Finite Group Databasebeta

← #30765009

Subgroups — #30765009

Order 17152.

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

normal subgroups only

This group has 562 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
1C2×#16863120C212yesmaximal
2#22558157C214yesmaximal, complemented
3C67×#2315C211yesmaximal, characteristic
4C2×C22⋊(C67⋊C8)—22no—
5#16863120—22no—
6#16863120C2214yes—
7C22×C22⋊C4×C67C411yescharacteristic
8C22×C22⋊C4×C67C411yescharacteristic
9C22×C22⋊C4×C67C2211yescharacteristic
10#385×C67—22no—
11#385×C67C412yes—
12#385×C67C2212yes—
13#380×C67—22no—
14C22×C67⋊C8—42no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—42no—
17C25×C67C2×C411yescharacteristic
18C23×C4×C67—21no—
19C23×C4×C67C2×C412yes—
20C23×C4×C67D412yes—
21C2×C4⋊C4×C67—22no—
22C2×C4⋊C4×C67—22no—
23C2×C22⋊C4×C67—21no—
24C2×C22⋊C4×C67—22no—
25C2×C22⋊C4×C67—21no—
26C2×C22⋊C4×C67—21no—
27C2×C22⋊C4×C67—42no—
28C2×C22⋊C4×C67—42no—
29C2×C22⋊C4×C67C814yescomplemented
30C2×C22⋊C4×C67C2×C412yes—
31C2×C22⋊C4×C67C2×C411yescharacteristic
32C2×C22⋊C4×C67D412yes—
33C2×C22⋊C4×C67C2311yescharacteristic
34#95×C67—22no—
35#95×C67—22no—
36C2×C67⋊C8—44no—
37C2×C67⋊C8—84no—
38C24×C67—22no—
39C24×C67—22no—
40C24×C67—82no—
41C24×C67C22⋊C411yescharacteristic
42C24×C67C22⋊C411yescharacteristic
43C24×C67C2×C812yes—
44C24×C67M1612yes—
45C24×C67C22×C411yescharacteristic
46C22×C4×C67—22no—
47C22×C4×C67—22no—
48C22×C4×C67—24no—
49C22×C4×C67—22no—
50C22×C4×C67—22no—
51C22×C4×C67—42no—
52C22×C4×C67—82no—
53C22×C4×C67—82no—
54C22×C4×C67—82no—
55C22×C4×C67C22⋊C412yes—
56C22×C4×C67C2×D412yes—
57C4⋊C4×C67—82no—
58C4⋊C4×C67—82no—
59C22⋊C4×C67—24no—
60C22⋊C4×C67—22no—
61C22⋊C4×C67—41no—
62C22⋊C4×C67—42no—
63C22⋊C4×C67—41no—
64C22⋊C4×C67—42no—
65C22⋊C4×C67—42no—
66C22⋊C4×C67C2×C814yescomplemented
67C23×C67—21no—
68C23×C67—21no—
69C23×C67—21no—
70C23×C67—21no—
71C23×C67—24no—
72C23×C67—24no—
73C23×C67—42no—
74C23×C67—42no—
75C23×C67—42no—
76C23×C67—82no—
77C23×C67—82no—
78C23×C67—82no—
79C23×C67—82no—
80C23×C67—84no—
81C23×C67C22⋊C812yes—
82C23×C67C22⋊C812yes—
83C23×C67#9912yes—
84C23×C67#10112yes—
85C23×C67C2×C22⋊C411yescharacteristic
86C23×C67C22×C811yescharacteristic
87C23×C67C2×M1611yescharacteristic
88C2×C4×C67—24no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—42no—
92C2×C4×C67—42no—
93C2×C4×C67—44no—
94C2×C4×C67—42no—
95C2×C4×C67—42no—
96C2×C4×C67—42no—
97C2×C4×C67—42no—
98C2×C4×C67—42no—
99C2×C4×C67—82no—
100C2×C4×C67—84no—
101C2×C4×C67—84no—
102C67⋊C8—84no—
103C22×C67—22no—
104C22×C67—22no—
105C22×C67—22no—
106C22×C67—22no—
107C22×C67—21no—
108C22×C67—21no—
109C22×C67—42no—
110C22×C67—42no—
111C22×C67—42no—
112C22×C67—41no—
113C22×C67—41no—
114C22×C67—42no—
115C22×C67—42no—
116C22×C67—42no—
117C22×C67—42no—
118C22×C67—84no—
119C22×C67—84no—
120C22×C67#32414yes—
