Finite Group Databasebeta

← #14534274

Subgroups — #14534274

Order 2176.

440 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#11762748C211yesmaximal, characteristic, complemented
2#11762726C212yesmaximal
3#11762710C211yesmaximal, characteristic
4#11762701C212yesmaximal, complemented
5#11762701C212yesmaximal, complemented
6#11762701C212yesmaximal, complemented
7#11762699C212yesmaximal, complemented
8C4⋊C4×Dic17C212yesmaximal, complemented
9C17×#398C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C2×C4⋊C4×C17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×#63227C2212yes—
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4×Dic17C2212yes—
23C2×C4×Dic17C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25C17×#95C2211yescharacteristic
26C17×#95C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31C23×D17C2312yes—
32C22×C4×C17C2311yescharacteristic
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C2×D68—22no—
41C2×D68—41no—
42C2×D68—41no—
43C4⋊C4×C17—24no—
44C4⋊C4×C17—41no—
45C4⋊C4×C17—41no—
46#63227—24no—
47#63227—24no—
48#63227—24no—
49#63227—24no—
50#63227—42no—
51#63227—42no—
52C4⋊Dic17—24no—
53C4⋊Dic17—41no—
54C4×Dic17—24no—
55C4×Dic17—41no—
56C4×Dic17—42no—
57C23×C17C2411yescharacteristic
58C22×D17—22no—
59C22×D17—22no—
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17#4112yes—
77C2×C4×C17#4114yes—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17#4114yes—
87D68—44no—
88#2593—171nomaximal, Sylow
89C22×C17C2×#4111yescharacteristic
90C22×C17C2×#4112yes—
91C22×C17C2×#4112yes—
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C2×D17—42no—
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—42no—
101C4×C17—24no—
102C4×C17—22no—
103C4×C17—42no—
104C4×C17—41no—
105C4×C17—41no—
106Dic17—24no—
107Dic17—42no—
108Dic17—42no—
109#400—171no—
110#398D1711yescharacteristic, complemented
111#394—172no—
112#390—171no—
113#388—172no—
114#388—172no—
115#388—172no—
116#388—172no—
117C4×C4⋊C4—172no—
118C2×C17#52912yes—
119C2×C17#53811yescharacteristic
120C2×C17#55312yes—
121C2×C17#55511yescharacteristic
122C2×C17#55911yescharacteristic
123D17—82no—
124C22×D4—171no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—171no—
127C2×C4⋊C4C2×D1712yes—
128C2×C4⋊C4C2×D1711yescharacteristic
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C22⋊C4—172no—
131C2×C22⋊C4—172no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4—172no—
135C2×C22⋊C4—172no—
136C2×C42—172no—
137C2×C42—171no—
138C2×C42—172no—
139#95—172no—
140#95—172no—
141#95—172no—
142#95—171no—
143#95C2×D1711yescharacteristic
144#95C2×D1712yes—
145C17#259311yescharacteristic, complemented, Sylow
146C24—172no—
147C2×D4—342no—
148C2×D4—681no—
149C2×D4—681no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1712yes—
156C22×C4C22×D1711yescharacteristic
157C22×C4C22×D1712yes—
158C4⋊C4—24no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—344no—
162C4⋊C4—681no—
163C22⋊C4—344no—
164C22⋊C4—344no—
165C22⋊C4—344no—
166C22⋊C4—344no—
167C22⋊C4—682no—
168C22⋊C4—682no—
169C42—344no—
170C42—681no—
171C42—682no—
172C23—342no—
173C23—342no—
174C23—342no—
175C23—342no—
176C23—342no—
177C23—342no—
178C23—342no—
179C23C23×D1711yescharacteristic
180D4—684no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—174no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4#6325412yes—
202C2×C4#6325614yes—
203C22—682no—
204C22—682no—
205C22—682no—
206C22—682no—
207C22—682no—
208C22—682no—
209C22—682no—
210C22C2×#6325411yescharacteristic
211C22#1058769111yescharacteristic
212C22C2×#6325612yes—
213C22#41×D1712yes—
214C22#1058769611yescharacteristic
215C4—24no—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—344no—
221C4—682no—
222C4—682no—
223C2—1362no—
224C2#1176334512yes—
225C2#1176339611yescharacteristic
226C2#1176351212yes—
227C2#1176352511yescharacteristic
228C2#1176354311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC22×D68C22
hypercenter#398D17
hypocenterC17#2593
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #11762748C2semidirect
4. #11762701C2semidirect
5. #11762701C2semidirect
6. #11762701C2semidirect
7. #11762699C2semidirect
8. C4⋊C4×Dic17C2semidirect
9. C17×#398C2semidirect
110. #398D17semidirect
145. C17#2593semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627262
#117627102
#117627012
#117626992
C4⋊C4×Dic172
C17×#3982
#259317