Finite Group Databasebeta

← C2×#22558155 #30764985

Subgroups — C2×#22558155 #30764985

Order 17152.

598 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16863122C212yesmaximal
2#22558155C214yesmaximal, direct product factor
3C2×#384×C67C211yesmaximal, characteristic
4C2×C22⋊(C67⋊C8)—22no—
5#16863122—22no—
6#16863122C2214yes—
7C22×C4⋊C4×C67—22no—
8C22×C4⋊C4×C67C411yescharacteristic
9C22×C4⋊C4×C67C2211yescharacteristic
10C22×C42×C67C411yescharacteristic
11#384×C67—22no—
12#384×C67C412yes—
13#384×C67C2212yes—
14C22×C67⋊C8—42no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—42no—
17C23×C4×C67—21no—
18C23×C4×C67—21no—
19C23×C4×C67C2×C411yescharacteristic
20C23×C4×C67D412yes—
21C2×C4⋊C4×C67—24no—
22C2×C4⋊C4×C67—21no—
23C2×C4⋊C4×C67—24no—
24C2×C4⋊C4×C67—41no—
25C2×C4⋊C4×C67—41no—
26C2×C4⋊C4×C67—41no—
27C2×C4⋊C4×C67—41no—
28C2×C4⋊C4×C67—48no—
29C2×C4⋊C4×C67C2×C412yes—
30C2×C4⋊C4×C67C2×C411yescharacteristic
31C2×C4⋊C4×C67D412yes—
32C2×C4⋊C4×C67C2311yescharacteristic
33C2×C42×C67—21no—
34C2×C42×C67—22no—
35C2×C42×C67C814yescomplemented
36C2×C42×C67C2×C412yes—
37C2×C67⋊C8—44no—
38C2×C67⋊C8—84no—
39C24×C67C22⋊C411yescharacteristic
40C22×C4×C67—24no—
41C22×C4×C67—24no—
42C22×C4×C67—24no—
43C22×C4×C67—21no—
44C22×C4×C67—21no—
45C22×C4×C67—22no—
46C22×C4×C67—22no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67—22no—
50C22×C4×C67—44no—
51C22×C4×C67—44no—
52C22×C4×C67—41no—
53C22×C4×C67—41no—
54C22×C4×C67C22⋊C411yescharacteristic
55C22×C4×C67C22⋊C412yes—
56C22×C4×C67C2×C812yes—
57C22×C4×C67M1612yes—
58C22×C4×C67C22×C411yescharacteristic
59C22×C4×C67C2×D412yes—
60C4⋊C4×C67—41no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—42no—
64C4⋊C4×C67—48no—
65C4⋊C4×C67—48no—
66C4⋊C4×C67—42no—
67C42×C67—24no—
68C42×C67—22no—
69C42×C67C2×C814yescomplemented
70C23×C67—22no—
71C23×C67—22no—
72C23×C67C22⋊C812yes—
73C23×C67#9911yescharacteristic
74C23×C67#9911yescharacteristic
75C23×C67#10112yes—
76C23×C67C2×C22⋊C411yescharacteristic
77C2×C4×C67—24no—
78C2×C4×C67—216no—
79C2×C4×C67—24no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—24no—
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—44no—
92C2×C4×C67—42no—
93C2×C4×C67—42no—
94C2×C4×C67—44no—
95C2×C4×C67—42no—
96C2×C4×C67—82no—
97C2×C4×C67—82no—
98C2×C4×C67—82no—
99C2×C4×C67C22⋊C812yes—
100C2×C4×C67C22×C811yescharacteristic
101C2×C4×C67C2×M1611yescharacteristic
102C67⋊C8—84no—
103C22×C67—24no—
104C22×C67—24no—
105C22×C67—21no—
106C22×C67—21no—
107C22×C67—21no—
108C22×C67—21no—
109C22×C67#32412yes—
110C22×C67#32412yes—
111C22×C67#35312yes—
112C22×C67#35612yes—
113C22×C67C2×C22⋊C811yescharacteristic
114C22×C67C2×#9911yescharacteristic
115C22×C67C2×#10111yescharacteristic
116C4×C67—28no—
117C4×C67—22no—
118C4×C67—21no—
119C4×C67—21no—
120C4×C67—42no—
121C4×C67—42no—
122C4×C67—82no—
123C2×#1325—671nomaximal, Sylow
124C2×C67—21no—
125C2×C67—22no—
126C2×C67—21no—
