Finite Group Databasebeta

← C71×#9440 #31830312

Subgroups — C71×#9440 #31830312

Order 18176.

434 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C71×#3166C211yesmaximal, characteristic
2C71×#2371C211yesmaximal, characteristic
3C71×#1534C211yesmaximal, characteristic
4C71×#1506C212yesmaximal
5C71×#1300C212yesmaximal
6#533×C71C2211yescharacteristic
7C2×#125×C71C2211yescharacteristic
8#501×C71C414yescomplemented
9#499×C71—22no—
10#432×C71C2212yes—
11#428×C71—22no—
12C4×M16×C71C2211yescharacteristic
13#387×C71—22no—
14#387×C71—22no—
15C4×C4⋊C4×C71—22no—
16#376×C71C2212yes—
17#330×C71—22no—
18#328×C71—22no—
19#328×C71—22no—
20C2×M16×C71—21no—
21C4⋊Q8×C71C2×C412yes—
22#125×C71—21no—
23#125×C71—22no—
24#125×C71C2×C412yes—
25#125×C71D412yes—
26C22⋊Q8×C71—22no—
27C4×Q8×C71—22no—
28#117×C71—21no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71D412yes—
32C2×C42×C71—22no—
33C2×C42×C71—22no—
34C2×C42×C71C2311yescharacteristic
35#107×C71—24no—
36#106×C71—24no—
37C4⋊C8×C71—22no—
38C22⋊C8×C71—22no—
39#97×C71—22no—
40#97×C71C2×C412yes—
41C4×C8×C71—21no—
42#95×C71—22no—
43#95×C71—22no—
44#95×C71—22no—
45#95×C71D412yes—
46#95×C71D412yes—
47C2×Q8×C71—22no—
48C22×C4×C71—22no—
49C22×C4×C71—22no—
50C22×C4×C71—22no—
51C22×C4×C71C2×D411yescharacteristic
52C22×C4×C71C2×D412yes—
53M16×C71—24no—
54C2×C8×C71—22no—
55C2×C8×C71—42no—
56C4⋊C4×C71—24no—
57C4⋊C4×C71—24no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71—44no—
62C4⋊C4×C71—42no—
63C4⋊C4×C71C22⋊C414yes—
64C22⋊C4×C71—41no—
65C42×C71—44no—
66C42×C71—44no—
67C42×C71—41no—
68C42×C71C22×C411yescharacteristic
69C42×C71C2×D411yescharacteristic
70C42×C71#4112yes—
71C23×C71#12011yescharacteristic
72Q8×C71—82no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71—24no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
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#12112yes—
88C2×C4×C71#12311yescharacteristic
89C8×C71—24no—
90C8×C71—82no—
91C22×C71—22no—
92C22×C71#38611yescharacteristic
93C22×C71#45612yes—
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—44no—
99C4×C71—44no—
100C4×C71—82no—
101C4×C71—82no—
102#9440C7111yesmaximal, characteristic, direct product factor, Sylow
103C2×C71—22no—
104C2×C71#188611yescharacteristic
105C2×C71#190612yes—
106#3166C2×C7111yescharacteristic
107#2371C2×C7111yescharacteristic
108#1534C2×C7111yescharacteristic
109#1506C2×C7112yes—
110#1300C2×C7112yes—
111C71#944011yescharacteristic, direct product factor, Sylow
112#533C22×C7111yescharacteristic
113C2×#125C22×C7111yescharacteristic
114#501C4×C7114yescomplemented
115#499—22no—
116#432C22×C7112yes—
117#428—22no—
118C4×M16C22×C7111yescharacteristic
119#387—22no—
120#387—22no—
121C4×C4⋊C4—22no—
122#376C22×C7112yes—
123#330—22no—
124#328—22no—
125#328—22no—
126C2×M16—21no—
127C4⋊Q8C2×C4×C7112yes—
128#125—21no—
129#125—22no—
130#125C2×C4×C7112yes—
131#125D4×C7112yes—
132C22⋊Q8—22no—
133C4×Q8—22no—
134#117—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4D4×C7112yes—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C23×C7111yescharacteristic
141#107—24no—
142#106—24no—
143C4⋊C8—22no—
144C22⋊C8—22no—
145#97—22no—
146#97C2×C4×C7112yes—
147C4×C8—21no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95D4×C7112yes—
152#95D4×C7112yes—
153C2×Q8—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4C2×D4×C7111yescharacteristic
158C22×C4C2×D4×C7112yes—
159M16—24no—
160C2×C8—22no—
161C2×C8—42no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4C22⋊C4×C7114yes—
170C22⋊C4—41no—
171C42—44no—
172C42—44no—
173C42—41no—
174C42C22×C4×C7111yescharacteristic
175C42C2×D4×C7111yescharacteristic
176C42#41×C7112yes—
177C23#120×C7111yescharacteristic
178Q8—82no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4C2×C22⋊C4×C7111yescharacteristic
192C2×C4C4×D4×C7112yes—
193C2×C4#121×C7112yes—
194C2×C4#123×C7111yescharacteristic
195C8—24no—
196C8—82no—
197C22—22no—
198C22#386×C7111yescharacteristic
199C22#456×C7112yes—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—44no—
206C4—82no—
207C4—82no—
208C2—22no—
209C2C71×#188611yescharacteristic
210C2C71×#190612yes—
subgroupG/H
center, socleC22×C71#386
commutatorC42C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC23×C71#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
8. #501×C71C4semidirect
102. #9440C71direct (N × H)
111. C71#9440direct (N × H)
114. #501C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#31662
C71×#23712
C71×#15342
C71×#15062
C71×#13002
#944071