Finite Group Databasebeta

← #16953012

Subgroups — #16953012

Order 4352.

420 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 111 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#14528780C211yesmaximal, characteristic
2C4×#10587507C211yesmaximal, characteristic
3#14526911C214yesmaximal
4C17×#2567C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587507C2212yes—
7C2×#10587507C2212yes—
8C4×#63223—21no—
9#11762578—21no—
10#11762578—21no—
11C2×C17×C4⋊Q8C411yescharacteristic
12C2×C4×Q8×C17C2211yescharacteristic
13C17×#395—21no—
14C17×#395—21no—
15C17×#391C2212yes—
16C17×#389—21no—
17C17×#389—21no—
18C17×#384C412yes—
19C4×C4⋊C4×C17C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17C2311yescharacteristic
22C17×C4⋊Q8—41no—
23C17×C4⋊Q8—41no—
24C4×Q8×C17—41no—
25C4×Q8×C17—41no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17C2×C412yes—
36C2×C42×C17D411yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×#63223—22no—
39C2×#63223—22no—
40C17×#95—21no—
41C17×#95—22no—
42C17×#95—22no—
43#10587507—22no—
44#10587507—24no—
45#10587507Q814yescomplemented
46C4⋊#3622—41no—
47C4⋊#3622—41no—
48#10587474—41no—
49C4×#3622—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×Q8×C17—42no—
53C2×Q8×C17—82no—
54C2×Q8×C17C2×Q812yes—
55C2×Q8×C17#4112yes—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C4⋊C4×C17—24no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—82no—
73C4⋊C4×C17—82no—
74C42×C17—21no—
75C42×C17—21no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—42no—
81#63223—24no—
82#63223—42no—
83#63223—42no—
84#63223—42no—
85C2×#3622—42no—
86C2×#3622—42no—
87#13302—171nomaximal, Sylow
88C23×C17C2×C22⋊C411yescharacteristic
89Q8×C17—42no—
90Q8×C17—42no—
91Q8×C17—84no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—24no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17#9911yescharacteristic
112C2×C4×C17#9911yescharacteristic
113C2×C4×C17C4×Q812yes—
114C2×C4×C17C22⋊Q812yes—
115C2×C4×C17#12411yescharacteristic
116C2×C4×C17C4⋊Q811yescharacteristic
117#3622—44no—
118#3622—82no—
119#2567D1711yescharacteristic
120#1801—171no—
121C4×#101—171no—
122#1394—174no—
123C22×C17—21no—
124C22×C17—21no—
125C22×C17#39111yescharacteristic
126C22×C17C2×#9911yescharacteristic
127C22×C17#41011yescharacteristic
128C4×C17—21no—
129C4×C17—22no—
130C4×C17—21no—
131C4×C17—22no—
132C4×C17—21no—
133C4×C17—44no—
134C4×C17—41no—
135C4×C17—41no—
136C4×C17—41no—
137C4×C17—82no—
138C4×C17#35612yes—
139C2×C4⋊Q8Dic1711yescharacteristic
140C2×C4×Q8C2×D1711yescharacteristic
141#423—341no—
142C2×#101—172no—
143C2×#101—172no—
144C4×M16—341no—
145#395—21no—
146#395—21no—
147#391C2×D1712yes—
148#389—21no—
149#389—21no—
150#384Dic1712yes—
151C4×C4⋊C4Dic1711yescharacteristic
152#343—341no—
153#343—341no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17#199611yescharacteristic
157C2×C17C2×#35611yescharacteristic
158C2×C17#213411yescharacteristic
159C22×Q8—21no—
160C22×Q8C22×D1711yescharacteristic
161C2×M16—342no—
162C2×M16—342no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C2×Dic1712yes—
172C2×C4⋊C4C2×Dic1711yescharacteristic
173C2×C4⋊C4C22⋊D1711yescharacteristic
174C2×C4⋊C4C22⋊D1711yescharacteristic
175C2×C42C2×Dic1711yescharacteristic
176C2×C42C2×Dic1712yes—
177C2×C42C22⋊D1711yescharacteristic
178C2×C42C22⋊D1711yescharacteristic
179C4⋊C8—681no—
180C4⋊C8—681no—
181#101—174no—
182#101—342no—
183#101—344no—
184#97—681no—
185C4×C8—681no—
186#95—21no—
187#95—22no—
188#95—22no—
189C17#1330211yescharacteristic, complemented, Sylow
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8Q8×D1712yes—
195C2×Q8#6325612yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4C22⋊Dic1711yescharacteristic
199C22×C4C22⋊Dic1711yescharacteristic
200C22×C4C22⋊Dic1711yescharacteristic
201C22×C4C22⋊Dic1711yescharacteristic
202C22×C4C22×Dic1711yescharacteristic
203C22×C4C2×C22⋊D1711yescharacteristic
204C22×C4C2×C22⋊D1711yescharacteristic
205M16—344no—
206M16—682no—
207M16—682no—
208M16—682no—
209C2×C8—682no—
210C2×C8—682no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23C2×C22⋊Dic1711yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#1058750511yescharacteristic
251C2×C4#1058750511yescharacteristic
252C2×C4#1058763011yescharacteristic
253C2×C4Q8×Dic1712yes—
254C2×C4#1058763212yes—
255C2×C4#1058763311yescharacteristic
256C8—684no—
257C8—1362no—
258C22—21no—
259C22—21no—
260C22#1176271611yescharacteristic
261C22C2×#1058750511yescharacteristic
262C22#1176278311yescharacteristic
263C4—21no—
264C4—22no—
265C4—21no—
266C4—21no—
267C4—22no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#1176260812yes—
274C2—21no—
275C2—21no—
276C2#1452977011yescharacteristic
277C2C2×#1176260811yescharacteristic
278C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2567C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#2567D17
hypocenterC17#13302
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
45. #10587507Q8semidirect
189. C17#13302semidirect

Maximal subgroups

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

subgroupindex
#145287802
C4×#105875072
#145269112
C17×#25672
#1330217