Finite Group Databasebeta

← #22568267

Subgroups — #22568267

Order 8576.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524043C211yesmaximal, characteristic, complemented
2#16863335C212yesmaximal, complemented
3#16863332C212yesmaximal
4#16863326C212yesmaximal, complemented
5#16863321C212yesmaximal
6#16863316C212yesmaximal, complemented
7#16863308C212yesmaximal
8#16863264C211yesmaximal, characteristic, complemented
9C2×#126×C67C211yesmaximal, characteristic
10C22×Dic134C2212yes—
11#14524043—24no—
12C2×C4⋊Dic67C2212yes—
13C2×C4⋊Dic67C2212yes—
14C2×Dic67⋊C4C2214yes—
15C2×C22⋊Dic67C2214yes—
16C67⋊#95C2212yes—
17C67⋊#95C2212yes—
18C67⋊#95C2212yes—
19C67⋊#95C2214yes—
20C67⋊#95C2214yes—
21#126×C67—24no—
22C2×C4⋊C4×C67C2211yescharacteristic
23C2×C4⋊C4×C67C2211yescharacteristic
24C2×C4⋊C4×C67C2211yescharacteristic
25C2×C22⋊C4×C67C2211yescharacteristic
26C2×C22⋊C4×C67C2211yescharacteristic
27C2×C22⋊C4×C67C2211yescharacteristic
28C2×C42×C67C2211yescharacteristic
29C2×Dic134—24no—
30C2×Dic134—44no—
31C22×Dic67C2314yes—
32C22×Dic67C2314yes—
33C4⋊Dic67—24no—
34C4⋊Dic67—24no—
35Dic67⋊C4—44no—
36C22⋊Dic67—44no—
37C24×C67C2311yescharacteristic
38C22×C4×C67C2311yescharacteristic
39C22×C4×C67C2311yescharacteristic
40C22×C4×C67C2311yescharacteristic
41C22×C4×C67C2311yescharacteristic
42C22×C4×C67C2311yescharacteristic
43C22×C4×C67C2311yescharacteristic
44C4⋊C4×C67—22no—
45C4⋊C4×C67—22no—
46C4⋊C4×C67—22no—
47C22⋊C4×C67—22no—
48C22⋊C4×C67—22no—
49C22⋊C4×C67—22no—
50C42×C67D414yes—
51C23×C67—21no—
52C23×C67—22no—
53C23×C67—22no—
54C23×C67—22no—
55C23×C67C2411yescharacteristic
56C2×C4×C67—22no—
57C2×C4×C67—22no—
58C2×C4×C67—22no—
59C2×C4×C67—21no—
60C2×C4×C67—21no—
61C2×C4×C67—21no—
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—22no—
65C2×C4×C67C2×D412yes—
66C2×C4×C67C2×D412yes—
67C2×C4×C67C2×D412yes—
68C2×Dic67—24no—
69C2×Dic67—24no—
70C2×Dic67—24no—
71C2×Dic67—24no—
72C2×Dic67—24no—
73C2×Dic67—24no—
74Dic134—48no—
75C22×C67—41no—
76C22×C67—41no—
77C22×C67—41no—
78C22×C67—44no—
79C22×C67C22×D411yescharacteristic
80C22×C67H(5,𝔽2)12yes—
81C22×C672−1+412yes—
82C22×C672−1+412yes—
83C4×C67—22no—
84C4×C67—22no—
85C4×C67—22no—
86C4×C67—41no—
87C4×C67—41no—
88C4×C67—41no—
89Dic67—44no—
90Dic67—44no—
91C2×C67—81no—
92C2×C67#53511yescharacteristic
93C2×C67#53611yescharacteristic
94C2×C67#53611yescharacteristic
95C2×C67#56314yes—
96#2821—671nomaximal, Sylow
97C67#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8—671no—
99C2×#126D6711yescharacteristic
100#400—672no—
101#399—672no—
102#398—672no—
103#397—672no—
104#395—672no—
105#393—672no—
106#383—671no—
107C22×Q8—672no—
108C4⋊Q8—1344no—
109#126—24no—
110C2×C4⋊C4—674no—
111C2×C4⋊C4—672no—
112C2×C4⋊C4—672no—
113C2×C4⋊C4C2×D6711yescharacteristic
114C2×C4⋊C4C2×D6711yescharacteristic
115C2×C4⋊C4C2×D6711yescharacteristic
116C2×C22⋊C4—674no—
117C2×C22⋊C4C2×D6711yescharacteristic
118C2×C22⋊C4C2×D6711yescharacteristic
119C2×C22⋊C4C2×D6711yescharacteristic
120C2×C42C2×D6711yescharacteristic
121#95—672no—
122#95—672no—
123#95—672no—
124#95—674no—
125#95—674no—
126C24C22×D6711yescharacteristic
127C2×Q8—2684no—
128C2×Q8—1344no—
129C22×C4—674no—
130C22×C4—674no—
131C22×C4C22×D6711yescharacteristic
132C22×C4C22×D6711yescharacteristic
133C22×C4C22×D6711yescharacteristic
134C22×C4C22×D6711yescharacteristic
135C22×C4C22×D6711yescharacteristic
136C22×C4C22×D6711yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—2684no—
141C4⋊C4—1344no—
142C4⋊C4—1344no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—2684no—
147C42C22⋊D6714yes—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23C23×D6711yescharacteristic
153Q8—2688no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—1344no—
164C2×C4—1344no—
165C2×C4—1344no—
166C2×C4—1344no—
167C2×C4—1344no—
168C2×C4—1344no—
169C2×C4C2×C22⋊D6712yes—
170C2×C4C2×C22⋊D6712yes—
171C2×C4C2×C22⋊D6712yes—
172C22—44no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C22×C22⋊D6711yescharacteristic
177C22#1452408912yes—
178C22#1452409012yes—
179C22#1452409012yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—2684no—
187C4—2684no—
188C2—81no—
189C2#1686396011yescharacteristic
190C2#1686396711yescharacteristic
191C2#1686396711yescharacteristic
192C2#1686414814yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#126×C67C2
hypercenterC2×#126D67
hypocenterC67#2821
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14524043C2semidirect
2. #16863335C2semidirect
4. #16863326C2semidirect
6. #16863316C2semidirect
8. #16863264C2semidirect
97. C67#2821semidirect