Finite Group Databasebeta

← #32989073

Subgroups — #32989073

Order 18688.

454 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 97 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#22618309C211yesmaximal, characteristic, complemented
2#22610759C211yesmaximal, characteristic
3#22608492C212yesmaximal
4#22607790C212yesmaximal, complemented
5C73×#1276C211yesmaximal, characteristic, complemented
6#17989137C2211yescharacteristic
7C2×#14556591—21no—
8#17988611C2211yescharacteristic
9C2×#14556519—21no—
10C2×#14556519—21no—
11#17988493—24no—
12#17988453—21no—
13#17988453C2212yes—
14#17988322—24no—
15#431×C73C2212yes—
16C43×C73C2211yescharacteristic
17C2×#11924028—21no—
18C22×Dic146—22no—
19C2×#11924012—21no—
20#14556591—42no—
21#14556591—42no—
22#14556591Q811yescharacteristic, complemented
23#14556588Q811yescharacteristic, complemented
24#14556579—21no—
25#14556565—22no—
26C4×D292—22no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—22no—
29C4⋊#10596831—22no—
30#14556519—42no—
31#14556519—42no—
32#14556498—24no—
33#14556498—22no—
34#14556498D414yes—
35C2×C42×C73—22no—
36C2×C42×C73D411yescharacteristic
37C2×C42×C73D411yescharacteristic
38C2×C42×C73C2311yescharacteristic
39C4⋊C8×C73—22no—
40C22⋊C8×C73—22no—
41C4×C8×C73—22no—
42#11924028—41no—
43#11924028—41no—
44#11924028—41no—
45#11924028—41no—
46C2×Dic146—21no—
47C2×Dic146—44no—
48C2×Dic146—48no—
49C2×D292—21no—
50C2×D292—22no—
51C2×C22⋊D73—41no—
52C22×Dic73—24no—
53C2×C4×D73—41no—
54#11924012—41no—
55#11924012—41no—
56C2×#10596831—41no—
57C4⋊Dic73—22no—
58C4⋊Dic73—44no—
59C4⋊Dic73—42no—
60C4⋊Dic73—42no—
61#11924005—42no—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73C2×D412yes—
65C22×C4×C73C2×D411yescharacteristic
66C2×C8×C73—42no—
67C42×C73—21no—
68C42×C73—21no—
69C42×C73—44no—
70C42×C73D812yes—
71C42×C73SD1612yes—
72C42×C73C2×Q811yescharacteristic
73C42×C73#4111yescharacteristic
74C42×C73#4112yes—
75C23×C73#12011yescharacteristic
76C22×D73—41no—
77C2×C4×C73—24no—
78C2×C4×C73—22no—
79C2×C4×C73—22no—
80C2×C4×C73—42no—
81C2×C4×C73—42no—
82C2×C4×C73C22⋊Q812yes—
83C2×C4×C73C22⋊Q811yescharacteristic
84C2×C4×C73#12412yes—
85C2×C4×C73#12411yescharacteristic
86C2×C4×C73C2×D811yescharacteristic
87C2×C4×C73C2×SD1611yescharacteristic
88C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C73#13711yescharacteristic
90C22⋊D73—81no—
91C22⋊D73—81no—
92C2×Dic73—48no—
93C2×Dic73—41no—
94C2×Dic73—44no—
95D292—41no—
96D292—41no—
97D292—82no—
98C4×D73—81no—
99C4×D73—81no—
100Dic146—41no—
101Dic146—41no—
102Dic146—88no—
103C8×C73—82no—
104#10596831—81no—
105C22×C73—22no—
106C22×C73#39311yescharacteristic
107C22×C73#44711yescharacteristic
108C22×C73#44811yescharacteristic
109C2×D73—81no—
110C2×D73—82no—
111C4×C73—24no—
112C4×C73—22no—
113C4×C73—22no—
114C4×C73—42no—
115C4×C73#47411yescharacteristic
116C4×C73#47611yescharacteristic
117C4×C73#48611yescharacteristic
118C4×C73#48911yescharacteristic
119Dic73—84no—
120Dic73—81no—
121#10362—731nomaximal, Sylow
122C2×C73—21no—
123C2×C73—21no—
124C2×C73#202311yescharacteristic
125C2×C73#203112yes—
126D73—161no—
