Finite Group Databasebeta

← #30766885

Subgroups — #30766885

Order 17152.

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

normal subgroups only

This group has 476 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#22560205C211yesmaximal, characteristic, complemented
2#22558533C211yesmaximal, characteristic, complemented
3#22558212C214yesmaximal, complemented
4C67×#2548C211yesmaximal, characteristic
5#16863449—21no—
6#16863420—21no—
7C2×#14523962—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C2×#127×C67C411yescharacteristic
12C2×#124×C67—21no—
13C2×C4×Q8×C67C2211yescharacteristic
14#394×C67C412yes—
15#394×C67C2212yes—
16#390×C67—21no—
17#390×C67—21no—
18#390×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—22no—
21C4⋊(C67⋊C8)—41no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523962—42no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—41no—
27C22×Q8×C67D411yescharacteristic
28C22×D4×C67—21no—
29C22×D4×C67—21no—
30C22×D4×C67C2×C412yes—
31#127×C67—41no—
32#127×C67—41no—
33#124×C67—42no—
34#124×C67—42no—
35C4×Q8×C67—41no—
36C4×Q8×C67—41no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C4⋊C4×C67C2311yescharacteristic
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67—22no—
45C2×C42×C67—21no—
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67D411yescharacteristic
48C2×C42×C67D411yescharacteristic
49#95×C67C2×C412yes—
50#11761612—42no—
51#11761612—42no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C24×C67—22no—
55C2×Q8×C67—41no—
56C2×Q8×C67—41no—
57C2×Q8×C67D812yes—
58C2×Q8×C67SD1612yes—
59C2×D4×C67—42no—
60C2×D4×C67—42no—
61C2×D4×C67—42no—
62C2×D4×C67—42no—
63C2×D4×C67—42no—
64C2×D4×C67—41no—
65C2×D4×C67—41no—
66C2×D4×C67—42no—
67C2×D4×C67—82no—
68C22×C4×C67—22no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—42no—
80C22⋊C4×C67—44no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—42no—
83C22⋊C4×C67—82no—
84C22⋊C4×C67—82no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—21no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—42no—
92C23×C67—42no—
93C23×C67—42no—
94C23×C67—42no—
95C23×C67—42no—
96C23×C67—42no—
97C23×C67—42no—
98C23×C67—42no—
99C23×C67C2×C22⋊C411yescharacteristic
100Q8×C67—42no—
101Q8×C67—42no—
102D4×C67—82no—
103D4×C67—84no—
104D4×C67—84no—
105D4×C67—82no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—22no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—42no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67#9911yescharacteristic
123C2×C4×C67#9911yescharacteristic
124C2×C4×C67#10212yes—
125C2×C4×C67#10212yes—
126C2×C4×C67C2×D811yescharacteristic
127C2×C4×C67C2×SD1611yescharacteristic
128C67⋊C8—82no—
129C67⋊C8—82no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67—82no—
133C22×C67—82no—
134C22×C67—82no—
135C22×C67—82no—
136C22×C67—82no—
137C22×C67—82no—
138C22×C67—82no—
139C22×C67C2×#9911yescharacteristic
140C22×C67C2×#10211yescharacteristic
141C22×C67#42111yescharacteristic
142C4×C67—21no—
143C4×C67—21no—
144C4×C67—22no—
145C4×C67—22no—
146C4×C67—21no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—82no—
151C4×C67#35512yes—
152#11882—671nomaximal, Sylow
153C2×C67—21no—
154C2×C67—21no—
155C2×C67—162no—
156C2×C67#151911yescharacteristic
157C2×C67#212211yescharacteristic
158C2×C67C2×#35511yescharacteristic
159#2548D6711yescharacteristic
160#1801—671no—
161#1461—671no—
162#1345—674no—
163C67#1188211yescharacteristic, complemented, Sylow
164C2×#127Dic6711yescharacteristic
165C2×#124—21no—
166C2×C4×Q8C2×D6711yescharacteristic
167#431—1341no—
168#423—1341no—
169C2×#101—1341no—
170#394Dic6712yes—
171#394C2×D6712yes—
172#390—21no—
173#390—21no—
174#390—21no—
175C4×C4⋊C4Dic6711yescharacteristic
176#343—672no—
177#324—672no—
178#324—1341no—
179C22×Q8C22⋊D6711yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C2×Dic6712yes—
183C2×M16—1342no—
184#127—41no—
185#127—41no—
186#124—42no—
187#124—42no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic6711yescharacteristic
192C2×C4⋊C4C22⋊D6711yescharacteristic
193C2×C4⋊C4C22×D6711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42—21no—
199C2×C42C2×Dic6711yescharacteristic
200C2×C42C22⋊D6711yescharacteristic
201C2×C42C22⋊D6711yescharacteristic
202C4⋊C8—2681no—
203C4⋊C8—2681no—
204C4⋊C8—2681no—
205#101—2682no—
206C22⋊C8—1342no—
207C4×C8—2681no—
208#95C2×Dic6712yes—
209C24—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1176161312yes—
213C2×Q8#1176161612yes—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—82no—
223C22×C4—22no—
224C22×C4C22⋊Dic6711yescharacteristic
225C22×C4C22⋊Dic6711yescharacteristic
226C22×C4C22⋊Dic6711yescharacteristic
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22×Dic6711yescharacteristic
229C22×C4C2×C22⋊D6711yescharacteristic
230C22×C4C2×C22⋊D6711yescharacteristic
231M16—2682no—
232M16—2682no—
233C2×C8—2682no—
234C2×C8—2682no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23C2×C22⋊Dic6711yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—84no—
262D4—82no—
263D4—84no—
264D4—82no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#1452395811yescharacteristic
282C2×C4#1452395811yescharacteristic
283C2×C4#1452396412yes—
284C2×C4#1452396412yes—
285C2×C4C2×#1176161311yescharacteristic
286C2×C4C2×#1176161611yescharacteristic
287C8—5362no—
288C8—5362no—
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22C2×#1452395811yescharacteristic
299C22C2×#1452396411yescharacteristic
300C22#1686341311yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4#1686319312yes—
311C2—21no—
312C2—21no—
313C2—162no—
314C2#2255877711yescharacteristic
315C2#2256185911yescharacteristic
316C2C2×#1686319311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2548C2
socleC22×C67C2×#99
prime radicalC2×#127×C67C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2548D67
hypocenterC67#11882
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #22560205C2semidirect
2. #22558533C2semidirect
3. #22558212C2semidirect
163. C67#11882semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225585332
#225582122
C67×#25482
#1188267