Finite Group Databasebeta

← #14536729

Subgroups — #14536729

Order 2176.

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

normal subgroups only

This group has 480 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#11762722C212yesmaximal, complemented
3#11762722C212yesmaximal, complemented
4#11762713C212yesmaximal, complemented
5#11762710C211yesmaximal, characteristic
6#11762687C212yesmaximal
7#11762683C212yesmaximal, complemented
8#11762680C212yesmaximal, complemented
9C17×#400C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C17×#95C2212yes—
24C17×#95C2211yescharacteristic
25C17×#95C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31C23×D17C2312yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C2×D68—41no—
41C2×D68—41no—
42C2×D68Q814yescomplemented
43C2×C4×D17—42no—
44C2×C4×D17—42no—
45C2×C4×D17—42no—
46C2×Dic34—24no—
47C2×Dic34—42no—
48C2×Dic34—42no—
49C4⋊C4×C17—41no—
50C4⋊C4×C17—41no—
51#63227—24no—
52#63227—24no—
53#63227—42no—
54#63227—42no—
55#63227—42no—
56C4⋊Dic17—41no—
57#63225—42no—
58C4×Dic17—41no—
59C23×C17C2411yescharacteristic
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17C2×Q814yes—
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—22no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17C2×Q812yes—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17#4114yes—
89D68—44no—
90C4×D17—48no—
91Dic34—48no—
92#2804—171nomaximal, Sylow
93C22×C17C22×Q811yescharacteristic
94C22×C17C2×#4112yes—
95C22×C17H(5,𝔽2)11yescharacteristic
96C22×C17H(5,𝔽2)11yescharacteristic
97C22×C17H(5,𝔽2)12yes—
98C2×D17—24no—
99C2×D17—24no—
100C2×D17—24no—
101C2×D17—42no—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C4×C17—22no—
106C4×C17—42no—
107C4×C17—41no—
108C4×C17—41no—
109C4×C17—42no—
110Dic17—24no—
111Dic17—42no—
112Dic17—42no—
113#400—171no—
114#400D1711yescharacteristic, complemented
115#393—172no—
116#393—172no—
117#391—172no—
118#390—171no—
119#386—172no—
120#385—172no—
121#385—172no—
122C2×C17#53812yes—
123C2×C17#54311yescharacteristic
124C2×C17#55412yes—
125C2×C17#55911yescharacteristic
126C2×C17H(3,𝔽4)11yescharacteristic
127D17—44no—
128C22×Q8—172no—
129C22×D4—171no—
130C23×C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—171no—
133C2×C4⋊C4C2×D1711yescharacteristic
134C2×C4⋊C4C2×D1711yescharacteristic
135C2×C22⋊C4—172no—
136C2×C22⋊C4—172no—
137C2×C22⋊C4—172no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C42—171no—
141#95—172no—
142#95—172no—
143#95—172no—
144#95—172no—
145#95—171no—
146#95C2×D1712yes—
147#95C2×D1712yes—
148#95C2×D1711yescharacteristic
149C17#280411yescharacteristic, complemented, Sylow
150C24—172no—
151C2×Q8—344no—
152C2×Q8—682no—
153C2×Q8—682no—
154C2×D4—174no—
155C2×D4—681no—
156C2×D4—681no—
157C22×C4—172no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—682no—
161C22×C4—682no—
162C22×C4—682no—
163C22×C4C22×D1711yescharacteristic
164C22×C4C22×D1712yes—
165C22×C4C22×D1711yescharacteristic
166C22×C4C22×D1712yes—
167C22×C4C22×D1711yescharacteristic
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—682no—
171C4⋊C4—681no—
172C22⋊C4—344no—
173C22⋊C4—344no—
174C22⋊C4—682no—
175C22⋊C4—682no—
176C22⋊C4—682no—
177C42—681no—
178C23—174no—
179C23—342no—
180C23—342no—
181C23—342no—
182C23—342no—
183C23—342no—
184C23—342no—
185C23C23×D1711yescharacteristic
186Q8—688no—
187D4—684no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—174no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—688no—
210C2×C4Q8×D1712yes—
211C22—344no—
212C22—344no—
213C22—344no—
214C22—682no—
215C22—682no—
216C22—682no—
217C22—682no—
218C22#1058769111yescharacteristic
219C22C2×Q8×D1711yescharacteristic
220C22#41×D1712yes—
221C22#1058769612yes—
222C22#1058769611yescharacteristic
223C4—22no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—42no—
228C4—344no—
229C4—682no—
230C4—682no—
231C2—684no—
232C2#1176339812yes—
233C2#1176343411yescharacteristic
234C2#1176351512yes—
235C2#1176354311yescharacteristic
236C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC22×D68C22
hypercenter#400D17
hypocenterC17#2804
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #11762748C2semidirect
2. #11762722C2semidirect
3. #11762722C2semidirect
4. #11762713C2semidirect
7. #11762683C2semidirect
8. #11762680C2semidirect
9. C17×#400C2semidirect
42. C2×D68Q8semidirect
114. #400D17semidirect
149. C17#2804semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627222
#117627132
#117627102
#117626872
#117626832
#117626802
C17×#4002
#280417