Finite Group Databasebeta

← #14534259

Subgroups — #14534259

Order 2176.

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

normal subgroups only

This group has 596 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×#10587616C211yesmaximal, characteristic
2C22×#63225C211yesmaximal, characteristic, complemented
3#11762749C212yesmaximal, complemented
4#11762727C212yesmaximal, complemented
5#11762700C214yesmaximal, complemented
6#11762686C212yesmaximal, complemented
7#11762686C212yesmaximal, complemented
8C4×C22⋊C4×C17C211yesmaximal, characteristic, complemented
9C23×C4×C17C2211yescharacteristic
10C22×C22⋊D17C2212yes—
11C23×Dic17C2211yescharacteristic
12C23×Dic17C2211yescharacteristic
13C22×D68C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C42×C17C2212yes—
16#10587616—24no—
17C2×#63227C2212yes—
18C2×#63227C2214yes—
19C2×#63227C2214yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225—24no—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C17×#95C2212yes—
25#10587502C2214yes—
26#10587502C2214yes—
27C24×C17C2311yescharacteristic
28C23×D17C2312yes—
29C22×C4×C17—42no—
30C22×C4×C17D414yes—
31C22×C4×C17C2314yes—
32C22×C4×C17C2311yescharacteristic
33C22×C4×C17C2311yescharacteristic
34C2×C22⋊D17—24no—
35C2×C22⋊D17—28no—
36C22×Dic17—22no—
37C22×Dic17—24no—
38C22×Dic17—22no—
39C22×Dic17—42no—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C2×D68—22no—
44C2×D68—42no—
45C22⋊C4×C17—24no—
46C42×C17—42no—
47#63227—24no—
48#63227—28no—
49#63227—44no—
50C4⋊Dic17—22no—
51#63225—24no—
52#63225—24no—
53C23×C17—22no—
54C23×C17—22no—
55C23×C17C2×D412yes—
56C23×C17#4114yes—
57C23×C17C2411yescharacteristic
58C22×D17—22no—
59C22×D17—24no—
60C22×D17—24no—
61C22×D17—24no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—24no—
65C2×C4×C17—24no—
66C2×C4×C17—24no—
67C2×C4×C17—44no—
68C2×C4×C17C2×D412yes—
69C2×C4×C17C2×D412yes—
70C22⋊D17—48no—
71C2×Dic17—28no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—24no—
77C2×Dic17—24no—
78C2×Dic17—44no—
79C2×Dic17#4114yes—
80D68—44no—
81#2591—171nomaximal, Sylow
82C22×C17—22no—
83C22×C17—22no—
84C22×C17—28no—
85C22×C17#12414yes—
86C22×C17C22×D411yescharacteristic
87C22×C17C2×#4112yes—
88C22×C17C2×#4112yes—
89C22×C17H(5,𝔽2)12yes—
90C2×D17—42no—
91C2×D17—42no—
92C2×D17—42no—
93C2×D17—48no—
94C4×C17—22no—
95C4×C17—22no—
96C4×C17—44no—
97Dic17—24no—
98Dic17—42no—
99Dic17—42no—
100C2×#121—171no—
101C22×C4⋊C4—171no—
102#400—172no—
103#394—172no—
104#388—174no—
105#386—172no—
106#386—172no—
107C4×C22⋊C4D1711yescharacteristic, complemented
108C2×C17—24no—
109C2×C17#52511yescharacteristic
110C2×C17C2×#12411yescharacteristic
111C2×C17#53411yescharacteristic
112C2×C17#55314yes—
113D17—82no—
114C22×D4—172no—
115C22×D4—171no—
116C23×C4—171no—
117C23×C4—171no—
118C23×C4C2×D1711yescharacteristic
119#121—344no—
120C2×C4⋊C4—172no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—171no—
123C2×C4⋊C4—344no—
124C2×C22⋊C4—172no—
125C2×C22⋊C4—174no—
126C2×C22⋊C4—174no—
127C2×C22⋊C4C2×D1712yes—
128C2×C42C2×D1712yes—
129#95—174no—
130#95—174no—
131#95C2×D1712yes—
132C17#259111yescharacteristic, complemented, Sylow
133C24—172no—
134C24C22×D1711yescharacteristic
135C2×D4—344no—
136C2×D4—348no—
137C2×D4—342no—
138C2×D4—682no—
139C22×C4—42no—
140C22×C4—172no—
141C22×C4—172no—
142C22×C4—172no—
143C22×C4—344no—
144C22×C4—342no—
145C22×C4—342no—
146C22×C4—682no—
147C22×C4D6814yes—
148C22×C4C22×D1714yes—
149C22×C4C22×D1711yescharacteristic
150C22×C4C22×D1711yescharacteristic
151C4⋊C4—344no—
152C4⋊C4—344no—
153C4⋊C4—342no—
154C22⋊C4—24no—
155C22⋊C4—348no—
156C22⋊C4—344no—
157C22⋊C4—684no—
158C42—42no—
159C23—22no—
160C23—22no—
161C23—342no—
162C23—344no—
163C23—344no—
164C23—344no—
165C23C2×D6812yes—
166C23#6325214yes—
167C23C23×D1711yescharacteristic
168D4—684no—
169D4—688no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—174no—
177C2×C4—348no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—344no—
183C2×C4—344no—
184C2×C4—684no—
185C2×C4C2×D6812yes—
186C2×C4C2×D6812yes—
187C22—28no—
188C22—22no—
189C22—22no—
190C22—688no—
191C22—682no—
192C22—682no—
193C22—682no—
194C22#1058756114yes—
195C22C22×D6811yescharacteristic
196C22C2×#6325212yes—
197C22#1058769112yes—
198C22#41×D1712yes—
199C4—22no—
200C4—22no—
201C4—44no—
202C4—344no—
203C4—682no—
204C4—682no—
205C2—24no—
206C2—1362no—
207C2#1176332111yescharacteristic
208C2C2×#1058756111yescharacteristic
209C2#1176337111yescharacteristic
210C2#1176350814yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC4×C22⋊C4×C17C2
prime radicalC2×#10587616C2
hypercenterC4×C22⋊C4D17
hypocenterC17#2591
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#63225C2semidirect
3. #11762749C2semidirect
4. #11762727C2semidirect
5. #11762700C2semidirect
6. #11762686C2semidirect
7. #11762686C2semidirect
8. C4×C22⋊C4×C17C2semidirect
107. C4×C22⋊C4D17semidirect
132. C17#2591semidirect

Maximal subgroups

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

subgroupindex
C2×#105876162
C22×#632252
#117627492
#117627272
#117627002
#117626862
C4×C22⋊C4×C172
#259117