Finite Group Databasebeta

← #32973718

Subgroups — #32973718

Order 18688.

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

normal subgroups only

This group has 262 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
1C2×#17988386C211yesmaximal, characteristic
2#22610325C211yesmaximal, characteristic
3C73×#1558C211yesmaximal, characteristic
4#17989065C411yescharacteristic
5#17988596—21no—
6C2×#14556510—21no—
7C2×#14556510C411yescharacteristic
8#17988386—21no—
9#17988386C412yescomplemented
10#17988347—21no—
11C2×C4⋊Q8×C73C2211yescharacteristic
12#432×C73—21no—
13#333×C73—21no—
14#333×C73C412yescomplemented
15C2×#11924012—21no—
16C2×#11924012—41no—
17Q8×Dic73—42no—
18#14556550—21no—
19#14556550—41no—
20#14556515—42no—
21#14556510—22no—
22#14556510—41no—
23#14556510—41no—
24C22×Q8×C73C2×C411yescharacteristic
25C22×Q8×C73D411yescharacteristic
26C4⋊Q8×C73—41no—
27C4⋊Q8×C73C2×C411yescharacteristic
28C4⋊Q8×C73C2×C411yescharacteristic
29C4⋊Q8×C73D411yescharacteristic
30C4⋊Q8×C73Q811yescharacteristic
31C2×C4⋊C4×C73—41no—
32C2×C42×C73D411yescharacteristic
33C4⋊C8×C73—22no—
34C22⋊C8×C73—41no—
35#97×C73—41no—
36#11924012—42no—
37#11924012—41no—
38#11924012—41no—
39#11924012—81no—
40C2×#10596831—41no—
41C2×#10596831—81no—
42C4⋊Dic73—22no—
43C4⋊Dic73—42no—
44C22⋊Dic73—41no—
45C4×Dic73—41no—
46C4×Dic73—42no—
47C2×Q8×C73—21no—
48C2×Q8×C73—21no—
49C2×Q8×C73—41no—
50C2×Q8×C73—41no—
51C2×Q8×C73—41no—
52C2×Q8×C73—41no—
53C2×Q8×C73C4211yescharacteristic
54C2×Q8×C73C22⋊C411yescharacteristic
55C2×Q8×C73C4⋊C411yescharacteristic
56C2×Q8×C73C4⋊C411yescharacteristic
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73C22⋊C411yescharacteristic
61C2×C8×C73—42no—
62C4⋊C4×C73—41no—
63C4⋊C4×C73—41no—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—41no—
66C42×C73—21no—
67C42×C73C22⋊C411yescharacteristic
68C42×C73C4⋊C411yescharacteristic
69C23×C73#9911yescharacteristic
70Q8×C73—41no—
71Q8×C73—41no—
72Q8×C73—41no—
73Q8×C73—41no—
74Q8×C73—42no—
75Q8×C73—81no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—42no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73#9511yescharacteristic
90C2×C4×C73#9911yescharacteristic
91C2×Dic73—42no—
92C8×C73—82no—
93#10596831—81no—
94#10596831—82no—
95C22×C73—41no—
96C22×C73#34211yescharacteristic
97C22×C73AΣL(1,𝔽16)11yescharacteristic
98C22×C73#35211yescharacteristic
99C4×C73—22no—
100C4×C73—21no—
101C4×C73—21no—
102C4×C73—41no—
103C4×C73—41no—
104C4×C73—41no—
105C4×C73—41no—
106C4×C73—41no—
107Dic73—82no—
108#7483—731nomaximal, Sylow
109C2×C73—21no—
110C2×C73—21no—
111C2×C73—21no—
112C2×C73#139011yescharacteristic
113C2×#356—731no—
114#1886—731no—
115#1558D7311yescharacteristic
116C73#748311yescharacteristic, complemented, Sylow
117C2×C4⋊Q8C2×D7311yescharacteristic
118#519—731no—
119#432—21no—
120#419—1461no—
121C2×#101—731no—
122C2×#101—1461no—
123#356—732no—
124#356—1461no—
125#339—1461no—
126#333—21no—
127#333Dic7312yescomplemented
128C22×Q8C4×D7311yescharacteristic
129C22×Q8D29211yescharacteristic
130C2×M16—2921no—
131C2×M16—1461no—
132C4⋊Q8—41no—
133C4⋊Q8Dic14611yescharacteristic
134C4⋊Q8C4×D7311yescharacteristic
135C4⋊Q8C2×Dic7311yescharacteristic
136C4⋊Q8C22⋊D7311yescharacteristic
137C4×Q8—2922no—
138#117—2921no—
139#117—1461no—
140C2×C4⋊C4—41no—
141C2×C42C22⋊D7311yescharacteristic
142C4⋊C8—22no—
143#103—2922no—
144#101—2921no—
145#101—2921no—
146#101—1462no—
147C22⋊C8—41no—
148#97—41no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8C4×Dic7311yescharacteristic
156C2×Q8#1192400511yescharacteristic
157C2×Q8#1192400711yescharacteristic
158C2×Q8C4⋊Dic7311yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4#1192400511yescharacteristic
163M16—2922no—
164M16—2921no—
165M16—2921no—
166M16—5841no—
167C2×C8—42no—
168C2×C8—2921no—
169C2×C8—5841no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—2922no—
175C4⋊C4—1462no—
176C22⋊C4—2921no—
177C42—21no—
178C42—2921no—
179C42—2922no—
180C42C22⋊Dic7311yescharacteristic
181C42#1192400711yescharacteristic
182C23#1455650511yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—41no—
187Q8—42no—
188Q8—81no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2922no—
203C2×C4#1455649811yescharacteristic
204C2×C4#1455650511yescharacteristic
205C8—82no—
206C8—5841no—
207C8—5842no—
208C22—41no—
209C22#1798835311yescharacteristic
210C22#1798837511yescharacteristic
211C22#1798837811yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—5842no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#2260805011yescharacteristic
subgroupG/H
centerC2#22608050
commutatorC2×Q8×C73C42
FrattiniC2×C4⋊Q8C2×D73
FittingC73×#1558C2
socleC2×C73#1390
prime radicalC23×C73#99
cosocleC2×C4⋊Q8×C73C22
hypercenter#1558D73
hypocenterC73#7483
LCSG ▷ C2×Q8×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1558

Complements

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

NH ≅ G/Ntype
9. #17988386C4semidirect
14. #333×C73C4semidirect
116. C73#7483semidirect
127. #333Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179883862
#226103252
C73×#15582
#748373