Finite Group Databasebeta

← #31891954

Subgroups — #31891954

Order 18176.

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

normal subgroups only

This group has 412 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#22597407C211yesmaximal, characteristic
2#22587131C212yesmaximal, complemented
3#22587002C211yesmaximal, characteristic
4#22586173C212yesmaximal, complemented
5C71×#1962C211yesmaximal, characteristic
6#17986077—22no—
7D4×C71⋊C8—22no—
8#17986052C2212yes—
9#17986052C2212yes—
10#17985966—22no—
11#17985751—22no—
12#17985749—22no—
13#17985748C412yescomplemented
14#17985743C411yescharacteristic, complemented
15#17985743C411yescharacteristic, complemented
16#532×C71C2211yescharacteristic
17#423×C71C2211yescharacteristic
18C2×#104×C71—22no—
19C43×C71C2211yescharacteristic
20C2×#11765849—22no—
21C22×C71⋊C8—22no—
22C4⋊(C71⋊C8)C2×C412yes—
23C4⋊(C71⋊C8)C2×C412yes—
24C4⋊(C71⋊C8)D412yes—
25C4⋊(C71⋊C8)D412yes—
26C22⋊(C71⋊C8)—24no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—22no—
30C4×C71⋊C8—22no—
31C2×#41×C71D412yes—
32C2×M16×C71—21no—
33C4⋊Q8×C71C2×C411yescharacteristic
34#127×C71C2×C411yescharacteristic
35#124×C71Q812yes—
36#121×C71—22no—
37C4×D4×C71—22no—
38C2×C42×C71—21no—
39C2×C42×C71—22no—
40C2×C42×C71C2311yescharacteristic
41C4⋊C8×C71—22no—
42#104×C71—44no—
43#11765849—82no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8—22no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—82no—
49#41×C71—42no—
50#41×C71—42no—
51C2×Q8×C71C4⋊C412yes—
52C2×D4×C71—22no—
53C2×D4×C71—22no—
54C2×D4×C71C4⋊C412yes—
55C22×C4×C71—22no—
56C22×C4×C71—41no—
57C22×C4×C71C2×D411yescharacteristic
58C22×C4×C71#4112yes—
59M16×C71—42no—
60C2×C8×C71—41no—
61C4⋊C4×C71—22no—
62C22⋊C4×C71—24no—
63C42×C71—22no—
64C42×C71—24no—
65C42×C71—42no—
66C42×C71—41no—
67C42×C71C22×C411yescharacteristic
68C42×C71C2×D412yes—
69C42×C71C2×Q811yescharacteristic
70C23×C71—22no—
71C23×C71#12311yescharacteristic
72Q8×C71—42no—
73D4×C71—42no—
74D4×C71—42no—
75D4×C71—42no—
76D4×C71—82no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—42no—
83C2×C4×C71—44no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71C2×C4⋊C411yescharacteristic
87C2×C4×C71C4×D412yes—
88C2×C4×C71#12011yescharacteristic
89C2×C4×C71C22⋊Q812yes—
90C8×C71—81no—
91C71⋊C8—42no—
92C71⋊C8—42no—
93C71⋊C8—42no—
94C71⋊C8—42no—
95C22×C71—21no—
96C22×C71—22no—
97C22×C71—42no—
98C22×C71—44no—
99C22×C71#38511yescharacteristic
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—44no—
103C4×C71—42no—
104C4×C71#44312yes—
105C4×C71#44412yes—
106C4×C71#45311yescharacteristic
107C4×C71#45511yescharacteristic
108#8934—711nomaximal, Sylow
109C2×C71—41no—
110C2×C71—82no—
111C2×C71#185611yescharacteristic
112C2×C71#186612yes—
113#2979—711no—
114#1962D7111yescharacteristic
115#1574—712no—
116#1543—711no—
117#1276—712no—
118C71#893411yescharacteristic, complemented, Sylow
119#532C2×D7111yescharacteristic
120#436—1422no—
121C8×D4—1422no—
122#431—712no—
123#431—712no—
124#423C2×D7111yescharacteristic
125C2×#104—22no—
126#408—1422no—
127C43C2×D7111yescharacteristic
128#335—1422no—
129#334—1422no—
130#333—712no—
131#331—711no—
132#331—711no—
133C2×#41C22⋊D7112yes—
134C2×M16—21no—
135C2×M16—1422no—
136C22×C8—1422no—
137C4⋊Q8C4×D7111yescharacteristic
138#127C4×D7111yescharacteristic
139#124Dic14212yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42C22×D7111yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—712no—
147C4⋊C8—712no—
148C4⋊C8—712no—
149C4⋊C8—712no—
150#104—44no—
151C22⋊C8—1424no—
152C22⋊C8—1422no—
153C22⋊C8—1422no—
154C4×C8—1422no—
155C4×C8—1422no—
156#41—42no—
157#41—42no—
158C2×Q8Dic71⋊C412yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4Dic71⋊C412yes—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4C2×C22⋊D7111yescharacteristic
165C22×C4#1176586412yes—
166M16—42no—
167M16—5682no—
168C2×C8—41no—
169C2×C8—5682no—
170C2×C8—1422no—
171C2×C8—1422no—
172C2×C8—1422no—
173C2×C8—1422no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—22no—
177C42—24no—
178C42—42no—
179C42—41no—
180C42C2×C4×D7111yescharacteristic
181C42D4×D7112yes—
182C42C2×Dic14211yescharacteristic
183C23—22no—
184C23#1455610911yescharacteristic
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4C2×Dic71⋊C411yescharacteristic
200C2×C4(C22⋊D71)⋊C412yes—
201C2×C4#1455609611yescharacteristic
202C2×C4#1455610512yes—
203C8—81no—
204C8—2842no—
205C8—2842no—
206C8—2842no—
207C8—2842no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22#1798587111yescharacteristic
213C4—21no—
214C4—22no—
215C4—44no—
216C4—42no—
217C4#1798611312yes—
218C4#1798611812yes—
219C4#1798616611yescharacteristic
220C4#1798617611yescharacteristic
221C2—41no—
222C2—82no—
223C2#2258854911yescharacteristic
224C2#2258861312yes—
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1962C2
socleC22×C71#385
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#1962D71
hypocenterC71#8934
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #22587131C2semidirect
4. #22586173C2semidirect
13. #17985748C4semidirect
14. #17985743C4semidirect
15. #17985743C4semidirect
118. C71#8934semidirect

Maximal subgroups

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

subgroupindex
#225974072
#225871312
#225870022
#225861732
C71×#19622
#893471