Finite Group Databasebeta

← #14536814

Subgroups — #14536814

Order 2176.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587610C211yesmaximal, characteristic, complemented
2C2×#10587616C211yesmaximal, characteristic
3#11762746C212yesmaximal, complemented
4#11762741C212yesmaximal, complemented
5#11762727C212yesmaximal, complemented
6#11762725C214yesmaximal, complemented
7#11762722C212yesmaximal, complemented
8C17×#381C211yesmaximal, characteristic, complemented
9C23×C4×C17C2211yescharacteristic
10C22×C22⋊D17C2212yes—
11C22×D68C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊Dic17C2212yes—
15#10587616—24no—
16C2×#63227C2212yes—
17C2×#63227C2214yes—
18C2×#63227C2214yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2214yes—
22#10587610—24no—
23C2×#63225C2212yes—
24C17×#95C2212yes—
25C17×#95C2212yes—
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—44no—
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C2×D68—22no—
40C2×D68—42no—
41C2×Dic34—22no—
42C2×Dic34—42no—
43C22⋊C4×C17—42no—
44C22⋊Dic17—24no—
45#63227—24no—
46#63227—44no—
47#63227—44no—
48C4⋊Dic17—22no—
49C4⋊Dic17—22no—
50C4⋊Dic17—44no—
51#63225—24no—
52C23×C17—22no—
53C23×C17—22no—
54C23×C17—22no—
55C23×C17C2×D412yes—
56C23×C17C2411yescharacteristic
57C22×D17—22no—
58C22×D17—24no—
59C22×D17—24no—
60C22×D17—24no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—24no—
64C2×C4×C17—24no—
65C2×C4×C17—24no—
66C2×C4×C17—44no—
67C2×C4×C17C2×D412yes—
68C2×C4×C17C2×D412yes—
69C22⋊D17—48no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—24no—
74C2×Dic17—24no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77D68—44no—
78Dic34—44no—
79#2816—171nomaximal, Sylow
80C22×C17—22no—
81C22×C17—22no—
82C22×C17—22no—
83C22×C17—44no—
84C22×C17C22×D411yescharacteristic
85C22×C17H(5,𝔽2)12yes—
86C22×C17H(5,𝔽2)12yes—
87C22×C172−1+412yes—
88C2×D17—42no—
89C2×D17—42no—
90C2×D17—42no—
91C2×D17—48no—
92C4×C17—22no—
93C4×C17—22no—
94C4×C17—44no—
95Dic17—42no—
96Dic17—42no—
97Dic17—42no—
98C2×C22⋊Q8—171no—
99C2×#121—171no—
100#399—172no—
101#398—172no—
102#394—174no—
103#394—172no—
104#393—172no—
105#381D1711yescharacteristic, complemented
106C2×C17—42no—
107C2×C17#53411yescharacteristic
108C2×C17#53511yescharacteristic
109C2×C17#53611yescharacteristic
110C2×C17#56214yes—
111D17—82no—
112C22×Q8—171no—
113C22×D4—172no—
114C22×D4—171no—
115C23×C4C2×D1711yescharacteristic
116C22⋊Q8—344no—
117#121—344no—
118C2×C4⋊C4—171no—
119C2×C4⋊C4—172no—
120C2×C4⋊C4—171no—
121C2×C4⋊C4—174no—
122C2×C22⋊C4—172no—
123C2×C22⋊C4—172no—
124C2×C22⋊C4—174no—
125C2×C22⋊C4—174no—
126C2×C22⋊C4C2×D1712yes—
127#95—174no—
128#95C2×D1712yes—
129#95C2×D1712yes—
130C17#281611yescharacteristic, complemented, Sylow
131C24—172no—
132C24C22×D1711yescharacteristic
133C2×Q8—342no—
134C2×Q8—682no—
135C2×D4—344no—
136C2×D4—342no—
137C2×D4—682no—
138C2×D4—684no—
139C22×C4—42no—
140C22×C4—172no—
141C22×C4—172no—
142C22×C4—172no—
143C22×C4D6814yes—
144C22×C4C22×D1714yes—
145C22×C4C22×D1711yescharacteristic
146C22×C4C22×D1711yescharacteristic
147C4⋊C4—342no—
148C4⋊C4—344no—
149C4⋊C4—342no—
150C4⋊C4—684no—
151C22⋊C4—42no—
152C22⋊C4—344no—
153C22⋊C4—344no—
154C22⋊C4—684no—
155C22⋊C4—684no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—342no—
160C23—344no—
161C23—344no—
162C23—344no—
163C23C2×D6812yes—
164C23C23×D1711yescharacteristic
165Q8—684no—
166D4—684no—
167D4—688no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—44no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—342no—
177C2×C4—344no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—344no—
181C2×C4C2×D6812yes—
182C2×C4C2×D6812yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—688no—
188C22—682no—
189C22—682no—
190C22—682no—
191C22C22×D6811yescharacteristic
192C22#1058769112yes—
193C22#1058769612yes—
194C22#1058769712yes—
195C4—22no—
196C4—22no—
197C4—44no—
198C4—682no—
199C4—682no—
200C4—682no—
201C2—42no—
202C2—1362no—
203C2#1176337111yescharacteristic
204C2#1176337811yescharacteristic
205C2#1176338511yescharacteristic
206C2#1176355614yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC2×#10587616C2
hypercenter#381D17
hypocenterC17#2816
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#10587610C2semidirect
3. #11762746C2semidirect
4. #11762741C2semidirect
5. #11762727C2semidirect
6. #11762725C2semidirect
7. #11762722C2semidirect
8. C17×#381C2semidirect
105. #381D17semidirect
130. C17#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
C2×#105876162
#117627462
#117627412
#117627272
#117627252
#117627222
C17×#3812
#281617