Finite Group Databasebeta

← #14533261

Subgroups — #14533261

Order 2176.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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×#10587632C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic
3#11762743C214yesmaximal, complemented
4#11762716C211yesmaximal, characteristic, complemented
5#11762683C214yesmaximal
6#11762679C211yesmaximal, characteristic, complemented
7#11762678C211yesmaximal, characteristic, complemented
8C4×#63227C211yesmaximal, characteristic
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×C4×D17C2211yescharacteristic
12C22×C4×D17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15#10587632—24no—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2214yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2214yes—
24C2×C4×Dic17C2211yescharacteristic
25C4⋊C4×D17—24no—
26C17×#95C2214yes—
27#10587502C2214yes—
28#10587502C2212yes—
29C23×D17C2311yescharacteristic
30C2×Q8×C17—22no—
31C2×Q8×C17—22no—
32C2×Q8×C17—22no—
33C22×C4×C17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2311yescharacteristic
40C22×Dic17C2312yes—
41C2×C4×D17—22no—
42C2×C4×D17—22no—
43C2×C4×D17—24no—
44C2×C4×D17—44no—
45C2×C4×D17D414yes—
46C4⋊C4×C17—22no—
47C42×C17—22no—
48#63227—24no—
49#63227Q818yescomplemented
50C4⋊Dic17—22no—
51C4⋊Dic17—22no—
52C4⋊Dic17—44no—
53#63225—24no—
54#63225—24no—
55#63225—44no—
56C4×Dic17—22no—
57C23×C17C2411yescharacteristic
58C22×D17—21no—
59C22×D17—21no—
60C22×D17—22no—
61C22×D17C2×D412yes—
62C22×D17C2×Q814yes—
63Q8×C17—44no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—24no—
71C2×C4×C17C2×D412yes—
72C2×C4×C17C2×Q814yes—
73C2×C4×C17#4112yes—
74C2×C4×C17#4112yes—
75C2×Dic17—21no—
76C2×Dic17—21no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—24no—
82C2×Dic17—24no—
83C2×Dic17C2×D412yes—
84C2×Dic17C2×Q814yes—
85C4×D17—28no—
86C4×D17—48no—
87#2502—171nomaximal, Sylow
88C22×C17C22×D411yescharacteristic
89C22×C17C22×Q812yes—
90C22×C17C2×#4111yescharacteristic
91C22×C17C2×#4111yescharacteristic
92C22×C17H(5,𝔽2)12yes—
93C2×D17—22no—
94C2×D17—22no—
95C2×D17—24no—
96C2×D17—24no—
97C2×D17C4⋊Q814yes—
98C4×C17—24no—
99C4×C17—42no—
100C4×C17—42no—
101C4×C17#12314yes—
102Dic17—22no—
103Dic17—24no—
104Dic17—42no—
105Dic17—42no—
106C2×C22⋊Q8—171no—
107C22×C4⋊C4—171no—
108#398—174no—
109#391—171no—
110#391D1711yescharacteristic, complemented
111#385—174no—
112#384—171no—
113#384—171no—
114C4×C22⋊C4—171no—
115C2×C17C2×#12311yescharacteristic
116C2×C17C2×C4⋊Q811yescharacteristic
117C2×C17#53511yescharacteristic
118C2×C17#55412yes—
119C2×C17#55412yes—
120D17—24no—
121C22×Q8C2×D1711yescharacteristic
122C23×C4—171no—
123C23×C4—172no—
124C22⋊Q8—344no—
125C2×C4⋊C4—171no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—174no—
130C2×C4⋊C4—174no—
131C2×C4⋊C4—344no—
132C2×C4⋊C4C2×D1711yescharacteristic
133C2×C22⋊C4—172no—
134C2×C22⋊C4—172no—
135C2×C42—171no—
136C2×C42C2×D1711yescharacteristic
137#95—172no—
138#95—174no—
139#95C2×D1714yes—
140C17#250211yescharacteristic, complemented, Sylow
141C24—171no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C22×C4—172no—
146C22×C4—171no—
147C22×C4—174no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—344no—
151C22×C4—342no—
152C22×C4—342no—
153C22×C4—684no—
154C22×C4C22×D1712yes—
155C22×C4C22×D1712yes—
156C22×C4C22×D1712yes—
157C22×C4C22×D1711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—344no—
160C4⋊C4—344no—
161C4⋊C4—342no—
162C4⋊C4—342no—
163C4⋊C4—684no—
164C4⋊C4—684no—
165C22⋊C4—178no—
166C22⋊C4—344no—
167C42—22no—
168C42—342no—
169C23—172no—
170C23—174no—
171C23—341no—
172C23—341no—
173C23—342no—
174C23C23×D1711yescharacteristic
175Q8—44no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—172no—
184C2×C4—174no—
185C2×C4—342no—
186C2×C4—341no—
187C2×C4—341no—
188C2×C4—342no—
189C2×C4—348no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—344no—
193C2×C4—344no—
194C2×C4—688no—
195C2×C4#6325212yes—
196C2×C4D4×D1712yes—
197C2×C4#6325412yes—
198C2×C4Q8×D1714yes—
199C22—174no—
200C22—344no—
201C22—344no—
202C22—342no—
203C22—342no—
204C22C2×#6325211yescharacteristic
205C22C2×D4×D1711yescharacteristic
206C22C2×#6325411yescharacteristic
207C22C2×Q8×D1712yes—
208C22#1058769612yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—344no—
213C4—342no—
214C4—682no—
215C4—682no—
216C4#1058756914yes—
217C2—344no—
218C2C2×#1058756911yescharacteristic
219C2C4⋊Q8×D1711yescharacteristic
220C2#1176337711yescharacteristic
221C2#1176351712yes—
222C2#1176351912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC23×D17C23
hypercenter#391D17
hypocenterC17#2502
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #11762743C2semidirect
4. #11762716C2semidirect
6. #11762679C2semidirect
7. #11762678C2semidirect
9. C17×#391C2semidirect
49. #63227Q8semidirect
110. #391D17semidirect
140. C17#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
C2×C4⋊C4×D172
#117627432
#117627162
#117626832
#117626792
#117626782
C4×#632272
C17×#3912
#250217