Subgroups — #35214233

Order 20864.

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	C₂²×#14579683	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23781420	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579683	—	2	2	no	—
6	C₂×#14579683	—	2	2	no	—
7	C₂×#14579683	C₄	1	4	yes	complemented
8	C₂²×C₁₆₃⋊C₈	—	2	2	no	—
9	C₂²×C₁₆₃⋊C₈	—	2	2	no	—
10	C₂²×C₁₆₃⋊C₈	C₄	1	4	yes	complemented
11	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
14	#117×C₁₆₃	—	2	2	no	—
15	#117×C₁₆₃	—	2	2	no	—
16	C₂×C₄⋊C₄×C₁₆₃	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₆₃	C₄	1	2	yes	—
18	C₂×C₄²×C₁₆₃	C₂²	1	2	yes	—
19	#14579683	—	2	8	no	—
20	C₂×C₁₆₃⋊C₈	—	2	8	no	—
21	C₂×C₁₆₃⋊C₈	—	2	2	no	—
22	C₂×C₁₆₃⋊C₈	—	2	2	no	—
23	C₂×C₁₆₃⋊C₈	—	2	2	no	—
24	C₂×C₁₆₃⋊C₈	—	2	2	no	—
25	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	—
26	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂⁴×C₁₆₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₆₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
33	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
35	C₂²×C₄×C₁₆₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₆₃	—	2	4	no	—
38	C₂²⋊C₄×C₁₆₃	—	2	4	no	—
39	C₄²×C₁₆₃	—	2	2	no	—
40	C₄²×C₁₆₃	—	2	2	no	—
41	C₁₆₃⋊C₈	—	2	4	no	—
42	C₁₆₃⋊C₈	—	2	4	no	—
43	C₂³×C₁₆₃	—	2	4	no	—
44	C₂³×C₁₆₃	C₄²	1	2	yes	—
45	C₂³×C₁₆₃	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₆₃	C₄⋊C₄	1	2	yes	—
47	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	—	2	4	no	—
49	C₂×C₄×C₁₆₃	—	2	8	no	—
50	C₂×C₄×C₁₆₃	C₄²	1	2	yes	—
51	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₆₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₆₃	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₁₆₃	—	2	4	no	—
64	C₂²×C₁₆₃	—	2	4	no	—
65	C₂²×C₁₆₃	—	2	4	no	—
66	C₂²×C₁₆₃	#95	1	4	yes	—
67	C₂²×C₁₆₃	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₆₃	#131	1	4	yes	—
71	C₄×C₁₆₃	—	4	4	no	—
72	C₄×C₁₆₃	#95	1	4	yes	—
73	C₄×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₁₆₃	—	2	4	no	—
78	C₂×C₁₆₃	C₂×#95	1	1	yes	characteristic
79	C₂×C₁₆₃	#405	1	2	yes	—
80	C₂×C₁₆₃	#408	1	2	yes	—
81	C₂×C₁₆₃	#424	1	2	yes	—
82	C₁₆₃	#1729	1	1	yes	characteristic, complemented, Sylow
83	#1729	—	163	1	no	maximal, Sylow
84	C₂²×M₁₆	—	163	1	no	—
85	C₂³×C₈	—	163	1	no	—
86	C₂×#117	D₁₆₃	1	1	yes	characteristic
87	#336	—	163	4	no	—
88	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
89	C₂×M₁₆	—	326	2	no	—
90	C₂×M₁₆	—	326	2	no	—
91	C₂×M₁₆	—	163	4	no	—
92	C₂²×C₈	—	326	2	no	—
93	C₂²×C₈	—	326	2	no	—
94	C₂²×C₈	—	163	4	no	—
95	C₂²×C₈	—	163	2	no	—
96	C₂²×C₈	—	163	2	no	—
97	#117	—	2	2	no	—
98	#117	—	2	2	no	—
99	C₂×C₄⋊C₄	Dic₁₆₃	1	2	yes	—
100	C₂×C₂²⋊C₄	Dic₁₆₃	1	2	yes	—
101	C₂×C₄²	C₂×D₁₆₃	1	2	yes	—
102	C₂⁴	C₂×Dic₁₆₃	1	1	yes	characteristic
103	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
104	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
105	C₂²×C₄	C₂×Dic₁₆₃	1	4	yes	—
106	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
108	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
109	C₂²×C₄	D₆₅₂	1	2	yes	—
110	C₂²×C₄	Dic₃₂₆	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
112	M₁₆	—	326	8	no	—
113	C₂×C₈	—	326	8	no	—
114	C₂×C₈	—	326	2	no	—
115	C₂×C₈	—	326	2	no	—
116	C₂×C₈	—	326	2	no	—
117	C₂×C₈	—	326	2	no	—
118	C₂×C₈	—	163	4	no	—
119	C₂×C₈	—	163	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₁₆₃	1	2	yes	—
126	C₂³	C₂²⋊Dic₁₆₃	1	2	yes	—
127	C₂³	C₄⋊Dic₁₆₃	1	2	yes	—
128	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	C₄×Dic₁₆₃	1	2	yes	—
132	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	4	yes	—
133	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	4	yes	—
134	C₂×C₄	C₂²⋊Dic₁₆₃	1	2	yes	—
135	C₂×C₄	Dic₁₆₃⋊C₄	1	4	yes	—
136	C₂×C₄	Dic₁₆₃⋊C₄	1	4	yes	—
137	C₂×C₄	C₄⋊Dic₁₆₃	1	2	yes	—
138	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
142	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
144	C₈	—	326	4	no	—
145	C₈	—	326	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	C₁₆₃⋊#95	1	4	yes	—
150	C₂²	C₂×C₄×Dic₁₆₃	1	1	yes	characteristic
151	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
152	C₂²	C₂×C₄⋊Dic₁₆₃	1	1	yes	characteristic
153	C₂²	#19104742	1	4	yes	—
154	C₄	—	4	4	no	—
155	C₄	C₁₆₃⋊#95	1	4	yes	—
156	C₄	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
157	C₄	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
158	C₄	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
159	C₄	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×C₁₆₃⋊#95	1	1	yes	characteristic
162	C₂	#23781621	1	2	yes	—
163	C₂	#23781632	1	2	yes	—
164	C₂	#23781692	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator	C₂²×C₁₆₃	C₂×C₄²
Fitting	C₂×#117×C₁₆₃	C₂
socle	C₂³×C₁₆₃	C₂²×C₄
prime radical	C₂⁴×C₁₆₃	C₂×C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×#117	D₁₆₃
hypocenter	C₁₆₃	#1729
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #23781420	C₂	semidirect
7. C₂×#14579683	C₄	semidirect
10. C₂²×C₁₆₃⋊C₈	C₄	semidirect
82. C₁₆₃	#1729	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
C₂³×C₁₆₃⋊C₈	2
#23781420	2
C₂×#117×C₁₆₃	2
#1729	163