C₁₆₃	#1728	semidirect
98. C₂×M₁₆	Dic₁₆₃	semidirect
124. C₂×C₈	C₂×Dic₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579683	2
C₂²×C₄×Dic₁₆₃	2
#23781419	2
C₂²×M₁₆×C₁₆₃	2
#1728	163