Finite Group Databasebeta

← #30749397

Subgroups — #30749397

Order 17152.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#22560201C211yesmaximal, characteristic
2#22558530C211yesmaximal, characteristic
3#22558112C212yesmaximal, complemented
4#22558067C212yesmaximal, complemented
5C67×#2349C211yesmaximal, characteristic
6#16863449C2212yes—
7#16863420C2212yes—
8C2×#14523960—21no—
9#16863165—21no—
10#16863120—22no—
11C2×C4×D4×C67—21no—
12C2×C4×D4×C67C2211yescharacteristic
13#389×C67—21no—
14#385×C67—22no—
15#382×C67C2211yescharacteristic
16#380×C67C412yes—
17C4×C4⋊C4×C67C2211yescharacteristic
18C2×#95×C67C412yes—
19C2×#11761612—22no—
20C4⋊(C67⋊C8)—22no—
21C4⋊(C67⋊C8)—22no—
22C4⋊(C67⋊C8)—22no—
23#14523960—42no—
24C22⋊(C67⋊C8)—22no—
25C4×C67⋊C8—22no—
26C22×D4×C67D411yescharacteristic
27C23×C4×C67—22no—
28C23×C4×C67C2×C412yes—
29C4×D4×C67—22no—
30C4×D4×C67—22no—
31C4×D4×C67—42no—
32C4×D4×C67—42no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67D411yescharacteristic
37C2×C22⋊C4×C67—22no—
38C2×C22⋊C4×C67D412yes—
39C2×C42×C67—21no—
40C2×C42×C67C2311yescharacteristic
41#95×C67—22no—
42#95×C67—21no—
43#95×C67C2×C411yescharacteristic
44#95×C67C2×C412yes—
45#95×C67C2×C411yescharacteristic
46#11761612—42no—
47#11761612—42no—
48C2×C67⋊C8—42no—
49C2×C67⋊C8—42no—
50C24×C67C22⋊C412yes—
51C2×D4×C67—22no—
52C2×D4×C67—22no—
53C2×D4×C67D812yes—
54C2×D4×C67SD1612yes—
55C22×C4×C67—21no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—21no—
59C22×C4×C67—22no—
60C22×C4×C67—44no—
61C22×C4×C67—44no—
62C22×C4×C67—82no—
63C22×C4×C67—82no—
64C22×C4×C67C22⋊C412yes—
65C22×C4×C67C22×C411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C4⋊C4×C67—22no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—82no—
73C22⋊C4×C67—24no—
74C22⋊C4×C67—44no—
75C42×C67—41no—
76C42×C67—41no—
77C42×C67#4112yes—
78C42×C67#4112yes—
79C23×C67—22no—
80C23×C67—22no—
81C23×C67—42no—
82C23×C67—42no—
83C23×C67#10214yes—
84C23×C67C2×C22⋊C411yescharacteristic
85D4×C67—24no—
86D4×C67—24no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—41no—
90C2×C4×C67—41no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—42no—
94C2×C4×C67—41no—
95C2×C4×C67—41no—
96C2×C4×C67—42no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—42no—
100C2×C4×C67—82no—
101C2×C4×C67—82no—
102C2×C4×C67—88no—
103C2×C4×C67#11711yescharacteristic
104C2×C4×C67#11711yescharacteristic
105C2×C4×C67#12312yes—
106C2×C4×C67#12312yes—
107C2×C4×C67C2×D811yescharacteristic
108C2×C4×C67C2×SD1611yescharacteristic
109C2×C4×C67#13712yes—
110C67⋊C8—82no—
111C67⋊C8—82no—
112C22×C67—21no—
113C22×C67—21no—
114C22×C67—24no—
115C22×C67—44no—
116C22×C67—44no—
117C22×C67—42no—
118C22×C67—42no—
119C22×C67#38111yescharacteristic
120C22×C67C2×#10211yescharacteristic
121C22×C67#41911yescharacteristic
122C4×C67—22no—
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—42no—
126C4×C67—42no—
127C4×C67—81no—
128C4×C67—82no—
129C4×C67#48012yes—
130C4×C67#48312yes—
131#8764—671nomaximal, Sylow
132C2×C67—21no—
133C2×C67—21no—
134C2×C67—44no—
135C2×C67#182311yescharacteristic
136C2×C67#182711yescharacteristic
137C2×C67#182911yescharacteristic
138#2349D6711yescharacteristic
139#1800—671no—
140#1460—671no—
141#1303—672no—
142#1286—672no—
143C67#876411yescharacteristic, complemented, Sylow
144C2×C4×D4—21no—
145C2×C4×D4C2×D6711yescharacteristic
146#431—672no—
147#423—672no—
148C2×#100—1341no—
149#389—21no—
150#385—22no—
151#382C2×D6711yescharacteristic
152#380Dic6712yes—
153C4×C4⋊C4C2×D6711yescharacteristic
154C2×#95Dic6712yes—
155#343—1341no—
156#323—1342no—
157C22×D4C22⋊D6711yescharacteristic
158C23×C4—22no—
159C23×C4C2×Dic6712yes—
160C2×M16—1342no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D6711yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊D6712yes—
171C2×C42—21no—
172C2×C42C22×D6711yescharacteristic
173C4⋊C8—1342no—
174C4⋊C8—1342no—
175C4⋊C8—1342no—
176#100—2682no—
177C22⋊C8—1342no—
178C4×C8—1342no—
179#95—22no—
180#95—21no—
181#95C2×Dic6711yescharacteristic
182#95C2×Dic6712yes—
183#95C2×Dic6711yescharacteristic
184C24C22⋊Dic6712yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4#1176161312yes—
188C2×D4#1176161612yes—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4C22⋊Dic6712yes—
199C22×C4C22×Dic6711yescharacteristic
200C22×C4C2×C22⋊D6711yescharacteristic
201C22×C4C2×C22⋊D6711yescharacteristic
202M16—2682no—
203M16—2682no—
204C2×C8—2682no—
205C2×C8—2682no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—24no—
212C22⋊C4—44no—
213C42—41no—
214C42—41no—
215C42#1176162812yes—
216C42#1176162812yes—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23#1452396414yes—
222C23C2×C22⋊Dic6711yescharacteristic
223D4—24no—
224D4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—88no—
241C2×C4#1452400211yescharacteristic
242C2×C4#1452400211yescharacteristic
243C2×C4#1452402612yes—
244C2×C4#1452402612yes—
245C2×C4C2×#1176161311yescharacteristic
246C2×C4C2×#1176161611yescharacteristic
247C2×C4#1452407412yes—
248C8—5362no—
249C8—5362no—
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22#1686325911yescharacteristic
258C22C2×#1452396411yescharacteristic
259C22#1686340511yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—82no—
266C4—81no—
267C4#1686370912yes—
268C4#1686372612yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2256029511yescharacteristic
273C2#2256031311yescharacteristic
274C2#2256032011yescharacteristic
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2349C2
socleC22×C67#381
prime radicalC22×D4×C67D4
cosocleC2×C42×C67C23
hypercenter#2349D67
hypocenterC67#8764
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22558112C2semidirect
4. #22558067C2semidirect
143. C67#8764semidirect

Maximal subgroups

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

subgroupindex
#225602012
#225585302
#225581122
#225580672
C67×#23492
#876467