Finite Group Databasebeta

← #31891828

Subgroups — #31891828

Order 18176.

428 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 119 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#22591526C211yesmaximal, characteristic
2#22589310C211yesmaximal, characteristic
3#22588508C212yesmaximal
4#22586222C212yesmaximal
5C2×#330×C71C211yesmaximal, characteristic
6C2×#14556062—22no—
7C2×#14556060—22no—
8C2×#14556039C2211yescharacteristic
9#17985898—22no—
10#17985885—22no—
11#17985885—22no—
12#17985869C2211yescharacteristic
13#17985865C2212yes—
14#17985761—22no—
15C2×#125×C71C2211yescharacteristic
16C2×#97×C71C2212yes—
17#330×C71C414yescomplemented
18C22×C71⋊C8—21no—
19C2×C4⋊Dic71—22no—
20C2×C4⋊Dic71—21no—
21C2×C4⋊Dic71—22no—
22C2×C4⋊Dic71D412yes—
23C2×C4⋊Dic71D412yes—
24C2×C4×Dic71—22no—
25#14556062—44no—
26#14556060—44no—
27#14556039—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C22×C8×C71—22no—
33#125×C71—22no—
34#125×C71C2×C412yes—
35#125×C71Q812yes—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71D412yes—
38C2×C42×C71C2311yescharacteristic
39#97×C71—22no—
40#97×C71C2×C414yes—
41C22×Dic71—22no—
42C22×Dic71—22no—
43C22×Dic71—41no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8—41no—
46C2×C71⋊C8—41no—
47C4⋊Dic71—42no—
48C4⋊Dic71—42no—
49C4⋊Dic71—42no—
50C4⋊Dic71—42no—
51C4⋊Dic71—42no—
52C4⋊Dic71—82no—
53C4×Dic71—82no—
54C22×C4×C71—22no—
55C22×C4×C71C2×D412yes—
56C22×C4×C71C2×D411yescharacteristic
57C2×C8×C71—24no—
58C2×C8×C71—24no—
59C2×C8×C71—24no—
60C4⋊C4×C71—24no—
61C4⋊C4×C71—24no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71C4⋊C414yes—
64C42×C71C22×C411yescharacteristic
65C42×C71C2×Q811yescharacteristic
66C42×C71#4112yes—
67C23×C71#12011yescharacteristic
68C2×C4×C71—22no—
69C2×C4×C71—21no—
70C2×C4×C71—22no—
71C2×C4×C71—21no—
72C2×C4×C71—24no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71C2×C4⋊C411yescharacteristic
76C2×C4×C71C4×D412yes—
77C2×C4×C71C22⋊Q812yes—
78C2×C4×C71#12311yescharacteristic
79C2×Dic71—42no—
80C2×Dic71—42no—
81C2×Dic71—42no—
82C2×Dic71—42no—
83C2×Dic71—42no—
84C2×Dic71—42no—
85C2×Dic71—82no—
86C2×Dic71—81no—
87C8×C71—28no—
88C71⋊C8—42no—
89C22×C71#38511yescharacteristic
90C22×C71#45312yes—
91C22×C71#45411yescharacteristic
92C22×C71#45612yes—
93C22×C71#45611yescharacteristic
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—44no—
99Dic71—82no—
100Dic71—82no—
101Dic71—161no—
102#8913—711nomaximal, Sylow
103C2×C71#165611yescharacteristic
104C2×C71#166012yes—
105C2×C71#166311yescharacteristic
106C2×C71#185711yescharacteristic
107C2×C71#185912yes—
108#2382—711no—
109#1992—711no—
110#1847—712no—
111C2×#330D7111yescharacteristic
112#1295—712no—
113C71#891311yescharacteristic, complemented, Sylow
114C2×#125C2×D7111yescharacteristic
115C2×#107—1422no—
116C2×#106—1422no—
117C2×#97—711no—
118C2×#97C2×D7112yes—
119#389—1422no—
120#387—1422no—
121#387—1422no—
122#384—711no—
123#382—712no—
124#340—1422no—
125#330Dic7114yescomplemented
126C22×C8—22no—
127C22×C8—1421no—
128#125—22no—
129#125Dic14212yes—
130#125C2×Dic7112yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—712no—
133C2×C4⋊C4—712no—
134C2×C4⋊C4—1422no—
135C2×C4⋊C4—1421no—
136C2×C4⋊C4—1422no—
137C2×C4⋊C4D28412yes—
138C2×C42—1422no—
139C2×C42C22×D7111yescharacteristic
140#107—2844no—
141#106—2844no—
142#97—22no—
143#97—1422no—
144#97C2×Dic7114yes—
145#95—1422no—
146#95—1422no—
147#95—1422no—
148#95—1422no—
149C22×C4—22no—
150C22×C4—2841no—
151C22×C4—1422no—
152C22×C4—1422no—
153C22×C4D4×D7112yes—
154C22×C4C2×D28411yescharacteristic
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—24no—
158C2×C8—2841no—
159C2×C8—2841no—
160C2×C8—1422no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—2842no—
165C4⋊C4—2842no—
166C4⋊C4—2842no—
167C4⋊C4—2842no—
168C4⋊C4—2842no—
169C4⋊C4—5682no—
170C4⋊C4C4⋊Dic7114yes—
171C42—5682no—
172C42C22×Dic7111yescharacteristic
173C42C2×Dic14211yescharacteristic
174C42#1176586412yes—
175C23#1455609811yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—2842no—
184C2×C4—2842no—
185C2×C4—2842no—
186C2×C4—2842no—
187C2×C4—2842no—
188C2×C4—2842no—
189C2×C4—5682no—
190C2×C4—5681no—
191C2×C4C2×C4⋊Dic7111yescharacteristic
192C2×C4D4×Dic7112yes—
193C2×C4#1455610512yes—
194C2×C4#1455611211yescharacteristic
195C8—28no—
196C8—2842no—
197C22#1798587411yescharacteristic
198C22#1798616812yes—
199C22#1798617411yescharacteristic
200C22#1798618312yes—
201C22#1798618411yescharacteristic
202C4—21no—
203C4—21no—
204C4—22no—
205C4—22no—
206C4—44no—
207C4—5682no—
208C4—5682no—
209C4—11361no—
210C2#2258763511yescharacteristic
211C2#2258765312yes—
212C2#2258766911yescharacteristic
213C2#2258855711yescharacteristic
214C2#2258856812yes—
subgroupG/H
centerC23#14556098
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#330×C71C2
socle, prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenterC2×#330D71
hypocenterC71#8913
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
17. #330×C71C4semidirect
113. C71#8913semidirect
125. #330Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915262
#225893102
#225885082
#225862222
C2×#330×C712
#891371