Finite Group Databasebeta

← #16945839

Subgroups — #16945839

Order 4352.

292 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 79 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#14539434C211yesmaximal, characteristic
2#14530600C211yesmaximal, characteristic, complemented
3#14526846C212yesmaximal, complemented
4#14526820C212yesmaximal, complemented
5C17×#2153C211yesmaximal, characteristic, complemented
6#11763414C412yes—
7#11763211C2211yescharacteristic
8#11763186C2211yescharacteristic
9#11763174C412yes—
10#11763062—21no—
11#11763045—21no—
12#11762922—21no—
13C4×D136—21no—
14#11762630C2212yes—
15#11762591—21no—
16#11762587—21no—
17C2×C17×M32—21no—
18C17×#405C2211yescharacteristic
19C17×#364—21no—
20C17×#364C2212yes—
21C2×M16×C17D411yescharacteristic
22C22×C8×C17D411yescharacteristic
23C17×#117C2×C411yescharacteristic
24#10587616—22no—
25C2×D136—21no—
26C2×D136—21no—
27#10587580—42no—
28#10587570—42no—
29#10587560C2×C412yes—
30C4×D68—22no—
31#10587556C2×C412yes—
32C17×M32—22no—
33C17×M32—41no—
34C2×C16×C17—41no—
35C17×#97C2×C411yescharacteristic
36C17×#97D411yescharacteristic
37C4×C8×C17D411yescharacteristic
38C4×C8×C17C2311yescharacteristic
39#10587492—22no—
40#10587492—22no—
41#10587489—21no—
42#10587488—21no—
43#10587483—22no—
44C22×C4×C17C22⋊C411yescharacteristic
45C2×C22⋊D17—42no—
46C2×D68—22no—
47C2×D68—42no—
48C2×C4×D17—42no—
49M16×C17—22no—
50C2×C8×C17—21no—
51C2×C8×C17—21no—
52C2×C8×C17C22⋊C411yescharacteristic
53C2×C8×C17C22⋊C411yescharacteristic
54C2×C8×C17C2×D411yescharacteristic
55C2×C8×C17C2×D411yescharacteristic
56C4⋊C4×C17—21no—
57C22⋊C4×C17—21no—
58C42×C17C22⋊C411yescharacteristic
59C42×C17C22×C411yescharacteristic
60#63227—42no—
61#63227—42no—
62C4⋊Dic17—22no—
63#63225—42no—
64D136—42no—
65D136—41no—
66D136—41no—
67C16×C17—42no—
68#63214—42no—
69#11989—171nomaximal, Sylow
70C23×C17#10011yescharacteristic
71C22×D17—42no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17#10011yescharacteristic
77C2×C4×C17C2×C22⋊C411yescharacteristic
78C22⋊D17—162no—
79C2×Dic17—42no—
80D68—42no—
81D68—42no—
82D68—162no—
83C4×D17—162no—
84C8×C17—21no—
85C8×C17—21no—
86C8×C17#13512yes—
87C8×C17AGL(1,ℤ/8)11yescharacteristic
88C8×C17#13711yescharacteristic
89#3176—171no—
90#2166—171no—
91#2153D1711yescharacteristic, complemented
92#1366—172no—
93#1357—172no—
94C22×C17—21no—
95C22×C17—41no—
96C22×C17C2×#10011yescharacteristic
97C2×D17—82no—
98C2×D17—82no—
99C2×D17—82no—
100C4×C17—21no—
101C4×C17—41no—
102C4×C17—41no—
103C4×C17#41611yescharacteristic
104C4×C17#41711yescharacteristic
105Dic17—82no—
106#540—172no—
107C2×M32—21no—
108#499—171no—
109#493—171no—
110#490—172no—
111#469—341no—
112#466—341no—
113#442—341no—
114C4×D8—341no—
115#405C2×D1711yescharacteristic
116#364—21no—
117#364—172no—
118#364C2×D1712yes—
119#349—341no—
120#347—341no—
121C2×C17—21no—
122C2×C17—41no—
123C2×C17#155111yescharacteristic
124D17—162no—
125C2×D8—341no—
126C2×D8—341no—
127C2×M16D6811yescharacteristic
128C22×C8C22⋊D1711yescharacteristic
129#127—172no—
130#125—172no—
131#123—682no—
132#121—342no—
133#121—682no—
134C4×D4—342no—
135#117C4×D1711yescharacteristic
136M32—22no—
137M32—41no—
138M32—342no—
139C2×C16—41no—
140#107—341no—
141#106—341no—
142#102—342no—
143#102—342no—
144#97C4×D1711yescharacteristic
145#97D6811yescharacteristic
146C4×C8C22⋊D1711yescharacteristic
147C4×C8C22×D1711yescharacteristic
148C17#1198911yescharacteristic, complemented, Sylow
149C2×D4—342no—
150C2×D4—682no—
151C2×D4—682no—
152C22×C4—682no—
153C22×C4#6322711yescharacteristic
154D8—682no—
155D8—681no—
156D8—681no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8#6322711yescharacteristic
161C2×C8#6322711yescharacteristic
162C2×C8C2×D6811yescharacteristic
163C2×C8C2×C22⋊D1711yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—342no—
166C4⋊C4—682no—
167C22⋊C4—21no—
168C22⋊C4—682no—
169C22⋊C4—682no—
170C42#6322711yescharacteristic
171C42C2×C4×D1711yescharacteristic
172C16—42no—
173C16—682no—
174C23—682no—
175C23#1058749411yescharacteristic
176D4—2722no—
177D4—2722no—
178D4—682no—
179D4—682no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—2722no—
185C2×C4—682no—
186C2×C4#1058749411yescharacteristic
187C2×C4C2×#6322711yescharacteristic
188C8—21no—
189C8—21no—
190C8#1058759012yes—
191C8#1058761811yescharacteristic
192C8#1058762811yescharacteristic
193C22—21no—
194C22—41no—
195C22—1362no—
196C22—1362no—
197C22—1362no—
198C22C2×#1058749411yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—1362no—
203C4#1176280611yescharacteristic
204C4#1176280811yescharacteristic
205C2—21no—
206C2—41no—
207C2—2722no—
208C2#1452747711yescharacteristic
subgroupG/H
centerC2#14527477
commutatorC42×C17C22×C4
FrattiniC4×C8C22×D17
FittingC17×#2153C2
socleC2×C17#1551
prime radical#14539434C2
cosocleC4×C8×C17C23
hypercenter#2153D17
hypocenterC17#11989
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #14530600C2semidirect
3. #14526846C2semidirect
4. #14526820C2semidirect
5. C17×#2153C2semidirect
91. #2153D17semidirect
148. C17#11989semidirect

Maximal subgroups

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

subgroupindex
#145394342
#145306002
#145268462
#145268202
C17×#21532
#1198917