Finite Group Databasebeta

← C2×#22624874 #34009856

Subgroups — C2×#22624874 #34009856

Order 18688.

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×#17989432C212yesmaximal
2#22624874C214yesmaximal, direct product factor
3C2×#17988456C211yesmaximal, characteristic
4C2×C22⋊#10596833—22no—
5#17989432—22no—
6#17989432C2214yes—
7C22×C4⋊Dic73C2211yescharacteristic
8C22×#11924007—22no—
9C22×C4×Dic73C411yescharacteristic
10#17988456—22no—
11#17988456C412yes—
12#17988456C2212yes—
13C22×C4⋊C4×C73C411yescharacteristic
14C22×#10596833—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—41no—
20C2×C4⋊Dic73—41no—
21C2×C4⋊Dic73C2×C411yescharacteristic
22C2×C4⋊Dic73D412yes—
23C2×C4⋊Dic73C2311yescharacteristic
24C2×#11924007—24no—
25C2×#11924007—24no—
26C2×#11924007—48no—
27C2×C4×Dic73—21no—
28C2×C4×Dic73—22no—
29C2×C4×Dic73C814yescomplemented
30C2×C4×Dic73C2×C412yes—
31C23×C4×C73—21no—
32C23×C4×C73C2×C411yescharacteristic
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73—41no—
35C2×C4⋊C4×C73—41no—
36C2×C4⋊C4×C73C2×C412yes—
37C2×#10596833—44no—
38C2×#10596833—84no—
39C22×Dic73—24no—
40C22×Dic73—24no—
41C22×Dic73—24no—
42C22×Dic73—22no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45C22×Dic73—44no—
46C22×Dic73C22⋊C412yes—
47C22×Dic73C2×D412yes—
48C4⋊Dic73—41no—
49C4⋊Dic73—42no—
50C4⋊Dic73—41no—
51#11924007—48no—
52#11924007—48no—
53C4×Dic73—24no—
54C4×Dic73—22no—
55C4×Dic73C2×C814yescomplemented
56C24×C73C22⋊C411yescharacteristic
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73—44no—
62C22×C4×C73—41no—
63C22×C4×C73—41no—
64C22×C4×C73C22⋊C411yescharacteristic
65C22×C4×C73C2×C812yes—
66C22×C4×C73M1612yes—
67C22×C4×C73C22×C411yescharacteristic
68C4⋊C4×C73—42no—
69C4⋊C4×C73—42no—
70C23×C73—22no—
71C23×C73—22no—
72C23×C73C22⋊C812yes—
73C23×C73#9911yescharacteristic
74C23×C73#9911yescharacteristic
75C23×C73#10112yes—
76C23×C73C2×C22⋊C411yescharacteristic
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73—22no—
80C2×C4×C73—22no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—44no—
86C2×C4×C73—42no—
87C2×C4×C73—82no—
88C2×C4×C73—82no—
89C2×C4×C73C22⋊C812yes—
90C2×C4×C73C22×C811yescharacteristic
91C2×C4×C73C2×M1611yescharacteristic
92C2×Dic73—24no—
93C2×Dic73—216no—
94C2×Dic73—24no—
95C2×Dic73—24no—
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—44no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73—82no—
102#10596833—84no—
103C22×C73—24no—
104C22×C73—24no—
105C22×C73—21no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73—21no—
109C22×C73#32412yes—
110C22×C73#32412yes—
111C22×C73#35312yes—
112C22×C73#35612yes—
113C22×C73C2×C22⋊C811yescharacteristic
114C22×C73C2×#9911yescharacteristic
115C22×C73C2×#10111yescharacteristic
116C4×C73—22no—
117C4×C73—21no—
118C4×C73—21no—
119C4×C73—82no—
120Dic73—28no—
121Dic73—42no—
122Dic73—42no—
123C2×#1325—731nomaximal, Sylow
124C2×C73—21no—
125C2×C73—22no—
126C2×C73—21no—
127C2×C73#132514yescomplemented
128C2×C73C2×#32411yescharacteristic
129C2×C73C2×#35311yescharacteristic
130C2×C73C2×#35611yescharacteristic
131C2×#384—731no—
132C2×#324—732no—
133#1325—734no—
134C73C2×#132511yescharacteristic, complemented, Sylow
135C22×C4⋊C4—731no—
136C22×C4⋊C4—1462no—
137C22×C4⋊C4#6440811yescharacteristic
138C22×C42—731no—
139C2×C22⋊C8—1462no—
140#384—732no—
141#384—732no—
142#384—1462no—
143#324—734no—
144#324—1462no—
145C23×C4—21no—
146C23×C4—732no—
147C23×C4—1461no—
148C23×C4C2×#6440811yescharacteristic
149C22×C8—2922no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—2921no—
154C2×C4⋊C4—2921no—
155C2×C4⋊C4—2928no—
156C2×C4⋊C4—732no—
157C2×C4⋊C4—731no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—1464no—
160C2×C4⋊C4—1464no—
161C2×C4⋊C4C2×#6440812yes—
162C2×C42—734no—
163C2×C42—732no—
164C2×C42—1461no—
165C2×C42—1462no—
166C22⋊C8—2922no—
167C22⋊C8—1464no—
168C24C22⋊#6440811yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—44no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—2924no—
177C22×C4—732no—
178C22×C4—732no—
179C22×C4—1462no—
180C22×C4—1462no—
181C22×C4—1464no—
182C22×C4—1464no—
183C22×C4—1464no—
184C22×C4—1462no—
185C22×C4C22⋊#6440811yescharacteristic
186C22×C4#1192403612yes—
187C22×C4#1192403812yes—
188C22×C4C22×#6440811yescharacteristic
189C2×C8—2924no—
190C2×C8—5844no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—2922no—
194C4⋊C4—2921no—
195C4⋊C4—2921no—
196C4⋊C4—2928no—
197C4⋊C4—2928no—
198C42—734no—
199C42—1462no—
200C42—1464no—
201C23—22no—
202C23—22no—
203C23#1455664812yes—
204C23#1455664911yescharacteristic
205C23#1455664911yescharacteristic
206C23#1455665212yes—
207C23C2×C22⋊#6440811yescharacteristic
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—2924no—
221C2×C4—2922no—
222C2×C4—2922no—
223C2×C4—5842no—
224C2×C4—1462no—
225C2×C4—1462no—
226C2×C4—1464no—
227C2×C4—14616no—
228C2×C4—1464no—
229C2×C4—1464no—
230C2×C4#1455664812yes—
231C2×C4C2×#1192403611yescharacteristic
232C2×C4C2×#1192403811yescharacteristic
233C8—5844no—
234C22—24no—
235C22—24no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#1798943312yes—
241C22#1798943312yes—
242C22#1798949112yes—
243C22#1798949712yes—
244C22C2×#1455664811yescharacteristic
245C22C2×#1455664911yescharacteristic
246C22C2×#1455665211yescharacteristic
247C4—21no—
248C4—21no—
249C4—22no—
250C4—82no—
251C4—2922no—
252C4—2922no—
253C4—1468no—
254C2—21no—
255C2—22no—
256C2—21no—
257C2#2262487414yesdirect product factor
258C2C2×#1798943311yescharacteristic
259C2C2×#1798949111yescharacteristic
260C2C2×#1798949711yescharacteristic
subgroupG/H
centerC23#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73C2×#1325
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624874C2direct (N × H)
29. C2×C4×Dic73C8semidirect
55. C4×Dic73C2×C8semidirect
127. C2×C73#1325semidirect
134. C73C2×#1325semidirect
257. C2#22624874direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179894322
#226248742
C2×#179884562
C2×#132573