Finite Group Databasebeta

← #31891960

Subgroups — #31891960

Order 18176.

412 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 107 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#22589169C211yesmaximal, characteristic
2#22587128C212yesmaximal, complemented
3#22587002C211yesmaximal, characteristic
4#22586172C212yesmaximal, complemented
5C71×#2979C211yesmaximal, characteristic
6#17986052C2212yes—
7#17986023C2211yescharacteristic
8C2×#14556055—22no—
9C42×Dic71C2211yescharacteristic
10#17985751—22no—
11#17985749—22no—
12#17985748C411yescharacteristic, complemented
13#17985748C411yescharacteristic, complemented
14#17985743C412yescomplemented
15#532×C71C2211yescharacteristic
16#436×C71—22no—
17C8×D4×C71—22no—
18#431×C71C2212yes—
19#408×C71—22no—
20C2×#11765849—21no—
21C2×C4×Dic71—21no—
22C2×C4×Dic71—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)C2×C412yes—
25C4⋊(C71⋊C8)C2×C412yes—
26#14556055—44no—
27C22⋊(C71⋊C8)—22no—
28C4×C71⋊C8—22no—
29C2×#41×C71D412yes—
30C2×M16×C71—22no—
31C22×C8×C71—22no—
32C4⋊Q8×C71C2×C411yescharacteristic
33#127×C71C2×C411yescharacteristic
34#124×C71Q812yes—
35#121×C71—22no—
36C4×D4×C71—22no—
37C2×C42×C71C2311yescharacteristic
38C4⋊C8×C71D412yes—
39C4⋊C8×C71D412yes—
40C22⋊C8×C71—24no—
41C22⋊C8×C71—22no—
42C4×C8×C71—22no—
43C22×Dic71—41no—
44#11765849—42no—
45C2×C71⋊C8—22no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—41no—
48C4×Dic71—22no—
49C4×Dic71—24no—
50C4×Dic71—42no—
51C4×Dic71—41no—
52#41×C71—42no—
53#41×C71—42no—
54C2×Q8×C71C4⋊C412yes—
55C2×D4×C71—22no—
56C2×D4×C71—22no—
57C2×D4×C71C4⋊C412yes—
58C22×C4×C71—22no—
59C22×C4×C71C2×D411yescharacteristic
60C22×C4×C71#4112yes—
61M16×C71—82no—
62C2×C8×C71—22no—
63C2×C8×C71—22no—
64C2×C8×C71—82no—
65C4⋊C4×C71—22no—
66C22⋊C4×C71—24no—
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—42no—
84C2×C4×C71C2×C4⋊C411yescharacteristic
85C2×C4×C71C4×D412yes—
86C2×C4×C71#12011yescharacteristic
87C2×C4×C71C22⋊Q812yes—
88C2×Dic71—42no—
89C2×Dic71—44no—
90C8×C71—42no—
91C8×C71—42no—
92C71⋊C8—42no—
93C71⋊C8—42no—
94C71⋊C8—81no—
95C22×C71—21no—
96C22×C71—22no—
97C22×C71—42no—
98C22×C71—44no—
99C22×C71#38511yescharacteristic
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—42no—
103C4×C71#44312yes—
104C4×C71#44412yes—
105C4×C71#45311yescharacteristic
106C4×C71#45511yescharacteristic
107Dic71—44no—
108#8935—711nomaximal, Sylow
109C2×C71—41no—
110C2×C71—82no—
111C2×C71#185611yescharacteristic
112C2×C71#186612yes—
113#2979D7111yescharacteristic
114#1962—711no—
115#1574—712no—
116#1543—711no—
117#1276—712no—
118C71#893511yescharacteristic, complemented, Sylow
119#532C2×D7111yescharacteristic
120#436—22no—
121C8×D4—22no—
122#431—712no—
123#431C2×D7112yes—
124#423—711no—
125C2×#104—1422no—
126#408—22no—
127C43—711no—
128#335—1422no—
129#334—1422no—
130#333—711no—
131#333—711no—
132#331—712no—
133C2×#41C22⋊D7112yes—
134C2×M16—22no—
135C2×M16—1421no—
136C22×C8—22no—
137C4⋊Q8C4×D7111yescharacteristic
138#127C4×D7111yescharacteristic
139#124Dic14212yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—1421no—
143C2×C42—1422no—
144C2×C42C22×D7111yescharacteristic
145C4⋊C8—712no—
146C4⋊C8—712no—
147C4⋊C8—1422no—
148C4⋊C8C22⋊D7112yes—
149C4⋊C8C22⋊D7112yes—
150#104—2844no—
151C22⋊C8—24no—
152C22⋊C8—22no—
153C22⋊C8—1422no—
154C4×C8—22no—
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—2841no—
164C22×C4C2×C22⋊D7111yescharacteristic
165C22×C4#1176586512yes—
166M16—82no—
167M16—2842no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—82no—
171C2×C8—2841no—
172C2×C8—1422no—
173C2×C8—1422no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—2842no—
177C42—2841no—
178C42—1422no—
179C42—1424no—
180C42C2×C4×D7111yescharacteristic
181C42C2×C22⋊D7112yes—
182C42C2×Dic14211yescharacteristic
183C23—22no—
184C23#1455611111yescharacteristic
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—42no—
197C2×C4—2842no—
198C2×C4—2844no—
199C2×C4C2×Dic71⋊C411yescharacteristic
200C2×C4C4×C22⋊D7112yes—
201C2×C4#1455609711yescharacteristic
202C2×C4#1455610712yes—
203C8—42no—
204C8—42no—
205C8—2842no—
206C8—2842no—
207C8—5681no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22#1798587211yescharacteristic
213C4—21no—
214C4—22no—
215C4—42no—
216C4—2844no—
217C4#1798611412yes—
218C4#1798611912yes—
219C4#1798616711yescharacteristic
220C4#1798617711yescharacteristic
221C2—41no—
222C2—82no—
223C2#2258855011yescharacteristic
224C2#2258861512yes—
subgroupG/H
centerC2×C4#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2979C2
socleC22×C71#385
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#2979D71
hypocenterC71#8935
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #22587128C2semidirect
4. #22586172C2semidirect
12. #17985748C4semidirect
13. #17985748C4semidirect
14. #17985743C4semidirect
118. C71#8935semidirect

Maximal subgroups

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

subgroupindex
#225891692
#225871282
#225870022
#225861722
C71×#29792
#893571