127C2×C67#132514yescomplemented
128C2×C67C2×#32411yescharacteristic
129C2×C67C2×#35311yescharacteristic
130C2×C67C2×#35611yescharacteristic
131C2×#384D6711yescharacteristic
132C2×#324—672no—
133#1325—674no—
134C67C2×#132511yescharacteristic, complemented, Sylow
135C22×C4⋊C4—22no—
136C22×C4⋊C4Dic6711yescharacteristic
137C22×C4⋊C4C2×D6711yescharacteristic
138C22×C42Dic6711yescharacteristic
139C2×C22⋊C8—1342no—
140#384—22no—
141#384Dic6712yes—
142#384C2×D6712yes—
143#324—674no—
144#324—1342no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×Dic6711yescharacteristic
148C23×C4C22⋊D6712yes—
149C22×C8—2682no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C2×Dic6712yes—
159C2×C4⋊C4C2×Dic6711yescharacteristic
160C2×C4⋊C4C22⋊D6712yes—
161C2×C4⋊C4C22×D6711yescharacteristic
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C67⋊C814yescomplemented
165C2×C42C2×Dic6712yes—
166C22⋊C8—2682no—
167C22⋊C8—1344no—
168C24C22⋊Dic6711yescharacteristic
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4C22⋊Dic6711yescharacteristic
184C22×C4C22⋊Dic6712yes—
185C22×C4C2×C67⋊C812yes—
186C22×C4#1176161212yes—
187C22×C4C22×Dic6711yescharacteristic
188C22×C4C2×C22⋊D6712yes—
189C2×C8—2684no—
190C2×C8—5364no—
191C4⋊C4—41no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—48no—
196C4⋊C4—48no—
197C4⋊C4—42no—
198C42—24no—
199C42—22no—
200C42C2×C67⋊C814yescomplemented
201C23—22no—
202C23—22no—
203C23C22⋊(C67⋊C8)12yes—
204C23#1452395811yescharacteristic
205C23#1452395811yescharacteristic
206C23#1452396212yes—
207C23C2×C22⋊Dic6711yescharacteristic
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—216no—
211C2×C4—24no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4C22⋊(C67⋊C8)12yes—
231C2×C4C22×C67⋊C811yescharacteristic
232C2×C4C2×#1176161211yescharacteristic
233C8—5364no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—24no—
239C22—24no—
240C22#1686312212yes—
241C22#1686312212yes—
242C22#1686318912yes—
243C22#1686319512yes—
244C22C2×C22⋊(C67⋊C8)11yescharacteristic
245C22C2×#1452395811yescharacteristic
246C22C2×#1452396211yescharacteristic
247C4—28no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C2—21no—
255C2—21no—
256C2—22no—
257C2#2255815514yesdirect product factor
258C2C2×#1686312211yescharacteristic
259C2C2×#1686318911yescharacteristic
260C2C2×#1686319511yescharacteristic
subgroupG/H
centerC23#14523958
commutatorC2×C4×C67C22×C8
FrattiniC2×C4⋊C4C22×D67
FittingC2×#384×C67C2
socleC23×C67#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#384D67
hypocenterC67C2×#1325
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. #22558155C2direct (N × H)
35. C2×C42×C67C8semidirect
69. C42×C67C2×C8semidirect
127. C2×C67#1325semidirect
134. C67C2×#1325semidirect
164. C2×C42C67⋊C8semidirect
200. C42C2×C67⋊C8semidirect
257. C2#22558155direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168631222
#225581552
C2×#384×C672
C2×#132567