Finite Group Databasebeta

← #16928708

Subgroups — #16928708

Order 4352.

396 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 115 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#14529754C211yesmaximal, characteristic
2#14528917C212yesmaximal
3C2×#11762551C211yesmaximal, characteristic
4#14526665C212yesmaximal
5C17×#2382C211yesmaximal, characteristic
6C2×#10587474C2211yescharacteristic
7C2×#10587474C2212yes—
8#11762679C2211yescharacteristic
9#11762571—22no—
10#11762571—22no—
11#11762571—22no—
12#11762551C414yescomplemented
13C2×C17×#125C2211yescharacteristic
14C17×#389—22no—
15C17×#387—22no—
16C17×#387—22no—
17C17×#382C2212yes—
18C17×#125—22no—
19C17×#125C2×C412yes—
20C17×#125Q812yes—
21C2×C4⋊C4×C17—22no—
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17D412yes—
24C2×C4⋊C4×C17D412yes—
25C2×C4⋊C4×C17D412yes—
26C2×C42×C17—22no—
27C2×C42×C17C2311yescharacteristic
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—22no—
30C22×#3622—21no—
31C22×#3622—22no—
32C17×#95—22no—
33C17×#95—22no—
34C17×#95—22no—
35C17×#95—22no—
36#10587474—22no—
37#10587474—22no—
38#10587474C2×C414yes—
39C22×C4×C17—22no—
40C22×C4×C17—22no—
41C22×C4×C17—22no—
42C22×C4×C17C2×D411yescharacteristic
43C22×C4×C17C2×D412yes—
44C22×Dic17—41no—
45C4⋊C4×C17—24no—
46C4⋊C4×C17—24no—
47C4⋊C4×C17—22no—
48C4⋊C4×C17—42no—
49C4⋊C4×C17—42no—
50C4⋊C4×C17—82no—
51C4⋊C4×C17C4⋊C414yes—
52C42×C17—82no—
53C42×C17C22×C411yescharacteristic
54C42×C17C2×Q811yescharacteristic
55C42×C17#4112yes—
56C4⋊Dic17—42no—
57C4⋊Dic17—44no—
58C2×#3622—22no—
59C2×#3622—24no—
60C2×#3622—41no—
61C2×#3622—42no—
62C2×#3622—42no—
63C2×#3622—41no—
64#8915—171nomaximal, Sylow
65C23×C17#12011yescharacteristic
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—21no—
70C2×C4×C17—24no—
71C2×C4×C17—42no—
72C2×C4×C17—42no—
73C2×C4×C17—42no—
74C2×C4×C17—42no—
75C2×C4×C17—42no—
76C2×C4×C17—42no—
77C2×C4×C17—42no—
78C2×C4×C17—42no—
79C2×C4×C17C2×C4⋊C411yescharacteristic
80C2×C4×C17C4×D412yes—
81C2×C4×C17C22⋊Q812yes—
82C2×C4×C17#12311yescharacteristic
83C2×Dic17—81no—
84C2×Dic17—82no—
85#3622—42no—
86#3622—44no—
87#2382D1711yescharacteristic
88#1992—171no—
89#1836—172no—
90C2×#330—171no—
91#1295—172no—
92C22×C17#38511yescharacteristic
93C22×C17#45411yescharacteristic
94C22×C17#45412yes—
95C22×C17#45512yes—
96C22×C17#45611yescharacteristic
97C4×C17—21no—
98C4×C17—21no—
99C4×C17—22no—
100C4×C17—22no—
101C4×C17—44no—
102C4×C17—82no—
103C4×C17—82no—
104Dic17—161no—
105C2×#125C2×D1711yescharacteristic
106C2×#97—172no—
107C2×#97—171no—
108#389—22no—
109#387—22no—
110#387—22no—
111#384—171no—
112#382C2×D1712yes—
113#340—342no—
114#340—342no—
115#340—342no—
116#330—174no—
117C2×C17#165811yescharacteristic
118C2×C17#166211yescharacteristic
119C2×C17#166412yes—
120C2×C17#185711yescharacteristic
121C2×C17#186012yes—
122C22×C8—341no—
123C22×C8—342no—
124#125—22no—
125#125Dic3412yes—
126#125C4×D1712yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—341no—
130C2×C4⋊C4—342no—
131C2×C4⋊C4C22⋊D1712yes—
132C2×C4⋊C4C22⋊D1712yes—
133C2×C4⋊C4C22⋊D1712yes—
134C2×C42—22no—
135C2×C42C22×D1711yescharacteristic
136#97—174no—
137#97—342no—
138#97—342no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143C17#891511yescharacteristic, complemented, Sylow
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—681no—
148C22×C4C2×C22⋊D1711yescharacteristic
149C22×C4C2×C22⋊D1712yes—
150C2×C8—342no—
151C2×C8—344no—
152C2×C8—681no—
153C2×C8—682no—
154C2×C8—682no—
155C2×C8—681no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—682no—
163C4⋊C4—684no—
164C4⋊C4#6322514yes—
165C42—82no—
166C42C2×Dic3411yescharacteristic
167C42C2×C4×D1711yescharacteristic
168C42#6325212yes—
169C23#1058763911yescharacteristic
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—1361no—
184C2×C4—1362no—
185C2×C4C2×#6322511yescharacteristic
186C2×C4#1058761012yes—
187C2×C4C4×C22⋊D1712yes—
188C2×C4#1058761511yescharacteristic
189C8—682no—
190C8—684no—
191C22#1176268211yescharacteristic
192C22#1176298211yescharacteristic
193C22#1176298212yes—
194C22#1176298712yes—
195C22#1176299211yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—82no—
202C4—82no—
203C4—2721no—
204C2#1452808611yescharacteristic
205C2#1452810711yescharacteristic
206C2#1452811812yes—
207C2#1452899911yescharacteristic
208C2#1452901712yes—
subgroupG/H
centerC23#10587639
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2382C2
socle, prime radicalC23×C17#120
cosocleC2×C42×C17C23
hypercenter#2382D17
hypocenterC17#8915
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
12. #11762551C4semidirect
143. C17#8915semidirect

Maximal subgroups

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

subgroupindex
#145297542
#145289172
C2×#117625512
#145266652
C17×#23822
#891517