Finite Group Databasebeta

← #30765133

Subgroups — #30765133

Order 17152.

382 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 115 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#22561830C212yesmaximal, complemented
2#22558162C212yesmaximal, complemented
3#22558145C212yesmaximal, complemented
4C67×#2929C211yesmaximal, characteristic
5C2×#14523983—22no—
6#16863178—22no—
7#16863178C2214yes—
8C2×#131×C67C2211yescharacteristic
9C23×C8×C67C411yescharacteristic
10#525×C67C411yescharacteristic
11#435×C67—22no—
12C8×D4×C67—22no—
13#424×C67C2211yescharacteristic
14#408×C67C411yescharacteristic
15#407×C67C2211yescharacteristic
16#405×C67C411yescharacteristic
17#14523983—24no—
18#14523983—42no—
19C2×C67⋊C16—42no—
20C2×#41×C67C2×C411yescharacteristic
21C23×C4×C67C2×C411yescharacteristic
22#131×C67—41no—
23#131×C67—41no—
24C2×M16×C67C2×C411yescharacteristic
25C2×M16×C67D412yes—
26C2×M16×C67C2311yescharacteristic
27C22×C8×C67—21no—
28C22×C8×C67—21no—
29C22×C8×C67—41no—
30C22×C8×C67—41no—
31C22×C8×C67C2×C411yescharacteristic
32C22×C8×C67C2×C411yescharacteristic
33C22×C8×C67D412yes—
34#123×C67C812yes—
35C22⋊Q8×C67—21no—
36#121×C67—21no—
37#120×C67C812yes—
38C4×D4×C67—22no—
39#117×C67C2×C411yescharacteristic
40C4⋊C8×C67—22no—
41C22⋊C8×C67—22no—
42C22⋊C8×C67—22no—
43#97×C67—21no—
44C4×C8×C67—21no—
45C67⋊C16—44no—
46C24×C67C2×C811yescharacteristic
47#41×C67—41no—
48#41×C67—41no—
49C2×Q8×C67M1611yescharacteristic
50C2×D4×C67—22no—
51C2×D4×C67C2×C812yes—
52C2×D4×C67M1611yescharacteristic
53C22×C4×C67—21no—
54C22×C4×C67—21no—
55C22×C4×C67—41no—
56C22×C4×C67—41no—
57C22×C4×C67C22⋊C412yes—
58C22×C4×C67C2×C811yescharacteristic
59C22×C4×C67C22×C411yescharacteristic
60M16×C67—42no—
61M16×C67—41no—
62M16×C67—41no—
63C2×C8×C67—21no—
64C2×C8×C67—21no—
65C2×C8×C67—22no—
66C2×C8×C67—22no—
67C2×C8×C67—21no—
68C2×C8×C67—21no—
69C2×C8×C67—42no—
70C2×C8×C67—41no—
71C2×C8×C67—41no—
72C2×C8×C67—41no—
73C2×C8×C67C22⋊C411yescharacteristic
74C2×C8×C67C22⋊C411yescharacteristic
75C2×C8×C67C2×D412yes—
76C4⋊C4×C67—22no—
77C4⋊C4×C67—21no—
78C22⋊C4×C67—22no—
79C22⋊C4×C67—22no—
80C22⋊C4×C67C2×C812yes—
81C42×C67—21no—
82C23×C67—21no—
83C23×C67—41no—
84C23×C67—42no—
85C23×C67C22⋊C812yes—
86C23×C67C22×C811yescharacteristic
87Q8×C67—41no—
88D4×C67—42no—
89D4×C67—41no—
90D4×C67—44no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—21no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67C22⋊C812yes—
102C2×C4×C67C2×C22⋊C411yescharacteristic
103C2×C4×C67C2×M1611yescharacteristic
104C8×C67—21no—
105C8×C67—22no—
106C8×C67—41no—
107C8×C67—41no—
108C8×C67#9911yescharacteristic
109C8×C67#9911yescharacteristic
110C22×C67—22no—
111C22×C67—21no—
112C22×C67—22no—
113C22×C67—42no—
114C22×C67—41no—
115C22×C67—41no—
116C22×C67—42no—
117C22×C67—42no—
118C22×C67—41no—
119C22×C67C2×C22⋊C811yescharacteristic
120C4×C67—21no—
121C4×C67—22no—
122C4×C67—42no—
123C4×C67—41no—
124C4×C67C2×#9911yescharacteristic
125C4×C67#41311yescharacteristic
126#11483—671nomaximal, Sylow
