Finite Group Databasebeta

← #30753686

Subgroups — #30753686

Order 17152.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#22569290C211yesmaximal, characteristic, complemented
2#22560205C211yesmaximal, characteristic, complemented
3#22558634C212yesmaximal
4#22558067C212yesmaximal, complemented
5C67×#2536C211yesmaximal, characteristic
6Q8×C67⋊C8—24no—
7#16863449—22no—
8#16863449C2212yes—
9#16863420C2212yes—
10C2×#14523962—21no—
11#16863165—21no—
12D4⋊(C67⋊C8)—24no—
13C2×C22⋊Q8×C67—21no—
14C2×C4×Q8×C67C2211yescharacteristic
15C2×C4×D4×C67C2211yescharacteristic
16#398×C67C412yes—
17#394×C67C412yes—
18#388×C67—22no—
19#382×C67C2211yescharacteristic
20#380×C67—21no—
21C2×#11761612—22no—
22C4⋊(C67⋊C8)—24no—
23C4⋊(C67⋊C8)—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26#14523962—42no—
27C22⋊(C67⋊C8)—22no—
28C4×C67⋊C8—24no—
29C4×C67⋊C8D414yes—
30C22×Q8×C67D411yescharacteristic
31C22×D4×C67C2×C411yescharacteristic
32C23×C4×C67—21no—
33C22⋊Q8×C67—42no—
34C22⋊Q8×C67—42no—
35C4×Q8×C67—22no—
36C4×Q8×C67—22no—
37C4×D4×C67—22no—
38C4×D4×C67—22no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C412yes—
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67—22no—
45C2×C22⋊C4×C67—22no—
46C2×C22⋊C4×C67—21no—
47C2×C42×C67D412yes—
48C2×C42×C67C2311yescharacteristic
49#95×C67—22no—
50#95×C67—21no—
51#95×C67C2×C412yes—
52#11761612—42no—
53#11761612—42no—
54C2×C67⋊C8—24no—
55C2×C67⋊C8—42no—
56C24×C67—21no—
57C2×Q8×C67—22no—
58C2×Q8×C67—22no—
59C2×Q8×C67D812yes—
60C2×Q8×C67SD1612yes—
61C2×D4×C67—22no—
62C2×D4×C67—42no—
63C22×C4×C67—22no—
64C22×C4×C67—21no—
65C22×C4×C67—42no—
66C22×C4×C67—41no—
67C22×C4×C67—41no—
68C22×C4×C67—42no—
69C22×C4×C67C22⋊C412yes—
70C22×C4×C67C22⋊C412yes—
71C22×C4×C67C22×C411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C4⋊C4×C67—24no—
75C4⋊C4×C67—22no—
76C4⋊C4×C67—22no—
77C4⋊C4×C67—44no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C22⋊C4×C67—44no—
81C22⋊C4×C67—44no—
82C22⋊C4×C67—41no—
83C22⋊C4×C67—41no—
84C42×C67—22no—
85C42×C67—22no—
86C42×C67C2×D412yes—
87C42×C67#4112yes—
88C23×C67—22no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67C2×C22⋊C411yescharacteristic
96Q8×C67—24no—
97Q8×C67—24no—
98D4×C67—24no—
99D4×C67—42no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—41no—
109C2×C4×C67—42no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—82no—
113C2×C4×C67—82no—
114C2×C4×C67#10214yes—
115C2×C4×C67C4×D412yes—
116C2×C4×C67#12011yescharacteristic
117C2×C4×C67#12112yes—
118C2×C4×C67#12311yescharacteristic
119C2×C4×C67C2×D811yescharacteristic
120C2×C4×C67C2×SD1611yescharacteristic
121C2×C4×C67#13512yes—
122C67⋊C8—44no—
123C67⋊C8—82no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—42no—
127C22×C67—42no—
128C22×C67—42no—
129C22×C67—81no—
130C22×C67—81no—
131C22×C67—81no—
132C22×C67—81no—
133C22×C67#38611yescharacteristic
134C22×C67C2×#10211yescharacteristic
135C22×C67#41611yescharacteristic
136C4×C67—22no—
137C4×C67—24no—
138C4×C67—21no—
139C4×C67—21no—
140C4×C67—42no—
141C4×C67—82no—
142C4×C67—81no—
143C4×C67#46512yes—
144C4×C67#46612yes—
145#9486—671nomaximal, Sylow
146C2×C67—21no—
147C2×C67—21no—
148C2×C67—82no—
149C2×C67#189111yescharacteristic
150C2×C67#189311yescharacteristic
151C2×C67#191211yescharacteristic
152#2982—671no—
153#2536D6711yescharacteristic
154#1801—671no—
155#1486—672no—
156#1286—672no—
157C67#948611yescharacteristic, complemented, Sylow
158C2×C22⋊Q8—21no—
159C2×C4×Q8C2×D6711yescharacteristic
160C2×C4×D4C2×D6711yescharacteristic
161C8×Q8—1344no—
162#431—672no—
163#431—1342no—
164#423—672no—
165C2×#101—1341no—
166#398Dic6712yes—
167#394Dic6712yes—
168#388—22no—
169#382C2×D6711yescharacteristic
170#380—21no—
171#343—1341no—
172D4⋊C8—1344no—
173C22×Q8C22⋊D6711yescharacteristic
174C22×D4C2×Dic6711yescharacteristic
175C23×C4—21no—
176C2×M16—1342no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic6711yescharacteristic
186C2×C4⋊C4C2×Dic6712yes—
187C2×C4⋊C4C22⋊D6711yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C42C22⋊D6712yes—
192C2×C42C22×D6711yescharacteristic
193C4⋊C8—1344no—
194C4⋊C8—1342no—
195C4⋊C8—1342no—
196C4⋊C8—1342no—
197#101—2682no—
198C22⋊C8—1342no—
199C4×C8—674no—
200C4×C8—1344no—
201#95—22no—
202#95—21no—
203#95C2×Dic6712yes—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1176161312yes—
208C2×Q8#1176161612yes—
209C2×D4—22no—
210C2×D4—42no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊Dic6712yes—
218C22×C4C22⋊Dic6712yes—
219C22×C4C22×Dic6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221C22×C4C2×C22⋊D6711yescharacteristic
222M16—2682no—
223M16—2682no—
224C2×C8—2682no—
225C2×C8—1344no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C42—22no—
237C42—22no—
238C42D4×D6712yes—
239C42#1176162712yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic6711yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—24no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#1452396414yes—
267C2×C4D4×Dic6712yes—
268C2×C4#1452401111yescharacteristic
269C2×C4#1452401512yes—
270C2×C4#1452402411yescharacteristic
271C2×C4C2×#1176161311yescharacteristic
272C2×C4C2×#1176161611yescharacteristic
273C2×C4#1452406312yes—
274C8—2684no—
275C8—5362no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#1686327611yescharacteristic
286C22C2×#1452396411yescharacteristic
287C22#1686339211yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#1686362312yes—
296C4#1686363012yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2256062811yescharacteristic
301C2#2256063811yescharacteristic
302C2#2256074711yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2536C2
socleC22×C67#386
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenter#2536D67
hypocenterC67#9486
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #22569290C2semidirect
2. #22560205C2semidirect
4. #22558067C2semidirect
157. C67#9486semidirect

Maximal subgroups

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

subgroupindex
#225692902
#225602052
#225586342
#225580672
C67×#25362
#948667