Finite Group Databasebeta

← #14529233

Subgroups — #14529233

Order 2176.

358 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#10587612C211yesmaximal, characteristic, complemented
2C2×#10587508C212yesmaximal, complemented
3#11762773C211yesmaximal, characteristic, complemented
4#11762568C212yesmaximal, complemented
5C17×#525C211yesmaximal, characteristic
6C2×#41×C17C2211yescharacteristic
7C23×C4×C17C2211yescharacteristic
8C17×#123—21no—
9C17×C22⋊Q8C412yescomplemented
10C17×#121C412yescomplemented
11C17×#120—21no—
12C4×D4×C17—22no—
13C17×#117C2211yescharacteristic
14C2×C22⋊Dic17—21no—
15#10587612—24no—
16C2×C4⋊Dic17—21no—
17C2×C4×Dic17C2212yes—
18C2×#63223C2212yes—
19#10587508—24no—
20C22⋊#3622—22no—
21C24×C17D411yescharacteristic
22#41×C17—22no—
23#41×C17—22no—
24C2×Q8×C17C2×C411yescharacteristic
25C2×D4×C17—22no—
26C2×D4×C17—21no—
27C2×D4×C17C2×C411yescharacteristic
28C22×C4×C17—22no—
29C22×C4×C17—22no—
30C22×C4×C17—21no—
31C22×C4×C17C2×C412yes—
32C22×C4×C17D411yescharacteristic
33C22×C4×C17D412yes—
34C22×C4×C17C2311yescharacteristic
35C22×Dic17—22no—
36C4⋊C4×C17—22no—
37C4⋊C4×C17C2×C412yes—
38C22⋊C4×C17—22no—
39C22⋊C4×C17—21no—
40C22⋊C4×C17—22no—
41C42×C17—21no—
42C22⋊Dic17—24no—
43C4⋊Dic17—24no—
44C4×Dic17—22no—
45C4×Dic17D414yescomplemented
46#63223—22no—
47#63223—22no—
48C2×#3622—22no—
49C23×C17—21no—
50C23×C17—21no—
51C23×C17—21no—
52C23×C17—42no—
53C23×C17C22⋊C412yes—
54C23×C17C2×D411yescharacteristic
55Q8×C17—22no—
56D4×C17—22no—
57D4×C17—44no—
58D4×C17—41no—
59D4×C17—41no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—24no—
63C2×C4×C17—24no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—42no—
67C2×C4×C17C22⋊C412yes—
68C2×C4×C17C22×C411yescharacteristic
69C2×C4×C17C2×D412yes—
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17#4112yes—
72C2×Dic17—24no—
73C2×Dic17—44no—
74#3622—42no—
75#1897—171nomaximal, Sylow
76C22×C17—22no—
77C22×C17—22no—
78C22×C17—22no—
79C22×C17—22no—
80C22×C17—42no—
81C22×C17—42no—
82C22×C17—41no—
83C22×C17—42no—
84C22×C17C2×C22⋊C411yescharacteristic
85C22×C17#12011yescharacteristic
86C22×C17#12311yescharacteristic
87C4×C17—22no—
88C4×C17—41no—
89C4×C17—42no—
90C4×C17C4×D412yes—
91C4×C17#12112yes—
92Dic17—44no—
93#525D1711yescharacteristic
94C2×#117—171no—
95C2×#104—172no—
96#407—171no—
97#339—172no—
98C2×C17—21no—
99C2×C17—21no—
100C2×C17—42no—
101C2×C17—81no—
102C2×C17#38611yescharacteristic
103C2×C17#42112yes—
104C2×#41C2×D1711yescharacteristic
105C23×C4C2×D1711yescharacteristic
106C2×M16—172no—
107#123—21no—
108C22⋊Q8Dic1712yescomplemented
109#121Dic1712yescomplemented
110#120—21no—
111C4×D4—22no—
112#117—344no—
113#117C2×D1711yescharacteristic
114C2×C4⋊C4—341no—
115C2×C22⋊C4—341no—
116C2×C42—172no—
117#104—344no—
118C22⋊C8—342no—
119C17#189711yescharacteristic, complemented, Sylow
120C24C22⋊D1711yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×Dic1711yescharacteristic
124C2×D4—22no—
125C2×D4—21no—
126C2×D4C2×Dic1711yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—342no—
131C22×C4C2×Dic1712yes—
132C22×C4C22⋊D1711yescharacteristic
133C22×C4C22⋊D1712yes—
134C22×C4C22×D1711yescharacteristic
135M16—342no—
136M16—342no—
137C2×C8—342no—
138C4⋊C4—22no—
139C4⋊C4—344no—
140C4⋊C4C2×Dic1712yes—
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—22no—
144C22⋊C4—344no—
145C42—21no—
146C42—174no—
147C42—342no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23C22⋊Dic1712yes—
153C23C2×C22⋊D1711yescharacteristic
154Q8—22no—
155D4—22no—
156D4—44no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—344no—
167C2×C4—684no—
168C2×C4C22⋊Dic1712yes—
169C2×C4D4×D1712yes—
170C2×C4#6325412yes—
171C2×C4C22×Dic1711yescharacteristic
172C2×C4C2×C22⋊D1711yescharacteristic
173C8—682no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—42no—
182C22#1058762111yescharacteristic
183C22#1058762311yescharacteristic
184C22C2×C22⋊Dic1711yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—684no—
189C4D4×Dic1712yes—
190C4#1058762512yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—81no—
195C2#1176268911yescharacteristic
196C2#1176282612yes—
subgroupG/H
centerC2×C4C2×C22⋊D17
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
Fitting, prime radicalC17×#525C2
socleC22×C17C2×C22⋊C4
cosocleC22×C4×C17C23
hypercenter#525D17
hypocenterC17#1897
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#10587612C2semidirect
2. C2×#10587508C2semidirect
3. #11762773C2semidirect
4. #11762568C2semidirect
9. C17×C22⋊Q8C4semidirect
10. C17×#121C4semidirect
45. C4×Dic17D4semidirect
108. C22⋊Q8Dic17semidirect
109. #121Dic17semidirect
119. C17#1897semidirect

Maximal subgroups

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

subgroupindex
C2×#105876122
C2×#105875082
#117627732
#117625682
C17×#5252
#189717