127#2862—731no—
128#1962—731no—
129#1517—732no—
130#1286—732no—
131#1276D7311yescharacteristic, complemented
132C73#1036211yescharacteristic, complemented, Sylow
133#532—731no—
134C2×C4⋊Q8—1461no—
135#431C2×D7312yes—
136#423—731no—
137C2×#104—1461no—
138C2×#104—1461no—
139#391—1464no—
140#382—732no—
141#382—1461no—
142C43C2×D7311yescharacteristic
143#327—1464no—
144C2×#41—1461no—
145C22×Q8—1462no—
146C2×M16—1461no—
147C4⋊Q8—2922no—
148C4⋊Q8—2922no—
149C4⋊Q8—731no—
150#127—731no—
151#124—1461no—
152#121—1462no—
153C4×D4—1462no—
154C2×C4⋊C4—1462no—
155C2×C4⋊C4—1462no—
156C2×C42—22no—
157C2×C42D29211yescharacteristic
158C2×C42D29211yescharacteristic
159C2×C42C22×D7311yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—1462no—
162#104—2922no—
163#104—2922no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—734no—
167#95—1464no—
168#95—1462no—
169#41—2921no—
170#41—2921no—
171#41—2921no—
172#41—2921no—
173C2×Q8—2924no—
174C2×Q8—2928no—
175C2×Q8—1461no—
176C2×D4—2921no—
177C2×D4—1462no—
178C2×D4—1461no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—2921no—
182C22×C4—1464no—
183C22×C4D4×D7312yes—
184C22×C4C2×D29211yescharacteristic
185M16—2921no—
186M16—2921no—
187C2×C8—42no—
188C2×C8—2921no—
189C4⋊C4—2924no—
190C4⋊C4—2922no—
191C4⋊C4—2922no—
192C4⋊C4—1462no—
193C22⋊C4—2922no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42D58412yes—
198C42#1192401712yes—
199C42Q8×D7311yescharacteristic
200C42#1192402711yescharacteristic
201C42#1192402812yes—
202C23—2921no—
203C23#1455656111yescharacteristic
204Q8—2921no—
205Q8—2921no—
206Q8—5848no—
207D4—2921no—
208D4—2921no—
209D4—5842no—
210D4—5841no—
211D4—5841no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—2928no—
218C2×C4—2924no—
219C2×C4—2921no—
220C2×C4—5841no—
221C2×C4—5841no—
222C2×C4#1455656912yes—
223C2×C4#1455657111yescharacteristic
224C2×C4#1455657712yes—
225C2×C4#1455657911yescharacteristic
226C2×C4C2×D58411yescharacteristic
227C2×C4C2×#1192401711yescharacteristic
228C2×C4#1455661911yescharacteristic
229C2×C4#1455662411yescharacteristic
230C8—82no—
231C8—5841no—
232C22—22no—
233C22—5841no—
234C22—5842no—
235C22#1798850011yescharacteristic
236C22#1798872311yescharacteristic
237C22#1798872811yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—5844no—
243C4—5841no—
244C4#1798886811yescharacteristic
245C4#1798887811yescharacteristic
246C4#1798893411yescharacteristic
247C4#1798894711yescharacteristic
248C2—21no—
249C2—21no—
250C2—11681no—
251C2#2261106811yescharacteristic
252C2#2261110412yes—
subgroupG/H
centerC22#17988500
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1276C2
socleC22×C73#393
prime radical#17989137C22
hypercenter#1276D73
hypocenterC73#10362
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #22618309C2semidirect
4. #22607790C2semidirect
5. C73×#1276C2semidirect
22. #14556591Q8semidirect
23. #14556588Q8semidirect
131. #1276D73semidirect
132. C73#10362semidirect

Maximal subgroups

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

subgroupindex
#226183092
#226107592
#226084922
#226077902
C73×#12762
#1036273