Finite Group Databasebeta

← #17952520

Subgroups — #17952520

Order 4352.

684 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 108 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#14544086C214yesmaximal, complemented
2#14544017C212yesmaximal
3C2×#11763420C211yesmaximal, characteristic
4#11763825C2214yes—
5Q8⋊#3624—28no—
6#11763655—22no—
7#11763653—22no—
8#11763420—28no—
9C2×#10587584—22no—
10C2×#10587556C411yescharacteristic
11C2×#10587575—22no—
12C2×#10587632C412yes—
13C2×Q8×Dic17C2212yes—
14C2×#10587558C2211yescharacteristic
15C2×C17×C4⋊Q8C411yescharacteristic
16C22×Q8×C17C2×C412yes—
17C22×C4×D17C2×C411yescharacteristic
18C17×C4⋊Q8—24no—
19C2×C4⋊C4×C17—22no—
20C2×C42×C17C2×C411yescharacteristic
21#10587632—24no—
22#10587632—24no—
23Q8×Dic17—24no—
24Q8×Dic17D418yes—
25C2×#63227—22no—
26C2×#63227D412yes—
27C2×C4⋊Dic17—22no—
28C2×C4⋊Dic17C2×C412yes—
29C2×#63225—22no—
30C2×#63225—22no—
31C2×#63225D412yes—
32C2×C4×Dic17—22no—
33C2×C4×Dic17C2311yescharacteristic
34#10587584—216no—
35#10587575—216no—
36#10587558—24no—
37#10587556—24no—
38C22⋊#3624—24no—
39C4⋊#3624—24no—
40C4×#3624—24no—
41C23×D17C22⋊C411yescharacteristic
42C2×Q8×C17—22no—
43C2×Q8×C17—28no—
44C2×Q8×C17C22⋊C414yes—
45C22×C4×C17—22no—
46C22×C4×C17C22⋊C412yes—
47C22×C4×C17C22×C411yescharacteristic
48C22×Dic17—22no—
49C22×Dic17C22⋊C411yescharacteristic
50C22×Dic17C2×D412yes—
51C2×C4×D17—22no—
52C2×C4×D17—24no—
53C2×#3624—44no—
54C4⋊C4×C17—28no—
55C42×C17—22no—
56#63227—28no—
57#63227—24no—
58C4⋊Dic17—22no—
59C4⋊Dic17—28no—
60C4⋊Dic17C22⋊C414yes—
61#63225—28no—
62#63225—28no—
63#63225—24no—
64C4×Dic17—28no—
65C4×Dic17C2×D414yes—
66#8327—171nomaximal, Sylow
67C23×C17C2×C22⋊C411yescharacteristic
68C22×D17—21no—
69C22×D17—22no—
70C22×D17—42no—
71Q8×C17—28no—
72Q8×C17—44no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17—44no—
76C2×C4×C17—42no—
77C2×C4×C17C2×C22⋊C412yes—
78C2×C4×C17#13114yes—
79C2×Dic17—21no—
80C2×Dic17—24no—
81C2×Dic17—22no—
82C2×Dic17—42no—
83C2×Dic17—44no—
84C2×Dic17#12014yes—
85C4×D17—44no—
86#3624—84no—
87C2×#541—171no—
88#1479—174no—
89#1458—172no—
90C22×C17—22no—
91C22×C17#37911yescharacteristic
92C22×C17#40812yes—
93C2×D17—42no—
94C2×D17—41no—
95C2×D17—82no—
96C4×C17—42no—
97C4×C17—44no—
98C4×C17#40814yes—
99Dic17—24no—
100Dic17—44no—
101Dic17—41no—
102#541—348no—
103C2×C4⋊Q8#59511yescharacteristic
104C2×#126—342no—
105C2×#125—171no—
106C2×#125—342no—
107C2×C22⋊Q8—172no—
108C2×C4×Q8—172no—
109C2×#117—171no—
110#431—174no—
111Q8⋊C8—348no—
112#324—342no—
113#323—342no—
114C2×C17—22no—
115C2×C17#178011yescharacteristic
116C2×C17#179612yes—
117D17—81no—
118C22×Q8C2×#59512yes—
119C23×C4—171no—
120C4⋊Q8—24no—
121#126—3416no—
122#125—344no—
123#125—3416no—
124C22⋊Q8—344no—
125C22⋊Q8—344no—
126C4×Q8—178no—
127C4×Q8—344no—
128#117—344no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—342no—
133C2×C4⋊C4—342no—
134C2×C4⋊C4—342no—
135C2×C22⋊C4—172no—
136C2×C22⋊C4—342no—
137C2×C42—171no—
138C2×C42—342no—
139C2×C42C2×#59511yescharacteristic
140C4⋊C8—344no—
141C22⋊C8—344no—
142C4×C8—344no—
143C17#832711yescharacteristic, complemented, Sylow
144C24—171no—
145C2×Q8—22no—
146C2×Q8—28no—
147C2×Q8C22⋊#59514yes—
148C22×C4—22no—
149C22×C4—172no—
150C22×C4—171no—
151C22×C4—342no—
152C22×C4—344no—
153C22×C4—342no—
154C22×C4C22⋊#59512yes—
155C22×C4C22×#59511yescharacteristic
156C2×C8—684no—
157C4⋊C4—28no—
158C4⋊C4—174no—
159C4⋊C4—344no—
160C4⋊C4—348no—
161C4⋊C4—348no—
162C4⋊C4—348no—
163C4⋊C4—342no—
164C22⋊C4—344no—
165C22⋊C4—348no—
166C42—22no—
167C42—174no—
168C42—348no—
169C23—341no—
170C23—342no—
171C23—682no—
172C23C2×C22⋊#59511yescharacteristic
173Q8—28no—
174Q8—44no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—174no—
180C2×C4—344no—
181C2×C4—341no—
182C2×C4—342no—
183C2×C4—682no—
184C2×C4—684no—
185C2×C4—684no—
186C2×C4#1058767814yes—
187C2×C4C2×C22⋊#59512yes—
188C8—1364no—
189C22—22no—
190C22—681no—
191C22—682no—
192C22—1362no—
193C22#1176375011yescharacteristic
194C22#1176378312yes—
195C4—44no—
196C4—42no—
197C4—344no—
198C4—681no—
199C4—684no—
200C4#1176378214yes—
201C2—22no—
202C2—1361no—
203C2#1454460811yescharacteristic
204C2#1454463612yes—
subgroupG/H
centerC22#11763750
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17#379
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#8327
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #14544086C2semidirect
143. C17#8327semidirect

Maximal subgroups

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

subgroupindex
#145440862
#145440172
C2×#117634202
#832717