Finite Group Databasebeta

← C137×#33983 #46883369

Subgroups — C137×#33983 #46883369

Order 35072.

462 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 202 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#3417×C137C211yesmaximal, characteristic, complemented
2#3415×C137C211yesmaximal, characteristic, complemented
3#2264×C137C213yesmaximal, complemented
4#2263×C137C213yesmaximal, complemented
5#2262×C137C213yesmaximal, complemented
6C2×SD64×C137C213yesmaximal, complemented
7#2258×C137C211yesmaximal, characteristic, complemented
8#578×C137—21no—
9#576×C137—21no—
10#510×C137—23no—
11#509×C137—23no—
12#508×C137C2213yes—
13#508×C137C2213yescomplemented
14C2×Q32×C137C2213yescomplemented
15C2×D16×C137C2213yes—
16#504×C137C2211yescharacteristic
17Q64×C137C2213yescomplemented
18SD64×C137C2213yescomplemented
19SD64×C137C2216yescomplemented
20SD64×C137C2211yescharacteristic, complemented
21D32×C137C2213yescomplemented
22M64×C137C2213yescomplemented
23C2×C32×C137C2213yescomplemented
242−1+4×C137—41no—
25H(5,𝔽2)×C137—41no—
26#137×C137—43no—
27AGL(1,ℤ/8)×C137—43no—
28#135×C137—23no—
29#135×C137—23no—
30C2×Q16×C137—23no—
31C2×D8×C137—23no—
32#131×C137D411yescharacteristic
33Q32×C137—23no—
34Q32×C137C2313yes—
35Q32×C137C2311yescharacteristic
36SD32×C137—23no—
37SD32×C137—23no—
38D16×C137—23no—
39D16×C137C2311yescharacteristic
40D16×C137C2313yes—
41M32×C137D413yes—
42C2×C16×C137C2313yes—
43C32×C137C2311yescharacteristic
44C32×C137C2313yes—
45#41×C137—43no—
46#41×C137—43no—
47#41×C137—81no—
48#41×C137—83no—
49#41×C137D811yescharacteristic
50C2×Q8×C137—43no—
51C2×Q8×C137—81no—
52C2×D4×C137—43no—
53C2×D4×C137—83no—
54Q16×C137—21no—
55Q16×C137—23no—
56Q16×C137—43no—
57SD16×C137—43no—
58SD16×C137—43no—
59D8×C137—23no—
60D8×C137—21no—
61D8×C137—43no—
62M16×C137D813yes—
63C2×C8×C137C2×D413yes—
64C16×C137C2×D413yes—
65C16×C137C2411yescharacteristic
66C23×C137—83no—
67Q8×C137—41no—
68Q8×C137—43no—
69Q8×C137—83no—
70Q8×C137D1611yescharacteristic
71D4×C137—43no—
72D4×C137—41no—
73D4×C137—83no—
74D4×C137—83no—
75D4×C137—83no—
76D4×C137D1613yes—
77C2×C4×C137—83no—
78C2×C4×C137—83no—
79C2×C4×C137—83no—
80C2×C4×C137C2×D813yes—
81C8×C137C2×D813yes—
82C8×C137C22×D411yescharacteristic
83C22×C137—81no—
84C22×C137—83no—
85C22×C137—323no—
86C22×C137C2×D1613yes—
87C4×C137—81no—
88C4×C137—83no—
89C4×C137C2×D1613yes—
90C4×C137C22×D811yescharacteristic
91C2×C137—23no—
92C2×C137—161no—
93C2×C137—163no—
94C2×C137C22×D1611yescharacteristic
95#33983C13711yesmaximal, characteristic, direct product factor, Sylow
96C137#3398311yescharacteristic, direct product factor, Sylow
97#3417C2×C13711yescharacteristic, complemented
98#3415C2×C13711yescharacteristic, complemented
99#2264C2×C13713yescomplemented
100#2263C2×C13713yescomplemented
101#2262C2×C13713yescomplemented
102C2×SD64C2×C13713yescomplemented
103#2258C2×C13711yescharacteristic, complemented
104#578—21no—
105#576—21no—
106#510—23no—
107#509—23no—
108#508C22×C13713yes—
