Finite Group Databasebeta

← #17953024

Subgroups — #17953024

Order 4352.

690 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 92 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#14544685C211yesmaximal, characteristic
2#14544681C211yesmaximal, characteristic
3#14544194C214yesmaximal
4C2×#11763420C211yesmaximal, characteristic
5#11763781C2212yes—
6#11763752C2212yes—
7#11763689—22no—
8#11763687—22no—
9#11763663—28no—
10#11763420—28no—
11C2×#10587584—22no—
12C2×#10587556C411yescharacteristic, complemented
13C2×#10587575—22no—
14C2×#10587632C2212yes—
15C2×Q8×Dic17C412yescomplemented
16C2×#10587558C2211yescharacteristic
17C2×C17×C4⋊Q8C411yescharacteristic, complemented
18C22×Q8×C17C2×C412yes—
19C22×C4×D17C2311yescharacteristic
20C2×C22⋊#595—22no—
21C2×#63247—22no—
22C17×C4⋊Q8—24no—
23C2×C4⋊C4×C17—22no—
24C2×C42×C17C2×C411yescharacteristic
25#10587632—24no—
26#10587632D418yes—
27Q8×Dic17—24no—
28Q8×Dic17—24no—
29C2×#63227—22no—
30C2×#63227D412yes—
31C2×C4⋊Dic17—22no—
32C2×C4⋊Dic17C2×C412yes—
33C2×#63225—22no—
34C2×#63225—22no—
35C2×#63225D412yes—
36C2×C4×Dic17—22no—
37C2×C4×Dic17C2×C411yescharacteristic
38#10587584—216no—
39#10587575—216no—
40#10587558—24no—
41#10587556—24no—
42(C2×#595)⋊C4—24no—
43#10587543—24no—
44C23×D17C2×D411yescharacteristic
45C22×#595—42no—
46C2×Q8×C17—22no—
47C2×Q8×C17—44no—
48C2×Q8×C17C22⋊C414yes—
49C22×C4×C17—22no—
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22×C411yescharacteristic
52C22×Dic17—22no—
53C22×Dic17C22⋊C412yes—
54C22×Dic17C2×D411yescharacteristic
55C2×C4×D17—42no—
56C2×C4×D17C2×D414yes—
57C22⋊#595—44no—
58#63247—44no—
59C2×#3624—42no—
60C4⋊C4×C17—28no—
61C42×C17—22no—
62#63227—24no—
63#63227—28no—
64C4⋊Dic17—28no—
65C4⋊Dic17—22no—
66C4⋊Dic17C22⋊C414yes—
67#63225—28no—
68#63225—24no—
69#63225—28no—
70C4×Dic17—28no—
71C4×Dic17—22no—
72#8521—171nomaximal, Sylow
73C23×C17C2×C22⋊C411yescharacteristic
74C22×D17—22no—
75C22×D17—42no—
76C22×D17#12012yes—
77C2×#595—82no—
78C2×#595—84no—
79Q8×C17—44no—
80Q8×C17—44no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—24no—
84C2×C4×C17—44no—
85C2×C4×C17—42no—
86C2×C4×C17C2×C22⋊C412yes—
87C2×Dic17—22no—
88C2×Dic17—24no—
89C2×Dic17—22no—
90C2×Dic17—44no—
91C2×Dic17—42no—
92C2×Dic17#12012yes—
93C4×D17—44no—
94#3624—82no—
95C2×#541—171no—
96#1822—171no—
97#1820—171no—
98#1511—174no—
99C22×C17—22no—
100C22×C17#37911yescharacteristic
101C22×C17#41012yes—
102C2×D17—24no—
103C2×D17—22no—
104C2×D17—82no—
105#595—162no—
106C4×C17—22no—
107C4×C17—42no—
108C4×C17—44no—
109Dic17—22no—
110Dic17—42no—
111Dic17—44no—
112#541—348no—
113C2×C4⋊Q8#59511yescharacteristic, complemented
114C2×#126—342no—
115C2×#125—171no—
116C2×#125—342no—
117C2×C22⋊Q8—172no—
118C2×C4×Q8—172no—
119C2×#117—171no—
120#407—172no—
121#381—172no—
122#343—342no—
123#342—342no—
124#328—348no—
125C2×C17—22no—
126C2×C17#178211yescharacteristic
127C2×C17#179612yes—
128D17—42no—
129C22×Q8C2×#59512yes—
130C23×C4—171no—
131C2×M16—342no—
132C4⋊Q8—24no—
133#126—3416no—
134#125—344no—
135#125—3416no—
136C22⋊Q8—178no—
137C22⋊Q8—344no—
138C4×Q8—344no—
139C4×Q8—344no—
140#117—344no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—342no—
145C2×C4⋊C4—342no—
146C2×C4⋊C4—342no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—342no—
149C2×C22⋊C4—342no—
150C2×C42—171no—
151C2×C42—342no—
152C2×C42C2×#59511yescharacteristic
153C22⋊C8—344no—
154#95—344no—
155C17#852111yescharacteristic, complemented, Sylow
156C24—171no—
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊#59514yes—
160C22×C4—22no—
161C22×C4—171no—
162C22×C4—174no—
163C22×C4—172no—
164C22×C4—342no—
165C22×C4—682no—
166C22×C4—682no—
167C22×C4C22⋊#59512yes—
168C22×C4C22×#59511yescharacteristic
169M16—684no—
170C2×C8—682no—
171C4⋊C4—28no—
172C4⋊C4—174no—
173C4⋊C4—344no—
174C4⋊C4—348no—
175C4⋊C4—348no—
176C4⋊C4—348no—
177C4⋊C4—342no—
178C22⋊C4—344no—
179C22⋊C4—348no—
180C22⋊C4—684no—
181C42—22no—
182C42—348no—
183C42—342no—
184C23—172no—
185C23—342no—
186C23—682no—
187C23C2×C22⋊#59511yescharacteristic
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—172no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—344no—
199C2×C4—684no—
200C2×C4—684no—
201C2×C4—682no—
202C2×C4—1362no—
203C2×C4—1364no—
204C2×C4C2×C22⋊#59512yes—
205C8—1362no—
206C22—22no—
207C22—342no—
208C22—344no—
209C22—1362no—
210C22#1176375011yescharacteristic
211C22#1176378912yes—
212C4—22no—
213C4—44no—
214C4—42no—
215C4—2722no—
216C4—342no—
217C4—684no—
218C4—682no—
219C2—22no—
220C2—682no—
221C2#1454461411yescharacteristic
222C2#1454463612yes—
subgroupG/H
centerC22#11763750
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17#379
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#8521
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. C2×#10587556C4semidirect
15. C2×Q8×Dic17C4semidirect
17. C2×C17×C4⋊Q8C4semidirect
113. C2×C4⋊Q8#595semidirect
155. C17#8521semidirect

Maximal subgroups

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

subgroupindex
#145446852
#145446812
#145441942
C2×#117634202
#852117