Finite Group Databasebeta

← #16953017

Subgroups — #16953017

Order 4352.

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

normal subgroups only

This group has 396 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#14528780C211yesmaximal, characteristic
2C4×#10587507C211yesmaximal, characteristic
3#14526911C212yesmaximal
4#14526911C212yesmaximal
5C17×#2565C211yesmaximal, characteristic
6#11762833—21no—
7C2×#10587507C2212yes—
8C2×#10587507C2212yes—
9C4×#63223—21no—
10#11762578—21no—
11#11762578—21no—
12C2×C17×#125C411yescharacteristic
13C2×C4×Q8×C17C2211yescharacteristic
14C17×#398—21no—
15C17×#398—21no—
16C17×#391C2211yescharacteristic
17C17×#391C2211yescharacteristic
18C17×#387—21no—
19C17×#387—21no—
20C17×#384C411yescharacteristic
21C17×#382C411yescharacteristic
22C4×C4⋊C4×C17C411yescharacteristic
23C22×Q8×C17C2311yescharacteristic
24C17×#125—41no—
25C17×#125—41no—
26C4×Q8×C17—41no—
27C4×Q8×C17—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17C2×C411yescharacteristic
40C2×C42×C17C2×C411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×#63223—22no—
44C2×#63223—22no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52#10587507—22no—
53#10587507—24no—
54#10587507Q814yes—
55C4⋊#3622—41no—
56C4⋊#3622—41no—
57#10587474—41no—
58C4×#3622—41no—
59C2×Q8×C17—41no—
60C2×Q8×C17—41no—
61C2×Q8×C17C2×Q812yes—
62C2×Q8×C17#4112yes—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—41no—
92C42×C17—41no—
93#63223—24no—
94#63223—42no—
95#63223—42no—
96#63223—42no—
97C2×#3622—42no—
98C2×#3622—42no—
99#13303—171nomaximal, Sylow
100C23×C17C2×C22⋊C411yescharacteristic
101Q8×C17—42no—
102Q8×C17—42no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17#9911yescharacteristic
129C2×C4×C17C4×Q812yes—
130C2×C4×C17C22⋊Q812yes—
131C2×C4×C17#12411yescharacteristic
132C2×C4×C17C4⋊Q811yescharacteristic
133#3622—44no—
134#3622—82no—
135#2565D1711yescharacteristic
136#1801—171no—
137C4×#101—171no—
138#1394—172no—
139#1394—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17#39111yescharacteristic
143C22×C17C2×#9911yescharacteristic
144C22×C17#41011yescharacteristic
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—22no—
148C4×C17—21no—
149C4×C17—22no—
150C4×C17—21no—
151C4×C17—42no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—81no—
156C4×C17—81no—
157C4×C17—81no—
158C2×#125Dic1711yescharacteristic
159C2×C4×Q8C2×D1711yescharacteristic
160#423—341no—
161C2×#101—172no—
162C2×#101—172no—
163C4×M16—341no—
164#398—21no—
165#398—21no—
166#391C2×D1711yescharacteristic
167#391C2×D1711yescharacteristic
168#387—21no—
169#387—21no—
170#384Dic1711yescharacteristic
171#382Dic1711yescharacteristic
172C4×C4⋊C4Dic1711yescharacteristic
173#343—341no—
174#343—341no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17#199611yescharacteristic
178C2×C17#213211yescharacteristic
179C2×C17#213311yescharacteristic
180C22×Q8C22×D1711yescharacteristic
181C2×M16—342no—
182C2×M16—342no—
183#125—41no—
184#125—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×Dic1711yescharacteristic
193C2×C4⋊C4C2×Dic1711yescharacteristic
194C2×C4⋊C4C2×Dic1711yescharacteristic
195C2×C4⋊C4C22⋊D1711yescharacteristic
196C2×C4⋊C4C22⋊D1711yescharacteristic
197C2×C42C2×Dic1711yescharacteristic
198C2×C42C2×Dic1711yescharacteristic
199C2×C42C2×Dic1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202C4⋊C8—681no—
203C4⋊C8—681no—
204#101—174no—
205#101—342no—
206#101—344no—
207#97—681no—
208C4×C8—681no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216C17#1330311yescharacteristic, complemented, Sylow
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8Q8×D1712yes—
220C2×Q8#6325612yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C22⋊Dic1711yescharacteristic
226C22×C4C22⋊Dic1711yescharacteristic
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4C22×Dic1711yescharacteristic
230C22×C4C2×C22⋊D1711yescharacteristic
231C22×C4C2×C22⋊D1711yescharacteristic
232M16—344no—
233M16—682no—
234M16—682no—
235M16—682no—
236C2×C8—682no—
237C2×C8—682no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—41no—
256C42—41no—
257C23C2×C22⋊Dic1711yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#1058750511yescharacteristic
285C2×C4#1058750511yescharacteristic
286C2×C4#1058763011yescharacteristic
287C2×C4Q8×Dic1712yes—
288C2×C4#1058763212yes—
289C2×C4#1058763311yescharacteristic
290C8—684no—
291C8—1362no—
292C22—21no—
293C22—21no—
294C22#1176271611yescharacteristic
295C22C2×#1058750511yescharacteristic
296C22#1176278311yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2#1452977011yescharacteristic
313C2#1453046911yescharacteristic
314C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2565C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#2565D17
hypocenterC17#13303
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2565

Complements

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

NH ≅ G/Ntype
216. C17#13303semidirect

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×#25652
#1330317