Finite Group Databasebeta

← #14533761

Subgroups — #14533761

Order 2176.

576 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10587633C211yesmaximal, characteristic
2C2×#10587633C211yesmaximal, characteristic
3C2×Q8×Dic17C211yesmaximal, characteristic, complemented
4#11762731C214yesmaximal, complemented
5#11762716C211yesmaximal, characteristic, complemented
6#11762711C211yesmaximal, characteristic
7#11762699C214yesmaximal, complemented
8C4×#63225C211yesmaximal, characteristic, complemented
9C2×C17×C4⋊Q8C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C22×D68C2211yescharacteristic
13C17×C4⋊Q8—24no—
14C2×C4⋊C4×C17C2214yes—
15C2×C42×C17C2211yescharacteristic
16#10587633—24no—
17#10587633—24no—
18Q8×Dic17—24no—
19C2×#63227C2212yes—
20C2×#63227C2214yes—
21C2×#63227C2214yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2214yes—
31C23×D17C2312yes—
32C2×Q8×C17—22no—
33C2×Q8×C17—22no—
34C2×Q8×C17—22no—
35C2×Q8×C17—22no—
36C2×Q8×C17—22no—
37C2×Q8×C17D414yes—
38C22×C4×C17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2311yescharacteristic
46C2×D68—22no—
47C2×D68—22no—
48C2×D68—22no—
49C4⋊C4×C17—28no—
50C42×C17—22no—
51#63227—28no—
52#63227—28no—
53#63227—24no—
54#63227—24no—
55C4⋊Dic17—22no—
56C4⋊Dic17—24no—
57#63225—24no—
58C4×Dic17—22no—
59C4×Dic17—24no—
60C4×Dic17D414yes—
61C23×C17C2411yescharacteristic
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—24no—
66C22×D17—24no—
67Q8×C17—28no—
68Q8×C17—44no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17C2×D412yes—
77C2×C4×C17C2×D414yes—
78C2×C4×C17C2×D412yes—
79C2×C4×C17#4112yes—
80C2×C4×C17#4114yes—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17C2×D412yes—
89C2×Dic17C2×D412yes—
90D68—44no—
91#2545—171nomaximal, Sylow
92C22×C17C22×D411yescharacteristic
93C22×C17C22×D411yescharacteristic
94C22×C17C2×#4111yescharacteristic
95C22×C17C2×#4112yes—
96C22×C172−1+412yes—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—44no—
101C2×D17—44no—
102C4×C17—24no—
103C4×C17—24no—
104C4×C17—42no—
105C4×C17#12114yes—
106Dic17—22no—
107Dic17—22no—
108Dic17—42no—
109Dic17—42no—
110C2×C4⋊Q8D1711yescharacteristic, complemented
111C2×#124—171no—
112C2×#124—171no—
113C2×C4×Q8—171no—
114#396—174no—
115#391—171no—
116#390—171no—
117#388—174no—
118C4×C4⋊C4—171no—
119C2×C17C2×#12111yescharacteristic
120C2×C17#53211yescharacteristic
121C2×C17#53611yescharacteristic
122C2×C17#54812yes—
123C2×C17#55612yes—
124D17—82no—
125C22×Q8C2×D1711yescharacteristic
126C22×Q8C2×D1711yescharacteristic
127C22×D4—171no—
128C4⋊Q8—24no—
129#124—344no—
130#124—344no—
131C4×Q8—344no—
132C2×C4⋊C4—171no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4C2×D1714yes—
136C2×C22⋊C4—172no—
137C2×C22⋊C4—174no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4—174no—
140C2×C42—171no—
141C2×C42—172no—
142C2×C42—171no—
143C2×C42C2×D1711yescharacteristic
144#95—172no—
145#95—174no—
146C17#254511yescharacteristic, complemented, Sylow
147C24—172no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8C22⋊D1714yes—
154C2×D4—342no—
155C2×D4—342no—
156C2×D4—342no—
157C22×C4—171no—
158C22×C4—171no—
159C22×C4—172no—
160C22×C4—172no—
161C22×C4C22×D1712yes—
162C22×C4C22×D1712yes—
163C22×C4C22×D1712yes—
164C22×C4C22×D1711yescharacteristic
165C4⋊C4—28no—
166C4⋊C4—344no—
167C4⋊C4—342no—
168C4⋊C4—344no—
169C22⋊C4—344no—
170C22⋊C4—344no—
171C22⋊C4—348no—
172C22⋊C4—348no—
173C42—22no—
174C42—174no—
175C42—344no—
176C42—342no—
177C23—342no—
178C23—342no—
179C23—344no—
180C23—344no—
181C23—342no—
182C23C23×D1711yescharacteristic
183Q8—28no—
184Q8—44no—
185D4—684no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—172no—
194C2×C4—172no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—344no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4D4×D1712yes—
203C2×C4#6325412yes—
204C2×C4#6325614yes—
205C2×C4C2×C22⋊D1714yes—
206C2×C4C2×C22⋊D1712yes—
207C22—684no—
208C22—684no—
209C22—682no—
210C22—682no—
211C22—682no—
212C22C2×D4×D1711yescharacteristic
213C22C2×#6325411yescharacteristic
214C22C2×#6325612yes—
215C22#1058769412yes—
216C22C22×C22⋊D1711yescharacteristic
217C4—24no—
218C4—24no—
219C4—42no—
220C4—342no—
221C4—342no—
222C4—682no—
223C4—682no—
224C4#1058762514yes—
225C2—1362no—
226C2C2×#1058762511yescharacteristic
227C2#1176336011yescharacteristic
228C2#1176338411yescharacteristic
229C2#1176347012yes—
230C2#1176353312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C4⋊Q8C2
prime radicalC22×D68C22
hypercenterC2×C4⋊Q8D17
hypocenterC17#2545
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8×Dic17C2semidirect
4. #11762731C2semidirect
5. #11762716C2semidirect
7. #11762699C2semidirect
8. C4×#63225C2semidirect
9. C2×C17×C4⋊Q8C2semidirect
110. C2×C4⋊Q8D17semidirect
146. C17#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#105876332
C2×Q8×Dic172
#117627312
#117627162
#117627112
#117626992
C4×#632252
C2×C17×C4⋊Q82
#254517