121C22×C67C2×C22⋊C811yescharacteristic
122C22×C67C2×#9911yescharacteristic
123C22×C67C2×#10111yescharacteristic
124C4×C67—44no—
125C4×C67—42no—
126C4×C67—42no—
127C4×C67—82no—
128#11453—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67—24no—
132C2×C67—42no—
133C2×C67—41no—
134C2×C67—41no—
135C2×C67C2×#32411yescharacteristic
136C2×C67#212112yes—
137#2315D6711yescharacteristic
138C2×#323—672no—
139#1326—674no—
140C67#1145311yescharacteristic, complemented, Sylow
141C22×C22⋊C4Dic6711yescharacteristic
142C22×C22⋊C4Dic6711yescharacteristic
143C22×C22⋊C4C2×D6711yescharacteristic
144C2×C22⋊C8—1342no—
145#385—22no—
146#385Dic6712yes—
147#385C2×D6712yes—
148#380—22no—
149#323—674no—
150#323—1342no—
151C25C2×Dic6711yescharacteristic
152C23×C4—21no—
153C23×C4C2×Dic6712yes—
154C23×C4C22⋊D6712yes—
155C22×C8—2682no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4C67⋊C814yescomplemented
165C2×C22⋊C4C2×Dic6712yes—
166C2×C22⋊C4C2×Dic6711yescharacteristic
167C2×C22⋊C4C22⋊D6712yes—
168C2×C22⋊C4C22×D6711yescharacteristic
169C22⋊C8—2682no—
170C22⋊C8—1344no—
171#95—22no—
172#95—22no—
173C24—22no—
174C24—22no—
175C24—82no—
176C24C22⋊Dic6711yescharacteristic
177C24C22⋊Dic6711yescharacteristic
178C24C2×C67⋊C812yes—
179C24#1176161212yes—
180C24C22×Dic6711yescharacteristic
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C22⋊Dic6712yes—
191C22×C4C2×C22⋊D6712yes—
192C2×C8—2684no—
193C2×C8—5364no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4C2×C67⋊C814yescomplemented
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—84no—
218C23C22⋊(C67⋊C8)12yes—
219C23C22⋊(C67⋊C8)12yes—
220C23#1452395812yes—
221C23#1452396212yes—
222C23C2×C22⋊Dic6711yescharacteristic
223C23C22×C67⋊C811yescharacteristic
224C23C2×#1176161211yescharacteristic
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—84no—
239C8—5364no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—21no—
245C22—21no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22#1686312214yes—
258C22C2×C22⋊(C67⋊C8)11yescharacteristic
259C22C2×#1452395811yescharacteristic
260C22C2×#1452396211yescharacteristic
261C4—44no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—42no—
269C2—41no—
270C2—41no—
271C2C2×#1686312211yescharacteristic
272C2#2256185512yes—
subgroupG/H
centerC22C2×#14523958
commutatorC23×C67C22×C8
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2315C2
socleC22×C67C2×#99
prime radicalC25×C67C2×C4
cosocleC2×C22⋊C4×C67C23
hypercenter#2315D67
hypocenterC67#11453
LCSG ▷ C23×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #22558157C2semidirect
29. C2×C22⋊C4×C67C8semidirect
66. C22⋊C4×C67C2×C8semidirect
140. C67#11453semidirect
164. C2×C22⋊C4C67⋊C8semidirect
203. C22⋊C4C2×C67⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#168631202
#225581572
C67×#23152
#1145367