Finite Group Databasebeta

← #30747589

Subgroups — #30747589

Order 17152.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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#22560205C211yesmaximal, characteristic, complemented
2#22558890C212yesmaximal, complemented
3#22558634C212yesmaximal
4#22558533C211yesmaximal, characteristic, complemented
5C2×#556×C67C211yesmaximal, characteristic
6#16863449C2212yes—
7#16863420C2212yes—
8C2×#14523962—21no—
9#16863165—21no—
10#16863145—24no—
11D4⋊(C67⋊C8)—24no—
12#16863122—22no—
13#556×C67—24no—
14#556×C67—24no—
15C2×C4⋊Q8×C67C2211yescharacteristic
16C2×#126×C67—22no—
17C2×#124×C67C412yes—
18C2×C22⋊Q8×C67—21no—
19C2×C4×Q8×C67C412yes—
20C2×C4×Q8×C67C2211yescharacteristic
21C2×C4×D4×C67C2211yescharacteristic
22C2×#117×C67—21no—
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)—22no—
27#14523962—42no—
28C22⋊(C67⋊C8)—22no—
29C4×C67⋊C8—22no—
30C22×Q8×C67C2×C412yes—
31C22×Q8×C67D411yescharacteristic
32C22×D4×C67C2×C411yescharacteristic
33C23×C4×C67—21no—
34C4⋊Q8×C67—22no—
35C4⋊Q8×C67D414yes—
36#126×C67—216no—
37#124×C67—24no—
38#124×C67—24no—
39C22⋊Q8×C67—28no—
40C4×Q8×C67—24no—
41C4×Q8×C67—24no—
42C4×Q8×C67—22no—
43C4×Q8×C67—22no—
44C4×D4×C67—22no—
45C4×D4×C67D414yes—
46#117×C67—28no—
47C2×C4⋊C4×C67—22no—
48C2×C4⋊C4×C67—21no—
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67C2×C411yescharacteristic
51C2×C4⋊C4×C67D411yescharacteristic
52C2×C4⋊C4×C67D412yes—
53C2×C22⋊C4×C67—22no—
54C2×C22⋊C4×C67—22no—
55C2×C22⋊C4×C67—21no—
56C2×C42×C67—22no—
57C2×C42×C67C2×C412yes—
58C2×C42×C67C2311yescharacteristic
59#11761612—42no—
60#11761612—42no—
61C2×C67⋊C8—42no—
62C2×C67⋊C8—42no—
63C24×C67—21no—
64C2×Q8×C67—22no—
65C2×Q8×C67—22no—
66C2×Q8×C67—22no—
67C2×Q8×C67—21no—
68C2×Q8×C67—21no—
69C2×Q8×C67—44no—
70C2×Q8×C67C22⋊C414yes—
71C2×D4×C67—21no—
72C2×D4×C67—42no—
73C2×D4×C67C22⋊C412yes—
74C22×C4×C67—21no—
75C22×C4×C67—22no—
76C22×C4×C67—24no—
77C22×C4×C67—42no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67C22⋊C412yes—
81C22×C4×C67C22⋊C412yes—
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C4⋊C4×C67—28no—
86C4⋊C4×C67—21no—
87C4⋊C4×C67—24no—
88C4⋊C4×C67—24no—
89C4⋊C4×C67—22no—
90C4⋊C4×C67—24no—
91C4⋊C4×C67C22⋊C412yes—
92C22⋊C4×C67—28no—
93C22⋊C4×C67—28no—
94C22⋊C4×C67—24no—
95C42×C67—28no—
96C42×C67—22no—
97C42×C67—22no—
98C42×C67C2×D412yes—
99C42×C67C2×D412yes—
100C23×C67—22no—
101C23×C67—42no—
102C23×C67—42no—
103C23×C67—41no—
104C23×C67—41no—
105C23×C67C2×C22⋊C411yescharacteristic
106Q8×C67—44no—
107Q8×C67—44no—
108Q8×C67—42no—
109Q8×C67—42no—
110D4×C67—24no—
111D4×C67—42no—
112C2×C4×C67—22no—
113C2×C4×C67—22no—
114C2×C4×C67—22no—
115C2×C4×C67—24no—
116C2×C4×C67—22no—
117C2×C4×C67—22no—
118C2×C4×C67—22no—
119C2×C4×C67—41no—
120C2×C4×C67—44no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—44no—
124C2×C4×C67—44no—
125C2×C4×C67—42no—
126C2×C4×C67C2×C22⋊C411yescharacteristic
127C2×C4×C67C2×C22⋊C411yescharacteristic
128C2×C4×C67#12012yes—
129C2×C4×C67#12012yes—
130C2×C4×C67#13512yes—
131C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C67#13711yescharacteristic
133C67⋊C8—82no—
134C67⋊C8—82no—
135C22×C67—21no—
136C22×C67—21no—
137C22×C67—42no—
138C22×C67—42no—
139C22×C67—42no—
140C22×C67—82no—
141C22×C67—82no—
142C22×C67#37911yescharacteristic
143C22×C67#41611yescharacteristic
144C22×C67#41711yescharacteristic
145C4×C67—22no—
146C4×C67—21no—
