Finite Group Databasebeta

← #30771818

Subgroups — #30771818

Order 17152.

544 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16863191C212yesmaximal
2#22560197C212yesmaximal
3#22558334C212yesmaximal
4C4×#124×C67C211yesmaximal, characteristic
5C2×#14523958C2214yes—
6#16863256—22no—
7#16863191—24no—
8#16863163—22no—
9C2×#124×C67C2211yescharacteristic
10C2×C4×Q8×C67C411yescharacteristic, complemented
11C2×C4×D4×C67C2211yescharacteristic
12#391×C67C411yescharacteristic, complemented
13#390×C67C2211yescharacteristic
14#388×C67—22no—
15#382×C67C411yescharacteristic, complemented
16C4×C22⋊C4×C67—22no—
17C43×C67C411yescharacteristic, complemented
18C2×C4⋊Dic67—42no—
19C2×C22⋊Dic67—24no—
20#14523958—24no—
21#14523958—24no—
22C67⋊#95—42no—
23C22×Q8×C67C2×C411yescharacteristic
24C22×D4×C67C2311yescharacteristic
25C23×C4×C67D412yes—
26#124×C67—22no—
27#124×C67D412yes—
28#124×C67Q812yes—
29C4×Q8×C67—41no—
30C4×Q8×C67—41no—
31C4×D4×C67—41no—
32C4×D4×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67C2×C411yescharacteristic
35C2×C4⋊C4×C67C2×C411yescharacteristic
36C2×C22⋊C4×C67—22no—
37C2×C22⋊C4×C67—22no—
38C2×C22⋊C4×C67D412yes—
39C2×C42×C67—21no—
40C2×C42×C67—21no—
41C2×C42×C67—21no—
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67C2×C411yescharacteristic
45#95×C67—22no—
46#95×C67—22no—
47C22×Dic67—44no—
48C4⋊Dic67—84no—
49C22⋊Dic67—44no—
50C22⋊Dic67—44no—
51C24×C67C2×D412yes—
52C2×Q8×C67—21no—
53C2×Q8×C67—41no—
54C2×Q8×C67—41no—
55C2×Q8×C67C4⋊C412yescomplemented
56C2×D4×C67—41no—
57C2×D4×C67—41no—
58C2×D4×C67C2×D411yescharacteristic
59C2×D4×C67C2×Q811yescharacteristic
60C2×D4×C67#4112yes—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—22no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—44no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C412yes—
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22×C411yescharacteristic
72C4⋊C4×C67—41no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—42no—
75C22⋊C4×C67—24no—
76C22⋊C4×C67—24no—
77C22⋊C4×C67—42no—
78C22⋊C4×C67—44no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—41no—
83C42×C67—42no—
84C42×C67—42no—
85C42×C67—41no—
86C42×C67—41no—
87C42×C67C4⋊C412yescomplemented
88C23×C67—22no—
89C23×C67—22no—
90C23×C67—42no—
91C23×C67—42no—
92C23×C67C2×C22⋊C411yescharacteristic
93C23×C67#12112yes—
94C23×C67C22⋊Q812yes—
95Q8×C67—42no—
96Q8×C67—42no—
97D4×C67—41no—
98D4×C67—41no—
99D4×C67—42no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—22no—
105C2×C4×C67—22no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—44no—
115C2×C4×C67—41no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—41no—
119C2×C4×C67—82no—
120C2×C4×C67#9911yescharacteristic
121C2×C4×C67#9911yescharacteristic
122C2×C4×C67C2×C4⋊C411yescharacteristic
123C2×C4×C67#11711yescharacteristic
124C2×Dic67—84no—
125C2×Dic67—84no—
126C2×Dic67—84no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—22no—
130C22×C67—22no—
131C22×C67—42no—
132C22×C67—42no—
133C22×C67—42no—
134C22×C67—42no—
135C22×C67—82no—
136C22×C67#38011yescharacteristic
137C22×C67C2×#9911yescharacteristic
138C22×C67#41011yescharacteristic
139C4×C67—21no—
140C4×C67—21no—
141C4×C67—21no—
142C4×C67—42no—
143C4×C67—42no—
144C4×C67—42no—
145C4×C67—42no—
146C4×C67—41no—
147C4×C67—81no—
148C4×C67#35412yescomplemented
149Dic67—164no—
150#12798—671nomaximal, Sylow
151C2×C67—21no—
152C2×C67—21no—
153C2×C67—42no—
154C2×C67—42no—
155C2×C67#179911yescharacteristic
