Finite Group Databasebeta

← #16950596

Subgroups — #16950596

Order 4352.

566 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528708C212yesmaximal
2#14526911C214yesmaximal, complemented
3C17×#2310C211yesmaximal, characteristic
4C2×#10587507—22no—
5C2×#10587507C2214yes—
6#11762773—22no—
7C23×Q8×C17C2211yescharacteristic
8C2×C17×#117C411yescharacteristic
9C2×C17×#117C411yescharacteristic
10C17×#391—22no—
11C17×#391C412yes—
12C17×#391C2212yes—
13C17×#381—22no—
14C22×Q8×C17—24no—
15C22×Q8×C17—42no—
16C22×Q8×C17—42no—
17C22×Q8×C17C2×C411yescharacteristic
18C22×Q8×C17D412yes—
19C22×Q8×C17C2311yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17D412yes—
22C17×#117—42no—
23C17×#117—42no—
24C2×C4⋊C4×C17—21no—
25C2×C4⋊C4×C17C2×C412yes—
26C2×C22⋊C4×C17—21no—
27C2×C22⋊C4×C17—21no—
28C2×C42×C17—21no—
29C2×C42×C17C2×C412yes—
30C2×#63223—24no—
31C17×#95—24no—
32C17×#95—24no—
33#10587507—28no—
34#10587507—44no—
35C22⋊#3622—44no—
36C24×C17C22⋊C411yescharacteristic
37C2×Q8×C17—24no—
38C2×Q8×C17—22no—
39C2×Q8×C17—24no—
40C2×Q8×C17—41no—
41C2×Q8×C17—41no—
42C2×Q8×C17—42no—
43C2×Q8×C17—41no—
44C2×Q8×C17—41no—
45C2×Q8×C17—416no—
46C2×Q8×C17#4114yes—
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—24no—
51C22×C4×C17—24no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—42no—
55C22×C4×C17C22⋊C412yes—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×C4×C17C2×D412yes—
59C4⋊C4×C17—42no—
60C4⋊C4×C17—42no—
61C22⋊C4×C17—42no—
62C22⋊C4×C17—42no—
63C42×C17—42no—
64C42×C17—42no—
65#63223—44no—
66#63223—44no—
67C2×#3622—44no—
68#12862—171nomaximal, Sylow
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—42no—
72C23×C17#9912yes—
73C23×C17C2×C22⋊C411yescharacteristic
74Q8×C17—44no—
75Q8×C17—48no—
76Q8×C17—44no—
77C2×C4×C17—28no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—48no—
88C2×C4×C17—44no—
89C2×C4×C17—44no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17#9912yes—
93C2×C4×C17#11712yes—
94C2×C4×C17#12314yes—
95#3622—84no—
96#2310D1711yescharacteristic
97#1784—172no—
98#1394—174no—
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—24no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17#38111yescharacteristic
105C22×C17C2×#9911yescharacteristic
106C22×C17C2×#9911yescharacteristic
107C4×C17—24no—
108C4×C17—22no—
109C4×C17—22no—
110C4×C17—42no—
111C4×C17—82no—
112C4×C17—82no—
113C23×Q8C2×D1711yescharacteristic
114C2×#117Dic1711yescharacteristic
115C2×#117Dic1711yescharacteristic
116C2×#101—174no—
117C2×#101—342no—
118#407—342no—
119#391—22no—
120#391Dic1712yes—
121#391C2×D1712yes—
122#381—22no—
123C2×C17—21no—
124C2×C17—21no—
125C2×C17—42no—
126C2×C17#181911yescharacteristic
127C2×C17#213312yes—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8C2×Dic1711yescharacteristic
132C22×Q8C22⋊D1712yes—
133C22×Q8C22×D1711yescharacteristic
134C23×C4C2×Dic1711yescharacteristic
135C23×C4C22⋊D1712yes—
136C2×M16—344no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C2×Dic1712yes—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C42—21no—
144C2×C42C2×Dic1712yes—
145#101—348no—
146#101—684no—
147C22⋊C8—684no—
148#95—24no—
149#95—24no—
150C17#1286211yescharacteristic, complemented, Sylow
151C24C22⋊Dic1711yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—24no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—416no—
161C2×Q8#6325214yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4C22⋊Dic1712yes—
171C22×C4C22⋊Dic1711yescharacteristic
172C22×C4C22×Dic1711yescharacteristic
173C22×C4C2×C22⋊D1712yes—
174M16—684no—
175M16—684no—
176C2×C8—684no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—42no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1058750512yes—
187C23C2×C22⋊Dic1711yescharacteristic
188Q8—44no—
189Q8—48no—
190Q8—44no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#1058750512yes—
207C2×C4#1058761212yes—
208C2×C4#1058761514yes—
209C8—1364no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22#1176267211yescharacteristic
216C22C2×#1058750511yescharacteristic
217C22C2×#1058750511yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—82no—
223C4—82no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#1452885811yescharacteristic
228C2#1453047312yes—
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2310C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#2310D17
hypocenterC17#12862
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #14526911C2semidirect
150. C17#12862semidirect

Maximal subgroups

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

subgroupindex
#145287082
#145269112
C17×#23102
#1286217