Finite Group Databasebeta

← #30751860

Subgroups — #30751860

Order 17152.

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

normal subgroups only

This group has 478 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#22568847C211yesmaximal, characteristic
2#22558881C212yesmaximal, complemented
3#22558523C211yesmaximal, characteristic
4#22558067C212yesmaximal, complemented
5C67×#2621C211yesmaximal, characteristic
6#16863457—24no—
7#16863449C2212yes—
8#16863420C2212yes—
9#16863360—21no—
10C4×#11761612—21no—
11#16863142—24no—
12#16863122—21no—
13#16863120—21no—
14C2×C4⋊Q8×C67C2211yescharacteristic
15C2×C22⋊Q8×C67—21no—
16C2×#117×C67C2211yescharacteristic
17#398×C67—21no—
18#398×C67C412yes—
19#397×C67—21no—
20#397×C67C412yes—
21#391×C67—21no—
22#382×C67C2211yescharacteristic
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)D414yes—
27C22⋊(C67⋊C8)—24no—
28C22⋊(C67⋊C8)—22no—
29#14523954—22no—
30C4×C67⋊C8—22no—
31C4×C67⋊C8—22no—
32C22×Q8×C67—21no—
33C23×C4×C67D411yescharacteristic
34C4⋊Q8×C67—22no—
35C4⋊Q8×C67—22no—
36C22⋊Q8×C67—44no—
37#117×C67—22no—
38#117×C67—22no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67C2×C412yes—
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67—22no—
46C2×C22⋊C4×C67C2×C411yescharacteristic
47C2×C22⋊C4×C67D411yescharacteristic
48C2×C42×C67D411yescharacteristic
49C2×C42×C67C2311yescharacteristic
50#95×C67—22no—
51#95×C67—22no—
52#95×C67—21no—
53#95×C67C2×C412yes—
54#11761612—42no—
55#11761612—42no—
56C2×C67⋊C8—24no—
57C2×C67⋊C8—42no—
58C24×C67C22⋊C411yescharacteristic
59C2×Q8×C67—41no—
60C2×Q8×C67—41no—
61C2×Q8×C67—42no—
62C2×Q8×C67—42no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67—22no—
68C22×C4×C67—22no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—22no—
76C4⋊C4×C67—22no—
77C4⋊C4×C67—24no—
78C4⋊C4×C67—44no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—42no—
81C22⋊C4×C67—22no—
82C22⋊C4×C67—22no—
83C22⋊C4×C67—44no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67C2×D412yes—
87C42×C67#4112yes—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67C2×C22⋊C411yescharacteristic
94Q8×C67—82no—
95Q8×C67—82no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67C4×D412yes—
114C2×C4×C67#12011yescharacteristic
115C2×C4×C67#12112yes—
116C2×C4×C67#12311yescharacteristic
117C2×C4×C67#13512yes—
118C2×C4×C67AGL(1,ℤ/8)12yes—
119C67⋊C8—44no—
120C67⋊C8—82no—
121C22×C67—21no—
122C22×C67—21no—
123C22×C67—42no—
124C22×C67—41no—
125C22×C67—41no—
126C22×C67—41no—
127C22×C67—81no—
128C22×C67#38611yescharacteristic
129C22×C67#41611yescharacteristic
130C22×C67#41811yescharacteristic
131C4×C67—22no—
132C4×C67—22no—
133C4×C67—41no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—82no—
137C4×C67—82no—
138C4×C67#44912yes—
139C4×C67#48212yes—
140#9196—671nomaximal, Sylow
141C2×C67—21no—
142C2×C67—21no—
143C2×C67—81no—
144C2×C67#187511yescharacteristic
145C2×C67#190911yescharacteristic
146C2×C67#191511yescharacteristic
147#2921—671no—
148#2621D6711yescharacteristic
149#1546—672no—
150#1458—671no—
151#1286—672no—
152C67#919611yescharacteristic, complemented, Sylow
153C2×C4⋊Q8C2×D6711yescharacteristic
154C2×C22⋊Q8—21no—
155C2×#117C2×D6711yescharacteristic
156#433—1344no—
157#431—672no—
158#423—672no—
159#407—1341no—
160C4×M16—1341no—
161#398—21no—
162#398Dic6712yes—
163#397—21no—
164#397Dic6712yes—
165#391—21no—
166#382C2×D6711yescharacteristic
167#332—1344no—
168#324—1341no—
169#323—1341no—
170C22×Q8—21no—
171C23×C4C22⋊D6711yescharacteristic
172C2×M16—1342no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C22⋊Q8—44no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×Dic6712yes—
182C2×C4⋊C4C2×Dic6711yescharacteristic
183C2×C4⋊C4C22⋊D6711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic6711yescharacteristic
186C2×C22⋊C4C22⋊D6711yescharacteristic
187C2×C42C22⋊D6711yescharacteristic
188C2×C42C22×D6711yescharacteristic
189C4⋊C8—674no—
190C4⋊C8—1342no—
191C4⋊C8—1342no—
192C22⋊C8—1342no—
193C22⋊C8—1344no—
194#97—1342no—
195C4×C8—1342no—
196C4×C8—1342no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95C2×Dic6712yes—
201C24C22⋊Dic6711yescharacteristic
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4C22⋊Dic6711yescharacteristic
213C22×C4C22⋊Dic6711yescharacteristic
214C22×C4C22⋊Dic6711yescharacteristic
215C22×C4C22×Dic6711yescharacteristic
216C22×C4C2×C22⋊D6711yescharacteristic
217C22×C4C2×C22⋊D6711yescharacteristic
218M16—2682no—
219M16—2682no—
220C2×C8—2682no—
221C2×C8—1344no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—44no—
231C42—21no—
232C42—21no—
233C42D4×D6712yes—
234C42#1176162712yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×C22⋊Dic6711yescharacteristic
241Q8—82no—
242Q8—82no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4D4×Dic6712yes—
261C2×C4#1452401111yescharacteristic
262C2×C4#1452401512yes—
263C2×C4#1452402411yescharacteristic
264C2×C4#1452406312yes—
265C2×C4#1452406912yes—
266C8—2684no—
267C8—5362no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1686327611yescharacteristic
276C22#1686339211yescharacteristic
277C22#1686340111yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1686353912yes—
286C4#1686371712yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2256053211yescharacteristic
291C2#2256073211yescharacteristic
292C2#2256076211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2621C2
socleC22×C67#386
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2621D67
hypocenterC67#9196
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. #22558881C2semidirect
4. #22558067C2semidirect
152. C67#9196semidirect

Maximal subgroups

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

subgroupindex
#225688472
#225588812
#225585232
#225580672
C67×#26212
#919667