Finite Group Databasebeta

← #30751890

Subgroups — #30751890

Order 17152.

516 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 119 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#22562812C211yesmaximal, characteristic
2#22558781C212yesmaximal
3C2×#16863122C211yesmaximal, characteristic
4#22558076C212yesmaximal
5C67×#2627C211yesmaximal, characteristic
6C2×C22⋊(C67⋊C8)C2212yes—
7#16863282—24no—
8#16863259—21no—
9#16863256C2212yes—
10C4×C22⋊Dic67—21no—
11#16863133—24no—
12#16863122—22no—
13C2×C22⋊Q8×C67C2211yescharacteristic
14C22×C4⋊C4×C67C2211yescharacteristic
15#400×C67C412yescomplemented
16#398×C67—22no—
17#395×C67C412yescomplemented
18#385×C67—21no—
19#381×C67—21no—
20#380×C67C2211yescharacteristic
21C22×C67⋊C8—22no—
22C2×C4⋊Dic67—22no—
23C2×C22⋊Dic67—22no—
24C2×C4×Dic67—22no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27C67⋊#95—22no—
28C67⋊#95—24no—
29C67⋊#95D414yes—
30C22×Q8×C67C2×C411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C23×C4×C67C2311yescharacteristic
34C22⋊Q8×C67—22no—
35C22⋊Q8×C67—22no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C412yes—
43C2×C4⋊C4×C67D412yes—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46#95×C67—22no—
47#95×C67—22no—
48#95×C67—22no—
49#95×C67C2×C412yes—
50C22×Dic67—24no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—82no—
53C4⋊Dic67—82no—
54C22⋊Dic67—42no—
55C22⋊Dic67—42no—
56C4×Dic67—82no—
57C24×C67C2×D411yescharacteristic
58C2×Q8×C67—22no—
59C2×Q8×C67—81no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—22no—
68C22×C4×C67—21no—
69C22×C4×C67—81no—
70C22×C4×C67—81no—
71C22×C4×C67C22⋊C412yes—
72C22×C4×C67C22⋊C412yes—
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D412yes—
75C22×C4×C67C2×D411yescharacteristic
76C22×C4×C67#4112yes—
77C4⋊C4×C67—24no—
78C4⋊C4×C67—24no—
79C4⋊C4×C67—22no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—81no—
82C4⋊C4×C67—82no—
83C22⋊C4×C67—41no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—42no—
86C23×C67—41no—
87C23×C67—41no—
88C23×C67C2×C22⋊C411yescharacteristic
89C23×C67#12011yescharacteristic
90C23×C67#12311yescharacteristic
91C23×C67#13512yes—
92C23×C67#13712yes—
93Q8×C67—82no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—42no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—42no—
113C2×C4×C67—81no—
114C2×C4×C67C4×D412yes—
115C2×C4×C67#12112yes—
116C2×Dic67—44no—
117C2×Dic67—44no—
118C2×Dic67—44no—
119C67⋊C8—82no—
120C22×C67—21no—
121C22×C67—21no—
122C22×C67—21no—
123C22×C67—21no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—42no—
127C22×C67—42no—
128C22×C67#38611yescharacteristic
129C22×C67#41611yescharacteristic
130C22×C67#41911yescharacteristic
131C22×C67#45212yes—
132C22×C67#48512yes—
133C4×C67—22no—
134C4×C67—22no—
135C4×C67—42no—
136C4×C67—42no—
137C4×C67—82no—
138C4×C67—81no—
139C4×C67—81no—
140Dic67—84no—
141#9202—671nomaximal, Sylow
142C2×C67—22no—
143C2×C67—21no—
144C2×C67—21no—
145C2×C67—22no—
146C2×C67#187311yescharacteristic
147C2×C67#191211yescharacteristic
148C2×C67#191311yescharacteristic
149#2627D6711yescharacteristic
150#2313—671no—
151#1520—672no—
152C2×#324—671no—
153#1289—672no—
154C67#920211yescharacteristic, complemented, Sylow
155C2×C22⋊Q8C2×D6711yescharacteristic
156C22×C4⋊C4C2×D6711yescharacteristic
157C2×C22⋊C8—672no—
158#400Dic6712yescomplemented
159#398—22no—
160#395Dic6712yescomplemented
161#387—1344no—
162#385—21no—
163#381—21no—
164#381—1341no—
165#380—672no—
166#380C2×D6711yescharacteristic
167C4×C22⋊C4—1341no—
168#328—1344no—
169#324—1342no—
170C22×Q8C2×Dic6711yescharacteristic
171C23×C4C22⋊D6711yescharacteristic
172C23×C4C22⋊D6711yescharacteristic
173C23×C4C22×D6711yescharacteristic
174C22×C8—1342no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—1342no—
183C2×C4⋊C4C2×Dic6711yescharacteristic
184C2×C4⋊C4C2×Dic6712yes—
185C2×C4⋊C4C22⋊D6712yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—1342no—
189C2×C42—1342no—
190C22⋊C8—1342no—
191C22⋊C8—1342no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—674no—
196#95—1342no—
197#95—1344no—
198#95C2×Dic6712yes—
199C24C2×C22⋊D6711yescharacteristic
200C2×Q8—22no—
201C2×Q8—81no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—1344no—
214C22×C4C22⋊Dic6712yes—
215C22×C4C22⋊Dic6712yes—
216C22×C4C22×Dic6711yescharacteristic
217C22×C4D4×D6712yes—
218C22×C4C2×C22⋊D6711yescharacteristic
219C22×C4#1176162712yes—
220C2×C8—2682no—
221C2×C8—5362no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—5362no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—2682no—
233C22⋊C4—2682no—
234C42—5362no—
235C23—41no—
236C23—41no—
237C23C2×C22⋊Dic6711yescharacteristic
238C23#1452401111yescharacteristic
239C23#1452402411yescharacteristic
240C23#1452406312yes—
241C23#1452407412yes—
242Q8—82no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—2684no—
264C2×C4—2684no—
265C2×C4—2684no—
266C2×C4D4×Dic6712yes—
267C2×C4#1452401512yes—
268C8—5362no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#1686327611yescharacteristic
278C22#1686339211yescharacteristic
279C22#1686340511yescharacteristic
280C22#1686355612yes—
281C22#1686373412yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—81no—
289C4—5364no—
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2#2256052211yescharacteristic
295C2#2256074711yescharacteristic
296C2#2256075211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2627C2
socleC22×C67#386
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenter#2627D67
hypocenterC67#9202
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
15. #400×C67C4semidirect
17. #395×C67C4semidirect
154. C67#9202semidirect
158. #400Dic67semidirect
160. #395Dic67semidirect

Maximal subgroups

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

subgroupindex
#225628122
#225587812
C2×#168631222
#225580762
C67×#26272
#920267