Finite Group Databasebeta

← #31892402

Subgroups — #31892402

Order 18176.

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

normal subgroups only

This group has 416 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#22597411C211yesmaximal, characteristic
2#22587002C211yesmaximal, characteristic
3#22586813C212yesmaximal, complemented
4#22586239C212yesmaximal, complemented
5C71×#3135C211yesmaximal, characteristic
6#17986071—24no—
7#17986056C2212yes—
8#17986052C2212yes—
9#17985966—22no—
10#17985748—21no—
11#17985743—21no—
12Q8⋊(C71⋊C8)—24no—
13#533×C71C2211yescharacteristic
14#532×C71C2211yescharacteristic
15#491×C71—21no—
16#490×C71—21no—
17#489×C71C414yes—
18#416×C71—22no—
19C2×#97×C71C2211yescharacteristic
20C2×#11765849—22no—
21C22×C71⋊C8—22no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)D414yes—
24C22⋊(C71⋊C8)—24no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27#14556039—22no—
28C4×C71⋊C8—22no—
29C2×#41×C71D412yes—
30C22×C8×C71—21no—
31C4⋊Q8×C71C2×C412yes—
32C4⋊Q8×C71D411yescharacteristic
33#127×C71D411yescharacteristic
34#125×C71—21no—
35#124×C71C2×C412yes—
36C22⋊Q8×C71—22no—
37#121×C71—22no—
38C4×Q8×C71—22no—
39C4×D4×C71—22no—
40#117×C71—21no—
41C2×C42×C71C2311yescharacteristic
42#103×C71—24no—
43#102×C71—24no—
44#97×C71—21no—
45#97×C71C2×C412yes—
46#11765849—82no—
47C2×C71⋊C8—24no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—82no—
50#41×C71—42no—
51#41×C71—42no—
52C2×Q8×C71—22no—
53C2×Q8×C71C22⋊C412yes—
54C2×D4×C71—22no—
55C2×D4×C71—22no—
56C2×D4×C71C22⋊C412yes—
57C22×C4×C71—22no—
58C22×C4×C71C2×D411yescharacteristic
59C22×C4×C71#4112yes—
60C2×C8×C71—22no—
61C2×C8×C71—42no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—42no—
66C22⋊C4×C71—24no—
67C22⋊C4×C71—41no—
68C42×C71—41no—
69C42×C71C22×C411yescharacteristic
70C42×C71C2×D412yes—
71C42×C71C2×D411yescharacteristic
72C23×C71—22no—
73C23×C71#12311yescharacteristic
74Q8×C71—42no—
75Q8×C71—44no—
76D4×C71—42no—
77D4×C71—82no—
78D4×C71—82no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71C2×C22⋊C411yescharacteristic
86C2×C4×C71C4×D412yes—
87C2×C4×C71#12011yescharacteristic
88C2×C4×C71#12112yes—
89C8×C71—42no—
90C71⋊C8—44no—
91C71⋊C8—82no—
92C22×C71—22no—
93C22×C71—21no—
94C22×C71—44no—
95C22×C71—42no—
96C22×C71#38611yescharacteristic
97C4×C71—22no—
98C4×C71—22no—
99C4×C71—21no—
100C4×C71—42no—
101C4×C71—82no—
102C4×C71#45512yes—
103#9005—711nomaximal, Sylow
104C2×C71—41no—
105C2×C71—82no—
106C2×C71#188112yes—
107C2×C71#188511yescharacteristic
108#3135D7111yescharacteristic
109#2980—711no—
110#1543—711no—
111#1497—712no—
112#1302—712no—
113C71#900511yescharacteristic, complemented, Sylow
114#533C2×D7111yescharacteristic
115#532C2×D7111yescharacteristic
116#491—21no—
117#490—21no—
118#489Dic7114yes—
119#435—1424no—
120#432—712no—
121#431—712no—
122#416—22no—
123#408—1422no—
124C2×#97C2×D7111yescharacteristic
125#333—1421no—
126#331—1421no—
127Q8⋊C8—1424no—
128C2×#41C22⋊D7112yes—
129C2×M16—1422no—
130C22×C8—21no—
131C22×C8—1422no—
132C4⋊Q8C2×Dic7112yes—
133C4⋊Q8C22⋊D7111yescharacteristic
134#127C22⋊D7111yescharacteristic
135#125—21no—
136#124C2×Dic7112yes—
137C22⋊Q8—22no—
138#121—22no—
139C4×Q8—22no—
140C4×D4—22no—
141#117—21no—
142C2×C42C22×D7111yescharacteristic
143C4⋊C8—714no—
144C4⋊C8—1422no—
145#103—24no—
146#102—24no—
147C22⋊C8—1422no—
148C22⋊C8—1424no—
149C22⋊C8—1422no—
150#97—21no—
151#97—1422no—
152#97C2×Dic7112yes—
153C4×C8—1422no—
154#41—42no—
155#41—42no—
156C2×Q8—22no—
157C2×Q8C22⋊Dic7112yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4C22⋊Dic7112yes—
161C22×C4—22no—
162C22×C4C2×C22⋊D7111yescharacteristic
163C22×C4#1176586412yes—
164M16—5682no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—2842no—
168C2×C8—5682no—
169C2×C8—1424no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C22⋊C4—24no—
175C22⋊C4—41no—
176C42—41no—
177C42C22×Dic7111yescharacteristic
178C42D4×D7112yes—
179C42C2×C22⋊D7111yescharacteristic
180C23—22no—
181C23#1455610911yescharacteristic
182Q8—42no—
183Q8—44no—
184D4—42no—
185D4—82no—
186D4—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4C2×C22⋊Dic7111yescharacteristic
194C2×C4D4×Dic7112yes—
195C2×C4#1455609611yescharacteristic
196C2×C4#1455610012yes—
197C8—42no—
198C8—2844no—
199C8—5682no—
200C22—22no—
201C22—21no—
202C22—44no—
203C22—42no—
204C22#1798587911yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—82no—
210C4#1798617612yes—
211C2—41no—
212C2—82no—
213C2#2258870412yes—
214C2#2258872911yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3135C2
socleC22×C71#386
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#3135D71
hypocenterC71#9005
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
3. #22586813C2semidirect
4. #22586239C2semidirect
113. C71#9005semidirect

Maximal subgroups

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

subgroupindex
#225974112
#225870022
#225868132
#225862392
C71×#31352
#900571