Finite Group Databasebeta

← #32998583

Subgroups — #32998583

Order 18688.

360 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 59 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#22618203C211yesmaximal, characteristic, complemented
2#22608968C212yesmaximal
3#22607957C212yesmaximal, complemented
4#22607875C211yesmaximal, characteristic, complemented
5C73×#2040C211yesmaximal, characteristic
6#17988903—22no—
7#17988529—22no—
8#17988526C2212yes—
9#17988508—22no—
10#17988507—21no—
11#17988453—21no—
12#17988321—22no—
13#17988313C2212yes—
14#537×C73C2211yescharacteristic
15C2×#104×C73—21no—
16#417×C73—21no—
17#382×C73C2211yescharacteristic
18#343×C73C2211yescharacteristic
19C22×Dic146—21no—
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73D412yes—
23C2×#11924007—22no—
24#14556522—42no—
25#14556512—42no—
26C22⋊#10596831—22no—
27#14556498—22no—
28#14556498—21no—
29#14556498—22no—
30#14556498—22no—
31#14556498D412yes—
32C2×#41×C73—21no—
33C2×M16×C73—21no—
34C22⋊Q8×C73—41no—
35#121×C73—22no—
36#121×C73—41no—
37C2×C4⋊C4×C73D411yescharacteristic
38C2×C4⋊C4×C73C2311yescharacteristic
39C2×C42×C73D411yescharacteristic
40#104×C73—42no—
41#103×C73—41no—
42#102×C73—41no—
43#95×C73—22no—
44C2×Dic146—82no—
45C2×Dic146—81no—
46C22×Dic73—22no—
47C22×Dic73—22no—
48C2×#10596831—42no—
49C4⋊Dic73—42no—
50C4⋊Dic73—81no—
51C4⋊Dic73—81no—
52#11924007—82no—
53#41×C73—41no—
54#41×C73—41no—
55#41×C73—41no—
56#41×C73—41no—
57C2×Q8×C73—21no—
58C2×D4×C73—21no—
59C2×D4×C73—22no—
60C2×D4×C73—41no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—41no—
64C22×C4×C73C2×D412yes—
65C22×C4×C73C2×D411yescharacteristic
66M16×C73—41no—
67M16×C73—41no—
68C2×C8×C73—41no—
69C4⋊C4×C73—22no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C22⋊C4×C73—42no—
73C42×C73—21no—
74C42×C73SD1612yes—
75C23×C73—41no—
76C23×C73#12011yescharacteristic
77Q8×C73—41no—
78Q8×C73—41no—
79D4×C73—41no—
80D4×C73—41no—
81D4×C73—82no—
82D4×C73—81no—
83D4×C73—81no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—22no—
87C2×C4×C73—22no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—42no—
91C2×C4×C73—42no—
92C2×C4×C73—41no—
93C2×C4×C73—41no—
94C2×C4×C73—81no—
95C2×C4×C73—81no—
96C2×C4×C73C2×SD1611yescharacteristic
97C2×C4×C73#13711yescharacteristic
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73—42no—
102C2×Dic73—42no—
103C2×Dic73—42no—
104Dic146—84no—
105C8×C73—81no—
106#10596831—82no—
107C22×C73—21no—
108C22×C73—21no—
109C22×C73—81no—
110C22×C73—81no—
111C22×C73—81no—
112C22×C73#44811yescharacteristic
113C22×C73#45411yescharacteristic
114C22×C73B(4,𝔽2)11yescharacteristic
115C4×C73—21no—
116C4×C73—21no—
117C4×C73—42no—
118C4×C73—42no—
119C4×C73—81no—
120C4×C73—81no—
121Dic73—82no—
122Dic73—82no—
123#12185—731nomaximal, Sylow
124C2×C73—21no—
125C2×C73—21no—
126C2×C73—161no—
127C2×C73#159911yescharacteristic
128C2×C73#220111yescharacteristic
129C2×C73#220211yescharacteristic
130#2844—731no—
131#2040D7311yescharacteristic
132#1616—732no—
133#1353—732no—
134#1324—731no—
135C73#1218511yescharacteristic, complemented, Sylow
136#537C2×D7311yescharacteristic
137#481—1462no—
138C2×#104—21no—
139#417—21no—
140#400—732no—
141#400—1462no—
142#395—1462no—
143#395—1461no—
144#382—1461no—
145#382C2×D7311yescharacteristic
146#343C2×D7311yescharacteristic
147#327—1462no—
148#324—732no—
149C2×#41—21no—
150C22×Q8—1461no—
151C2×M16—21no—
152C22⋊Q8—41no—
153#121—22no—
154#121—41no—
155C2×C4⋊C4—732no—
156C2×C4⋊C4—1461no—
157C2×C4⋊C4—1461no—
158C2×C4⋊C4—1462no—
159C2×C4⋊C4C22⋊D7311yescharacteristic
160C2×C4⋊C4C22×D7311yescharacteristic
161C2×C42C22⋊D7311yescharacteristic
162#106—2922no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—2922no—
167C22⋊C8—1462no—
168#95—22no—
169#95—732no—
170#95—1462no—
171#95—1461no—
172#95—1462no—
173#95—1462no—
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—5842no—
180C2×Q8—5841no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—1462no—
188C22×C4—1462no—
189C22×C4D4×D7312yes—
190C22×C4C2×C22⋊D7311yescharacteristic
191M16—41no—
192M16—41no—
193C2×C8—41no—
194C2×C8—2922no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—2922no—
199C4⋊C4—5842no—
200C4⋊C4—5841no—
201C4⋊C4—5841no—
202C22⋊C4—42no—
203C42—21no—
204C42#1192401612yes—
205C23—41no—
206C23#1455655911yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—5844no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—2922no—
230C2×C4—2922no—
231C2×C4—2922no—
232C2×C4—2922no—
233C2×C4C2×#1192401611yescharacteristic
234C2×C4#1455662111yescharacteristic
235C8—81no—
236C8—5842no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#1798872511yescharacteristic
243C22#1798875911yescharacteristic
244C22#1798877311yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—5842no—
252C4—5842no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#2260887011yescharacteristic
257C2#2261194311yescharacteristic
258C2#2261194711yescharacteristic
subgroupG/H
centerC22#17988773
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2040C2
socleC22×C73B(4,𝔽2)
prime radical#537×C73C22
hypercenter#2040D73
hypocenterC73#12185
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #22618203C2semidirect
3. #22607957C2semidirect
4. #22607875C2semidirect
135. C73#12185semidirect

Maximal subgroups

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

subgroupindex
#226182032
#226089682
#226079572
#226078752
C73×#20402
#1218573