Finite Group Databasebeta

← #30749321

Subgroups — #30749321

Order 17152.

450 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#22558543C212yesmaximal
2#22558065C214yesmaximal, complemented
3C67×#2296C211yesmaximal, characteristic
4#16863453C2214yes—
5#16863122—22no—
6#16863120—22no—
7C2×#117×C67C411yescharacteristic
8C2×#117×C67C2211yescharacteristic
9C22×C42×C67C411yescharacteristic
10#389×C67—22no—
11#385×C67—22no—
12C4×C4⋊C4×C67C412yes—
13C4×C4⋊C4×C67C2212yes—
14C4⋊(C67⋊C8)—24no—
15C22⋊(C67⋊C8)—24no—
16#14523954—24no—
17C23×C4×C67—21no—
18C23×C4×C67C2×C411yescharacteristic
19#117×C67—24no—
20#117×C67—42no—
21C2×C4⋊C4×C67—21no—
22C2×C4⋊C4×C67—22no—
23C2×C4⋊C4×C67—22no—
24C2×C4⋊C4×C67D412yes—
25C2×C22⋊C4×C67—21no—
26C2×C22⋊C4×C67D412yes—
27C2×C42×C67—21no—
28C2×C42×C67—42no—
29C2×C42×C67C2×C411yescharacteristic
30C2×C42×C67C2×C412yes—
31C2×C42×C67C2×C412yes—
32C2×C42×C67C2311yescharacteristic
33#95×C67—22no—
34#95×C67—22no—
35C2×C67⋊C8—44no—
36C24×C67C22⋊C411yescharacteristic
37C22×C4×C67—22no—
38C22×C4×C67—22no—
39C22×C4×C67—22no—
40C22×C4×C67—22no—
41C22×C4×C67—22no—
42C22×C4×C67—44no—
43C22×C4×C67—42no—
44C22×C4×C67C22⋊C411yescharacteristic
45C22×C4×C67C22⋊C412yes—
46C22×C4×C67C22×C411yescharacteristic
47C22×C4×C67C2×D412yes—
48C4⋊C4×C67—24no—
49C4⋊C4×C67—44no—
50C4⋊C4×C67—44no—
51C4⋊C4×C67—42no—
52C22⋊C4×C67—24no—
53C22⋊C4×C67—42no—
54C42×C67—22no—
55C42×C67—42no—
56C42×C67—42no—
57C42×C67—42no—
58C42×C67#4114yes—
59C23×C67—21no—
60C23×C67—21no—
61C23×C67—42no—
62C23×C67#10112yes—
63C23×C67C2×C22⋊C411yescharacteristic
64C2×C4×C67—28no—
65C2×C4×C67—21no—
66C2×C4×C67—21no—
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—24no—
70C2×C4×C67—22no—
71C2×C4×C67—42no—
72C2×C4×C67—42no—
73C2×C4×C67—48no—
74C2×C4×C67—42no—
75C2×C4×C67—42no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67—42no—
79C2×C4×C67#10012yes—
80C2×C4×C67#11712yes—
81C2×C4×C67#12314yes—
82C67⋊C8—84no—
83C22×C67—22no—
84C22×C67—22no—
85C22×C67—42no—
86C22×C67—42no—
87C22×C67—42no—
88C22×C67#38111yescharacteristic
89C22×C67C2×#10011yescharacteristic
90C22×C67C2×#10111yescharacteristic
91C4×C67—22no—
92C4×C67—22no—
93C4×C67—24no—
94C4×C67—44no—
95C4×C67—42no—
96C4×C67—82no—
97#8743—671nomaximal, Sylow
98C2×C67—21no—
99C2×C67—21no—
100C2×C67—42no—
101C2×C67#182111yescharacteristic
102C2×C67#182712yes—
103#2296D6711yescharacteristic
104#1464—672no—
105#1285—674no—
106C67#874311yescharacteristic, complemented, Sylow
107C2×#117Dic6711yescharacteristic
108C2×#117C2×D6711yescharacteristic
109C22×C42Dic6711yescharacteristic
110#432—674no—
111#389—22no—
112#385—22no—
113C4×C4⋊C4Dic6712yes—
114C4×C4⋊C4C2×D6712yes—
115#324—1342no—
116#323—1342no—
117C23×C4—21no—
118C23×C4C2×Dic6711yescharacteristic
119#117—24no—
120#117—42no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4C22⋊D6712yes—
125C2×C22⋊C4—21no—
126C2×C22⋊C4C22⋊D6712yes—
127C2×C42—21no—
128C2×C42—42no—
129C2×C42C2×Dic6711yescharacteristic
130C2×C42C2×Dic6712yes—
131C2×C42C2×Dic6712yes—
132C2×C42C22×D6711yescharacteristic
133C4⋊C8—1344no—
134C22⋊C8—1344no—
135#97—1344no—
136#95—22no—
137#95—22no—
138C24C22⋊Dic6711yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4C22⋊Dic6711yescharacteristic
147C22×C4C22⋊Dic6712yes—
148C22×C4C22×Dic6711yescharacteristic
149C22×C4C2×C22⋊D6712yes—
150C2×C8—2684no—
151C4⋊C4—24no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C22⋊C4—24no—
156C22⋊C4—42no—
157C42—22no—
158C42—42no—
159C42—42no—
160C42—42no—
161C42#1176162814yes—
162C23—21no—
163C23—21no—
164C23—42no—
165C23#1452396212yes—
166C23C2×C22⋊Dic6711yescharacteristic
167C2×C4—28no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—48no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4#1452396012yes—
183C2×C4#1452400212yes—
184C2×C4#1452402614yes—
185C8—5364no—
186C22—22no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22#1686325911yescharacteristic
192C22C2×#1452396011yescharacteristic
193C22C2×#1452396211yescharacteristic
194C4—22no—
195C4—22no—
196C4—24no—
197C4—44no—
198C4—42no—
199C4—82no—
200C2—21no—
201C2—21no—
202C2—42no—
203C2#2256028911yescharacteristic
204C2#2256031312yes—
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2296C2
socleC22×C67#381
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2296D67
hypocenterC67#8743
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #22558065C2semidirect
106. C67#8743semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225580652
C67×#22962
#874367