Finite Group Databasebeta

← #16927775

Subgroups — #16927775

Order 4352.

404 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 99 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#14531281C211yesmaximal, characteristic
2#14527096C212yesmaximal
3#14526617C211yesmaximal, characteristic, complemented
4#14526612C212yesmaximal, complemented
5C17×#1276C211yesmaximal, characteristic, complemented
6C2×#10587558C2211yescharacteristic
7#11762862C2212yes—
8#11762700—22no—
9#11762693—22no—
10C4×#63225C412yes—
11C4×#63227C412yes—
12#11762655C2211yescharacteristic
13#11762534—22no—
14#11762532—22no—
15C17×#431C2212yes—
16C43×C17C2211yescharacteristic
17C22×C4×D17C2×C411yescharacteristic
18C2×C42×C17—22no—
19C2×C42×C17C2×C412yes—
20C2×C42×C17C2311yescharacteristic
21C2×#63227—22no—
22C2×#63227D412yes—
23C2×#63225—22no—
24C2×#63225D412yes—
25C2×C4×Dic17C2×C412yes—
26C2×C4×Dic17C2×C411yescharacteristic
27#10587558—22no—
28#10587558—21no—
29#10587558—21no—
30C17×C4⋊C8—22no—
31C17×C22⋊C8—22no—
32C4×C8×C17—22no—
33#10587502—22no—
34#10587502—22no—
35#10587502—22no—
36C22⋊#3622—22no—
37C4⋊#3622—22no—
38C4×#3622—22no—
39C23×D17C22⋊C411yescharacteristic
40C22×C4×C17—22no—
41C22×C4×C17C22×C411yescharacteristic
42C22×C4×C17C2×D412yes—
43C22×Dic17—22no—
44C22×Dic17C22⋊C411yescharacteristic
45C22×Dic17C22⋊C412yes—
46C2×C4×D17—21no—
47C2×C4×D17—21no—
48C2×C4×D17—22no—
49C2×C4×D17—22no—
50C2×C8×C17—42no—
51C42×C17—22no—
52C42×C17—22no—
53C42×C17—44no—
54C42×C17#4112yes—
55C42×C17#4111yescharacteristic
56C42×C17#4111yescharacteristic
57#63227—22no—
58#63227—22no—
59#63227—44no—
60#63225—22no—
61#63225—22no—
62#63225—44no—
63C4×Dic17—21no—
64C4×Dic17—21no—
65C4×Dic17—22no—
66C4×Dic17—22no—
67C2×#3622—42no—
68#8732—171nomaximal, Sylow
69C23×C17C2×C22⋊C411yescharacteristic
70C22×D17—21no—
71C22×D17—22no—
72C22×D17—42no—
73C2×C4×C17—24no—
74C2×C4×C17—42no—
75C2×C4×C17—42no—
76C2×C4×C17—42no—
77C2×C4×C17#11711yescharacteristic
78C2×C4×C17#11711yescharacteristic
79C2×C4×C17#12312yes—
80C2×C4×C17#12312yes—
81C2×C4×C17#13112yes—
82C2×C4×C17#13112yes—
83C2×Dic17—21no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—42no—
87C2×Dic17—42no—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C4×D17—41no—
91C4×D17—42no—
92C4×D17—41no—
93C8×C17—82no—
94#3622—82no—
95#2298—171no—
96#1458—172no—
97#1276—171no—
98#1276D1711yescharacteristic, complemented
99#1274—172no—
100C22×C17—22no—
101C22×C17#38111yescharacteristic
102C22×C17#40812yes—
103C2×D17—41no—
104C2×D17—42no—
105C2×D17—82no—
106C4×C17—22no—
107C4×C17—22no—
108C4×C17—44no—
109C4×C17#43311yescharacteristic
110C4×C17#43312yes—
111C4×C17#43311yescharacteristic
112Dic17—41no—
113Dic17—42no—
114Dic17—82no—
115C2×#117—171no—
116#431—172no—
117#431C2×D1712yes—
118#388—342no—
119#387—342no—
120C4×C4⋊C4—172no—
121C4×C22⋊C4—172no—
122#376—171no—
123C43C2×D1711yescharacteristic
124#324—342no—
125#323—342no—
126C2×C17—22no—
127C2×C17#181811yescharacteristic
128C2×C17#182712yes—
129D17—81no—
130C23×C4—171no—
131#117—342no—
132#117—341no—
133#117—341no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—342no—
136C2×C22⋊C4—172no—
137C2×C22⋊C4—342no—
138C2×C42—22no—
139C2×C42—171no—
140C2×C42—172no—
141C2×C42C4×D1712yes—
142C2×C42C22×D1711yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—342no—
145C22⋊C8—22no—
146C22⋊C8—342no—
147C4×C8—22no—
148C4×C8—342no—
149#95—342no—
150#95—342no—
151#95—342no—
152C17#873211yescharacteristic, complemented, Sylow
153C24—171no—
154C22×C4—22no—
155C22×C4—171no—
156C22×C4—172no—
157C22×C4—341no—
158C22×C4—341no—
159C22×C4—342no—
160C22×C4—342no—
161C22×C4—342no—
162C22×C4C2×C4×D1711yescharacteristic
163C22×C4D4×D1712yes—
164C2×C8—42no—
165C2×C8—682no—
166C4⋊C4—342no—
167C4⋊C4—342no—
168C4⋊C4—684no—
169C22⋊C4—342no—
170C22⋊C4—342no—
171C22⋊C4—684no—
172C42—22no—
173C42—22no—
174C42—44no—
175C42—341no—
176C42—341no—
177C42—342no—
178C42—342no—
179C42#6325212yes—
180C42#6325411yescharacteristic
181C42#6325611yescharacteristic
182C23—341no—
183C23—342no—
184C23—682no—
185C23C22⋊C4×D1711yescharacteristic
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—341no—
193C2×C4—681no—
194C2×C4—682no—
195C2×C4—681no—
196C2×C4—682no—
197C2×C4—682no—
198C2×C4—682no—
199C2×C4—682no—
200C2×C4#1058755811yescharacteristic
201C2×C4#1058756912yes—
202C2×C4#1058757811yescharacteristic
203C2×C4#1058758012yes—
204C2×C4#1058758712yes—
205C2×C4#1058759312yes—
206C8—82no—
207C8—1362no—
208C22—22no—
209C22—681no—
210C22—682no—
211C22—1362no—
212C22#1176267111yescharacteristic
213C22#1176277412yes—
214C4—22no—
215C4—22no—
216C4—44no—
217C4—681no—
218C4—682no—
219C4—1362no—
220C4#1176286811yescharacteristic
221C4#1176286912yes—
222C4#1176287111yescharacteristic
223C2—22no—
224C2—1361no—
225C2#1452885411yescharacteristic
226C2#1452888712yes—
subgroupG/H
centerC2×C4#10587578
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1276C2
socleC22×C17#381
prime radicalC23×D17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1276D17
hypocenterC17#8732
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
3. #14526617C2semidirect
4. #14526612C2semidirect
5. C17×#1276C2semidirect
98. #1276D17semidirect
152. C17#8732semidirect

Maximal subgroups

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

subgroupindex
#145312812
#145270962
#145266172
#145266122
C17×#12762
#873217