Finite Group Databasebeta

← C2×#14543727 #17958196

Subgroups — C2×#14543727 #17958196

Order 4352.

598 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 120 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×#11763654C212yesmaximal
2#14543727C214yesmaximal, direct product factor
3C2×#11762678C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5#11763654—22no—
6#11763654C2214yes—
7C22×C4⋊Dic17C2211yescharacteristic
8C22×#63225—22no—
9C22×C4×Dic17C411yescharacteristic
10#11762678—22no—
11#11762678C412yes—
12#11762678C2212yes—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17—21no—
15C23×C4×C17C2×C411yescharacteristic
16C23×Dic17—21no—
17C23×Dic17D412yes—
18C22×#3624—42no—
19C2×C4⋊C4×C17—21no—
20C2×C4⋊C4×C17—41no—
21C2×C4⋊C4×C17—41no—
22C2×C4⋊C4×C17C2×C412yes—
23C2×C4⋊Dic17—41no—
24C2×C4⋊Dic17—41no—
25C2×C4⋊Dic17C2×C411yescharacteristic
26C2×C4⋊Dic17D412yes—
27C2×C4⋊Dic17C2311yescharacteristic
28C2×#63225—24no—
29C2×#63225—24no—
30C2×#63225—48no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17—22no—
33C2×C4×Dic17C814yescomplemented
34C2×C4×Dic17C2×C412yes—
35C22⋊#3624—24no—
36C22⋊#3624—42no—
37C24×C17C22⋊C411yescharacteristic
38C22×C4×C17—21no—
39C22×C4×C17—21no—
40C22×C4×C17—22no—
41C22×C4×C17—22no—
42C22×C4×C17—44no—
43C22×C4×C17—41no—
44C22×C4×C17—41no—
45C22×C4×C17C22⋊C411yescharacteristic
46C22×C4×C17C2×C812yes—
47C22×C4×C17M1612yes—
48C22×C4×C17C22×C411yescharacteristic
49C22×Dic17—24no—
50C22×Dic17—24no—
51C22×Dic17—24no—
52C22×Dic17—22no—
53C22×Dic17—22no—
54C22×Dic17—22no—
55C22×Dic17—44no—
56C22×Dic17C22⋊C412yes—
57C22×Dic17C2×D412yes—
58C2×#3624—44no—
59C2×#3624—84no—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—42no—
62C4⋊Dic17—41no—
63C4⋊Dic17—42no—
64C4⋊Dic17—41no—
65#63225—48no—
66#63225—48no—
67C4×Dic17—24no—
68C4×Dic17—22no—
69C4×Dic17C2×C814yescomplemented
70C2×#1325—171nomaximal, Sylow
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C22⋊C812yes—
74C23×C17#9911yescharacteristic
75C23×C17#9911yescharacteristic
76C23×C17#10112yes—
77C23×C17C2×C22⋊C411yescharacteristic
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—44no—
87C2×C4×C17—42no—
88C2×C4×C17—82no—
89C2×C4×C17—82no—
90C2×C4×C17C22⋊C812yes—
91C2×C4×C17C22×C811yescharacteristic
92C2×C4×C17C2×M1611yescharacteristic
93C2×Dic17—216no—
94C2×Dic17—24no—
95C2×Dic17—24no—
96C2×Dic17—24no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—44no—
100C2×Dic17—42no—
101C2×Dic17—42no—
102C2×Dic17—82no—
103#3624—84no—
104C2×#384—171no—
105C2×#324—172no—
106#1325—174no—
107C22×C17—24no—
108C22×C17—24no—
109C22×C17—21no—
110C22×C17—21no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17#32412yes—
114C22×C17#32412yes—
115C22×C17#35312yes—
116C22×C17#35612yes—
117C22×C17C2×C22⋊C811yescharacteristic
118C22×C17C2×#9911yescharacteristic
119C22×C17C2×#10111yescharacteristic
120C4×C17—22no—
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—82no—
124Dic17—28no—
125Dic17—42no—
126Dic17—42no—
127C22×C4⋊C4—171no—
128C22×C4⋊C4—342no—
129C22×C4⋊C4#59511yescharacteristic
130C22×C42—171no—
131C2×C22⋊C8—342no—
132#384—172no—
133#384—172no—
134#384—342no—
135#324—174no—
136#324—342no—
137C2×C17—21no—
138C2×C17—22no—
139C2×C17—21no—
140C2×C17#132514yescomplemented
141C2×C17C2×#32411yescharacteristic
142C2×C17C2×#35311yescharacteristic
143C2×C17C2×#35611yescharacteristic
144C23×C4—21no—
145C23×C4—172no—
146C23×C4—341no—
147C23×C4C2×#59511yescharacteristic
148C22×C8—682no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—171no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4—344no—
156C2×C4⋊C4—344no—
157C2×C4⋊C4—681no—
158C2×C4⋊C4—681no—
159C2×C4⋊C4—688no—
160C2×C4⋊C4C2×#59512yes—
161C2×C42—174no—
162C2×C42—172no—
163C2×C42—342no—
164C2×C42—341no—
165C22⋊C8—344no—
166C22⋊C8—682no—
167C17C2×#132511yescharacteristic, complemented, Sylow
168C24C22⋊#59511yescharacteristic
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—44no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—172no—
177C22×C4—172no—
178C22×C4—344no—
179C22×C4—344no—
180C22×C4—342no—
181C22×C4—344no—
182C22×C4—342no—
183C22×C4—342no—
184C22×C4—684no—
185C22×C4#6324212yes—
186C22×C4#6324312yes—
187C22×C4C22⋊#59511yescharacteristic
188C22×C4C22×#59511yescharacteristic
189C2×C8—684no—
190C2×C8—1364no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—682no—
194C4⋊C4—681no—
195C4⋊C4—681no—
196C4⋊C4—688no—
197C4⋊C4—688no—
198C42—174no—
199C42—342no—
200C42—344no—
201C23—22no—
202C23—22no—
203C23#1058754312yes—
204C23#1058754512yes—
205C23#1058755111yescharacteristic
206C23#1058755111yescharacteristic
207C23C2×C22⋊#59511yescharacteristic
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—344no—
221C2×C4—3416no—
222C2×C4—344no—
223C2×C4—344no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—684no—
227C2×C4—682no—
228C2×C4—682no—
229C2×C4—1362no—
230C2×C4#1058754312yes—
231C2×C4C2×#6324211yescharacteristic
232C2×C4C2×#6324311yescharacteristic
233C8—1364no—
234C22—24no—
235C22—24no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#1176365512yes—
241C22#1176365512yes—
242C22#1176371312yes—
243C22#1176371912yes—
244C22C2×#1058754311yescharacteristic
245C22C2×#1058755111yescharacteristic
246C22C2×#1058754511yescharacteristic
247C4—21no—
248C4—21no—
249C4—22no—
250C4—82no—
251C4—348no—
252C4—682no—
253C4—682no—
254C2—21no—
255C2—21no—
256C2—22no—
257C2#1454372714yesdirect product factor
258C2C2×#1176365511yescharacteristic
259C2C2×#1176371311yescharacteristic
260C2C2×#1176371911yescharacteristic
subgroupG/H
centerC23#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17C2×#1325
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543727C2direct (N × H)
33. C2×C4×Dic17C8semidirect
69. C4×Dic17C2×C8semidirect
140. C2×C17#1325semidirect
167. C17C2×#1325semidirect
257. C2#14543727direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117636542
#145437272
C2×#117626782
C2×#132517