Finite Group Databasebeta

← #32999314

Subgroups — #32999314

Order 18688.

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

normal subgroups only

This group has 324 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#22612148C212yesmaximal, complemented
2#22611447C211yesmaximal, characteristic
3#22607925C212yesmaximal, complemented
4#22607867C211yesmaximal, characteristic
5C73×#3195C211yesmaximal, characteristic
6#17988636C2211yescharacteristic
7#17988635—21no—
8#17988551—21no—
9#17988409—22no—
10#17988403—22no—
11#17988369C2212yes—
12#17988345C2211yescharacteristic
13C2×C22⋊Q8×C73—21no—
14#483×C73—22no—
15#470×C73—21no—
16#470×C73C2212yes—
17#451×C73—22no—
18#407×C73C2211yescharacteristic
19C2×#11924012—21no—
20C2×#11924012—21no—
21#14556531—22no—
22#14556527—22no—
23#14556526—42no—
24C22⋊#10596831—42no—
25C22×Q8×C73—21no—
26C23×C4×C73D411yescharacteristic
27C2×Q16×C73—21no—
28C2×Q16×C73D412yes—
29C2×M16×C73D411yescharacteristic
30C2×M16×C73C2311yescharacteristic
31C22⋊Q8×C73—22no—
32C22⋊Q8×C73—22no—
33C22⋊Q8×C73—41no—
34C22⋊Q8×C73—41no—
35C2×C4⋊C4×C73—21no—
36C2×C4⋊C4×C73—41no—
37C2×C22⋊C4×C73—41no—
38#106×C73—21no—
39#106×C73D412yes—
40#103×C73—22no—
41#103×C73—22no—
42C22⋊C8×C73—22no—
43#11924012—41no—
44#11924012—41no—
45#11924012—41no—
46#11924012—41no—
47C2×#10596831—22no—
48C2×#10596831—41no—
49#11924003—42no—
50C24×C73SD1611yescharacteristic
51C2×Q8×C73—22no—
52C2×Q8×C73—41no—
53C2×Q8×C73—81no—
54C2×Q8×C73—81no—
55C22×C4×C73—21no—
56C22×C4×C73—21no—
57C22×C4×C73—41no—
58C22×C4×C73—41no—
59C22×C4×C73—41no—
60C22×C4×C73—41no—
61C22×C4×C73SD1611yescharacteristic
62C22×C4×C73C2×D411yescharacteristic
63Q16×C73—42no—
64Q16×C73—81no—
65M16×C73—41no—
66M16×C73—41no—
67C2×C8×C73—21no—
68C2×C8×C73C2×D412yes—
69C4⋊C4×C73—22no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—42no—
72C4⋊C4×C73—42no—
73C22⋊C4×C73—42no—
74C22⋊C4×C73—42no—
75C23×C73—21no—
76C23×C73—41no—
77C23×C73—42no—
78C23×C73C2×SD1611yescharacteristic
79Q8×C73—42no—
80Q8×C73—42no—
81Q8×C73—82no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—42no—
89C2×C4×C73—42no—
90C2×C4×C73—81no—
91C2×C4×C73—82no—
92C2×C4×C73—82no—
93C2×C4×C73#12011yescharacteristic
94C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
95C8×C73—22no—
96C8×C73—41no—
97#10596831—42no—
98#10596831—81no—
99C22×C73—22no—
100C22×C73—21no—
101C22×C73—41no—
102C22×C73—41no—
103C22×C73—42no—
104C22×C73—41no—
105C22×C73—81no—
106C22×C73#45011yescharacteristic
107C4×C73—21no—
108C4×C73—41no—
109C4×C73—82no—
110C4×C73—82no—
111C4×C73#45311yescharacteristic
112C4×C73#45811yescharacteristic
113#12299—731nomaximal, Sylow
114C2×C73—21no—
115C2×C73—42no—
116C2×C73—41no—
117C2×C73#161311yescharacteristic
118#3195D7311yescharacteristic
119#2239—732no—
120#2091—731no—
121#1342—732no—
122#1320—731no—
123C73#1229911yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—21no—
125#483—22no—
126#470—21no—
127#470C2×D7312yes—
128#451—22no—
129#430—731no—
130#430—1461no—
131#407—1461no—
132#407C2×D7311yescharacteristic
133#365—1462no—
134#362—1462no—
135#349—732no—
136#338—731no—
137C22×Q8—21no—
138C23×C4C22⋊D7311yescharacteristic
139C2×Q16—21no—
140C2×Q16C22⋊D7312yes—
141C2×M16—1461no—
142C2×M16—1461no—
143C2×M16C22⋊D7311yescharacteristic
144C2×M16C22×D7311yescharacteristic
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152M32—1462no—
153#108—2922no—
154#108—1462no—
155#106—21no—
156#106C22⋊D7312yes—
157#103—22no—
158#103—22no—
159C22⋊C8—22no—
160C22⋊C8—2922no—
161C24#1192401611yescharacteristic
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—81no—
165C2×Q8—81no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4#1192401611yescharacteristic
173C22×C4C2×C22⋊D7311yescharacteristic
174Q16—42no—
175Q16—81no—
176M16—41no—
177M16—41no—
178M16—2921no—
179M16—2921no—
180M16—2921no—
181M16—2921no—
182C2×C8—21no—
183C2×C8—2921no—
184C2×C8—1462no—
185C2×C8C2×C22⋊D7312yes—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C16—2922no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1192401611yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#1455656011yescharacteristic
212C2×C4#1455661711yescharacteristic
213C8—22no—
214C8—41no—
215C8—2922no—
216C8—5841no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1798873811yescharacteristic
225C4—21no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4#1798875511yescharacteristic
230C4#1798877711yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2260895111yescharacteristic
subgroupG/H
centerC2#22608951
commutator, cosocleC2×M16×C73C23
FrattiniC2×M16C22×D73
FittingC73×#3195C2
socleC2×C73#1613
prime radicalC24×C73SD16
hypercenter#3195D73
hypocenterC73#12299
LCSG ▷ C2×M16×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. #22612148C2semidirect
3. #22607925C2semidirect
123. C73#12299semidirect

Maximal subgroups

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

subgroupindex
#226121482
#226114472
#226079252
#226078672
C73×#31952
#1229973