Finite Group Databasebeta

← #30766509

Subgroups — #30766509

Order 17152.

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

normal subgroups only

This group has 402 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#22560286C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3#22558209C214yesmaximal
4C67×#2621C211yesmaximal, characteristic
5#16863453—21no—
6#16863259—21no—
7#16863163—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×C22⋊Q8×C67—21no—
13C2×#117×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67C412yescomplemented
16#397×C67—21no—
17#397×C67C2212yes—
18#391×C67—21no—
19#382×C67C411yescharacteristic, complemented
20C2×C22⋊Dic67—22no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23#14523954—41no—
24C67⋊#95—41no—
25C67⋊#95—41no—
26C22×Q8×C67—21no—
27C23×C4×C67D411yescharacteristic
28C4⋊Q8×C67—41no—
29C4⋊Q8×C67—41no—
30C22⋊Q8×C67—44no—
31#117×C67—41no—
32#117×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67C2×C412yes—
37C2×C4⋊C4×C67C2×C411yescharacteristic
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67D411yescharacteristic
41C2×C22⋊C4×C67C2311yescharacteristic
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67D411yescharacteristic
44#95×C67—21no—
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48C22×Dic67—42no—
49C2×C67⋊C8—42no—
50C22⋊Dic67—42no—
51C22⋊Dic67—42no—
52C24×C67C2×D411yescharacteristic
53C2×Q8×C67—42no—
54C2×Q8×C67—42no—
55C2×Q8×C67—41no—
56C2×Q8×C67—41no—
57C22×C4×C67—22no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—41no—
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22×C411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69C4⋊C4×C67—41no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—44no—
74C4⋊C4×C67—42no—
75C22⋊C4×C67—41no—
76C22⋊C4×C67—41no—
77C22⋊C4×C67—44no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C23×C67—21no—
83C23×C67—21no—
84C23×C67—41no—
85C23×C67—41no—
86C23×C67C2×C22⋊C411yescharacteristic
87C23×C67AGL(1,ℤ/8)12yes—
88Q8×C67—82no—
89Q8×C67—82no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67#9911yescharacteristic
112C2×C4×C67#9911yescharacteristic
113C2×Dic67—82no—
114C2×Dic67—82no—
115C2×Dic67—82no—
116C67⋊C8—82no—
117C22×C67—21no—
118C22×C67—21no—
119C22×C67—22no—
120C22×C67—42no—
121C22×C67—41no—
122C22×C67—41no—
123C22×C67—81no—
124C22×C67C2×#9911yescharacteristic
125C22×C67#41811yescharacteristic
126C22×C67#42111yescharacteristic
127C4×C67—21no—
128C4×C67—21no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—41no—
133C4×C67—41no—
134C4×C67—82no—
135C4×C67—82no—
136Dic67—162no—
137#11801—671nomaximal, Sylow
138C2×C67—21no—
139C2×C67—21no—
140C2×C67—42no—
141C2×C67#151011yescharacteristic
142C2×C67#212311yescharacteristic
143C2×C67#212911yescharacteristic
144#2621D6711yescharacteristic
145#1820—671no—
146#1464—671no—
147#1344—674no—
148C67#1180111yescharacteristic, complemented, Sylow
149C2×C4⋊Q8Dic6711yescharacteristic, complemented
150C2×C22⋊Q8—21no—
151C2×#117C2×D6711yescharacteristic
152#432—1341no—
153#398—21no—
154#398Dic6712yescomplemented
155#397—21no—
156#397C2×D6712yes—
157#391—21no—
158#382Dic6711yescharacteristic, complemented
159#381—1341no—
160#342—672no—
161#342—1341no—
162#324—1341no—
163#323—672no—
164C22×Q8—21no—
165C23×C4C22⋊D6711yescharacteristic
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C22⋊Q8—44no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×Dic6712yes—
175C2×C4⋊C4C2×Dic6711yescharacteristic
176C2×C4⋊C4C22⋊D6711yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1342no—
179C2×C22⋊C4C22⋊D6711yescharacteristic
180C2×C22⋊C4C22×D6711yescharacteristic
181C2×C42C2×Dic6711yescharacteristic
182C2×C42C22⋊D6711yescharacteristic
183C4⋊C8—2681no—
184C22⋊C8—1342no—
185#97—2681no—
186#95—21no—
187#95—22no—
188#95—22no—
189#95—2681no—
190#95—2681no—
191#95C2×Dic6712yes—
192C24C2×C22⋊D6711yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—2682no—
204C22×C4C22⋊Dic6711yescharacteristic
205C22×C4C22⋊Dic6711yescharacteristic
206C22×C4C22⋊Dic6711yescharacteristic
207C22×C4C22⋊Dic6711yescharacteristic
208C22×C4C22×Dic6711yescharacteristic
209C22×C4C2×C22⋊D6711yescharacteristic
210C2×C8—2682no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—2682no—
221C22⋊C4—2682no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×C22⋊Dic6711yescharacteristic
231C23#1452406912yes—
232Q8—82no—
233Q8—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—5362no—
256C2×C4—5362no—
257C2×C4—5362no—
258C2×C4#1452395811yescharacteristic
259C2×C4#1452395811yescharacteristic
260C8—5362no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1452395811yescharacteristic
269C22#1686340111yescharacteristic
270C22#1686341311yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4—10722no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2255873811yescharacteristic
285C2#2256186311yescharacteristic
286C2#2256188411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2621C2
socleC22×C67C2×#99
prime radicalC24×C67C2×D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2621D67
hypocenterC67#11801
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C67C4semidirect
15. #398×C67C4semidirect
19. #382×C67C4semidirect
148. C67#11801semidirect
149. C2×C4⋊Q8Dic67semidirect
154. #398Dic67semidirect
158. #382Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602862
#225585432
#225582092
C67×#26212
#1180167