Finite Group Databasebeta

← #32997505

Subgroups — #32997505

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#22620591C211yesmaximal, characteristic
2#22611746C211yesmaximal, characteristic
3#22611695C211yesmaximal, characteristic
4#22607992C211yesmaximal, characteristic, complemented
5#22607969C211yesmaximal, characteristic, complemented
6#22607969C211yesmaximal, characteristic, complemented
7C73×#1366C211yesmaximal, characteristic, complemented
8#17989200C412yes—
9M32×D73—21no—
10#17988967C2211yescharacteristic
11#17988957C412yes—
12#17988846—21no—
13#17988829—21no—
14#17988693—21no—
15#17988681—21no—
16#17988541C2211yescharacteristic
17#17988408C2211yescharacteristic
18#17988408C2211yescharacteristic
19#17988407—21no—
20#17988368—21no—
21#17988364—21no—
22#499×C73C2211yescharacteristic
23#364×C73C2211yescharacteristic
24#364×C73C2211yescharacteristic
25C2×Dic292—21no—
26C2×Dic292—21no—
27C2×#11924011D411yescharacteristic
28C2×C8×D73D411yescharacteristic
29#14556591C2×C412yes—
30#14556586—42no—
31#14556582—42no—
32#14556571—22no—
33#14556556—22no—
34#14556548C2×C411yescharacteristic
35#14556531—21no—
36#14556531—21no—
37#14556530—41no—
38C16×D73—41no—
39#14556514—22no—
40#14556514—22no—
41#14556502C2×C411yescharacteristic
42#14556501D411yescharacteristic
43C8×Dic73D411yescharacteristic
44#125×C73C2×C412yes—
45M32×C73—21no—
46M32×C73—21no—
47#107×C73—21no—
48#106×C73—21no—
49C4×C8×C73C2311yescharacteristic
50C2×Dic146—22no—
51C2×C4×D73C22⋊C411yescharacteristic
52Dic292—41no—
53Dic292—41no—
54Dic292—41no—
55Dic292—41no—
56#11924011—21no—
57#11924011—21no—
58C2×#10596831C22⋊C411yescharacteristic
59C2×#10596831C22⋊C411yescharacteristic
60C8×D73—21no—
61C8×D73—21no—
62C4⋊Dic73—42no—
63C4⋊Dic73—42no—
64#11924007—21no—
65#11924007—42no—
66#11924005—21no—
67#11924005—42no—
68C4×Dic73—42no—
69C4×Dic73C22⋊C411yescharacteristic
70#11924003—41no—
71#11924003—41no—
72C2×C8×C73C2×D411yescharacteristic
73C2×C8×C73C2×D411yescharacteristic
74C4⋊C4×C73—22no—
75C4⋊C4×C73—42no—
76C42×C73C22×C411yescharacteristic
77C16×C73—41no—
78C16×C73—41no—
79C22×D73#10011yescharacteristic
80C2×C4×C73—21no—
81C2×C4×C73—42no—
82C2×C4×C73C2×C22⋊C411yescharacteristic
83C2×Dic73—21no—
84C2×Dic73—42no—
85C2×Dic73#10011yescharacteristic
86C4×D73—21no—
87C4×D73—21no—
88Dic146—42no—
89Dic146—42no—
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#1366D7311yescharacteristic, complemented
115#1358—731no—
116#1358—731no—
117C73#1199011yescharacteristic, complemented, Sylow
118#541—732no—
119C2×M32—1461no—
120#499C2×D7311yescharacteristic
121#494—731no—
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—1461no—
136C2×Q16—1461no—
137C2×M16—731no—
138C22×C8—731no—
139C4⋊Q8—732no—
140#126—2922no—
141#125—2922no—
142#125C4×D7312yes—
143C22⋊Q8—1462no—
144C4×Q8—1462no—
145#117—731no—
146M32—21no—
147M32—21no—
148M32—2921no—
149M32—1461no—
150M32—1461no—
151C2×C16—2921no—
152#107—21no—
153#106—21no—
154#103—1462no—
155#103—1462no—
156#97—731no—
157#97—731no—
158C4×C8—731no—
159C4×C8C22×D7311yescharacteristic
160C2×Q8—1462no—
161C22×C4—731no—
162Q16—2921no—
163Q16—2921no—
164Q16—2921no—
165Q16—2921no—
166M16—1461no—
167M16—1461no—
168C2×C8—731no—
169C2×C8—731no—
170C2×C8—1461no—
171C2×C8—1461no—
172C2×C8D4×D7311yescharacteristic
173C2×C8D4×D7311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—2922no—
177C4⋊C4—2922no—
178C4⋊C4—2922no—
179C4⋊C4—1461no—
180C22⋊C4—2922no—
181C22⋊C4—1461no—
182C42—2922no—
183C42—731no—
184C42C2×C4×D7311yescharacteristic
185C16—41no—
186C16—41no—
187C16—2921no—
188C16—2921no—
189C23—731no—
190Q8—2922no—
191Q8—2922no—
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#1455661211yescharacteristic
203C8#1455661411yescharacteristic
204C8#1455661811yescharacteristic
205C8#1455662311yescharacteristic
206C22—2921no—
207C22—1461no—
208C22#100×D7311yescharacteristic
209C4—41no—
210C4—82no—
211C4—2921no—
212C4—5842no—
213C4—1461no—
214C4#1798858211yescharacteristic
215C4#1798858711yescharacteristic
216C2—21no—
217C2—2921no—
218C2#2260862511yescharacteristic
subgroupG/H
centerC2#22608625
commutatorC42×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#1366C2
socleC2×C73#1551
prime radicalC22×D73#100
cosocleC4×C8×C73C23
hypercenter#1366D73
hypocenterC73#11990
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
4. #22607992C2semidirect
5. #22607969C2semidirect
6. #22607969C2semidirect
7. C73×#1366C2semidirect
114. #1366D73semidirect
117. C73#11990semidirect

Maximal subgroups

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

subgroupindex
#226205912
#226117462
#226116952
#226079922
#226079692
C73×#13662
#1199073