Finite Group Databasebeta

← #34009983

Subgroups — #34009983

Order 18688.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#22625168C212yesmaximal
2#22624874C212yesmaximal, complemented
3#22624861C212yesmaximal, complemented
4#22612638C211yesmaximal, characteristic
5#17989555—22no—
6#17989432C2214yes—
7#17989431—22no—
8C2×#14556580C411yescharacteristic
9C2×C4×D292C2211yescharacteristic
10#17988494C2211yescharacteristic
11#17988488C411yescharacteristic
12#17988478—22no—
13#17988458—22no—
14#17988456C2211yescharacteristic
15#17988433C411yescharacteristic
16C2×C4×Q8×C73C411yescharacteristic
17C2×#11924036—42no—
18#14556648—42no—
19C22⋊#10596833—24no—
20C22×D292C2×C411yescharacteristic
21C22×C4×D73D412yes—
22#14556580—22no—
23#14556580C814yescomplemented
24C4×D292—42no—
25C2×C4⋊Dic73C2311yescharacteristic
26C2×#11924007—22no—
27C2×#11924005—22no—
28C2×#11924005—22no—
29C2×#11924005D412yes—
30C2×C4×Dic73C2×C411yescharacteristic
31C2×C4×Dic73C2×C411yescharacteristic
32#14556498—22no—
33#14556498—21no—
34#14556498—21no—
35C22×Q8×C73C2×C411yescharacteristic
36C4×Q8×C73—42no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C42×C73—21no—
40C2×C42×C73C2×C411yescharacteristic
41#11924036—84no—
42C2×#10596833—44no—
43C23×D73C22⋊C412yes—
44C2×D292—41no—
45C2×D292—41no—
46C2×D292C2×C812yes—
47C2×D292M1612yes—
48C22×Dic73—21no—
49C22×Dic73—21no—
50C22×Dic73C2×D412yes—
51C2×C4×D73—22no—
52C2×C4×D73—22no—
53C2×C4×D73—82no—
54C4⋊Dic73—41no—
55#11924007—82no—
56#11924005—28no—
57#11924005—42no—
58#11924005—82no—
59C4×Dic73—21no—
60C4×Dic73—41no—
61C4×Dic73C2×C812yescomplemented
62C2×Q8×C73—21no—
63C2×Q8×C73—41no—
64C2×Q8×C73—41no—
65C2×Q8×C73C2×C812yescomplemented
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22×C411yescharacteristic
71C4⋊C4×C73—41no—
72C4⋊C4×C73—42no—
73C42×C73—22no—
74C42×C73—42no—
75C23×C73C2×C22⋊C411yescharacteristic
76C22×D73—22no—
77C22×D73—22no—
78C22×D73—42no—
79C22×D73—42no—
80C22×D73C22⋊C814yes—
81Q8×C73—42no—
82Q8×C73—42no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—22no—
86C2×C4×C73—21no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73—42no—
92C2×C4×C73—41no—
93C2×C4×C73#9912yes—
94C2×C4×C73C22×C811yescharacteristic
95C2×C4×C73C2×M1611yescharacteristic
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—41no—
99C2×Dic73—41no—
100C2×Dic73—42no—
101C2×Dic73—41no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104D292—42no—
105D292—42no—
106C4×D73—44no—
107C4×D73—82no—
108#10596833—84no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73C2×C22⋊C811yescharacteristic
112C22×C73C2×#9911yescharacteristic
113C22×C73#41311yescharacteristic
114C2×D73—24no—
115C2×D73—42no—
116C2×D73—42no—
117C2×D73—44no—
118C2×D73—82no—
119C4×C73—22no—
120C4×C73—21no—
121C4×C73—21no—
122C4×C73—41no—
123C4×C73—42no—
124C4×C73—81no—
125Dic73—42no—
126Dic73—42no—
127Dic73—81no—
128#11481—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73#145911yescharacteristic
132C2×C73#212711yescharacteristic
133C2×C73#213511yescharacteristic
134D73—44no—
135#2326—731no—
136#1459—732no—
137#1325—732no—
138#1319—732no—
139C73#1148111yescharacteristic, complemented, Sylow
140C2×#124—731no—
141C2×C4×Q8#6440811yescharacteristic
142C2×C4×D4—731no—
143C2×C22⋊C8—1462no—
144#391—731no—
145#390—731no—
146#388—1462no—
147#385—1462no—
148#384—731no—
149#376—731no—
150#324—734no—
151#323—1462no—
152C22×Q8C2×#6440811yescharacteristic
153C22×D4—731no—
154C23×C4—732no—
155C22×C8—2922no—
156#124—734no—
157#124—1462no—
158C4×Q8—42no—
159C4×D4—2922no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—1462no—
163C2×C4⋊C4C2×#6440811yescharacteristic
164C2×C22⋊C4—732no—
165C2×C22⋊C4—1462no—
166C2×C22⋊C4—1462no—
167C2×C42—21no—
168C2×C42—731no—
169C2×C42—731no—
170C2×C42C2×#6440811yescharacteristic
171C22⋊C8—2922no—
172C22⋊C8—1464no—
173#95—1462no—
174#95—1461no—
175#95—1461no—
176C24—732no—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1192403612yescomplemented
181C2×D4—2921no—
182C2×D4—2921no—
183C2×D4—732no—
184C2×D4—732no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—5842no—
188C22×C4—732no—
189C22×C4—1461no—
190C22×C4—1461no—
191C22×C4—1462no—
192C22×C4—1462no—
193C22×C4C22⋊#6440811yescharacteristic
194C22×C4C22⋊#6440811yescharacteristic
195C22×C4C22×#6440811yescharacteristic
196C2×C8—2924no—
197C2×C8—5844no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—2921no—
201C4⋊C4—5842no—
202C22⋊C4—2922no—
203C22⋊C4—5842no—
204C22⋊C4—1468no—
205C42—22no—
206C42—42no—
207C42—2921no—
208C42—732no—
209C42—1461no—
210C23—2922no—
211C23—2922no—
212C23—734no—
213C23—1462no—
214C23—1462no—
215C23C2×C22⋊#6440811yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—2922no—
219D4—2922no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—2924no—
231C2×C4—2921no—
232C2×C4—2921no—
233C2×C4—2922no—
234C2×C4—2921no—
235C2×C4—2921no—
236C2×C4—2921no—
237C2×C4—5842no—
238C2×C4—1462no—
239C2×C4—1462no—
240C2×C4#1455664912yes—
241C2×C4C2×#1192403611yescharacteristic
242C2×C4D73⋊M1611yescharacteristic
243C8—5844no—
244C22—21no—
245C22—21no—
246C22—2924no—
247C22—2922no—
248C22—2922no—
249C22—5842no—
250C22—1464no—
251C22#1798955511yescharacteristic
252C22C2×#1455664911yescharacteristic
253C22#1798957311yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—2922no—
261C4—2922no—
262C4—5841no—
263C2—21no—
264C2—21no—
265C2—2924no—
266C2#2262516611yescharacteristic
267C2#2262620611yescharacteristic
268C2#2262623311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×C4×Q8#64408
hypocenterC73#11481
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22624874C2semidirect
3. #22624861C2semidirect
23. #14556580C8semidirect
61. C4×Dic73C2×C8semidirect
65. C2×Q8×C73C2×C8semidirect
139. C73#11481semidirect
180. C2×Q8#11924036semidirect

Maximal subgroups

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

subgroupindex
#226251682
#226248742
#226248612
#226126382
#1148173