Finite Group Databasebeta

← #30748744

Subgroups — #30748744

Order 17152.

460 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#16863468C211yesmaximal, characteristic
2#22560245C212yesmaximal, complemented
3#22558547C211yesmaximal, characteristic
4#22558085C212yesmaximal, complemented
5C67×#1465C211yesmaximal, characteristic
6C22×#11761612C411yescharacteristic
7C23×C67⋊C8C2211yescharacteristic
8#16863468—22no—
9#16863468—22no—
10C2×#14523978—22no—
11C2×C4⋊(C67⋊C8)C411yescharacteristic
12#16863360C2212yes—
13C2×C22⋊(C67⋊C8)C411yescharacteristic
14C2×C22⋊(C67⋊C8)C2211yescharacteristic
15C2×#14523954C411yescharacteristic
16#16863160—22no—
17#16863122—21no—
18#16863120—21no—
19C2×C4×D4×C67C2211yescharacteristic
20#407×C67C2212yes—
21#324×C67—21no—
22#323×C67—21no—
23C2×#11761612—21no—
24C2×#11761612—21no—
25C2×#11761612—22no—
26C2×#11761612—42no—
27C22×C67⋊C8—42no—
28C22×C67⋊C8C2×C411yescharacteristic
29C22×C67⋊C8C2×C411yescharacteristic
30C22×C67⋊C8C2×C411yescharacteristic
31C22×C67⋊C8C2×C411yescharacteristic
32C22×C67⋊C8D412yes—
33C22×C67⋊C8D412yes—
34#14523978—44no—
35C4⋊(C67⋊C8)—22no—
36C22⋊(C67⋊C8)—22no—
37C22⋊(C67⋊C8)—22no—
38C22⋊(C67⋊C8)—22no—
39C22⋊(C67⋊C8)—22no—
40#14523954—21no—
41#14523954—21no—
42C22×D4×C67Q811yescharacteristic
43C23×C4×C67C2×C411yescharacteristic
44C23×C4×C67C2311yescharacteristic
45C2×M16×C67—22no—
46C4×D4×C67—22no—
47C4×D4×C67—22no—
48C2×C4⋊C4×C67D411yescharacteristic
49C2×C22⋊C4×C67D411yescharacteristic
50C2×C22⋊C4×C67Q811yescharacteristic
51C2×C42×C67C2×C411yescharacteristic
52C22⋊C8×C67—22no—
53C22⋊C8×C67—22no—
54#11761612—42no—
55#11761612—42no—
56#11761612—42no—
57#11761612—42no—
58C2×C67⋊C8—24no—
59C2×C67⋊C8—22no—
60C2×C67⋊C8—21no—
61C2×C67⋊C8—21no—
62C2×C67⋊C8—41no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—41no—
65C2×C67⋊C8—41no—
66C2×C67⋊C8—42no—
67C2×C67⋊C8C22⋊C411yescharacteristic
68C2×C67⋊C8C22⋊C412yes—
69C2×C67⋊C8C22⋊C411yescharacteristic
70C24×C67C4⋊C411yescharacteristic
71C24×C67C2×Q811yescharacteristic
72C2×D4×C67—22no—
73C2×D4×C67—42no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—41no—
77C22×C4×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67C4⋊C411yescharacteristic
81C22×C4×C67C4⋊C411yescharacteristic
82C22×C4×C67C4⋊C411yescharacteristic
83C22×C4×C67C22×C411yescharacteristic
84C22×C4×C67C2×D412yes—
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67#4112yes—
87M16×C67—42no—
88M16×C67—42no—
89C2×C8×C67—42no—
90C4⋊C4×C67—22no—
91C22⋊C4×C67—22no—
92C22⋊C4×C67—22no—
93C42×C67—21no—
94C42×C67—21no—
95C23×C67—22no—
96C23×C67—21no—
97C23×C67—22no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67C2×C4⋊C411yescharacteristic
103C23×C67C22⋊Q812yes—
104D4×C67—44no—
105C2×C4×C67—22no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—22no—
113C2×C4×C67—41no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67C2×C22⋊C411yescharacteristic
121C2×C4×C67#11711yescharacteristic
122C2×C4×C67#12112yes—
123C8×C67—82no—
124C67⋊C8—22no—
125C67⋊C8—21no—
126C67⋊C8—41no—
127C67⋊C8—41no—
128C67⋊C8—82no—
129C67⋊C8#9912yes—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67—22no—
133C22×C67—22no—
134C22×C67—22no—
135C22×C67—44no—
136C22×C67—41no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—82no—
140C22×C67#38011yescharacteristic
141C22×C67#43012yes—
142C4×C67—22no—
143C4×C67—21no—
144C4×C67—41no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67C2×#9911yescharacteristic
