Finite Group Databasebeta

← #22565689

Subgroups — #22565689

Order 8576.

596 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14524017C211yesmaximal, characteristic
2#16863335C212yesmaximal, complemented
3#16863313C212yesmaximal, complemented
4#16863286C214yesmaximal, complemented
5#16863275C212yesmaximal, complemented
6#16863275C212yesmaximal, complemented
7C4×C22⋊Dic67C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C67C211yesmaximal, characteristic, complemented
9C22×D268C2211yescharacteristic
10C22×C22⋊D67C2212yes—
11#14524017—24no—
12C2×C4⋊Dic67C2211yescharacteristic
13C2×C22⋊Dic67C2212yes—
14C2×(C2×D67)⋊C4C2214yes—
15C2×(C2×D67)⋊C4C2212yes—
16C2×(C2×D67)⋊C4C2214yes—
17C2×C4×Dic67C2212yes—
18C67⋊#95C2212yes—
19C67⋊#95C2214yes—
20C67⋊#95C2214yes—
21C23×C4×C67C2211yescharacteristic
22C23×C4×C67C2211yescharacteristic
23C23×C4×C67C2211yescharacteristic
24C2×C4⋊C4×C67—24no—
25C2×C4⋊C4×C67C2212yes—
26C2×C4⋊C4×C67C2212yes—
27C23×D67C2312yes—
28C2×D268—22no—
29C2×D268—42no—
30C2×C22⋊D67—24no—
31C2×C22⋊D67—28no—
32C22×Dic67C2312yes—
33C22×Dic67C2314yes—
34C4⋊Dic67—22no—
35C22⋊Dic67—24no—
36(C2×D67)⋊C4—24no—
37(C2×D67)⋊C4—28no—
38(C2×D67)⋊C4—44no—
39C4×Dic67—42no—
40C24×C67C2311yescharacteristic
41C22×C4×C67—22no—
42C22×C4×C67—24no—
43C22×C4×C67—22no—
44C22×C4×C67—42no—
45C22×C4×C67—42no—
46C22×C4×C67D414yes—
47C22×C4×C67C2312yes—
48C22×C4×C67C2311yescharacteristic
49C22×C4×C67C2311yescharacteristic
50C22×C4×C67C2312yes—
51C4⋊C4×C67—24no—
52C4⋊C4×C67—24no—
53C23×C67—22no—
54C23×C67—22no—
55C23×C67C2×D412yes—
56C23×C67#4114yes—
57C23×C67C2411yescharacteristic
58C22×D67—22no—
59C22×D67—24no—
60C22×D67—24no—
61C22×D67—24no—
62C2×C4×C67—28no—
63C2×C4×C67—22no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—24no—
69C2×C4×C67—44no—
70C2×C4×C67—44no—
71C2×C4×C67C2×D412yes—
72C2×C4×C67C2×D412yes—
73C2×C4×C67#4114yes—
74C22⋊D67—48no—
75C2×Dic67—22no—
76C2×Dic67—24no—
77C2×Dic67—24no—
78C2×Dic67—24no—
79C2×Dic67—24no—
80D268—44no—
81C22×C67—22no—
82C22×C67—22no—
83C22×C67—28no—
84C22×C67#12414yes—
85C22×C67C22×D411yescharacteristic
86C22×C67C2×#4112yes—
87C22×C67C2×#4112yes—
88C22×C67H(5,𝔽2)12yes—
89C2×D67—42no—
90C2×D67—42no—
91C2×D67—42no—
92C2×D67—48no—
93C4×C67—24no—
94C4×C67—22no—
95C4×C67—22no—
96C4×C67—42no—
97Dic67—42no—
98Dic67—44no—
99C2×C67—24no—
100C2×C67#52511yescharacteristic
101C2×C67C2×#12411yescharacteristic
102C2×C67#53411yescharacteristic
103C2×C67#55314yes—
104D67—82no—
105#2591—671nomaximal, Sylow
106C67#259111yescharacteristic, complemented, Sylow
107C2×#121—671no—
108C22×C4⋊C4D6711yescharacteristic, complemented
109#400—672no—
110#394—672no—
111#388—674no—
112#386—672no—
113#386—672no—
114C4×C22⋊C4—671no—
115C22×D4—672no—
116C22×D4—671no—
117C23×C4C2×D6711yescharacteristic
118C23×C4C2×D6711yescharacteristic
119C23×C4C2×D6711yescharacteristic
120#121—1344no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—671no—
123C2×C4⋊C4C2×D6712yes—
124C2×C4⋊C4C2×D6712yes—
125C2×C22⋊C4—674no—
126C2×C22⋊C4—672no—
127C2×C22⋊C4—672no—
128C2×C22⋊C4—674no—
129C2×C42—672no—
130#95—672no—
131#95—674no—
132#95—674no—
133C24—672no—
134C24C22×D6711yescharacteristic
135C2×D4—2682no—
136C2×D4—1344no—
137C2×D4—1348no—
138C2×D4—1342no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—672no—
145C22×C4—674no—
146C22×C4C22⋊D6714yes—
147C22×C4C22×D6712yes—
148C22×C4C22×D6711yescharacteristic
149C22×C4C22×D6711yescharacteristic
150C22×C4C22×D6712yes—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—1342no—
154C22⋊C4—2684no—
155C22⋊C4—1344no—
156C22⋊C4—1344no—
157C22⋊C4—1348no—
158C42—2682no—
159C23—22no—
160C23—22no—
161C23—1342no—
162C23—1344no—
163C23—1344no—
164C23—1344no—
165C23C2×C22⋊D6712yes—
166C23#1176162914yes—
167C23C23×D6711yescharacteristic
168D4—2684no—
169D4—2688no—
170C2×C4—28no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—1342no—
180C2×C4—1344no—
181C2×C4—1344no—
182C2×C4—1344no—
183C2×C4—1344no—
184C2×C4C2×C22⋊D6712yes—
185C2×C4C2×C22⋊D6712yes—
186C2×C4#1176162814yes—
187C22—28no—
188C22—22no—
189C22—22no—
190C22—2688no—
191C22—2682no—
192C22—2682no—
193C22—2682no—
194C22#1452403214yes—
195C22C22×C22⋊D6711yescharacteristic
196C22C2×#1176162812yes—
197C22C2×#1176162912yes—
198C22#1452408812yes—
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—2684no—
204C4—2682no—
205C2—24no—
206C2—5362no—
207C2#1686390711yescharacteristic
208C2C2×#1452403211yescharacteristic
209C2#1686395511yescharacteristic
210C2#1686409414yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C4⋊C4×C67C2
hypercenterC22×C4⋊C4D67
hypocenterC67#2591
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16863335C2semidirect
3. #16863313C2semidirect
4. #16863286C2semidirect
5. #16863275C2semidirect
6. #16863275C2semidirect
7. C4×C22⋊Dic67C2semidirect
8. C22×C4⋊C4×C67C2semidirect
106. C67#2591semidirect
108. C22×C4⋊C4D67semidirect