Finite Group Databasebeta

← #17956547

Subgroups — #17956547

Order 4352.

506 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 124 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#14544700C211yesmaximal, characteristic
2#14544569C211yesmaximal, characteristic
3#14543662C214yesmaximal
4#14537181C211yesmaximal, characteristic
5C2×#10587544C2212yes—
6C2×#10587544C2212yes—
7C2×#10587542—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675—21no—
11C2×#10587574—21no—
12C2×#10587574—21no—
13C2×#10587574C412yes—
14#11762679C411yescharacteristic
15#11762678—21no—
16#11762678—21no—
17#11762678C2212yes—
18C42×Dic17C2211yescharacteristic
19C17×#384C411yescharacteristic
20C22×Dic34—22no—
21C22×#3624—22no—
22C22×#3624—22no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—21no—
25C2×C4⋊C4×C17C2×C412yes—
26C2×C42×C17C2×C411yescharacteristic
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17C2×C412yes—
30C2×#63225—22no—
31C2×#63225—22no—
32C2×#63225—22no—
33C2×C4×Dic17C2×C411yescharacteristic
34C2×C4×Dic17D411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36C2×C4×Dic17D411yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17C2311yescharacteristic
39#10587574—44no—
40#10587574—44no—
41#10587574—44no—
42#10587544—22no—
43#10587544—24no—
44#10587544Q814yescomplemented
45#10587542—42no—
46C22×C4×C17—22no—
47C22×C4×C17C22⋊C411yescharacteristic
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—22no—
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C22⋊C411yescharacteristic
53C22×Dic17C2×D411yescharacteristic
54C22×Dic17C2×D411yescharacteristic
55C2×Dic34—44no—
56C2×Dic34—44no—
57C2×Dic34—44no—
58C2×#3624—24no—
59C2×#3624—42no—
60C2×#3624—42no—
61C2×#3624—42no—
62C2×#3624—42no—
63C2×#3624—42no—
64C4⋊C4×C17—42no—
65C4⋊C4×C17—42no—
66C4⋊C4×C17—42no—
67C42×C17—22no—
68C4⋊Dic17—42no—
69C4⋊Dic17—42no—
70C4⋊Dic17—42no—
71#63225—44no—
72#63225—44no—
73#63225—44no—
74C4×Dic17—22no—
75C4×Dic17—22no—
76C4×Dic17—22no—
77C4×Dic17—22no—
78C4×Dic17SD1612yes—
79C4×Dic17Q1612yes—
80C4×Dic17C2×Q812yes—
81C4×Dic17#4112yes—
82#9903—171nomaximal, Sylow
83C23×C17C2×C22⋊C411yescharacteristic
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17#10112yes—
94C2×C4×C17#10312yes—
95C2×C4×C17C4×Q812yes—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—42no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17#10012yes—
104C2×Dic17#10312yes—
105C2×Dic17C22⋊Q812yes—
106C2×Dic17#12411yescharacteristic
107C2×Dic17C4⋊Q811yescharacteristic
108C2×Dic17C2×SD1611yescharacteristic
109C2×Dic17C2×Q1611yescharacteristic
110C2×Dic17AGL(1,ℤ/8)12yes—
111C2×Dic17AGL(1,ℤ/8)12yes—
112C2×Dic17#13712yes—
113Dic34—88no—
114#3624—44no—
115#3624—82no—
116#2868—171no—
117#1831—171no—
118#1767—171no—
119#1296—174no—
120C22×C17#39111yescharacteristic
121C22×C17C2×#10011yescharacteristic
122C22×C17C2×#10111yescharacteristic
123C22×C17C2×#10311yescharacteristic
124C22×C17#41811yescharacteristic
125C22×C17#41811yescharacteristic
126C22×C17#41911yescharacteristic
127C4×C17—22no—
128C4×C17—22no—
129C4×C17—22no—
130C4×C17—82no—
131Dic17—22no—
132Dic17—22no—
133Dic17—82no—