109#508C22×C13713yescomplemented
110C2×Q32C22×C13713yescomplemented
111C2×D16C22×C13713yes—
112#504C22×C13711yescharacteristic
113Q64C22×C13713yescomplemented
114SD64C22×C13713yescomplemented
115SD64C22×C13716yescomplemented
116SD64C22×C13711yescharacteristic, complemented
117D32C22×C13713yescomplemented
118M64C22×C13713yescomplemented
119C2×C32C22×C13713yescomplemented
1202−1+4—41no—
121H(5,𝔽2)—41no—
122#137—43no—
123AGL(1,ℤ/8)—43no—
124#135—23no—
125#135—23no—
126C2×Q16—23no—
127C2×D8—23no—
128#131D4×C13711yescharacteristic
129Q32—23no—
130Q32C23×C13713yes—
131Q32C23×C13711yescharacteristic
132SD32—23no—
133SD32—23no—
134D16—23no—
135D16C23×C13711yescharacteristic
136D16C23×C13713yes—
137M32D4×C13713yes—
138C2×C16C23×C13713yes—
139C32C23×C13711yescharacteristic
140C32C23×C13713yes—
141#41—43no—
142#41—43no—
143#41—81no—
144#41—83no—
145#41D8×C13711yescharacteristic
146C2×Q8—43no—
147C2×Q8—81no—
148C2×D4—43no—
149C2×D4—83no—
150Q16—21no—
151Q16—23no—
152Q16—43no—
153SD16—43no—
154SD16—43no—
155D8—23no—
156D8—21no—
157D8—43no—
158M16D8×C13713yes—
159C2×C8C2×D4×C13713yes—
160C16C2×D4×C13713yes—
161C16C24×C13711yescharacteristic
162C23—83no—
163Q8—41no—
164Q8—43no—
165Q8—83no—
166Q8D16×C13711yescharacteristic
167D4—43no—
168D4—41no—
169D4—83no—
170D4—83no—
171D4—83no—
172D4D16×C13713yes—
173C2×C4—83no—
174C2×C4—83no—
175C2×C4—83no—
176C2×C4C2×D8×C13713yes—
177C8C2×D8×C13713yes—
178C8C22×D4×C13711yescharacteristic
179C22—81no—
180C22—83no—
181C22—323no—
182C22C2×D16×C13713yes—
183C4—81no—
184C4—83no—
185C4C2×D16×C13713yes—
186C4C22×D8×C13711yescharacteristic
187C2—23no—
188C2—163no—
189C2—161no—
190C2C22×D16×C13711yescharacteristic
subgroupG/H
center, socleC2×C137C22×D16
commutator, Frattini, cosocleC16C24×C137
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ #41×C137 ◁ #131×C137 ◁ #504×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3417×C137C2semidirect
2. #3415×C137C2semidirect
3. #2264×C137C2semidirect
4. #2263×C137C2semidirect
5. #2262×C137C2semidirect
6. C2×SD64×C137C2semidirect
7. #2258×C137C2semidirect
13. #508×C137C22semidirect
14. C2×Q32×C137C22semidirect
17. Q64×C137C22semidirect
18. SD64×C137C22semidirect
19. SD64×C137C22semidirect
20. SD64×C137C22semidirect
21. D32×C137C22semidirect
22. M64×C137C22semidirect
23. C2×C32×C137C22semidirect
95. #33983C137direct (N × H)
96. C137#33983direct (N × H)
97. #3417C2×C137semidirect
98. #3415C2×C137semidirect
99. #2264C2×C137semidirect
100. #2263C2×C137semidirect
101. #2262C2×C137semidirect
102. C2×SD64C2×C137semidirect
103. #2258C2×C137semidirect
109. #508C22×C137semidirect
110. C2×Q32C22×C137semidirect
113. Q64C22×C137semidirect
114. SD64C22×C137semidirect
115. SD64C22×C137semidirect
116. SD64C22×C137semidirect
117. D32C22×C137semidirect
118. M64C22×C137semidirect
119. C2×C32C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3417×C1372
#3415×C1372
#2264×C1372
#2263×C1372
#2262×C1372
C2×SD64×C1372
#2258×C1372
#33983137