Finite Group Databasebeta

← C71×#7500 #31828372

Subgroups — C71×#7500 #31828372

Order 18176.

294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2126C212yesmaximal
2C71×#1962C211yesmaximal, characteristic
3#532×C71C2211yescharacteristic
4#423×C71C411yescharacteristic, complemented
5C2×#104×C71—22no—
6#410×C71—22no—
7C43×C71C411yescharacteristic, complemented
8#354×C71C414yescomplemented
9#353×C71—22no—
10C2×#41×C71D412yes—
11C2×M16×C71—21no—
12C4⋊Q8×C71Q811yescharacteristic
13#127×C71D411yescharacteristic
14#124×C71C2×C412yes—
15#121×C71—41no—
16C4×D4×C71—41no—
17#117×C71—42no—
18C2×C42×C71—21no—
19C2×C42×C71—41no—
20C2×C42×C71C2×C411yescharacteristic
21C4⋊C8×C71—41no—
22#104×C71—44no—
23#99×C71—24no—
24#99×C71—42no—
25#41×C71—42no—
26#41×C71—42no—
27C2×Q8×C71C4⋊C412yes—
28C2×D4×C71—22no—
29C2×D4×C71—41no—
30C2×D4×C71C22⋊C412yes—
31C22×C4×C71—21no—
32C22×C4×C71—22no—
33C22×C4×C71—41no—
34C22×C4×C71C22⋊C411yescharacteristic
35M16×C71—42no—
36C2×C8×C71—41no—
37C4⋊C4×C71—41no—
38C4⋊C4×C71—82no—
39C22⋊C4×C71—42no—
40C22⋊C4×C71—44no—
41C42×C71—21no—
42C42×C71—22no—
43C42×C71—41no—
44C42×C71—44no—
45C42×C71—82no—
46C42×C71C4211yescharacteristic
47C42×C71C4⋊C411yescharacteristic
48C23×C71—22no—
49C23×C71#9911yescharacteristic
50Q8×C71—42no—
51D4×C71—42no—
52D4×C71—82no—
53D4×C71—82no—
54C2×C4×C71—21no—
55C2×C4×C71—21no—
56C2×C4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—22no—
59C2×C4×C71—42no—
60C2×C4×C71—42no—
61C2×C4×C71—44no—
62C2×C4×C71—42no—
63C2×C4×C71—42no—
64C2×C4×C71—84no—
65C2×C4×C71#9511yescharacteristic
66C2×C4×C71#9911yescharacteristic
67C8×C71—81no—
68C22×C71—21no—
69C22×C71—41no—
70C22×C71—42no—
71C22×C71—82no—
72C22×C71#34211yescharacteristic
73C4×C71—22no—
74C4×C71—21no—
75C4×C71—41no—
76C4×C71—44no—
77C4×C71—42no—
78C4×C71—164no—
79C4×C71#35311yescharacteristic
80C4×C71#35411yescharacteristic
81#7500C7111yesmaximal, characteristic, direct product factor, Sylow
82C2×C71—21no—
83C2×C71—41no—
84C2×C71—82no—
85C2×C71#139311yescharacteristic
86#2126C2×C7112yes—
87#1962C2×C7111yescharacteristic
88C71#750011yescharacteristic, direct product factor, Sylow
89#532C22×C7111yescharacteristic
90#423C4×C7111yescharacteristic, complemented
91C2×#104—22no—
92#410—22no—
93C43C4×C7111yescharacteristic, complemented
94#354C4×C7114yescomplemented
95#353—22no—
96C2×#41D4×C7112yes—
97C2×M16—21no—
98C4⋊Q8Q8×C7111yescharacteristic
99#127D4×C7111yescharacteristic
100#124C2×C4×C7112yes—
101#121—41no—
102C4×D4—41no—
103#117—42no—
104C2×C42—21no—
105C2×C42—41no—
106C2×C42C2×C4×C7111yescharacteristic
107C4⋊C8—41no—
108#104—44no—
109#99—24no—
110#99—42no—
111#41—42no—
112#41—42no—
113C2×Q8C4⋊C4×C7112yes—
114C2×D4—22no—
115C2×D4—41no—
116C2×D4C22⋊C4×C7112yes—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4C22⋊C4×C7111yescharacteristic
121M16—42no—
122C2×C8—41no—
123C4⋊C4—41no—
124C4⋊C4—82no—
125C22⋊C4—42no—
126C22⋊C4—44no—
127C42—21no—
128C42—22no—
129C42—41no—
130C42—44no—
131C42—82no—
132C42C42×C7111yescharacteristic
133C42C4⋊C4×C7111yescharacteristic
134C23—22no—
135C23#99×C7111yescharacteristic
136Q8—42no—
137D4—42no—
138D4—82no—
139D4—82no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—84no—
151C2×C4#95×C7111yescharacteristic
152C2×C4#99×C7111yescharacteristic
153C8—81no—
154C22—21no—
155C22—41no—
156C22—42no—
157C22—82no—
158C22#342×C7111yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—44no—
163C4—42no—
164C4—164no—
165C4#353×C7111yescharacteristic
166C4#354×C7111yescharacteristic
167C2—21no—
168C2—41no—
169C2—82no—
170C2C71×#139311yescharacteristic
subgroupG/H
centerC4×C71#354
commutatorC42C42×C71
Frattini, cosocle#532C22×C71
socleC2×C71#1393
prime radical#532×C71C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C71 ◁ C2×C4×C71 ◁ C22×C4×C71 ◁ #532×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. #423×C71C4semidirect
7. C43×C71C4semidirect
8. #354×C71C4semidirect
81. #7500C71direct (N × H)
88. C71#7500direct (N × H)
90. #423C4×C71semidirect
93. C43C4×C71semidirect
94. #354C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#21262
C71×#19622
#750071