Finite Group Databasebeta

← #16925762

Subgroups — #16925762

Order 4352.

684 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 147 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#14527179C214yesmaximal, complemented
2#14527098C212yesmaximal
3C2×C17×#541C211yesmaximal, characteristic
4#11762862C2214yes—
5Q8⋊#3622—28no—
6#11762535—22no—
7#11762533—22no—
8C17×#541—28no—
9C2×C17×C4⋊Q8C411yescharacteristic
10C2×C17×#126—22no—
11C2×C17×#125—22no—
12C2×C17×#125C411yescharacteristic
13C2×C17×C22⋊Q8C412yes—
14C2×C4×Q8×C17C2212yes—
15C2×C17×#117C2211yescharacteristic
16C22×Q8×C17C2×C412yes—
17C23×C4×C17C2×C411yescharacteristic
18C17×C4⋊Q8—24no—
19C17×#126—216no—
20C17×#125—216no—
21C17×#125—24no—
22C17×C22⋊Q8—24no—
23C17×C22⋊Q8—24no—
24C4×Q8×C17—24no—
25C4×Q8×C17D418yes—
26C17×#117—24no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17D412yes—
35C2×C42×C17—22no—
36C2×C42×C17C2×C411yescharacteristic
37C2×C42×C17C2311yescharacteristic
38C22⋊#3622—24no—
39C4⋊#3622—24no—
40C4×#3622—24no—
41C24×C17C22⋊C411yescharacteristic
42C2×Q8×C17—22no—
43C2×Q8×C17—28no—
44C2×Q8×C17C22⋊C414yes—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17—24no—
49C22×C4×C17C22⋊C411yescharacteristic
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22×C411yescharacteristic
52C22×C4×C17C2×D412yes—
53C4⋊C4×C17—28no—
54C4⋊C4×C17—28no—
55C4⋊C4×C17—28no—
56C4⋊C4×C17—22no—
57C4⋊C4×C17—24no—
58C4⋊C4×C17—28no—
59C4⋊C4×C17C22⋊C414yes—
60C22⋊C4×C17—28no—
61C22⋊C4×C17—24no—
62C42×C17—22no—
63C42×C17—28no—
64C42×C17C2×D414yes—
65C2×#3622—44no—
66#8327—171nomaximal, Sylow
67C23×C17—21no—
68C23×C17—22no—
69C23×C17—42no—
70C23×C17C2×C22⋊C411yescharacteristic
71Q8×C17—28no—
72Q8×C17—44no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17—24no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—42no—
79C2×C4×C17—44no—
80C2×C4×C17—44no—
81C2×C4×C17—44no—
82C2×C4×C17—42no—
83C2×C4×C17C2×C22⋊C412yes—
84C2×C4×C17#12014yes—
85C2×C4×C17#13114yes—
86#3622—84no—
87C2×#541D1711yescharacteristic
88#1479—174no—
89#1458—172no—
90C22×C17—22no—
91C22×C17—42no—
92C22×C17—41no—
93C22×C17—82no—
94C22×C17#37911yescharacteristic
95C22×C17#40812yes—
96C4×C17—24no—
97C4×C17—42no—
98C4×C17—44no—
99C4×C17—41no—
100C4×C17—44no—
101C4×C17#40814yes—
102#541—28no—
103C2×C4⋊Q8Dic1711yescharacteristic
104C2×#126—22no—
105C2×#125—22no—
106C2×#125Dic1711yescharacteristic
107C2×C22⋊Q8Dic1712yes—
108C2×C4×Q8C2×D1712yes—
109C2×#117C2×D1711yescharacteristic
110#431—174no—
111Q8⋊C8—348no—
112#324—342no—
113#323—342no—
114C2×C17—22no—
115C2×C17—81no—
116C2×C17#178011yescharacteristic
117C2×C17#179612yes—
118C22×Q8C2×Dic1712yes—
119C23×C4C2×Dic1711yescharacteristic
120C4⋊Q8—24no—
121#126—216no—
122#125—216no—
123#125—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C4×Q8—24no—
127C4×Q8C22⋊D1718yes—
128#117—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C2×Dic1712yes—
134C2×C4⋊C4C22⋊D1712yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4C22⋊D1712yes—
137C2×C42—22no—
138C2×C42C2×Dic1711yescharacteristic
139C2×C42C22×D1711yescharacteristic
140C4⋊C8—344no—
141C22⋊C8—344no—
142C4×C8—344no—
143C17#832711yescharacteristic, complemented, Sylow
144C24C22⋊Dic1711yescharacteristic
145C2×Q8—22no—
146C2×Q8—28no—
147C2×Q8C22⋊Dic1714yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4C22⋊Dic1711yescharacteristic
153C22×C4C22⋊Dic1712yes—
154C22×C4C22×Dic1711yescharacteristic
155C22×C4C2×C22⋊D1712yes—
156C2×C8—684no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4C22⋊Dic1714yes—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C42—22no—
167C42—28no—
168C42C2×C22⋊D1714yes—
169C23—21no—
170C23—22no—
171C23—42no—
172C23C2×C22⋊Dic1711yescharacteristic
173Q8—28no—
174Q8—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4#1058763414yes—
186C2×C4C2×C22⋊Dic1712yes—
187C2×C4#1058763914yes—
188C8—1364no—
189C22—22no—
190C22—41no—
191C22—42no—
192C22—82no—
193C22#1176266511yescharacteristic
194C22#1176277612yes—
195C4—24no—
196C4—41no—
197C4—44no—
198C4—44no—
199C4—42no—
200C4#1176277614yes—
201C2—22no—
202C2—81no—
203C2#1452869611yescharacteristic
204C2#1452875912yes—
subgroupG/H
centerC22#11762665
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#541C2
socleC22×C17#379
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenterC2×#541D17
hypocenterC17#8327
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. #14527179C2semidirect
143. C17#8327semidirect

Maximal subgroups

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

subgroupindex
#145271792
#145270982
C2×C17×#5412
#832717