134Dic17#47612yes—
135Dic17#47712yes—
136Dic17#48912yes—
137Dic17#50112yes—
138C2×C4⋊Q8—172no—
139C2×C4⋊Q8—341no—
140C2×C4⋊Q8—341no—
141C2×C4⋊C8—172no—
142C2×C4⋊C8—172no—
143C2×#97—341no—
144#384—172no—
145#384—171no—
146#384—341no—
147#384—341no—
148#384#59511yescharacteristic
149C43—171no—
150#336—341no—
151#336—341no—
152#336—341no—
153C2×C17#154811yescharacteristic
154C2×C17#154911yescharacteristic
155C2×C17#155511yescharacteristic
156C2×C17#155611yescharacteristic
157C2×C17#198711yescharacteristic
158C2×C17#199311yescharacteristic
159C2×C17#199711yescharacteristic
160C22×Q8—342no—
161C22×C8—342no—
162C22×C8—342no—
163C4⋊Q8—684no—
164C4⋊Q8—684no—
165C4⋊Q8—684no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4—342no—
170C2×C4⋊C4—342no—
171C2×C4⋊C4—342no—
172C2×C4⋊C4—341no—
173C2×C4⋊C4—341no—
174C2×C4⋊C4C2×#59512yes—
175C2×C42—171no—
176C2×C42—171no—
177C2×C42—171no—
178C2×C42—171no—
179C2×C42—171no—
180C2×C42—171no—
181C2×C42C2×#59511yescharacteristic
182C4⋊C8—174no—
183C4⋊C8—342no—
184C4⋊C8—344no—
185#97—682no—
186C17#990311yescharacteristic, complemented, Sylow
187C2×Q8—684no—
188C2×Q8—684no—
189C2×Q8—684no—
190C22×C4—22no—
191C22×C4—171no—
192C22×C4—171no—
193C22×C4—171no—
194C22×C4—171no—
195C22×C4—342no—
196C22×C4C22⋊#59511yescharacteristic
197C22×C4C22⋊#59511yescharacteristic
198C22×C4C22×#59511yescharacteristic
199C2×C8—344no—
200C2×C8—682no—
201C2×C8—682no—
202C2×C8—682no—
203C2×C8—682no—
204C2×C8—682no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—682no—
209C4⋊C4—682no—
210C4⋊C4—682no—
211C4⋊C4—684no—
212C4⋊C4—684no—
213C4⋊C4—684no—
214C42—22no—
215C42—172no—
216C42—172no—
217C42—172no—
218C42—172no—
219C42—342no—
220C42—342no—
221C42—342no—
222C42—342no—
223C23C2×C22⋊#59511yescharacteristic
224Q8—1368no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—172no—
235C2×C4—171no—
236C2×C4—171no—
237C2×C4—171no—
238C2×C4—171no—
239C2×C4—172no—
240C2×C4—172no—
241C2×C4—172no—
242C2×C4—172no—
243C2×C4—172no—
244C2×C4—341no—
245C2×C4—341no—
246C2×C4—341no—
247C2×C4—341no—
248C2×C4—682no—
249C2×C4—682no—
250C2×C4—682no—
251C2×C4#1058754512yes—
252C2×C4#1058754912yes—
253C2×C4Q8×#59512yes—
254C8—684no—
255C8—1362no—
256C22(C2×#595)⋊Q811yescharacteristic
257C22D17⋊#10011yescharacteristic
258C22C2×#1058754511yescharacteristic
259C22C2×#1058754911yescharacteristic
260C22#1176380111yescharacteristic
261C22#1176380111yescharacteristic
262C22#1176380211yescharacteristic
263C4—22no—
264C4—22no—
265C4—22no—
266C4—82no—
267C4—172no—
268C4—172no—
269C4—172no—
270C4—172no—
271C4—342no—
272C4—342no—
273C4—1362no—
274C2#1454429811yescharacteristic
275C2#1454429911yescharacteristic
276C2#1454431211yescharacteristic
277C2#1454431311yescharacteristic
278C2#1454496211yescharacteristic
279C2#1454496811yescharacteristic
280C2#1454497711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#384#595
hypocenterC17#9903
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
44. #10587544Q8semidirect
186. C17#9903semidirect

Maximal subgroups

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

subgroupindex
#145447002
#145445692
#145436622
#145371812
#990317