127C2×C67—21no—
128C2×C67—42no—
129C2×C67—42no—
130C2×C67—41no—
131C2×C67#145911yescharacteristic
132#2929D6711yescharacteristic
133#2115—672no—
134#1328—672no—
135#1320—672no—
136C67#1148311yescharacteristic, complemented, Sylow
137C2×#131C2×D6711yescharacteristic
138C23×C8Dic6711yescharacteristic
139#525Dic6711yescharacteristic
140C2×M32—1342no—
141#435—22no—
142C8×D4—22no—
143#424C2×D6711yescharacteristic
144#408Dic6711yescharacteristic
145#407C2×D6711yescharacteristic
146#405Dic6711yescharacteristic
147#349—674no—
148#349—1342no—
149C2×#41C2×Dic6711yescharacteristic
150C23×C4C2×Dic6711yescharacteristic
151#131—41no—
152#131—41no—
153C2×M16C2×Dic6711yescharacteristic
154C2×M16C22⋊D6712yes—
155C2×M16C22×D6711yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8C2×Dic6711yescharacteristic
161C22×C8C2×Dic6711yescharacteristic
162C22×C8C22⋊D6712yes—
163#123C67⋊C812yes—
164C22⋊Q8—21no—
165#121—21no—
166#120C67⋊C812yes—
167C4×D4—22no—
168#117C2×Dic6711yescharacteristic
169M32—2682no—
170M32—1344no—
171C2×C16—2682no—
172C4⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175#97—21no—
176C4×C8—21no—
177C24C2×C67⋊C811yescharacteristic
178#41—41no—
179#41—41no—
180C2×Q8#1176161211yescharacteristic
181C2×D4—22no—
182C2×D4C2×C67⋊C812yes—
183C2×D4#1176161211yescharacteristic
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4C22⋊Dic6712yes—
189C22×C4C2×C67⋊C811yescharacteristic
190C22×C4C22×Dic6711yescharacteristic
191M16—42no—
192M16—41no—
193M16—41no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8C22⋊Dic6711yescharacteristic
205C2×C8C22⋊Dic6711yescharacteristic
206C2×C8C2×C22⋊D6712yes—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4C2×C67⋊C812yes—
212C42—21no—
213C16—2684no—
214C23—21no—
215C23—41no—
216C23—42no—
217C23C22⋊(C67⋊C8)12yes—
218C23C22×C67⋊C811yescharacteristic
219Q8—41no—
220D4—42no—
221D4—41no—
222D4—44no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4C22⋊(C67⋊C8)12yes—
234C2×C4C2×C22⋊Dic6711yescharacteristic
235C2×C4C2×#1176161211yescharacteristic
236C8—22no—
237C8—21no—
238C8—41no—
239C8—41no—
240C8#1452395811yescharacteristic
241C8#1452395811yescharacteristic
242C22—22no—
243C22—21no—
244C22—22no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—42no—
250C22—41no—
251C22C2×C22⋊(C67⋊C8)11yescharacteristic
252C4—21no—
253C4—22no—
254C4—42no—
255C4—41no—
256C4C2×#1452395811yescharacteristic
257C4#1686338011yescharacteristic
258C2—21no—
259C2—42no—
260C2—41no—
261C2—42no—
262C2#2255852711yescharacteristic
subgroupG/H
centerC8#14523958
commutatorC23×C67C22×C8
FrattiniC2×M16C22×D67
FittingC67×#2929C2
socleC2×C67#1459
prime radical#525×C67C4
cosocleC2×M16×C67C23
hypercenter#2929D67
hypocenterC67#11483
LCSG ▷ C23×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #22561830C2semidirect
2. #22558162C2semidirect
3. #22558145C2semidirect
136. C67#11483semidirect

Maximal subgroups

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

subgroupindex
#225618302
#225581622
#225581452
C67×#29292
#1148367