Finite Group Databasebeta

← C71×#13658 #31834530

Subgroups — C71×#13658 #31834530

Order 18176.

194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2263C211yesmaximal, characteristic
2C71×#2179C211yesmaximal, characteristic, complemented
3C71×#2177C211yesmaximal, characteristic, complemented
4C71×#1426C211yesmaximal, characteristic, complemented
5C71×#1419C211yesmaximal, characteristic, complemented
6C71×#1418C211yesmaximal, characteristic, complemented
7C71×#1398C211yesmaximal, characteristic, complemented
8#508×C71C2211yescharacteristic
9C2×D16×C71C2211yescharacteristic
10#443×C71—21no—
11C4×D8×C71—21no—
12SD64×C71C412yescomplemented
13D32×C71C412yescomplemented
14M64×C71C2211yescharacteristic
15M64×C71C2211yescharacteristic
16#368×C71C2211yescharacteristic
17#367×C71C2211yescharacteristic
18#359×C71—21no—
19#357×C71—21no—
20#346×C71C2211yescharacteristic
21#135×C71—21no—
22C2×D8×C71—21no—
23#131×C71—41no—
24C4×D4×C71—41no—
25Q32×C71C2×C411yescharacteristic, complemented
26SD32×C71—21no—
27D16×C71—21no—
28D16×C71C2×C412yes—
29D16×C71C2×C411yescharacteristic, complemented
30C2×C16×C71D411yescharacteristic
31C2×C16×C71C2311yescharacteristic
32#108×C71—21no—
33#107×C71—21no—
34#104×C71—41no—
35#102×C71—41no—
36C4×C8×C71D411yescharacteristic
37C32×C71—21no—
38C32×C71C2×C412yescomplemented
39#41×C71—41no—
40C2×D4×C71—41no—
41C22×C4×C71—81no—
42Q16×C71—21no—
43SD16×C71—41no—
44D8×C71—22no—
45D8×C71—21no—
46D8×C71—41no—
47M16×C71—41no—
48M16×C71—81no—
49C2×C8×C71—81no—
50C2×C8×C71D811yescharacteristic
51C2×C8×C71C2×D411yescharacteristic
52C4⋊C4×C71—41no—
53C22⋊C4×C71—81no—
54C42×C71D811yescharacteristic
55C16×C71—21no—
56C16×C71C22×C411yescharacteristic
57C16×C71#4111yescharacteristic
58C23×C71—81no—
59Q8×C71—41no—
60D4×C71—42no—
61D4×C71—41no—
62D4×C71—81no—
63D4×C71—81no—
64C2×C4×C71—21no—
65C2×C4×C71—81no—
66C2×C4×C71—81no—
67C2×C4×C71—162no—
68C2×C4×C71C2×D811yescharacteristic
69C8×C71—21no—
70C8×C71—81no—
71C8×C71C4×D411yescharacteristic
72C8×C71#13511yescharacteristic
73C22×C71—81no—
74C22×C71—82no—
75C22×C71—161no—
76C22×C71—161no—
77C22×C71#50911yescharacteristic
78C4×C71—41no—
79C4×C71—81no—
80C4×C71—161no—
81C4×C71C4×D811yescharacteristic
82C4×C71#50911yescharacteristic
83#13658C7111yesmaximal, characteristic, direct product factor, Sylow
84C2×C71—21no—
85C2×C71—161no—
86C2×C71—162no—
87C2×C71#217711yescharacteristic
88#2263C2×C7111yescharacteristic
89#2179C2×C7111yescharacteristic, complemented
90#2177C2×C7111yescharacteristic, complemented
91#1426C2×C7111yescharacteristic, complemented
92#1419C2×C7111yescharacteristic, complemented
93#1418C2×C7111yescharacteristic, complemented
94#1398C2×C7111yescharacteristic, complemented
95C71#1365811yescharacteristic, direct product factor, Sylow
96#508C22×C7111yescharacteristic
97C2×D16C22×C7111yescharacteristic
98#443—21no—
99C4×D8—21no—
100SD64C4×C7112yescomplemented
101D32C4×C7112yescomplemented
102M64C22×C7111yescharacteristic
103M64C22×C7111yescharacteristic
104#368C22×C7111yescharacteristic
105#367C22×C7111yescharacteristic
106#359—21no—
107#357—21no—
108#346C22×C7111yescharacteristic
109#135—21no—
110C2×D8—21no—
111#131—41no—
112C4×D4—41no—
113Q32C2×C4×C7111yescharacteristic, complemented
114SD32—21no—
115D16—21no—
116D16C2×C4×C7112yes—
117D16C2×C4×C7111yescharacteristic, complemented
118C2×C16D4×C7111yescharacteristic
119C2×C16C23×C7111yescharacteristic
120#108—21no—
121#107—21no—
122#104—41no—
123#102—41no—
124C4×C8D4×C7111yescharacteristic
125C32—21no—
126C32C2×C4×C7112yescomplemented
127#41—41no—
128C2×D4—41no—
129C22×C4—81no—
130Q16—21no—
131SD16—41no—
132D8—22no—
133D8—21no—
134D8—41no—
135M16—41no—
136M16—81no—
137C2×C8—81no—
138C2×C8D8×C7111yescharacteristic
139C2×C8C2×D4×C7111yescharacteristic
140C4⋊C4—41no—
141C22⋊C4—81no—
142C42D8×C7111yescharacteristic
143C16—21no—
144C16C22×C4×C7111yescharacteristic
145C16#41×C7111yescharacteristic
146C23—81no—
147Q8—41no—
148D4—42no—
149D4—41no—
150D4—81no—
151D4—81no—
152C2×C4—21no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4—162no—
156C2×C4C2×D8×C7111yescharacteristic
157C8—21no—
158C8—81no—
159C8C4×D4×C7111yescharacteristic
160C8#135×C7111yescharacteristic
161C22—81no—
162C22—82no—
163C22—161no—
164C22—161no—
165C22#509×C7111yescharacteristic
166C4—41no—
167C4—81no—
168C4—161no—
169C4C4×D8×C7111yescharacteristic
170C4#509×C7111yescharacteristic
171C2—21no—
172C2—162no—
173C2—161no—
174C2C71×#217711yescharacteristic
subgroupG/H
center, socleC2×C71#2177
commutatorC16C22×C4×C71
Frattini, cosocleC2×C16C23×C71
prime radicalC71×#2263C2
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C2×C4×C71 ◁ C4×C8×C71 ◁ #346×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2179C2semidirect
3. C71×#2177C2semidirect
4. C71×#1426C2semidirect
5. C71×#1419C2semidirect
6. C71×#1418C2semidirect
7. C71×#1398C2semidirect
12. SD64×C71C4semidirect
13. D32×C71C4semidirect
25. Q32×C71C2×C4semidirect
29. D16×C71C2×C4semidirect
38. C32×C71C2×C4semidirect
83. #13658C71direct (N × H)
89. #2179C2×C71semidirect
90. #2177C2×C71semidirect
91. #1426C2×C71semidirect
92. #1419C2×C71semidirect
93. #1418C2×C71semidirect
94. #1398C2×C71semidirect
95. C71#13658direct (N × H)
100. SD64C4×C71semidirect
101. D32C4×C71semidirect
113. Q32C2×C4×C71semidirect
117. D16C2×C4×C71semidirect
126. C32C2×C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#22632
C71×#21792
C71×#21772
C71×#14262
C71×#14192
C71×#14182
C71×#13982
#1365871