147C4×C67—42no—
148C4×C67—44no—
149C4×C67—42no—
150C4×C67—42no—
151C4×C67#41312yes—
152C4×C67#44912yes—
153C4×C67#45212yes—
154#8394—671nomaximal, Sylow
155C2×C67—21no—
156C2×C67—21no—
157C2×C67—82no—
158C2×C67#178511yescharacteristic
159C2×C67#178811yescharacteristic
160C2×C67#179611yescharacteristic
161C2×#556D6711yescharacteristic
162#1801—671no—
163#1548—672no—
164#1486—672no—
165#1461—671no—
166C67#839411yescharacteristic, complemented, Sylow
167#556—24no—
168#556—24no—
169C2×C4⋊Q8C2×D6711yescharacteristic
170C2×#126—22no—
171C2×#124Dic6712yes—
172C2×C22⋊Q8—21no—
173C2×C4×Q8Dic6712yes—
174C2×C4×Q8C2×D6711yescharacteristic
175C2×C4×D4C2×D6711yescharacteristic
176C2×#117—21no—
177#431—672no—
178#423—672no—
179C2×#101—1341no—
180#343—1341no—
181#333—1344no—
182D4⋊C8—1344no—
183#324—1342no—
184C22×Q8C2×Dic6712yes—
185C22×Q8C22⋊D6711yescharacteristic
186C22×D4C2×Dic6711yescharacteristic
187C23×C4—21no—
188C2×M16—1342no—
189C4⋊Q8—22no—
190C4⋊Q8C22⋊D6714yes—
191#126—216no—
192#124—24no—
193#124—24no—
194C22⋊Q8—28no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4C22⋊D6714yes—
201#117—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic6711yescharacteristic
206C2×C4⋊C4C22⋊D6711yescharacteristic
207C2×C4⋊C4C22⋊D6712yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42C2×Dic6712yes—
213C2×C42C22×D6711yescharacteristic
214C4⋊C8—1342no—
215C4⋊C8—1342no—
216C4⋊C8—1342no—
217#101—2682no—
218C22⋊C8—1342no—
219C4×C8—1342no—
220C24—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—44no—
227C2×Q8C22⋊Dic6714yes—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊Dic6712yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C22⋊Dic6712yes—
238C22×C4C22⋊Dic6712yes—
239C22×C4C22×Dic6711yescharacteristic
240C22×C4C2×C22⋊D6711yescharacteristic
241C22×C4C2×C22⋊D6711yescharacteristic
242M16—2682no—
243M16—2682no—
244C2×C8—2682no—
245C2×C8—2682no—
246C4⋊C4—28no—
247C4⋊C4—21no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4C22⋊Dic6712yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42C2×C22⋊D6712yes—
260C42C2×C22⋊D6712yes—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic6711yescharacteristic
267Q8—44no—
268Q8—42no—
269Q8—42no—
270Q8—44no—
271D4—24no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4C2×C22⋊Dic6711yescharacteristic
288C2×C4C2×C22⋊Dic6711yescharacteristic
289C2×C4#1452401212yes—
290C2×C4#1452401212yes—
291C2×C4#1452406312yes—
292C2×C4#1452406811yescharacteristic
293C2×C4#1452407311yescharacteristic
294C8—5362no—
295C8—5362no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22#1686325211yescharacteristic
304C22#1686339211yescharacteristic
305C22#1686339711yescharacteristic
306C4—21no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4#1686338012yes—
313C4#1686354112yes—
314C4#1686355812yes—
315C2—21no—
316C2—21no—
317C2—82no—
318C2#2256013711yescharacteristic
319C2#2256014911yescharacteristic
320C2#2256018411yescharacteristic
subgroupG/H
centerC22#16863252
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC2×#556×C67C2
socleC22×C67#379
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenterC2×#556D67
hypocenterC67#8394
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #22560205C2semidirect
2. #22558890C2semidirect
4. #22558533C2semidirect
166. C67#8394semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225588902
#225586342
#225585332
C2×#556×C672
#839467