Finite Group Databasebeta

← #30749310

Subgroups — #30749310

Order 17152.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#22560292C211yesmaximal, characteristic
2#22560286C211yesmaximal, characteristic
3#22558062C214yesmaximal
4C67×#2298C211yesmaximal, characteristic
5#16863360C2212yes—
6#16863259C2212yes—
7#16863165—22no—
8#16863163—22no—
9C2×#117×C67C2211yescharacteristic
10#388×C67—22no—
11#387×C67—22no—
12C4×C4⋊C4×C67C412yescomplemented
13C4×C22⋊C4×C67C2212yes—
14#376×C67C411yescharacteristic, complemented
15C43×C67C411yescharacteristic, complemented
16C2×#11761612—22no—
17C2×C22⋊Dic67—22no—
18C22⋊(C67⋊C8)—24no—
19C67⋊#95—24no—
20C23×C4×C67C2311yescharacteristic
21#117×C67—24no—
22C2×C4⋊C4×C67—22no—
23C2×C4⋊C4×C67D412yes—
24C2×C22⋊C4×C67—22no—
25C2×C22⋊C4×C67D412yes—
26C2×C42×C67—22no—
27C2×C42×C67C2×C411yescharacteristic
28C2×C42×C67C2×C411yescharacteristic
29C2×C42×C67C2×C412yes—
30C2×C42×C67C2×C412yes—
31#95×C67—22no—
32#95×C67—22no—
33#95×C67—22no—
34C22×Dic67—42no—
35#11761612—44no—
36C2×C67⋊C8—42no—
37C22⋊Dic67—44no—
38C24×C67C2×D411yescharacteristic
39C22×C4×C67—22no—
40C22×C4×C67—22no—
41C22×C4×C67—42no—
42C22×C4×C67C22⋊C412yes—
43C22×C4×C67C22⋊C412yes—
44C22×C4×C67C22×C411yescharacteristic
45C22×C4×C67C2×D411yescharacteristic
46C22×C4×C67#4114yes—
47C4⋊C4×C67—24no—
48C4⋊C4×C67—44no—
49C22⋊C4×C67—24no—
50C22⋊C4×C67—44no—
51C42×C67—22no—
52C42×C67—22no—
53C42×C67—42no—
54C42×C67—42no—
55C42×C67—44no—
56C23×C67—22no—
57C23×C67—42no—
58C23×C67C2×C22⋊C411yescharacteristic
59C23×C67#12312yes—
60C2×C4×C67—22no—
61C2×C4×C67—22no—
62C2×C4×C67—24no—
63C2×C4×C67—22no—
64C2×C4×C67—22no—
65C2×C4×C67—44no—
66C2×C4×C67—42no—
67C2×C4×C67—42no—
68C2×C4×C67—42no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—42no—
72C2×C4×C67—42no—
73C2×C4×C67#11712yes—
74C2×C4×C67#12312yes—
75C2×Dic67—82no—
76C2×Dic67—84no—
77C67⋊C8—82no—
78C22×C67—22no—
79C22×C67—22no—
80C22×C67—24no—
81C22×C67—82no—
82C22×C67#38111yescharacteristic
83C22×C67#41012yes—
84C4×C67—22no—
85C4×C67—22no—
86C4×C67—44no—
87C4×C67—42no—
88C4×C67—42no—
89C4×C67—82no—
90Dic67—162no—
91#8740—671nomaximal, Sylow
92C2×C67—22no—
93C2×C67—42no—
94C2×C67#182011yescharacteristic
95C2×C67#182712yes—
96#2298D6711yescharacteristic
97#1822—671no—
98#1820—671no—
99#1284—674no—
100C67#874011yescharacteristic, complemented, Sylow
101C2×#117C2×D6711yescharacteristic
102#407—672no—
103#388—22no—
104#387—22no—
105#381—672no—
106C4×C4⋊C4Dic6712yescomplemented
107C4×C22⋊C4C2×D6712yes—
108#376Dic6711yescharacteristic, complemented
109C43Dic6711yescharacteristic, complemented
110#343—1342no—
111#342—1342no—
112C23×C4C22×D6711yescharacteristic
113C2×M16—1342no—
114#117—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4C22⋊D6712yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—1342no—
119C2×C22⋊C4C22⋊D6712yes—
120C2×C42—22no—
121C2×C42C2×Dic6711yescharacteristic
122C2×C42C2×Dic6711yescharacteristic
123C2×C42C2×Dic6712yes—
124C2×C42C2×Dic6712yes—
125C22⋊C8—1344no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—1344no—
130C24C2×C22⋊D6711yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—2682no—
135C22×C4C22⋊Dic6712yes—
136C22×C4C22⋊Dic6712yes—
137C22×C4C22×Dic6711yescharacteristic
138C22×C4C2×C22⋊D6711yescharacteristic
139C22×C4#1176162814yes—
140M16—2684no—
141C2×C8—2682no—
142C4⋊C4—24no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—2684no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—42no—
151C42—44no—
152C23—22no—
153C23—42no—
154C23C2×C22⋊Dic6711yescharacteristic
155C23#1452402612yes—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—5362no—
170C2×C4—5364no—
171C2×C4#1452400212yes—
172C2×C4#1452402612yes—
173C8—5362no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#1686325911yescharacteristic
179C22#1686337012yes—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—10722no—
187C2—22no—
188C2—42no—
189C2#2256028611yescharacteristic
190C2#2256031312yes—
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2298C2
socleC22×C67#381
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenter#2298D67
hypocenterC67#8740
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
12. C4×C4⋊C4×C67C4semidirect
14. #376×C67C4semidirect
15. C43×C67C4semidirect
100. C67#8740semidirect
106. C4×C4⋊C4Dic67semidirect
108. #376Dic67semidirect
109. C43Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225602862
#225580622
C67×#22982
#874067