Finite Group Databasebeta

← #32998559

Subgroups — #32998559

Order 18688.

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

normal subgroups only

This group has 372 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#22615173C211yesmaximal, characteristic, complemented
2#22608968C212yesmaximal
3#22607957C212yesmaximal, complemented
4#22607877C211yesmaximal, characteristic, complemented
5C73×#2039C211yesmaximal, characteristic
6#17988903—22no—
7#17988526—21no—
8#17988526C2212yes—
9#17988486—22no—
10#17988452—22no—
11C4×C4⋊Dic73—21no—
12#17988321—22no—
13#17988313C2212yes—
14#537×C73C2211yescharacteristic
15C2×#104×C73—21no—
16#417×C73—21no—
17#384×C73C2211yescharacteristic
18#343×C73C2211yescharacteristic
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73D412yes—
23C2×C4×Dic73—22no—
24#14556522—42no—
25#14556512—42no—
26C22⋊#10596831—22no—
27#14556498—21no—
28#14556498—22no—
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×C73—22no—
38C2×C4⋊C4×C73D411yescharacteristic
39C2×C4⋊C4×C73C2311yescharacteristic
40C2×C42×C73D411yescharacteristic
41#104×C73—42no—
42#103×C73—41no—
43#102×C73—41no—
44C22×Dic73—22no—
45C22×Dic73—22no—
46C2×#10596831—42no—
47C4⋊Dic73—42no—
48C4⋊Dic73—42no—
49C4⋊Dic73—42no—
50C4⋊Dic73—81no—
51C4×Dic73—44no—
52#41×C73—41no—
53#41×C73—41no—
54#41×C73—41no—
55#41×C73—41no—
56C2×Q8×C73—21no—
57C2×D4×C73—21no—
58C2×D4×C73—22no—
59C2×D4×C73—41no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—41no—
63C22×C4×C73C2×D412yes—
64C22×C4×C73C2×D411yescharacteristic
65M16×C73—41no—
66M16×C73—41no—
67C2×C8×C73—41no—
68C4⋊C4×C73—22no—
69C4⋊C4×C73—41no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—44no—
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—22no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—22no—
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—
104C8×C73—81no—
105#10596831—82no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73—81no—
109C22×C73—81no—
110C22×C73—81no—
111C22×C73#44811yescharacteristic
112C22×C73#45411yescharacteristic
113C22×C73B(4,𝔽2)11yescharacteristic
114C4×C73—24no—
115C4×C73—21no—
116C4×C73—21no—
117C4×C73—42no—
118C4×C73—81no—
119C4×C73—81no—
120Dic73—82no—
121Dic73—82no—
122#12181—731nomaximal, Sylow
123C2×C73—21no—
124C2×C73—21no—
125C2×C73—161no—
126C2×C73#159911yescharacteristic
127C2×C73#219811yescharacteristic
128C2×C73#220411yescharacteristic
129#2571—731no—
130#2039D7311yescharacteristic
131#1616—732no—
132#1353—732no—
133#1325—731no—
134C73#1218111yescharacteristic, complemented, Sylow
135#537C2×D7311yescharacteristic
136#481—1462no—
137C2×#104—21no—
138#417—21no—
139#400—732no—
140#400—1461no—
141#389—1462no—
142#384C2×D7311yescharacteristic
143#382—1462no—
144C4×C4⋊C4—1461no—
145#343C2×D7311yescharacteristic
146#327—1462no—
147#324—732no—
148C2×#41—21no—
149C2×M16—21no—
150C22⋊Q8—41no—
151#121—22no—
152#121—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—732no—
155C2×C4⋊C4—1461no—
156C2×C4⋊C4—1461no—
157C2×C4⋊C4—1461no—
158C2×C4⋊C4C22⋊D7311yescharacteristic
159C2×C4⋊C4C22×D7311yescharacteristic
160C2×C42—1462no—
161C2×C42C22⋊D7311yescharacteristic
162#106—2922no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—2922no—
167C22⋊C8—1462no—
168#95—732no—
169#95—1461no—
170#95—1462no—
171#95—1462no—
172#95—1462no—
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—1462no—
185C22×C4—1462no—
186C22×C4D4×D7312yes—
187C22×C4C2×C22⋊D7311yescharacteristic
188M16—41no—
189M16—41no—
190C2×C8—41no—
191C2×C8—2922no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—44no—
196C4⋊C4—2922no—
197C4⋊C4—2922no—
198C4⋊C4—2922no—
199C4⋊C4—5841no—
200C22⋊C4—42no—
201C42—21no—
202C42—2924no—
203C42#1192401612yes—
204C23—41no—
205C23#1455655911yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—2922no—
226C2×C4—2922no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—2922no—
230C2×C4—2922no—
231C2×C4C2×#1192401611yescharacteristic
232C2×C4#1455662111yescharacteristic
233C8—81no—
234C8—5842no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#1798872511yescharacteristic
241C22#1798875911yescharacteristic
242C22#1798877311yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4—5842no—
250C4—5842no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2#2260887011yescharacteristic
255C2#2261192611yescharacteristic
256C2#2261195611yescharacteristic
subgroupG/H
centerC22#17988773
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2039C2
socleC22×C73B(4,𝔽2)
prime radical#537×C73C22
hypercenter#2039D73
hypocenterC73#12181
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #22615173C2semidirect
3. #22607957C2semidirect
4. #22607877C2semidirect
134. C73#12181semidirect

Maximal subgroups

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

subgroupindex
#226151732
#226089682
#226079572
#226078772
C73×#20392
#1218173