Finite Group Databasebeta

← #32989114

Subgroups — #32989114

Order 18688.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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#22610912C211yesmaximal, characteristic
2#22608429C212yesmaximal
3#22607834C212yesmaximal
4#22607760C211yesmaximal, characteristic
5C73×#2399C211yesmaximal, characteristic
6#17989144C2211yescharacteristic
7#17988640C2212yes—
8#17988611C2211yescharacteristic
9#17988347—21no—
10#17988347—21no—
11#17988323—22no—
12#17988323—22no—
13C2×C4×Q8×C73—21no—
14#398×C73—22no—
15#395×C73—22no—
16C4×C4⋊C4×C73—21no—
17C4×C4⋊C4×C73C2212yes—
18#376×C73C2211yescharacteristic
19C2×#11924012—21no—
20#14556591Q812yescomplemented
21#14556583—21no—
22#14556570—22no—
23C4×Dic146—22no—
24#14556550—21no—
25C4⋊#10596831—22no—
26C4⋊#10596831—22no—
27C22⋊#10596831—22no—
28C4×#10596831—22no—
29C22×Q8×C73—21no—
30C4×Q8×C73—42no—
31C4×Q8×C73—42no—
32C2×C4⋊C4×C73—22no—
33C2×C4⋊C4×C73—22no—
34C2×C4⋊C4×C73—21no—
35C2×C4⋊C4×C73—22no—
36C2×C42×C73—22no—
37C2×C42×C73D411yescharacteristic
38C2×C42×C73D411yescharacteristic
39C2×C42×C73C2311yescharacteristic
40#95×C73—21no—
41#95×C73—21no—
42#95×C73—22no—
43#95×C73D412yes—
44#95×C73D412yes—
45C2×Dic146—22no—
46#11924012—41no—
47#11924012—41no—
48C2×#10596831—42no—
49C2×#10596831—41no—
50C4⋊Dic73—22no—
51C4⋊Dic73—22no—
52#11924007—42no—
53C22⋊Dic73—41no—
54C4×Dic73—41no—
55C2×Q8×C73—42no—
56C2×Q8×C73—44no—
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73C2×D412yes—
62C22×C4×C73C2×D411yescharacteristic
63C4⋊C4×C73—44no—
64C4⋊C4×C73—44no—
65C4⋊C4×C73—44no—
66C4⋊C4×C73—42no—
67C42×C73—21no—
68C42×C73—21no—
69C42×C73—42no—
70C42×C73—42no—
71C42×C73SD1612yes—
72C42×C73Q1612yes—
73C42×C73C2×Q811yescharacteristic
74C42×C73#4112yes—
75C42×C73#4111yescharacteristic
76C23×C73#12011yescharacteristic
77Q8×C73—48no—
78C2×C4×C73—24no—
79C2×C4×C73—22no—
80C2×C4×C73—22no—
81C2×C4×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73—42no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—42no—
89C2×C4×C73C22⋊Q812yes—
90C2×C4×C73C22⋊Q811yescharacteristic
91C2×C4×C73#12412yes—
92C2×C4×C73#12411yescharacteristic
93C2×C4×C73C2×SD1611yescharacteristic
94C2×C4×C73C2×Q1611yescharacteristic
95C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C73#13711yescharacteristic
97C2×Dic73—42no—
98Dic146—82no—
99#10596831—82no—
100#10596831—81no—
101C22×C73—22no—
102C22×C73#39311yescharacteristic
103C22×C73#45011yescharacteristic
104C22×C73#45111yescharacteristic
105C4×C73—22no—
106C4×C73—22no—
107C4×C73—42no—
108C4×C73—44no—
109C4×C73—42no—
110C4×C73—82no—
111C4×C73#47511yescharacteristic
112C4×C73#47711yescharacteristic
113C4×C73#48711yescharacteristic
114C4×C73#48911yescharacteristic
115Dic73—82no—
116#10371—731nomaximal, Sylow
117C2×C73—21no—
118C2×C73—21no—
119C2×C73#202411yescharacteristic
120C2×C73#203112yes—
121#2399D7311yescharacteristic
122#1995—731no—
123#1503—732no—
124#1303—732no—
125#1274—731no—
126C73#1037111yescharacteristic, complemented, Sylow
127#533—731no—
128C2×C4×Q8—21no—
129#431—732no—
130#423—731no—
131#398—22no—
132#395—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×D7312yes—
135#376C2×D7311yescharacteristic
136#339—1461no—
137#339—1461no—
138#328—1462no—
139#328—1462no—
140C22×Q8—21no—
141C2×M16—1461no—
142C4⋊Q8—732no—
143#125—1461no—
144C22⋊Q8—1462no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—1462no—
148#117—1461no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22⋊D7311yescharacteristic
155C2×C42C22⋊D7311yescharacteristic
156C2×C42C22×D7311yescharacteristic
157C4⋊C8—1462no—
158C4⋊C8—1462no—
159C22⋊C8—1462no—
160C4×C8—1462no—
161#95—21no—
162#95—21no—
163#95—22no—
164#95C22⋊D7312yes—
165#95C22⋊D7312yes—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—1462no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4C2×C22⋊D7312yes—
174C22×C4C2×C22⋊D7311yescharacteristic
175M16—2921no—
176M16—2921no—
177C2×C8—2922no—
178C2×C8—2921no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—2922no—
184C4⋊C4—1462no—
185C4⋊C4—1462no—
186C22⋊C4—2921no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42—2921no—
192C42#1192401612yes—
193C42#1192401812yes—
194C42Q8×D7311yescharacteristic
195C42#1192402911yescharacteristic
196C42#1192402912yes—
197C23#1455656011yescharacteristic
198Q8—48no—
199Q8—5842no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—2922no—
212C2×C4#1455656712yes—
213C2×C4#1455656711yescharacteristic
214C2×C4#1455658012yes—
215C2×C4#1455658011yescharacteristic
216C2×C4C2×#1192401611yescharacteristic
217C2×C4C2×#1192401811yescharacteristic
218C2×C4#1455661711yescharacteristic
219C2×C4#1455662211yescharacteristic
220C8—5842no—
221C8—5841no—
222C22—22no—
223C22#1798849811yescharacteristic
224C22#1798873811yescharacteristic
225C22#1798874311yescharacteristic
226C4—22no—
227C4—22no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4—5842no—
233C4#1798886911yescharacteristic
234C4#1798887911yescharacteristic
235C4#1798893911yescharacteristic
236C4#1798894911yescharacteristic
237C2—21no—
238C2—21no—
239C2#2261106911yescharacteristic
240C2#2261110012yes—
subgroupG/H
centerC2×C4#14556580
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2399C2
socleC22×C73#393
prime radicalC23×C73#120
hypercenter#2399D73
hypocenterC73#10371
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
20. #14556591Q8semidirect
126. C73#10371semidirect

Maximal subgroups

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

subgroupindex
#226109122
#226084292
#226078342
#226077602
C73×#23992
#1037173