Finite Group Databasebeta

← #14535753

Subgroups — #14535753

Order 2176.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587632C212yesmaximal, complemented
2#11762749C212yesmaximal, complemented
3#11762739C211yesmaximal, characteristic, complemented
4#11762736C212yesmaximal, complemented
5#11762726C211yesmaximal, characteristic
6#11762700C212yesmaximal, complemented
7#11762687C212yesmaximal
8#11762683C212yesmaximal, complemented
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D68C2211yescharacteristic
12C22×C4×D17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15#10587632—28no—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C17×#95C2212yes—
26C17×#95C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C23×D17C2312yes—
33C2×Q8×C17—41no—
34C2×Q8×C17—41no—
35C2×Q8×C17D414yescomplemented
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×D68—24no—
45C2×D68—41no—
46C2×C4×D17—24no—
47C2×C4×D17—42no—
48C2×C4×D17—42no—
49C4⋊C4×C17—41no—
50C42×C17—41no—
51#63227—24no—
52#63227—24no—
53#63227—24no—
54#63227—24no—
55#63227—42no—
56C4⋊Dic17—24no—
57C4⋊Dic17—41no—
58#63225—24no—
59#63225—24no—
60C23×C17C2411yescharacteristic
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C22×D17#4114yes—
68Q8×C17—44no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D414yes—
81C2×C4×C17C2×D412yes—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91D68—44no—
92C4×D17—48no—
93#2728—171nomaximal, Sylow
94C22×C17C22×D411yescharacteristic
95C22×C17C2×#4112yes—
96C22×C17H(5,𝔽2)11yescharacteristic
97C22×C17H(5,𝔽2)11yescharacteristic
98C22×C172−1+412yes—
99C2×D17—24no—
100C2×D17—24no—
101C2×D17—24no—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C2×D17—42no—
106C4×C17—24no—
107C4×C17—22no—
108C4×C17—42no—
109C4×C17—41no—
110C4×C17—41no—
111Dic17—42no—
112Dic17—42no—
113Dic17—42no—
114C2×C22⋊Q8—172no—
115#400—172no—
116#398—171no—
117#397—172no—
118#394—171no—
119#391D1711yescharacteristic, complemented
120#388—172no—
121#386—172no—
122#385—172no—
123C2×C17#53411yescharacteristic
124C2×C17#53912yes—
125C2×C17#54812yes—
126C2×C17#55111yescharacteristic
127C2×C17#56311yescharacteristic
128D17—44no—
129C22×Q8C2×D1711yescharacteristic
130C22×D4—171no—
131C23×C4—172no—
132C22⋊Q8—348no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—172no—
141C2×C22⋊C4—172no—
142C2×C22⋊C4—172no—
143C2×C42C2×D1711yescharacteristic
144#95—172no—
145#95—172no—
146#95—172no—
147#95—171no—
148#95—171no—
149#95C2×D1712yes—
150#95C2×D1712yes—
151C17#272811yescharacteristic, complemented, Sylow
152C24—172no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D6814yescomplemented
156C2×D4—344no—
157C2×D4—681no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—172no—
161C22×C4—344no—
162C22×C4—682no—
163C22×C4—682no—
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1712yes—
166C22×C4C22×D1711yescharacteristic
167C22×C4C22×D1712yes—
168C22×C4C22×D1711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—344no—
171C4⋊C4—344no—
172C4⋊C4—344no—
173C4⋊C4—681no—
174C22⋊C4—344no—
175C22⋊C4—344no—
176C22⋊C4—344no—
177C22⋊C4—344no—
178C22⋊C4—682no—
179C42—41no—
180C23—174no—
181C23—342no—
182C23—342no—
183C23—342no—
184C23—342no—
185C23—342no—
186C23—342no—
187C23C23×D1711yescharacteristic
188Q8—44no—
189D4—684no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
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—342no—
210C2×C4—688no—
211C2×C4C2×D6812yes—
212C2×C4C2×D6814yes—
213C22—344no—
214C22—344no—
215C22—344no—
216C22—682no—
217C22—682no—
218C22—682no—
219C22—682no—
220C22C22×D6811yescharacteristic
221C22#1058769111yescharacteristic
222C22#1058769111yescharacteristic
223C22#1058769412yes—
224C22#41×D1712yes—
225C4—24no—
226C4—22no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—682no—
231C4—682no—
232C4—682no—
233C2—684no—
234C2#1176337111yescharacteristic
235C2#1176340512yes—
236C2#1176347112yes—
237C2#1176348911yescharacteristic
238C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC22×D68C22
hypercenter#391D17
hypocenterC17#2728
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#10587632C2semidirect
2. #11762749C2semidirect
3. #11762739C2semidirect
4. #11762736C2semidirect
6. #11762700C2semidirect
8. #11762683C2semidirect
9. C17×#391C2semidirect
35. C2×Q8×C17D4semidirect
119. #391D17semidirect
151. C17#2728semidirect
155. C2×Q8D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
#117627492
#117627392
#117627362
#117627262
#117627002
#117626872
#117626832
C17×#3912
#272817