Finite Group Databasebeta

← #31894990

Subgroups — #31894990

Order 18176.

534 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 103 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#22591517C211yesmaximal, characteristic
2#22587054C211yesmaximal, characteristic
3#22586891C212yesmaximal, complemented
4#22586205C212yesmaximal, complemented
5C71×#3138C211yesmaximal, characteristic
6#17986056C2212yes—
7#17985879—24no—
8#17985879—22no—
9#17985867C2212yes—
10#17985743—22no—
11#17985736—24no—
12#533×C71C2211yescharacteristic
13C2×#127×C71C2211yescharacteristic
14#490×C71—22no—
15#486×C71C414yescomplemented
16#418×C71—22no—
17C4×M16×C71C2211yescharacteristic
18C23×Dic71—22no—
19C2×C22⋊Dic71—24no—
20C2×C22⋊Dic71—22no—
21C4⋊(C71⋊C8)—22no—
22C22⋊(C71⋊C8)—22no—
23#14556039—22no—
24C71⋊#95—22no—
25C71⋊#95—22no—
26C71⋊#95D414yes—
27C22×D4×C71—22no—
28C22×D4×C71D412yes—
29C2×M16×C71—21no—
30C4⋊Q8×C71C2×C412yescomplemented
31#127×C71—22no—
32#127×C71C2×C412yes—
33#127×C71D412yes—
34#125×C71—21no—
35C22⋊Q8×C71—22no—
36C4×Q8×C71—22no—
37#117×C71—21no—
38C2×C42×C71C2311yescharacteristic
39#102×C71—28no—
40#97×C71—21no—
41C4×C8×C71C2×C412yescomplemented
42C22×Dic71—24no—
43C22×Dic71—84no—
44C22×Dic71—82no—
45C2×C71⋊C8—42no—
46C22⋊Dic71—84no—
47C22⋊Dic71—82no—
48C24×C71—22no—
49C2×Q8×C71—22no—
50C2×D4×C71—24no—
51C2×D4×C71—24no—
52C2×D4×C71—22no—
53C2×D4×C71—42no—
54C2×D4×C71—42no—
55C2×D4×C71—82no—
56C2×D4×C71—82no—
57C2×D4×C71C22⋊C414yes—
58C22×C4×C71C2×D412yes—
59C22×C4×C71C2×D411yescharacteristic
60M16×C71—42no—
61C2×C8×C71—22no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—42no—
65C22⋊C4×C71—41no—
66C42×C71—41no—
67C42×C71C22×C411yescharacteristic
68C42×C71C2×D411yescharacteristic
69C42×C71#4112yes—
70C23×C71—24no—
71C23×C71—44no—
72C23×C71—44no—
73C23×C71—42no—
74C23×C71—42no—
75C23×C71#12011yescharacteristic
76Q8×C71—82no—
77D4×C71—44no—
78D4×C71—82no—
79D4×C71—84no—
80D4×C71—84no—
81C2×C4×C71—22no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71C2×C22⋊C411yescharacteristic
85C2×C4×C71C4×D412yes—
86C2×C4×C71#12112yes—
87C2×C4×C71#12311yescharacteristic
88C2×Dic71—48no—
89C2×Dic71—44no—
90C2×Dic71—88no—
91C8×C71—42no—
92C71⋊C8—82no—
93C22×C71—22no—
94C22×C71—44no—
95C22×C71—48no—
96C22×C71—82no—
97C22×C71—82no—
98C22×C71—84no—
99C22×C71#38611yescharacteristic
100C22×C71#45512yes—
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—22no—
104C4×C71—22no—
105C4×C71—82no—
106Dic71—84no—
107#9425—711nomaximal, Sylow
108C2×C71—22no—
109C2×C71—84no—
110C2×C71#188511yescharacteristic
111C2×C71#190412yes—
112#3138D7111yescharacteristic
113#2380—711no—
114#1555—711no—
115#1515—712no—
116#1288—712no—
117C71#942511yescharacteristic, complemented, Sylow
118#533C2×D7111yescharacteristic
119C2×#127C2×D7111yescharacteristic
120#490—22no—
121#486Dic7114yescomplemented
122#432—712no—
123#418—22no—
124C4×M16C2×D7111yescharacteristic
125#386—1424no—
126#386—1422no—
127#383—712no—
128#331—1422no—
129#328—1424no—
130C22×D4—22no—
131C22×D4C22⋊D7112yes—
132C23×C4—1422no—
133C2×M16—21no—
134C4⋊Q8C2×Dic7112yescomplemented
135#127—22no—
136#127C2×Dic7112yes—
137#127C22⋊D7112yes—
138#125—21no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117—21no—
142C2×C22⋊C4—1424no—
143C2×C22⋊C4—1422no—
144C2×C42C22×D7111yescharacteristic
145C4⋊C8—1422no—
146#102—28no—
147C22⋊C8—1422no—
148#97—21no—
149#97—1422no—
150C4×C8C2×Dic7112yescomplemented
151#95—714no—
152#95—1422no—
153#95—1422no—
154C24—22no—
155C2×Q8—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4C22⋊Dic7114yes—
164C22×C4—5684no—
165C22×C4—5682no—
166C22×C4—1424no—
167C22×C4D4×D7112yes—
168C22×C4C2×C22⋊D7111yescharacteristic
169M16—42no—
170C2×C8—22no—
171C2×C8—2842no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—5684no—
177C22⋊C4—5682no—
178C42—41no—
179C42C22×Dic7111yescharacteristic
180C42C2×C22⋊D7111yescharacteristic
181C42#1176586412yes—
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1455609611yescharacteristic
188Q8—82no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2848no—
197C2×C4—2844no—
198C2×C4—5688no—
199C2×C4C2×C22⋊Dic7111yescharacteristic
200C2×C4D4×Dic7112yes—
201C2×C4#1455610012yes—
202C2×C4#1455610911yescharacteristic
203C8—42no—
204C8—5682no—
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1798587911yescharacteristic
212C22#1798617612yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—82no—
218C4—5684no—
219C2—22no—
220C2—84no—
221C2#2258872911yescharacteristic
222C2#2258882912yes—
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3138C2
socleC22×C71#386
prime radicalC2×#127×C71C22
cosocleC2×C42×C71C23
hypercenter#3138D71
hypocenterC71#9425
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
3. #22586891C2semidirect
4. #22586205C2semidirect
15. #486×C71C4semidirect
30. C4⋊Q8×C71C2×C4semidirect
41. C4×C8×C71C2×C4semidirect
117. C71#9425semidirect
121. #486Dic71semidirect
134. C4⋊Q8C2×Dic71semidirect
150. C4×C8C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915172
#225870542
#225868912
#225862052
C71×#31382
#942571