Finite Group Databasebeta

← #30772106

Subgroups — #30772106

Order 17152.

614 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560137C212yesmaximal
2#22558336C214yesmaximal, complemented
3C67×#2311C211yesmaximal, characteristic
4C2×#14523962C2214yes—
5C2×#14523960—22no—
6#16863360—22no—
7C22×#41×C67C2211yescharacteristic
8C2×#117×C67C411yescharacteristic
9C22×C4⋊C4×C67C411yescharacteristic
10#391×C67C2212yes—
11#390×C67C412yes—
12#386×C67—22no—
13#380×C67—22no—
14C2×#11761612—24no—
15#14523962—28no—
16#14523960—44no—
17C22⋊(C67⋊C8)—44no—
18C2×#41×C67—24no—
19C2×#41×C67—42no—
20C2×#41×C67—42no—
21C22×Q8×C67C2311yescharacteristic
22C22×D4×C67C2×C411yescharacteristic
23C22×D4×C67D412yes—
24C23×C4×C67—21no—
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67D412yes—
27#117×C67—42no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67—42no—
31C2×C4⋊C4×C67C2×C412yes—
32C2×C22⋊C4×C67—21no—
33C2×C22⋊C4×C67—24no—
34C2×C42×C67C2×C412yes—
35#95×C67—24no—
36#11761612—44no—
37#11761612—44no—
38C2×C67⋊C8—44no—
39C24×C67C22⋊C412yes—
40C24×C67C22⋊C411yescharacteristic
41#41×C67—416no—
42C2×Q8×C67—41no—
43C2×Q8×C67—41no—
44C2×Q8×C67#4114yes—
45C2×D4×C67—24no—
46C2×D4×C67—22no—
47C2×D4×C67—24no—
48C2×D4×C67—41no—
49C2×D4×C67—41no—
50C2×D4×C67—42no—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67—24no—
55C22×C4×C67—24no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—42no—
59C22×C4×C67—42no—
60C22×C4×C67—82no—
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22×C411yescharacteristic
63C22×C4×C67C2×D412yes—
64C4⋊C4×C67—42no—
65C4⋊C4×C67—42no—
66C4⋊C4×C67—42no—
67C22⋊C4×C67—42no—
68C22⋊C4×C67—84no—
69C42×C67—42no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67—22no—
73C23×C67—22no—
74C23×C67—22no—
75C23×C67—41no—
76C23×C67—41no—
77C23×C67—42no—
78C23×C67—42no—
79C23×C67#9912yes—
80C23×C67C2×C22⋊C411yescharacteristic
81Q8×C67—44no—
82D4×C67—48no—
83D4×C67—44no—
84C2×C4×C67—28no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—48no—
92C2×C4×C67—42no—
93C2×C4×C67—44no—
94C2×C4×C67—42no—
95C2×C4×C67—42no—
96C2×C4×C67—42no—
97C2×C4×C67—84no—
98C2×C4×C67#9912yes—
99C2×C4×C67#11712yes—
100C2×C4×C67#12314yes—
101C67⋊C8—84no—
102C22×C67—21no—
103C22×C67—21no—
104C22×C67—22no—
105C22×C67—42no—
106C22×C67—42no—
107C22×C67—44no—
108C22×C67—41no—
109C22×C67—41no—
110C22×C67—42no—
111C22×C67—42no—
112C22×C67—42no—
113C22×C67—82no—
114C22×C67#38111yescharacteristic
115C22×C67C2×#9911yescharacteristic
116C22×C67C2×#9911yescharacteristic
117C4×C67—24no—
118C4×C67—22no—
119C4×C67—22no—
120C4×C67—82no—
121C4×C67—82no—
122#12863—671nomaximal, Sylow
123C2×C67—21no—
124C2×C67—21no—
125C2×C67—42no—
126C2×C67—82no—
127C2×C67#181911yescharacteristic
128C2×C67#213411yescharacteristic
129C2×C67#213511yescharacteristic
130#2311D6711yescharacteristic
131#1785—672no—
132#1394—674no—
133C67#1286311yescharacteristic, complemented, Sylow
134C22×#41C2×D6711yescharacteristic
135C2×#117Dic6711yescharacteristic
136C22×C4⋊C4Dic6711yescharacteristic
137C2×#101—674no—
138C2×#100—1342no—
139#407—1342no—
140#391C2×D6712yes—
141#390Dic6712yes—
142#386—22no—
143#380—22no—
144C2×#41—24no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8C22×D6711yescharacteristic
148C22×D4C2×Dic6711yescharacteristic
149C22×D4C22⋊D6712yes—
150C23×C4—21no—
151C23×C4C2×Dic6711yescharacteristic
152C23×C4C22⋊D6712yes—
153C2×M16—1344no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×Dic6712yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C42C2×Dic6712yes—
162#101—1348no—
163#100—2684no—
164C22⋊C8—2684no—
165#95—24no—
166C24C22⋊Dic6712yes—
167C24C22⋊Dic6711yescharacteristic
168#41—416no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8#1176162814yes—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—24no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic6711yescharacteristic
189C22×C4C22×Dic6711yescharacteristic
190C22×C4C2×C22⋊D6712yes—
191M16—2684no—
192M16—2684no—
193C2×C8—2684no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—84no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23#1452395812yes—
210C23C2×C22⋊Dic6711yescharacteristic
211Q8—44no—
212D4—48no—
213D4—44no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4#1452395812yes—
229C2×C4#1452400212yes—
230C2×C4#1452402614yes—
231C8—5364no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—42no—
236C22—42no—
237C22—44no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22#1686325911yescharacteristic
245C22C2×#1452395811yescharacteristic
246C22C2×#1452395811yescharacteristic
247C4—24no—
248C4—22no—
249C4—22no—
250C4—82no—
251C4—82no—
252C2—21no—
253C2—21no—
254C2—42no—
255C2—82no—
256C2#2256028311yescharacteristic
257C2#2256190111yescharacteristic
258C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC67×#2311C2
socleC22×C67C2×#99
prime radicalC22×#41×C67C22
cosocleC22×Q8×C67C23
hypercenter#2311D67
hypocenterC67#12863
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #22558336C2semidirect
133. C67#12863semidirect

Maximal subgroups

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

subgroupindex
#225601372
#225583362
C67×#23112
#1286367