156C2×C67C2×#35411yescharacteristic
157C2×C67#212611yescharacteristic
158C4×#124D6711yescharacteristic
159C2×#354—672no—
160#1799—672no—
161#1393—672no—
162C67#1279811yescharacteristic, complemented, Sylow
163C2×#124C2×D6711yescharacteristic
164C2×C4×Q8Dic6711yescharacteristic, complemented
165C2×C4×D4C2×D6711yescharacteristic
166C2×#99—674no—
167#391Dic6711yescharacteristic, complemented
168#390C2×D6711yescharacteristic
169#388—22no—
170#382Dic6711yescharacteristic, complemented
171#380—1342no—
172C4×C22⋊C4—22no—
173C43Dic6711yescharacteristic, complemented
174#354—1344no—
175#342—1342no—
176C22×Q8C2×Dic6711yescharacteristic
177C22×D4C22×D6711yescharacteristic
178C23×C4C22⋊D6712yes—
179#124—22no—
180#124Dic13412yes—
181#124D26812yes—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—2682no—
188C2×C4⋊C4C2×Dic6711yescharacteristic
189C2×C4⋊C4C2×Dic6711yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1344no—
193C2×C22⋊C4C22⋊D6712yes—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×Dic6711yescharacteristic
198C2×C42C2×Dic6711yescharacteristic
199C2×C42C2×Dic6711yescharacteristic
200#99—1344no—
201#99—1344no—
202#95—22no—
203#95—22no—
204#95—2682no—
205C24C2×C22⋊D6712yes—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊Dic6712yescomplemented
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C2×D26811yescharacteristic
213C2×D4C2×Dic13411yescharacteristic
214C2×D4#1176162812yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—44no—
222C22×C4—2684no—
223C22×C4C22⋊Dic6712yes—
224C22×C4C22⋊Dic6711yescharacteristic
225C22×C4C22⋊Dic6711yescharacteristic
226C22×C4C22×Dic6711yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—5364no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—2684no—
236C22⋊C4—2684no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42C4⋊Dic6712yescomplemented
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23C2×C22⋊Dic6711yescharacteristic
251C23#1452401712yes—
252C23#1452402212yes—
253Q8—42no—
254Q8—42no—
255D4—41no—
256D4—41no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—5364no—
279C2×C4—5364no—
280C2×C4—5364no—
281C2×C4#1452395811yescharacteristic
282C2×C4#1452395811yescharacteristic
283C2×C4C2×C4⋊Dic6711yescharacteristic
284C2×C4#1452400211yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22#1686325611yescharacteristic
295C22C2×#1452395811yescharacteristic
296C22#1686337011yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—81no—
306C4—10724no—
307C4#1686319112yescomplemented
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2256019711yescharacteristic
313C2C2×#1686319111yescharacteristic
314C2#2256187411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×D4C22×D67
FittingC4×#124×C67C2
socleC22×C67C2×#99
prime radical, cosocleC22×D4×C67C23
hypercenterC4×#124D67
hypocenterC67#12798
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
10. C2×C4×Q8×C67C4semidirect
12. #391×C67C4semidirect
15. #382×C67C4semidirect
17. C43×C67C4semidirect
55. C2×Q8×C67C4⋊C4semidirect
87. C42×C67C4⋊C4semidirect
148. C4×C67#354semidirect
162. C67#12798semidirect
164. C2×C4×Q8Dic67semidirect
167. #391Dic67semidirect
170. #382Dic67semidirect
173. C43Dic67semidirect
209. C2×Q8C4⋊Dic67semidirect
245. C42C4⋊Dic67semidirect
307. C4#16863191semidirect

Maximal subgroups

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

subgroupindex
C2×#168631912
#225601972
#225583342
C4×#124×C672
#1279867