Finite Group Databasebeta

← #14533951

Subgroups — #14533951

Order 2176.

532 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 153 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×C4×D68C211yesmaximal, characteristic
2#11762748C212yesmaximal, complemented
3#11762726C212yesmaximal
4#11762699C214yesmaximal, complemented
5#11762666C212yesmaximal, complemented
6#11762666C212yesmaximal, complemented
7C4×C4⋊Dic17C211yesmaximal, characteristic, complemented
8C2×C17×#125C211yesmaximal, characteristic, complemented
9C22×D68C2211yescharacteristic
10C22×C4×D17C2212yes—
11C17×#125—24no—
12C2×C4⋊C4×C17C2212yes—
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2212yes—
18C2×#63227C2214yes—
19C2×#63227C2214yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2212yes—
22C2×C4×Dic17C2212yes—
23C4×D68—24no—
24#10587502C2214yes—
25#10587502C2214yes—
26#10587502C2212yes—
27C23×D17C2312yes—
28C22×C4×C17C2312yes—
29C22×C4×C17C2312yes—
30C22×C4×C17C2311yescharacteristic
31C22×C4×C17C2311yescharacteristic
32C22×C4×C17C2311yescharacteristic
33C22×Dic17C2312yes—
34C22×Dic17C2314yes—
35C2×D68—24no—
36C2×D68—22no—
37C2×C4×D17—24no—
38C2×C4×D17—24no—
39C2×C4×D17—24no—
40C4⋊C4×C17—24no—
41C4⋊C4×C17—24no—
42C4⋊C4×C17—22no—
43C4⋊C4×C17—22no—
44C42×C17D414yes—
45#63227—28no—
46#63227—24no—
47#63227—44no—
48C4⋊Dic17—22no—
49C4⋊Dic17—24no—
50C4×Dic17—42no—
51C23×C17C2411yescharacteristic
52C22×D17—22no—
53C22×D17—22no—
54C22×D17—22no—
55C22×D17—22no—
56C22×D17—22no—
57C22×D17—22no—
58C22×D17#4114yes—
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—24no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17C2×D412yes—
68C2×C4×C17C2×D412yes—
69C2×C4×C17C2×D412yes—
70C2×C4×C17#4114yes—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—24no—
75C2×Dic17—24no—
76C2×Dic17—24no—
77D68—28no—
78C4×D17—48no—
79#2562—171nomaximal, Sylow
80C22×C17C22×D411yescharacteristic
81C22×C17C2×#4112yes—
82C22×C17C2×#4112yes—
83C22×C17H(5,𝔽2)11yescharacteristic
84C22×C172−1+411yescharacteristic
85C2×D17—24no—
86C2×D17—24no—
87C2×D17—24no—
88C2×D17—42no—
89C2×D17—42no—
90C2×D17—42no—
91C2×D17—42no—
92C4×C17—22no—
93C4×C17—22no—
94C4×C17—42no—
95C4×C17—42no—
96C4×C17#12414yes—
97Dic17—42no—
98Dic17—44no—
99C2×#125D1711yescharacteristic, complemented
100C2×C4×D4—171no—
101#400—172no—
102#394—172no—
103#388—174no—
104#380—172no—
105#380—172no—
106C4×C4⋊C4—171no—
107C2×C17#52511yescharacteristic
108C2×C17C2×#12411yescharacteristic
109C2×C17#53711yescharacteristic
110C2×C17#55212yes—
111C2×C17#55312yes—
112D17—44no—
113C22×D4—171no—
114C23×C4—172no—
115#125—24no—
116C4×D4—344no—
117C2×C4⋊C4—171no—
118C2×C4⋊C4—172no—
119C2×C4⋊C4C2×D1711yescharacteristic
120C2×C4⋊C4C2×D1711yescharacteristic
121C2×C4⋊C4C2×D1712yes—
122C2×C4⋊C4C2×D1712yes—
123C2×C22⋊C4—172no—
124C2×C22⋊C4—174no—
125C2×C22⋊C4—174no—
126C2×C42—172no—
127C2×C42C2×D1711yescharacteristic
128#95—172no—
129#95—174no—
130#95—174no—
131C17#256211yescharacteristic, complemented, Sylow
132C24—172no—
133C2×D4—344no—
134C2×D4—342no—
135C22×C4—172no—
136C22×C4—174no—
137C22×C4—344no—
138C22×C4—344no—
139C22×C4—344no—
140C22×C4C22×D1712yes—
141C22×C4C22×D1712yes—
142C22×C4C22×D1711yescharacteristic
143C22×C4C22×D1711yescharacteristic
144C22×C4C22×D1711yescharacteristic
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—344no—
150C4⋊C4—342no—
151C22⋊C4—348no—
152C22⋊C4—344no—
153C22⋊C4—684no—
154C42—682no—
155C42C22⋊D1714yes—
156C23—174no—
157C23—342no—
158C23—342no—
159C23—342no—
160C23—342no—
161C23—342no—
162C23—342no—
163C23C23×D1711yescharacteristic
164D4—348no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—344no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—344no—
177C2×C4—344no—
178C2×C4—342no—
179C2×C4—688no—
180C2×C4#6325614yes—
181C2×C4C2×C22⋊D1712yes—
182C2×C4C2×C22⋊D1712yes—
183C2×C4C2×C22⋊D1712yes—
184C22—344no—
185C22—344no—
186C22—344no—
187C22—682no—
188C22—682no—
189C22—682no—
190C22—682no—
191C22C2×#6325612yes—
192C22#41×D1712yes—
193C22#1058769611yescharacteristic
194C22#1058769711yescharacteristic
195C22C22×C22⋊D1711yescharacteristic
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—684no—
201C4—682no—
202C4#1058763314yes—
203C2—684no—
204C2#1176331611yescharacteristic
205C2C2×#1058763311yescharacteristic
206C2#1176338711yescharacteristic
207C2#1176349712yes—
208C2#1176351212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#125C2
prime radicalC22×D68C22
hypercenterC2×#125D17
hypocenterC17#2562
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #11762748C2semidirect
4. #11762699C2semidirect
5. #11762666C2semidirect
6. #11762666C2semidirect
7. C4×C4⋊Dic17C2semidirect
8. C2×C17×#125C2semidirect
99. C2×#125D17semidirect
131. C17#2562semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D682
#117627482
#117627262
#117626992
#117626662
C4×C4⋊Dic172
C2×C17×#1252
#256217