Finite Group Databasebeta

← #31891826

Subgroups — #31891826

Order 18176.

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

normal subgroups only

This group has 428 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#22589310C211yesmaximal, characteristic
2#22588509C212yesmaximal
3C2×#17985741C211yesmaximal, characteristic
4#22586221C212yesmaximal
5C71×#2382C211yesmaximal, characteristic
6C2×#14556061—22no—
7C2×#14556059—22no—
8C2×#14556039C2211yescharacteristic
9C2×#14556039C2212yes—
10#17985869C2211yescharacteristic
11#17985761—22no—
12#17985741C414yescomplemented
13C2×#125×C71C2211yescharacteristic
14#389×C71—22no—
15#387×C71—22no—
16#387×C71—22no—
17#382×C71C2212yes—
18C22×C71⋊C8—22no—
19C22×C71⋊C8—21no—
20C2×C4⋊Dic71—21no—
21C2×C4⋊Dic71—22no—
22#14556061—44no—
23#14556059—44no—
24#14556039—22no—
25#14556039—22no—
26#14556039C2×C414yes—
27#125×C71—22no—
28#125×C71C2×C412yes—
29#125×C71Q812yes—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D412yes—
35C2×C42×C71—22no—
36C2×C42×C71C2311yescharacteristic
37#95×C71—22no—
38#95×C71—22no—
39#95×C71—22no—
40#95×C71—22no—
41C22×Dic71—41no—
42C2×C71⋊C8—24no—
43C2×C71⋊C8—24no—
44C2×C71⋊C8—24no—
45C2×C71⋊C8—22no—
46C2×C71⋊C8—41no—
47C2×C71⋊C8—41no—
48C4⋊Dic71—42no—
49C4⋊Dic71—42no—
50C4⋊Dic71—42no—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71C2×D412yes—
56C4⋊C4×C71—24no—
57C4⋊C4×C71—24no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—42no—
60C4⋊C4×C71—42no—
61C4⋊C4×C71—82no—
62C4⋊C4×C71C4⋊C414yes—
63C42×C71—82no—
64C42×C71C22×C411yescharacteristic
65C42×C71C2×Q811yescharacteristic
66C42×C71#4112yes—
67C23×C71#12011yescharacteristic
68C2×C4×C71—22no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—24no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71C2×C4⋊C411yescharacteristic
82C2×C4×C71C4×D412yes—
83C2×C4×C71C22⋊Q812yes—
84C2×C4×C71#12311yescharacteristic
85C2×Dic71—82no—
86C2×Dic71—81no—
87C71⋊C8—28no—
88C71⋊C8—42no—
89C22×C71#38511yescharacteristic
90C22×C71#45312yes—
91C22×C71#45411yescharacteristic
92C22×C71#45611yescharacteristic
93C22×C71#45612yes—
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—44no—
99C4×C71—82no—
100C4×C71—82no—
101Dic71—161no—
102#8913—711nomaximal, Sylow
103C2×C71#165611yescharacteristic
104C2×C71#166012yes—
105C2×C71#166311yescharacteristic
106C2×C71#185711yescharacteristic
107C2×C71#185912yes—
108#2382D7111yescharacteristic
109#1992—711no—
110#1847—712no—
111C2×#330—711no—
112#1295—712no—
113C71#891311yescharacteristic, complemented, Sylow
114C2×#125C2×D7111yescharacteristic
115C2×#107—1422no—
116C2×#106—1422no—
117C2×#97—712no—
118C2×#97—711no—
119#389—22no—
120#387—22no—
121#387—22no—
122#384—711no—
123#382C2×D7112yes—
124#340—1422no—
125#330—714no—
126C22×C8—1421no—
127C22×C8—1422no—
128#125—22no—
129#125Dic14212yes—
130#125C4×D7112yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—1421no—
134C2×C4⋊C4—1422no—
135C2×C4⋊C4C22⋊D7112yes—
136C2×C4⋊C4C22⋊D7112yes—
137C2×C4⋊C4C22⋊D7112yes—
138C2×C42—22no—
139C2×C42C22×D7111yescharacteristic
140#107—2844no—
141#106—2844no—
142#97—714no—
143#97—1422no—
144#97—1422no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—2841no—
153C22×C4C2×C22⋊D7111yescharacteristic
154C22×C4C2×C22⋊D7112yes—
155C2×C8—2841no—
156C2×C8—2841no—
157C2×C8—1422no—
158C2×C8—1424no—
159C2×C8—1424no—
160C2×C8—1424no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—2842no—
168C4⋊C4—2842no—
169C4⋊C4—2842no—
170C4⋊C4Dic71⋊C414yes—
171C42—82no—
172C42C2×C4×D7111yescharacteristic
173C42C2×Dic14211yescharacteristic
174C42#1176586512yes—
175C23#1455609711yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—5682no—
190C2×C4—5681no—
191C2×C4C2×Dic71⋊C411yescharacteristic
192C2×C4C4×C22⋊D7112yes—
193C2×C4#1455610712yes—
194C2×C4#1455611111yescharacteristic
195C8—2842no—
196C8—1428no—
197C22#1798587211yescharacteristic
198C22#1798616712yes—
199C22#1798617211yescharacteristic
200C22#1798618211yescharacteristic
201C22#1798618212yes—
202C4—21no—
203C4—21no—
204C4—22no—
205C4—22no—
206C4—44no—
207C4—82no—
208C4—82no—
209C4—11361no—
210C2#2258763311yescharacteristic
211C2#2258765112yes—
212C2#2258766811yescharacteristic
213C2#2258855511yescharacteristic
214C2#2258856612yes—
subgroupG/H
centerC23#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2382C2
socle, prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenter#2382D71
hypocenterC71#8913
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
12. #17985741C4semidirect
113. C71#8913semidirect

Maximal subgroups

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

subgroupindex
#225893102
#225885092
C2×#179857412
#225862212
C71×#23822
#891371