Finite Group Databasebeta

← C71×#8741 #31829613

Subgroups — C71×#8741 #31829613

Order 18176.

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2298C211yesmaximal, characteristic
2C71×#1822C211yesmaximal, characteristic
3C71×#1820C211yesmaximal, characteristic
4C71×#1284C214yesmaximal
5C2×#117×C71C2211yescharacteristic
6#407×C71C2212yes—
7#388×C71—22no—
8#387×C71—22no—
9#381×C71C2212yes—
10C4×C4⋊C4×C71C412yescomplemented
11C4×C22⋊C4×C71C2212yes—
12#376×C71C411yescharacteristic, complemented
13C43×C71C411yescharacteristic, complemented
14#343×C71—22no—
15#342×C71—22no—
16C23×C4×C71C2311yescharacteristic
17C2×M16×C71—22no—
18#117×C71—24no—
19C2×C4⋊C4×C71—22no—
20C2×C4⋊C4×C71D412yes—
21C2×C22⋊C4×C71—22no—
22C2×C22⋊C4×C71—22no—
23C2×C22⋊C4×C71D412yes—
24C2×C42×C71—22no—
25C2×C42×C71C2×C411yescharacteristic
26C2×C42×C71C2×C411yescharacteristic
27C2×C42×C71C2×C412yes—
28C2×C42×C71C2×C412yes—
29C22⋊C8×C71—24no—
30#95×C71—22no—
31#95×C71—22no—
32#95×C71—22no—
33#95×C71—24no—
34C24×C71C2×D411yescharacteristic
35C22×C4×C71—22no—
36C22×C4×C71—22no—
37C22×C4×C71—42no—
38C22×C4×C71—42no—
39C22×C4×C71C22⋊C412yes—
40C22×C4×C71C22⋊C412yes—
41C22×C4×C71C22×C411yescharacteristic
42C22×C4×C71C2×D411yescharacteristic
43C22×C4×C71#4114yes—
44M16×C71—44no—
45C2×C8×C71—42no—
46C4⋊C4×C71—24no—
47C4⋊C4×C71—44no—
48C22⋊C4×C71—24no—
49C22⋊C4×C71—44no—
50C22⋊C4×C71—44no—
51C42×C71—22no—
52C42×C71—22no—
53C42×C71—42no—
54C42×C71—42no—
55C42×C71—44no—
56C23×C71—22no—
57C23×C71—42no—
58C23×C71C2×C22⋊C411yescharacteristic
59C23×C71#12312yes—
60C2×C4×C71—22no—
61C2×C4×C71—22no—
62C2×C4×C71—24no—
63C2×C4×C71—22no—
64C2×C4×C71—22no—
65C2×C4×C71—44no—
66C2×C4×C71—42no—
67C2×C4×C71—42no—
68C2×C4×C71—42no—
69C2×C4×C71—42no—
70C2×C4×C71—42no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71—82no—
74C2×C4×C71—84no—
75C2×C4×C71#11712yes—
76C2×C4×C71#12312yes—
77C8×C71—82no—
78C22×C71—22no—
79C22×C71—22no—
80C22×C71—24no—
81C22×C71—82no—
82C22×C71#38111yescharacteristic
83C22×C71#41012yes—
84C4×C71—22no—
85C4×C71—22no—
86C4×C71—44no—
87C4×C71—42no—
88C4×C71—42no—
89C4×C71—82no—
90C4×C71—162no—
91#8741C7111yesmaximal, characteristic, direct product factor, Sylow
92C2×C71—22no—
93C2×C71—42no—
94C2×C71#182011yescharacteristic
95C2×C71#182712yes—
96#2298C2×C7111yescharacteristic
97#1822C2×C7111yescharacteristic
98#1820C2×C7111yescharacteristic
99#1284C2×C7114yes—
100C71#874111yescharacteristic, direct product factor, Sylow
101C2×#117C22×C7111yescharacteristic
102#407C22×C7112yes—
103#388—22no—
104#387—22no—
105#381C22×C7112yes—
106C4×C4⋊C4C4×C7112yescomplemented
107C4×C22⋊C4C22×C7112yes—
108#376C4×C7111yescharacteristic, complemented
109C43C4×C7111yescharacteristic, complemented
110#343—22no—
111#342—22no—
112C23×C4C23×C7111yescharacteristic
113C2×M16—22no—
114#117—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4D4×C7112yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4D4×C7112yes—
120C2×C42—22no—
121C2×C42C2×C4×C7111yescharacteristic
122C2×C42C2×C4×C7111yescharacteristic
123C2×C42C2×C4×C7112yes—
124C2×C42C2×C4×C7112yes—
125C22⋊C8—24no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—24no—
130C24C2×D4×C7111yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4C22⋊C4×C7112yes—
136C22×C4C22⋊C4×C7112yes—
137C22×C4C22×C4×C7111yescharacteristic
138C22×C4C2×D4×C7111yescharacteristic
139C22×C4#41×C7114yes—
140M16—44no—
141C2×C8—42no—
142C4⋊C4—24no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—42no—
151C42—44no—
152C23—22no—
153C23—42no—
154C23C2×C22⋊C4×C7111yescharacteristic
155C23#123×C7112yes—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—82no—
170C2×C4—84no—
171C2×C4#117×C7112yes—
172C2×C4#123×C7112yes—
173C8—82no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#381×C7111yescharacteristic
179C22#410×C7112yes—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—162no—
187C2—22no—
188C2—42no—
189C2C71×#182011yescharacteristic
190C2C71×#182712yes—
subgroupG/H
center, socleC22×C71#381
commutatorC22×C4C22×C4×C71
Frattini, cosocleC23×C4C23×C71
prime radicalC24×C71C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
10. C4×C4⋊C4×C71C4semidirect
12. #376×C71C4semidirect
13. C43×C71C4semidirect
91. #8741C71direct (N × H)
100. C71#8741direct (N × H)
106. C4×C4⋊C4C4×C71semidirect
108. #376C4×C71semidirect
109. C43C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#22982
C71×#18222
C71×#18202
C71×#12842
#874171