Finite Group Databasebeta

← #31890908

Subgroups — #31890908

Order 18176.

404 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 115 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#22586654C212yesmaximal
2#22586173C212yesmaximal, complemented
3#22586169C212yesmaximal, complemented
4C71×#2298C211yesmaximal, characteristic
5#17986052C2214yes—
6#17985725—22no—
7#17985723—22no—
8C2×#117×C71C2211yescharacteristic
9#388×C71—22no—
10#387×C71—22no—
11C4×C4⋊C4×C71C412yes—
12C4×C22⋊C4×C71C412yes—
13#376×C71C2211yescharacteristic
14C43×C71C2211yescharacteristic
15C4⋊(C71⋊C8)—24no—
16C22⋊(C71⋊C8)—24no—
17C4×C71⋊C8—24no—
18C23×C4×C71C2×C411yescharacteristic
19#117×C71—22no—
20#117×C71—22no—
21C2×C4⋊C4×C71—22no—
22C2×C4⋊C4×C71D412yes—
23C2×C22⋊C4×C71—22no—
24C2×C22⋊C4×C71D412yes—
25C2×C42×C71—22no—
26C2×C42×C71C2×C412yes—
27C2×C42×C71C2×C412yes—
28C2×C42×C71C2×C411yescharacteristic
29C2×C42×C71C2311yescharacteristic
30#95×C71—22no—
31#95×C71—22no—
32#95×C71—22no—
33C2×C71⋊C8—44no—
34C24×C71C22⋊C411yescharacteristic
35C22×C4×C71—22no—
36C22×C4×C71—21no—
37C22×C4×C71—21no—
38C22×C4×C71—24no—
39C22×C4×C71—22no—
40C22×C4×C71C22⋊C411yescharacteristic
41C22×C4×C71C22⋊C412yes—
42C22×C4×C71C22×C411yescharacteristic
43C22×C4×C71C2×D412yes—
44C4⋊C4×C71—24no—
45C4⋊C4×C71—44no—
46C22⋊C4×C71—24no—
47C22⋊C4×C71—44no—
48C42×C71—21no—
49C42×C71—21no—
50C42×C71—24no—
51C42×C71—24no—
52C42×C71—44no—
53C42×C71#4112yes—
54C42×C71#4112yes—
55C23×C71—21no—
56C23×C71—22no—
57C23×C71—42no—
58C23×C71C2×C22⋊C411yescharacteristic
59C2×C4×C71—24no—
60C2×C4×C71—22no—
61C2×C4×C71—21no—
62C2×C4×C71—22no—
63C2×C4×C71—42no—
64C2×C4×C71—42no—
65C2×C4×C71—42no—
66C2×C4×C71—42no—
67C2×C4×C71—42no—
68C2×C4×C71—42no—
69C2×C4×C71—42no—
70C2×C4×C71—42no—
71C2×C4×C71—42no—
72C2×C4×C71#11711yescharacteristic
73C2×C4×C71#11711yescharacteristic
74C2×C4×C71#12314yes—
75C2×C4×C71#13114yes—
76C71⋊C8—84no—
77C22×C71—22no—
78C22×C71—41no—
79C22×C71—42no—
80C22×C71—82no—
81C22×C71#38111yescharacteristic
82C22×C71#40812yes—
83C4×C71—24no—
84C4×C71—44no—
85C4×C71—41no—
86C4×C71—42no—
87C4×C71—82no—
88C4×C71#43312yes—
89C4×C71#43312yes—
90#8733—711nomaximal, Sylow
91C2×C71—22no—
92C2×C71—81no—
93C2×C71#181811yescharacteristic
94C2×C71#182712yes—
95#2298D7111yescharacteristic
96#1458—712no—
97#1276—712no—
98#1274—712no—
99C71#873311yescharacteristic, complemented, Sylow
100C2×#117C2×D7111yescharacteristic
101#431—714no—
102#388—22no—
103#387—22no—
104C4×C4⋊C4Dic7112yes—
105C4×C22⋊C4Dic7112yes—
106#376C2×D7111yescharacteristic
107C43C2×D7111yescharacteristic
108#324—1422no—
109#323—1422no—
110C23×C4C2×Dic7111yescharacteristic
111#117—22no—
112#117—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C22⋊D7112yes—
115C2×C22⋊C4—22no—
116C2×C22⋊C4C22⋊D7112yes—
117C2×C42—22no—
118C2×C42C2×Dic7112yes—
119C2×C42C2×Dic7112yes—
120C2×C42C2×Dic7111yescharacteristic
121C2×C42C22×D7111yescharacteristic
122C4⋊C8—1424no—
123C22⋊C8—1424no—
124C4×C8—1424no—
125#95—22no—
126#95—22no—
127#95—22no—
128C24C22⋊Dic7111yescharacteristic
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4C22⋊Dic7111yescharacteristic
135C22×C4C22⋊Dic7112yes—
136C22×C4C22×Dic7111yescharacteristic
137C22×C4C2×C22⋊D7112yes—
138C2×C8—2844no—
139C4⋊C4—24no—
140C4⋊C4—44no—
141C22⋊C4—24no—
142C22⋊C4—44no—
143C42—21no—
144C42—21no—
145C42—24no—
146C42—24no—
147C42—44no—
148C42#1176586512yes—
149C42#1176586512yes—
150C23—21no—
151C23—22no—
152C23—42no—
153C23C2×C22⋊Dic7111yescharacteristic
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4#1455608711yescharacteristic
168C2×C4#1455608711yescharacteristic
169C2×C4#1455611114yes—
170C2×C4#1455613514yes—
171C8—5684no—
172C22—22no—
173C22—42no—
174C22—41no—
175C22—82no—
176C22#1798586211yescharacteristic
177C22#1798596612yes—
178C4—24no—
179C4—44no—
180C4—42no—
181C4—41no—
182C4—82no—
183C4#1798606012yes—
184C4#1798606012yes—
185C2—22no—
186C2—81no—
187C2#2258841111yescharacteristic
188C2#2258844412yes—
subgroupG/H
centerC2×C4#14556087
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2298C2
socleC22×C71#381
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2298D71
hypocenterC71#8733
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #376 ◁ #2298

Complements

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

NH ≅ G/Ntype
2. #22586173C2semidirect
3. #22586169C2semidirect
99. C71#8733semidirect

Maximal subgroups

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

subgroupindex
#225866542
#225861732
#225861692
C71×#22982
#873371