Finite Group Databasebeta

← #32997503

Subgroups — #32997503

Order 18688.

268 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 69 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#22620587C211yesmaximal, characteristic
2#22611746C211yesmaximal, characteristic
3#22611695C211yesmaximal, characteristic
4#22607993C211yesmaximal, characteristic, complemented
5#22607991C211yesmaximal, characteristic, complemented
6#22607970C211yesmaximal, characteristic, complemented
7C73×#1358C211yesmaximal, characteristic, complemented
8#17989198C412yes—
9M32×D73—21no—
10#17988989C2211yescharacteristic
11#17988955C412yes—
12#17988842—21no—
13#17988825—21no—
14#17988694—21no—
15Q16×Dic73—21no—
16#17988541C2211yescharacteristic
17#17988408C2211yescharacteristic
18#17988408C2211yescharacteristic
19#17988407—21no—
20#17988368—21no—
21#17988364—21no—
22#494×C73C2211yescharacteristic
23#364×C73C2211yescharacteristic
24#364×C73C2211yescharacteristic
25C2×#11924011D411yescharacteristic
26C2×C8×D73D411yescharacteristic
27#14556586—42no—
28#14556583C2×C412yes—
29#14556581—42no—
30#14556567—22no—
31Q8×Dic73—22no—
32#14556548C2×C411yescharacteristic
33#14556531—21no—
34#14556531—21no—
35#14556530—41no—
36C16×D73—41no—
37#14556525—21no—
38#14556523—21no—
39#14556515—22no—
40#14556515—22no—
41#14556502C2×C411yescharacteristic
42#14556501D411yescharacteristic
43C8×Dic73D411yescharacteristic
44C2×Q16×C73—21no—
45C2×Q16×C73—21no—
46C4⋊Q8×C73C2×C412yes—
47M32×C73—21no—
48M32×C73—21no—
49C4×C8×C73C2311yescharacteristic
50C2×C4×D73C22⋊C411yescharacteristic
51#11924011—21no—
52#11924011—21no—
53C2×#10596831C22⋊C411yescharacteristic
54C2×#10596831C22⋊C411yescharacteristic
55C8×D73—21no—
56C8×D73—21no—
57C4⋊Dic73—22no—
58C4⋊Dic73—42no—
59#11924007—21no—
60#11924007—42no—
61#11924007—42no—
62#11924005—21no—
63#11924005—42no—
64C4×Dic73—42no—
65C4×Dic73C22⋊C411yescharacteristic
66#11924003—41no—
67#11924003—41no—
68C2×Q8×C73—22no—
69Q16×C73—41no—
70Q16×C73—41no—
71Q16×C73—41no—
72Q16×C73—41no—
73C2×C8×C73C2×D411yescharacteristic
74C2×C8×C73C2×D411yescharacteristic
75C4⋊C4×C73—42no—
76C42×C73C22×C411yescharacteristic
77C16×C73—41no—
78C16×C73—41no—
79C22×D73#10011yescharacteristic
80Q8×C73—42no—
81Q8×C73—42no—
82C2×C4×C73—21no—
83C2×C4×C73—42no—
84C2×C4×C73C2×C22⋊C411yescharacteristic
85C2×Dic73—21no—
86C2×Dic73—42no—
87C2×Dic73#10011yescharacteristic
88C4×D73—21no—
89C4×D73—21no—
90C8×C73#13511yescharacteristic
91C8×C73#13511yescharacteristic
92C8×C73AGL(1,ℤ/8)11yescharacteristic
93C8×C73#13711yescharacteristic
94#10596831—21no—
95#10596831—21no—
96C22×C73C2×#10011yescharacteristic
97C2×D73—21no—
98C2×D73—41no—
99C4×C73—41no—
100C4×C73—82no—
101C4×C73#41611yescharacteristic
102C4×C73#41711yescharacteristic
103Dic73—21no—
104Dic73—41no—
105Dic73—82no—
106#11990—731nomaximal, Sylow
107C2×C73—21no—
108C2×C73#155111yescharacteristic
109D73—41no—
110#3177—731no—
111#2166—731no—
112#2153—731no—
113#1366—731no—
114#1366—731no—
115#1358—731no—
116#1358D7311yescharacteristic, complemented
117C73#1199011yescharacteristic, complemented, Sylow
118#541—732no—
119C2×M32—1461no—
120#499—731no—
121#494C2×D7311yescharacteristic
122#491—732no—
123#470—1461no—
124#467—1461no—
125#441—1461no—
126C4×Q16—1461no—
127#405—731no—
128#364—731no—
129#364—731no—
130#364—1461no—
131#364C2×D7311yescharacteristic
132#364C2×D7311yescharacteristic
133#349—1461no—
134#347—1461no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×M16—731no—
138C22×C8—731no—
139C4⋊Q8C4×D7312yes—
140#126—2922no—
141#125—2922no—
142#125—732no—
143C22⋊Q8—1462no—
144C4×Q8—1462no—
145#117—731no—
146M32—21no—
147M32—21no—
148M32—2921no—
149M32—1461no—
150M32—1461no—
151C2×C16—2921no—
152#107—1461no—
153#106—1461no—
154#103—1462no—
155#103—1462no—
156#97—731no—
157#97—731no—
158C4×C8—731no—
159C4×C8C22×D7311yescharacteristic
160C2×Q8—22no—
161C22×C4—731no—
162Q16—41no—
163Q16—41no—
164Q16—41no—
165Q16—41no—
166M16—1461no—
167M16—1461no—
168C2×C8—731no—
169C2×C8—731no—
170C2×C8—1461no—
171C2×C8—1461no—
172C2×C8D4×D7311yescharacteristic
173C2×C8D4×D7311yescharacteristic
174C4⋊C4—42no—
175C4⋊C4—2922no—
176C4⋊C4—2922no—
177C4⋊C4—2922no—
178C4⋊C4—1461no—
179C4⋊C4—1462no—
180C22⋊C4—2922no—
181C22⋊C4—1461no—
182C42—2922no—
183C42—731no—
184C42C2×C4×D7311yescharacteristic
185C16—41no—
186C16—41no—
187C16—2921no—
188C16—2921no—
189C23—731no—
190Q8—42no—
191Q8—42no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—2922no—
195C2×C4—731no—
196C2×C4—1461no—
197C2×C4—1461no—
198C2×C4—1461no—
199C2×C4C22⋊C4×D7311yescharacteristic
200C8—1461no—
201C8—1461no—
202C8#1455661011yescharacteristic
203C8#1455661211yescharacteristic
204C8#1455661511yescharacteristic
205C8#1455662011yescharacteristic
206C22—2921no—
207C22—1461no—
208C22#100×D7311yescharacteristic
209C4—41no—
210C4—82no—
211C4—2921no—
212C4—5842no—
213C4—1461no—
214C4#1798858011yescharacteristic
215C4#1798858511yescharacteristic
216C2—21no—
217C2—2921no—
218C2#2260862311yescharacteristic
subgroupG/H
centerC2#22608623
commutatorC42×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#1358C2
socleC2×C73#1551
prime radicalC22×D73#100
cosocleC4×C8×C73C23
hypercenter#1358D73
hypocenterC73#11990
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
4. #22607993C2semidirect
5. #22607991C2semidirect
6. #22607970C2semidirect
7. C73×#1358C2semidirect
116. #1358D73semidirect
117. C73#11990semidirect

Maximal subgroups

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

subgroupindex
#226205872
#226117462
#226116952
#226079932
#226079912
#226079702
C73×#13582
#1199073