Finite Group Databasebeta

← #14529243

Subgroups — #14529243

Order 2176.

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

normal subgroups only

This group has 378 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×#10587505C212yesmaximal
3#11762672C211yesmaximal, characteristic, complemented
4#11762576C212yesmaximal, complemented
5C2×C17×#123C211yesmaximal, characteristic
6C22×D4×C17C2211yescharacteristic
7C23×C4×C17C2211yescharacteristic
8C17×#123—22no—
9C17×#123—21no—
10C17×#123—21no—
11C2×C4⋊C4×C17C412yescomplemented
12C2×C22⋊C4×C17C412yescomplemented
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2212yes—
16#10587612—24no—
17C2×C4⋊Dic17—21no—
18C2×C4×Dic17—21no—
19#10587505—24no—
20#10587502—22no—
21C24×C17C2×C411yescharacteristic
22C24×C17C2311yescharacteristic
23C2×D4×C17—21no—
24C2×D4×C17—21no—
25C2×D4×C17—22no—
26C2×D4×C17—22no—
27C22×C4×C17—22no—
28C22×C4×C17—22no—
29C22×C4×C17C2×C411yescharacteristic
30C22×C4×C17C2×C412yes—
31C22×C4×C17C2×C412yes—
32C22×C4×C17D412yes—
33C22×C4×C17D411yescharacteristic
34C22×C4×C17D411yescharacteristic
35C22×Dic17—22no—
36C22×Dic17—22no—
37C4⋊C4×C17—22no—
38C4⋊C4×C17—22no—
39C22⋊C4×C17—21no—
40C22⋊C4×C17—21no—
41C22⋊C4×C17—22no—
42C22⋊C4×C17—22no—
43C22⋊Dic17—22no—
44C22⋊Dic17—22no—
45C22⋊Dic17—22no—
46C22⋊Dic17D414yes—
47C4⋊Dic17—24no—
48C4×Dic17—24no—
49C23×C17—21no—
50C23×C17—22no—
51C23×C17—42no—
52C23×C17—42no—
53C23×C17C22×C411yescharacteristic
54C23×C17C2×D412yes—
55C23×C17C2×D411yescharacteristic
56C23×C17C2×D411yescharacteristic
57C23×C17#4112yes—
58D4×C17—41no—
59D4×C17—42no—
60D4×C17—41no—
61C2×C4×C17—24no—
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17—24no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—42no—
71C2×C4×C17C22⋊C412yes—
72C2×C4×C17C22⋊C412yes—
73C2×Dic17—24no—
74C2×Dic17—44no—
75C2×Dic17—44no—
76C2×Dic17—42no—
77#1899—171nomaximal, Sylow
78C22×C17—22no—
79C22×C17—22no—
80C22×C17—22no—
81C22×C17—42no—
82C22×C17—42no—
83C22×C17—41no—
84C22×C17—42no—
85C22×C17—82no—
86C22×C17C2×C22⋊C411yescharacteristic
87C22×C17C4×D412yes—
88C22×C17#12011yescharacteristic
89C22×C17#12112yes—
90C22×C17#12311yescharacteristic
91C4×C17—22no—
92C4×C17—22no—
93C4×C17—41no—
94C4×C17—42no—
95Dic17—44no—
96Dic17—82no—
97C2×#123D1711yescharacteristic
98C2×#117—171no—
99C2×#99—172no—
100#381—171no—
101#342—172no—
102C2×C17—21no—
103C2×C17—22no—
104C2×C17—22no—
105C2×C17—21no—
106C2×C17—81no—
107C2×C17#38611yescharacteristic
108C2×C17#41012yes—
109C22×D4C2×D1711yescharacteristic
110C23×C4C2×D1711yescharacteristic
111#123—22no—
112#123—21no—
113#123—21no—
114#117—344no—
115C2×C4⋊C4—341no—
116C2×C4⋊C4Dic1712yescomplemented
117C2×C22⋊C4—172no—
118C2×C22⋊C4—172no—
119C2×C22⋊C4Dic1712yescomplemented
120C2×C22⋊C4C2×D1711yescharacteristic
121C2×C42—341no—
122#99—344no—
123#95—342no—
124C17#189911yescharacteristic, complemented, Sylow
125C24C2×Dic1711yescharacteristic
126C24C22×D1711yescharacteristic
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—342no—
134C22×C4—342no—
135C22×C4C2×Dic1711yescharacteristic
136C22×C4C2×Dic1712yes—
137C22×C4C2×Dic1712yes—
138C22×C4C22⋊D1712yes—
139C22×C4C22⋊D1711yescharacteristic
140C22×C4C22⋊D1711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—344no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—174no—
149C22⋊C4—342no—
150C22⋊C4—342no—
151C22⋊C4—342no—
152C42—344no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23D4×D1712yes—
158C23#6325412yes—
159C23C22×Dic1711yescharacteristic
160C23C2×C22⋊D1711yescharacteristic
161C23C2×C22⋊D1711yescharacteristic
162D4—41no—
163D4—41no—
164D4—42no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—344no—
176C2×C4—684no—
177C2×C4—684no—
178C2×C4—682no—
179C2×C4C22⋊Dic1712yes—
180C2×C4C22⋊Dic1712yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—42no—
188C22—82no—
189C22D4×Dic1712yes—
190C22#1058762111yescharacteristic
191C22#1058762311yescharacteristic
192C22#1058762512yes—
193C22C2×C22⋊Dic1711yescharacteristic
194C4—22no—
195C4—22no—
196C4—41no—
197C4—42no—
198C4—684no—
199C4—1362no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2#1176268911yescharacteristic
206C2#1176278312yes—
subgroupG/H
centerC22C2×C22⋊Dic17
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC2×C17×#123C2
socleC22×C17C2×C22⋊C4
prime radicalC22×D4×C17C22
cosocleC24×C17C23
hypercenterC2×#123D17
hypocenterC17#1899
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#10587612C2semidirect
3. #11762672C2semidirect
4. #11762576C2semidirect
11. C2×C4⋊C4×C17C4semidirect
12. C2×C22⋊C4×C17C4semidirect
116. C2×C4⋊C4Dic17semidirect
119. C2×C22⋊C4Dic17semidirect
124. C17#1899semidirect

Maximal subgroups

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

subgroupindex
C2×#105876122
C2×#105875052
#117626722
#117625762
C2×C17×#1232
#189917