Finite Group Databasebeta

← #30765474

Subgroups — #30765474

Order 17152.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#22558553C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558143C214yesmaximal
4C67×#2546C211yesmaximal, characteristic
5#16863453—21no—
6#16863449—21no—
7#16863122—21no—
8#16863122—21no—
9#16863122C2212yes—
10#16863122C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C4⋊Q8×C67C412yes—
13C2×C4×Q8×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67—21no—
16#391×C67C2212yes—
17#389×C67—21no—
18#389×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic
20C4⋊(C67⋊C8)—41no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24#14523954—41no—
25C4×C67⋊C8—41no—
26C22×Q8×C67C2×C412yes—
27C22×Q8×C67C2×C411yescharacteristic
28C4⋊Q8×C67—41no—
29C4⋊Q8×C67—41no—
30C4⋊Q8×C67—44no—
31C4×Q8×C67—41no—
32C4×Q8×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C4⋊C4×C67C2311yescharacteristic
41C2×C42×C67C2×C411yescharacteristic
42C2×C42×C67C2×C412yes—
43C2×C42×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—21no—
46#95×C67—22no—
47#95×C67—22no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—42no—
50C2×Q8×C67—22no—
51C2×Q8×C67—22no—
52C2×Q8×C67—24no—
53C2×Q8×C67—41no—
54C2×Q8×C67—42no—
55C2×Q8×C67—42no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67C22⋊C411yescharacteristic
59C22×C4×C67C22⋊C411yescharacteristic
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22×C411yescharacteristic
63C22×C4×C67C2×D411yescharacteristic
64C22×C4×C67C2×D411yescharacteristic
65C4⋊C4×C67—24no—
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—41no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—44no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—44no—
74C42×C67—21no—
75C42×C67—21no—
76C42×C67—21no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—42no—
81C23×C67C2×C22⋊C411yescharacteristic
82Q8×C67—44no—
83Q8×C67—84no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—22no—
89C2×C4×C67—24no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—42no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67#9911yescharacteristic
105C2×C4×C67#9911yescharacteristic
106C2×C4×C67#10012yes—
107C2×C4×C67#13112yes—
108C67⋊C8—82no—
109C67⋊C8—82no—
110C22×C67—21no—
111C22×C67—21no—
112C22×C67#40811yescharacteristic
113C22×C67C2×#9911yescharacteristic
114C22×C67C2×#10011yescharacteristic
115C4×C67—21no—
116C4×C67—22no—
117C4×C67—22no—
118C4×C67—44no—
119C4×C67—41no—
120C4×C67—41no—
121C4×C67—41no—
122C4×C67—41no—
123C4×C67—82no—
124C4×C67#35612yes—
125#11568—671nomaximal, Sylow
126C2×C67—21no—
127C2×C67—21no—
128C2×C67#147111yescharacteristic
129C2×C67#212911yescharacteristic
130C2×C67C2×#35611yescharacteristic
131#2546D6711yescharacteristic
132#1467—671no—
133#1461—671no—
134#1319—674no—
135C67#1156811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8Dic6711yescharacteristic
137C2×C4⋊Q8Dic6712yes—
138C2×C4×Q8C2×D6711yescharacteristic
139#432—1341no—
140#431—1341no—
141#398—21no—
142#398—21no—
143#391C2×D6712yes—
144#389—21no—
145#389—21no—
146C4×C4⋊C4Dic6711yescharacteristic
147#324—672no—
148#324—672no—
149#324—1341no—
150#324—1341no—
151C22×Q8C2×Dic6712yes—
152C22×Q8C2×Dic6711yescharacteristic
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—44no—
156C4×Q8—41no—
157C4×Q8—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D6711yescharacteristic
164C2×C4⋊C4C22⋊D6711yescharacteristic
165C2×C4⋊C4C22×D6711yescharacteristic
166C2×C42C2×Dic6711yescharacteristic
167C2×C42C2×Dic6712yes—
168C2×C42C22⋊D6711yescharacteristic
169C2×C42C22⋊D6711yescharacteristic
170C4⋊C8—2681no—
171C4⋊C8—2681no—
172C22⋊C8—1342no—
173C22⋊C8—1342no—
174#97—2681no—
175C4×C8—2681no—
176#95—21no—
177#95—22no—
178#95—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C22⋊Dic6711yescharacteristic
188C22×C4C22⋊Dic6711yescharacteristic
189C22×C4C22⋊Dic6711yescharacteristic
190C22×C4C22⋊Dic6711yescharacteristic
191C22×C4C22×Dic6711yescharacteristic
192C22×C4C2×C22⋊D6711yescharacteristic
193C22×C4C2×C22⋊D6711yescharacteristic
194C2×C8—2682no—
195C2×C8—2682no—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—42no—
212C23C2×C22⋊Dic6711yescharacteristic
213Q8—44no—
214Q8—84no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4#1452395811yescharacteristic
236C2×C4#1452395811yescharacteristic
237C2×C4#1452396012yes—
238C2×C4#1452405012yes—
239C8—5362no—
240C8—5362no—
241C22—21no—
242C22—21no—
243C22#1686336311yescharacteristic
244C22C2×#1452395811yescharacteristic
245C22C2×#1452396011yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—44no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4#1686319512yes—
256C2—21no—
257C2—21no—
258C2#2255856911yescharacteristic
259C2#2256188411yescharacteristic
260C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2546C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2546D67
hypocenterC67#11568
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
135. C67#11568semidirect

Maximal subgroups

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

subgroupindex
#225585532
#225585332
#225581432
C67×#25462
#1156867