Finite Group Databasebeta

← #14533985

Subgroups — #14533985

Order 2176.

548 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 161 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#11762731C214yesmaximal, complemented
3#11762710C211yesmaximal, characteristic
4#11762710C211yesmaximal, characteristic
5#11762680C214yesmaximal, complemented
6#11762678C211yesmaximal, characteristic, complemented
7#11762674C211yesmaximal, characteristic, complemented
8C4×C4⋊Dic17C211yesmaximal, characteristic, complemented
9C2×C17×#125C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C17×#125—24no—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16C2×C4⋊C4×C17C2212yes—
17C2×C42×C17C2211yescharacteristic
18C2×#63227C2214yes—
19C2×#63227C2212yes—
20C2×#63227C2214yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2214yes—
24C2×C4×Dic17C2211yescharacteristic
25C2×C4×Dic17C2211yescharacteristic
26C4×D68—24no—
27#10587502C2214yes—
28#10587502C2212yes—
29C23×D17C2312yes—
30C22×C4×C17C2312yes—
31C22×C4×C17C2312yes—
32C22×C4×C17C2311yescharacteristic
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×Dic17C2312yes—
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C2×D68—22no—
39C2×D68—22no—
40C2×D68Q814yes—
41C2×C4×D17—24no—
42C2×C4×D17—44no—
43C4⋊C4×C17—24no—
44C4⋊C4×C17—24no—
45C4⋊C4×C17—22no—
46C4⋊C4×C17—22no—
47C42×C17D414yes—
48#63227—24no—
49#63227—28no—
50#63227—44no—
51C4⋊Dic17—22no—
52C4⋊Dic17—24no—
53#63225—44no—
54C4×Dic17—22no—
55C4×Dic17—22no—
56C23×C17C2411yescharacteristic
57C22×D17—22no—
58C22×D17—22no—
59C22×D17—24no—
60C22×D17—24no—
61C22×D17C2×Q814yes—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—24no—
67C2×C4×C17—24no—
68C2×C4×C17C2×D412yes—
69C2×C4×C17C2×D412yes—
70C2×C4×C17C2×D412yes—
71C2×C4×C17C2×Q812yes—
72C2×C4×C17#4112yes—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—24no—
79C2×Dic17—22no—
80C2×Dic17#4114yes—
81D68—28no—
82C4×D17—48no—
83#2566—171nomaximal, Sylow
84C22×C17C22×D411yescharacteristic
85C22×C17C22×Q811yescharacteristic
86C22×C17C2×#4111yescharacteristic
87C22×C17C2×#4112yes—
88C22×C17H(5,𝔽2)12yes—
89C2×D17—24no—
90C2×D17—28no—
91C2×D17—42no—
92C2×D17—42no—
93C2×D17—44no—
94C4×C17—22no—
95C4×C17—22no—
96C4×C17—42no—
97C4×C17—42no—
98C4×C17C22⋊Q814yes—
99Dic17—24no—
100Dic17—42no—
101Dic17—42no—
102C2×#125D1711yescharacteristic, complemented
103C2×C4×D4—171no—
104#396—174no—
105#390—171no—
106#390—171no—
107#385—174no—
108#384—171no—
109#382—171no—
110C4×C4⋊C4—171no—
111C2×C17C2×C22⋊Q811yescharacteristic
112C2×C17#53211yescharacteristic
113C2×C17#53511yescharacteristic
114C2×C17#55312yes—
115C2×C17#55412yes—
116D17—44no—
117C22×D4—171no—
118C23×C4—172no—
119#125—24no—
120C4×D4—344no—
121C2×C4⋊C4—171no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—174no—
124C2×C4⋊C4C2×D1711yescharacteristic
125C2×C4⋊C4C2×D1711yescharacteristic
126C2×C4⋊C4C2×D1712yes—
127C2×C4⋊C4C2×D1712yes—
128C2×C22⋊C4—172no—
129C2×C22⋊C4—174no—
130C2×C22⋊C4—174no—
131C2×C42—171no—
132C2×C42—171no—
133C2×C42C2×D1711yescharacteristic
134#95—172no—
135#95—174no—
136C17#256611yescharacteristic, complemented, Sylow
137C24—172no—
138C2×D4—174no—
139C2×D4—342no—
140C2×D4—342no—
141C22×C4—172no—
142C22×C4—172no—
143C22×C4—172no—
144C22×C4—344no—
145C22×C4—684no—
146C22×C4C22×D1712yes—
147C22×C4C22×D1712yes—
148C22×C4C22×D1711yescharacteristic
149C22×C4C22×D1711yescharacteristic
150C22×C4C22×D1711yescharacteristic
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—344no—
156C4⋊C4—342no—
157C4⋊C4—684no—
158C22⋊C4—348no—
159C22⋊C4—344no—
160C22⋊C4—684no—
161C42—342no—
162C42—342no—
163C42C22⋊D1714yes—
164C23—174no—
165C23—342no—
166C23—344no—
167C23—344no—
168C23—342no—
169C23C23×D1711yescharacteristic
170D4—348no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—174no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—344no—
184C2×C4—342no—
185C2×C4—688no—
186C2×C4Q8×D1712yes—
187C2×C4#6325612yes—
188C2×C4C2×C22⋊D1712yes—
189C2×C4C2×C22⋊D1712yes—
190C2×C4C2×C22⋊D1712yes—
191C22—348no—
192C22—344no—
193C22—684no—
194C22—682no—
195C22—682no—
196C22C2×Q8×D1711yescharacteristic
197C22C2×#6325611yescharacteristic
198C22#41×D1712yes—
199C22#1058769612yes—
200C22C22×C22⋊D1711yescharacteristic
201C4—22no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—344no—
206C4—682no—
207C4—682no—
208C4#1058763214yes—
209C2—684no—
210C2C2×#1058763211yescharacteristic
211C2#1176335511yescharacteristic
212C2#1176337311yescharacteristic
213C2#1176351212yes—
214C2#1176351512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#125C2
prime radicalC22×D68C22
hypercenterC2×#125D17
hypocenterC17#2566
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #11762731C2semidirect
5. #11762680C2semidirect
6. #11762678C2semidirect
7. #11762674C2semidirect
8. C4×C4⋊Dic17C2semidirect
9. C2×C17×#125C2semidirect
102. C2×#125D17semidirect
136. C17#2566semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D682
#117627312
#117627102
#117626802
#117626782
#117626742
C4×C4⋊Dic172
C2×C17×#1252
#256617