148C4×C67#41011yescharacteristic
149#8613—671nomaximal, Sylow
150C2×C67—22no—
151C2×C67—42no—
152C2×C67—81no—
153C2×C67#179911yescharacteristic
154C2×C67#181111yescharacteristic
155C2×C67#181311yescharacteristic
156C2×#435—671no—
157#1810—672no—
158#1465—671no—
159#1465D6711yescharacteristic
160#1293—672no—
161C67#861311yescharacteristic, complemented, Sylow
162C22×M16—671no—
163C23×C8—671no—
164C2×C4×D4C2×D6711yescharacteristic
165#435—1342no—
166#435—1342no—
167C2×#108—1342no—
168C2×C4⋊C8—671no—
169#407—672no—
170#407C2×D6712yes—
171C2×C22⋊C8—671no—
172C2×C22⋊C8—671no—
173C2×#97—671no—
174#341—1342no—
175#324—21no—
176#324—1341no—
177#323—21no—
178#323—1341no—
179C22×D4Dic13411yescharacteristic
180C23×C4C4×D6711yescharacteristic
181C23×C4C22×D6711yescharacteristic
182C2×M16—22no—
183C2×M16—2682no—
184C2×M16—1342no—
185C2×M16—1341no—
186C2×M16—1341no—
187C22×C8—2682no—
188C22×C8—671no—
189C22×C8—672no—
190C22×C8—671no—
191C22×C8—671no—
192C22×C8—671no—
193C22×C8—672no—
194C4×D4—22no—
195C4×D4—22no—
196C2×C4⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4Dic13411yescharacteristic
198C2×C22⋊C4C22⋊D6711yescharacteristic
199C2×C42C4×D6711yescharacteristic
200#108—2684no—
201C4⋊C8—1342no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—1342no—
205C22⋊C8—1342no—
206C22⋊C8—1342no—
207C22⋊C8—1342no—
208#97—1341no—
209#97—1341no—
210C24Dic67⋊C411yescharacteristic
211C24C2×Dic13411yescharacteristic
212C2×D4—22no—
213C2×D4—42no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4Dic67⋊C411yescharacteristic
221C22×C4Dic67⋊C411yescharacteristic
222C22×C4Dic67⋊C411yescharacteristic
223C22×C4C2×C4×D6711yescharacteristic
224C22×C4D4×D6712yes—
225C22×C4C2×C22⋊D6711yescharacteristic
226C22×C4#1176162712yes—
227M16—42no—
228M16—42no—
229M16—2682no—
230M16—2682no—
231M16—2682no—
232M16—2682no—
233C2×C8—42no—
234C2×C8—2681no—
235C2×C8—2681no—
236C2×C8—2682no—
237C2×C8—2682no—
238C2×C8—2681no—
239C2×C8—671no—
240C2×C8—671no—
241C2×C8—672no—
242C2×C8—1341no—
243C2×C8—1341no—
244C2×C8—1344no—
245C2×C8—1342no—
246C4⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C42—21no—
250C42—21no—
251C23—22no—
252C23—21no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C2×Dic67⋊C411yescharacteristic
259C23#1452402012yes—
260D4—44no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4C22⋊C4×D6711yescharacteristic
277C2×C4#1452399911yescharacteristic
278C2×C4#1452401512yes—
279C8—82no—
280C8—2681no—
281C8—2681no—
282C8—5362no—
283C8—672no—
284C8—1341no—
285C8—1342no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—44no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—82no—
296C22#1686325411yescharacteristic
297C22#1686344412yes—
298C4—22no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4#99×D6711yescharacteristic
304C4#1686336811yescharacteristic
305C2—22no—
306C2—42no—
307C2—81no—
308C2#2256019511yescharacteristic
309C2#2256024911yescharacteristic
310C2#2256025711yescharacteristic
subgroupG/H
centerC22#16863254
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#1465C2
socleC22×C67#380
prime radicalC22×D4×C67Q8
cosocleC23×C4×C67C23
hypercenter#1465D67
hypocenterC67#8613
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #22560245C2semidirect
4. #22558085C2semidirect
161. C67#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#168634682
#225602452
#225585472
#225580852
C67×#14652
#861367