Finite Group Databasebeta

← #14533736

Subgroups — #14533736

Order 2176.

676 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 177 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×#10587574C211yesmaximal, characteristic, complemented
2C2×#10587582C214yesmaximal, complemented
3C2×#10587579C211yesmaximal, characteristic
4#11762744C214yesmaximal, complemented
5#11762711C211yesmaximal, characteristic
6#11762710C211yesmaximal, characteristic
7#11762678C211yesmaximal, characteristic, complemented
8C4×#63225C211yesmaximal, characteristic, complemented
9C17×#384C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2214yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2214yes—
18C2×#63227C2214yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2214yes—
23C2×#63225C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25C2×C4×Dic17C2211yescharacteristic
26#10587582—216no—
27#10587579—24no—
28#10587574—24no—
29#10587502C2212yes—
30C23×D17C2312yes—
31C22×C4×C17C2312yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×Dic17C2312yes—
36C22×Dic17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2311yescharacteristic
39C2×D68—22no—
40C2×D68—22no—
41C2×D68Q814yescomplemented
42C2×C4×D17—24no—
43C2×C4×D17—28no—
44C2×Dic34—24no—
45C2×Dic34—28no—
46C4⋊C4×C17—22no—
47C4⋊C4×C17—22no—
48C4⋊C4×C17—28no—
49C42×C17—22no—
50#63227—28no—
51#63227—24no—
52#63227—28no—
53C4⋊Dic17—22no—
54#63225—24no—
55#63225—28no—
56#63225D418yescomplemented
57C4×Dic17—22no—
58C4×Dic17D414yes—
59C23×C17C2411yescharacteristic
60C22×D17—22no—
61C22×D17—24no—
62C22×D17—22no—
63C22×D17—24no—
64C22×D17C2×Q814yes—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—24no—
69C2×C4×C17—24no—
70C2×C4×C17—22no—
71C2×C4×C17C2×D414yes—
72C2×C4×C17C2×D412yes—
73C2×C4×C17C2×Q812yes—
74C2×C4×C17#4112yes—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—24no—
80C2×Dic17C2×D412yes—
81C2×Dic17C2×D414yes—
82C2×Dic17C2×D412yes—
83C2×Dic17C2×D414yes—
84D68—28no—
85C4×D17—48no—
86Dic34—48no—
87#2542—171nomaximal, Sylow
88C22×C17C22×D411yescharacteristic
89C22×C17C22×D412yes—
90C22×C17C22×Q811yescharacteristic
91C22×C17C2×#4111yescharacteristic
92C22×C172−1+412yes—
93C2×D17—24no—
94C2×D17—28no—
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—44no—
98C4×C17—24no—
99C4×C17—42no—
100C4×C17—42no—
101C4×C17C22⋊Q814yescomplemented
102Dic17—22no—
103Dic17—24no—
104Dic17—42no—
105Dic17#12714yes—
106C2×C4⋊Q8—171no—
107C2×C22⋊Q8—174no—
108C2×C4×D4—171no—
109#399—174no—
110#390—171no—
111#390—171no—
112#384—171no—
113#384D1711yescharacteristic, complemented
114C4×C4⋊C4—171no—
115C2×C17C2×C22⋊Q811yescharacteristic
116C2×C17C2×#12711yescharacteristic
117C2×C17#53611yescharacteristic
118C2×C17#54712yes—
119C2×C17D4×Q812yes—
120D17—44no—
121C22×Q8—172no—
122C22×D4—171no—
123C23×C4—172no—
124C4⋊Q8—344no—
125C22⋊Q8—3416no—
126C4×D4—344no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4—174no—
129C2×C4⋊C4—171no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4C2×D1711yescharacteristic
132C2×C4⋊C4C2×D1714yes—
133C2×C4⋊C4C2×D1711yescharacteristic
134C2×C22⋊C4—172no—
135C2×C22⋊C4—174no—
136C2×C22⋊C4—174no—
137C2×C42—171no—
138C2×C42—171no—
139C2×C42C2×D1711yescharacteristic
140#95—172no—
141C17#254211yescharacteristic, complemented, Sylow
142C24—172no—
143C2×Q8—348no—
144C2×Q8—344no—
145C2×D4—174no—
146C2×D4—342no—
147C2×D4—342no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—344no—
153C22×C4—348no—
154C22×C4C22×D1712yes—
155C22×C4C22×D1712yes—
156C22×C4C22×D1712yes—
157C22×C4C22×D1711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—178no—
162C4⋊C4—348no—
163C4⋊C4—344no—
164C4⋊C4—342no—
165C22⋊C4—344no—
166C22⋊C4—348no—
167C22⋊C4—348no—
168C42—22no—
169C42—174no—
170C42—342no—
171C23—174no—
172C23—342no—
173C23—344no—
174C23—342no—
175C23—344no—
176C23C23×D1711yescharacteristic
177Q8—688no—
178D4—348no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—172no—
186C2×C4—174no—
187C2×C4—172no—
188C2×C4—174no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—344no—
193C2×C4—342no—
194C2×C4—688no—
195C2×C4#6325212yes—
196C2×C4D4×D1714yes—
197C2×C4D4×D1712yes—
198C2×C4Q8×D1712yes—
199C22—348no—
200C22—344no—
201C22—684no—
202C22—682no—
203C22—682no—
204C22C2×#6325211yescharacteristic
205C22C2×D4×D1712yes—
206C22C2×D4×D1711yescharacteristic
207C22C2×Q8×D1711yescharacteristic
208C22#1058769412yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—174no—
213C4—344no—
214C4—342no—
215C4—682no—
216C4#1058758214yescomplemented
217C2—684no—
218C2C2×#1058758211yescharacteristic
219C2#127×D1711yescharacteristic
220C2#1176338411yescharacteristic
221C2#1176346012yes—
222C2#1176347412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#384C2
prime radicalC22×D68C22
hypercenter#384D17
hypocenterC17#2542
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
2. C2×#10587582C2semidirect
4. #11762744C2semidirect
7. #11762678C2semidirect
8. C4×#63225C2semidirect
9. C17×#384C2semidirect
41. C2×D68Q8semidirect
56. #63225D4semidirect
101. C4×C17C22⋊Q8semidirect
113. #384D17semidirect
141. C17#2542semidirect
216. C4#10587582semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875822
C2×#105875792
#117627442
#117627112
#117627102
#117626782
C4×#632252
C17×#3842
#254217