Finite Group Databasebeta

← #16944973

Subgroups — #16944973

Order 4352.

378 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533996C211yesmaximal, characteristic
2C4×#10587507C211yesmaximal, characteristic
3#14526812C214yesmaximal
4C17×#1467C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6C2×#10587507—21no—
7C4×#63223—21no—
8#11762748C412yes—
9#11762748C2212yes—
10#11762716—21no—
11#11762715—21no—
12#11762715—21no—
13#11762674—21no—
14C4×C4⋊Dic17C411yescharacteristic
15#11762578C2212yes—
16C2×C4×Q8×C17C2211yescharacteristic
17C17×#432—21no—
18C17×#324—21no—
19C17×#324C2212yes—
20C22×Q8×C17D411yescharacteristic
21C22×Dic34—21no—
22C4×Q8×C17—41no—
23C4×Q8×C17—41no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17C2×C411yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C412yes—
33C2×C4×Dic17—21no—
34C2×#63223—22no—
35#10587555—41no—
36#10587555—41no—
37C17×C4⋊C8—41no—
38C17×C22⋊C8—22no—
39C17×#97—41no—
40#10587507—24no—
41#10587502—22no—
42#10587502—22no—
43#10587502—22no—
44#10587502—22no—
45#10587502C2×C412yes—
46#10587474—41no—
47C4×#3622—41no—
48C2×Q8×C17—22no—
49C2×Q8×C17—22no—
50C2×Q8×C17—41no—
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22×C411yescharacteristic
56C22×C4×C17C2×D411yescharacteristic
57C22×C4×C17C2×D411yescharacteristic
58C22×Dic17—22no—
59C22×Dic17—22no—
60C2×Dic34—44no—
61C2×Dic34—41no—
62C2×Dic34—41no—
63C2×C8×C17—42no—
64C4⋊C4×C17—41no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—41no—
67C42×C17—21no—
68C42×C17—21no—
69C42×C17—21no—
70C42×C17—21no—
71C42×C17—21no—
72C42×C17—21no—
73C4⋊Dic17—42no—
74C4⋊Dic17—42no—
75C4⋊Dic17—42no—
76C4×Dic17—42no—
77#63223—24no—
78#63223—42no—
79C2×#3622—42no—
80#11794—171nomaximal, Sylow
81C23×C17C2×C22⋊C411yescharacteristic
82Q8×C17—44no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17#9911yescharacteristic
98C2×C4×C17#9911yescharacteristic
99C2×C4×C17#10412yes—
100C2×C4×C17#13712yes—
101C2×Dic17—44no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106Dic34—82no—
107Dic34—82no—
108C8×C17—82no—
109#3622—44no—
110#2568—171no—
111C4×#101—171no—
112#1467D1711yescharacteristic
113#1353—174no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17C2×#9911yescharacteristic
117C22×C17#41911yescharacteristic
118C22×C17C2×#10411yescharacteristic
119C4×C17—21no—
120C4×C17—22no—
121C4×C17—22no—
122C4×C17—41no—
123C4×C17—41no—
124C4×C17—41no—
125C4×C17—41no—
126C4×C17#35512yes—
127Dic17—82no—
128Dic17—82no—
129C2×C4⋊Q8—171no—
130C2×C4×Q8C2×D1711yescharacteristic
131#432—21no—
132C2×#101—341no—
133C4×M16—341no—
134#400—172no—
135#400—172no—
136#391—341no—
137#391—341no—
138#391—341no—
139#382—341no—
140C4×C4⋊C4—171no—
141#343—172no—
142#324—21no—
143#324C2×D1712yes—
144C2×C17—21no—
145C2×C17—21no—
146C2×C17#150811yescharacteristic
147C2×C17#212311yescharacteristic
148C2×C17C2×#35511yescharacteristic
149C22×Q8—341no—
150C22×Q8D6811yescharacteristic
151C2×M16—342no—
152C4⋊Q8—681no—
153C4⋊Q8—681no—
154C4×Q8—41no—
155C4×Q8—41no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—341no—
158C2×C4⋊C4—341no—
159C2×C4⋊C4C4×D1711yescharacteristic
160C2×C4⋊C4C22⋊D1711yescharacteristic
161C2×C4⋊C4C22×D1711yescharacteristic
162C2×C42—341no—
163C2×C42C4×D1711yescharacteristic
164C2×C42D6811yescharacteristic
165C2×C42C22⋊D1711yescharacteristic
166C4⋊C8—41no—
167#101—344no—
168C22⋊C8—22no—
169#97—41no—
170#97—681no—
171C4×C8—681no—
172#95—172no—
173#95—342no—
174#95—342no—
175#95—342no—
176#95—342no—
177C17#1179411yescharacteristic, complemented, Sylow
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—684no—
182C2×Q8—681no—
183C2×Q8—681no—
184C22×C4—342no—
185C22×C4—342no—
186C22×C4#6322711yescharacteristic
187C22×C4#6322711yescharacteristic
188C22×C4#6322711yescharacteristic
189C22×C4#6322711yescharacteristic
190C22×C4C2×C4×D1711yescharacteristic
191C22×C4C2×D6811yescharacteristic
192C22×C4C2×C22⋊D1711yescharacteristic
193M16—344no—
194M16—682no—
195C2×C8—42no—
196C2×C8—682no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—682no—
201C4⋊C4—682no—
202C4⋊C4—682no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—682no—
210C23C2×#6322711yescharacteristic
211Q8—44no—
212Q8—1362no—
213Q8—1362no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—682no—
229C2×C4—684no—
230C2×C4—682no—
231C2×C4—682no—
232C2×C4—682no—
233C2×C4#1058747711yescharacteristic
234C2×C4#1058747711yescharacteristic
235C2×C4#1058749612yes—
236C2×C4#1058759512yes—
237C8—82no—
238C8—684no—
239C22—21no—
240C22—21no—
241C22C2×#1058747711yescharacteristic
242C22#1176281911yescharacteristic
243C22C2×#1058749611yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—1362no—
252C4—1362no—
253C4#1176260512yes—
254C2—21no—
255C2—21no—
256C2#1452730511yescharacteristic
257C2#1453043711yescharacteristic
258C2C2×#1176260511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1467C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1467D17
hypocenterC17#11794
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
177. C17#11794semidirect

Maximal subgroups

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

subgroupindex
#145339962
C4×#105875072
#145268122
C17×#14672
#1179417