Finite Group Databasebeta

← C137×#20302 #46869688

Subgroups — C137×#20302 #46869688

Order 35072.

690 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 198 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#3325×C137C214yesmaximal, complemented
2#1709×C137C211yesmaximal, characteristic, complemented
3#1697×C137C214yesmaximal, complemented
4#1634×C137C212yesmaximal, complemented
5#1619×C137C212yesmaximal, complemented
6#1589×C137C212yesmaximal, complemented
7#555×C137C2212yes—
8D42×C137C2212yescomplemented
9#494×C137C2211yescharacteristic, complemented
10#493×C137C2211yescharacteristic
11#486×C137C2214yescomplemented
12#482×C137—24no—
13#476×C137C2214yescomplemented
14#469×C137—24no—
15#459×C137C2214yes—
16#447×C137—24no—
17C4×D8×C137C2212yes—
18#436×C137C2214yescomplemented
19#418×C137—24no—
20#335×C137C2212yescomplemented
21#333×C137C2212yescomplemented
22#331×C137C2212yescomplemented
23D4⋊C8×C137C2214yescomplemented
24C82×C137C2211yescharacteristic, complemented
25C22×D4×C137—24no—
26C2×Q16×C137—41no—
27C2×D8×C137—24no—
28C2×D8×C137—41no—
29C2×D8×C137D412yes—
30C2×M16×C137—24no—
31C4⋊Q8×C137C2312yes—
32#127×C137C2312yes—
33#124×C137—42no—
34#121×C137—24no—
35#121×C137—24no—
36#120×C137—42no—
37C4×D4×C137C2314yes—
38#117×C137—24no—
39#107×C137D412yes—
40#106×C137—24no—
41C4⋊C8×C137C2314yes—
42#102×C137—24no—
43#102×C137—24no—
44#102×C137—24no—
45#102×C137—24no—
46#102×C137D414yes—
47C22⋊C8×C137—24no—
48C4×C8×C137C2311yescharacteristic
49C4×C8×C137C2312yes—
50C24×C137—42no—
51C2×Q8×C137—22no—
52C2×D4×C137—24no—
53C2×D4×C137—22no—
54C2×D4×C137—44no—
55C2×D4×C137—48no—
56C2×D4×C137—42no—
57C2×D4×C137—84no—
58C2×D4×C137C2×D414yes—
59C22×C4×C137—24no—
60Q16×C137—82no—
61D8×C137—42no—
62D8×C137—82no—
63D8×C137—84no—
64M16×C137—84no—
65C2×C8×C137—21no—
66C2×C8×C137—24no—
67C2×C8×C137C2×D412yes—
68C2×C8×C137C2×D412yes—
69C4⋊C4×C137—24no—
70C4⋊C4×C137C2×D414yes—
71C22⋊C4×C137—24no—
72C22⋊C4×C137—42no—
73C22⋊C4×C137—44no—
74C42×C137—42no—
75C42×C137C2411yescharacteristic
76C23×C137—24no—
77C23×C137—42no—
78C23×C137—88no—
79C23×C137—162no—
80Q8×C137—82no—
81D4×C137—24no—
82D4×C137—24no—
83D4×C137—44no—
84D4×C137—44no—
85D4×C137—82no—
86D4×C137—88no—
87D4×C137—162no—
88C2×C4×C137—24no—
89C2×C4×C137—44no—
90C2×C4×C137—44no—
91C2×C4×C137—42no—
92C2×C4×C137C22×D412yes—
93C2×C4×C137H(5,𝔽2)11yescharacteristic
94C8×C137—22no—
95C8×C137—22no—
96C8×C137—22no—
97C8×C137—44no—
98C22×C137—44no—
99C22×C137—44no—
100C22×C137—44no—
101C22×C137—84no—
102C22×C137—82no—
103C22×C137—168no—
104C22×C137D4211yescharacteristic
105C4×C137—21no—
106C4×C137—44no—
107C4×C137—82no—
108C4×C137#57512yes—
109C4×C137#57612yes—
110C2×C137—84no—
111C2×C137—162no—
112C2×C137#327312yes—
113C2×C137#329211yescharacteristic
114#20302C13711yesmaximal, characteristic, direct product factor, Sylow
115C137#2030211yescharacteristic, direct product factor, Sylow
116#3325C2×C13714yescomplemented
117#1709C2×C13711yescharacteristic, complemented
118#1697C2×C13714yescomplemented
119#1634C2×C13712yescomplemented
120#1619C2×C13712yescomplemented
121#1589C2×C13712yescomplemented
122#555C22×C13712yes—
123D42C22×C13712yescomplemented
124#494C22×C13711yescharacteristic, complemented
125#493C22×C13711yescharacteristic
126#486C22×C13714yescomplemented
127#482—24no—
128#476C22×C13714yescomplemented
129#469—24no—
130#459C22×C13714yes—
131#447—24no—
132C4×D8C22×C13712yes—
133#436C22×C13714yescomplemented
134#418—24no—
135#335C22×C13712yescomplemented
136#333C22×C13712yescomplemented
137#331C22×C13712yescomplemented
138D4⋊C8C22×C13714yescomplemented
139C82C22×C13711yescharacteristic, complemented
140C22×D4—24no—
141C2×Q16—41no—
142C2×D8—24no—
143C2×D8—41no—
144C2×D8D4×C13712yes—
145C2×M16—24no—
146C4⋊Q8C23×C13712yes—
147#127C23×C13712yes—
148#124—42no—
149#121—24no—
150#121—24no—
151#120—42no—
152C4×D4C23×C13714yes—
153#117—24no—
154#107D4×C13712yes—
155#106—24no—
156C4⋊C8C23×C13714yes—
157#102—24no—
158#102—24no—
159#102—24no—
160#102—24no—
161#102D4×C13714yes—
162C22⋊C8—24no—
163C4×C8C23×C13711yescharacteristic
164C4×C8C23×C13712yes—
165C24—42no—
166C2×Q8—22no—
167C2×D4—24no—
168C2×D4—22no—
169C2×D4—44no—
170C2×D4—48no—
171C2×D4—42no—
172C2×D4—84no—
173C2×D4C2×D4×C13714yes—
174C22×C4—24no—
175Q16—82no—
176D8—42no—
177D8—82no—
178D8—84no—
179M16—84no—
180C2×C8—21no—
181C2×C8—24no—
182C2×C8C2×D4×C13712yes—
183C2×C8C2×D4×C13712yes—
184C4⋊C4—24no—
185C4⋊C4C2×D4×C13714yes—
186C22⋊C4—24no—
187C22⋊C4—42no—
188C22⋊C4—44no—
189C42—42no—
190C42C24×C13711yescharacteristic
191C23—24no—
192C23—42no—
193C23—88no—
194C23—162no—
195Q8—82no—
196D4—24no—
197D4—24no—
198D4—44no—
199D4—44no—
200D4—82no—
201D4—88no—
202D4—162no—
203C2×C4—24no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4C22×D4×C13712yes—
208C2×C4H(5,𝔽2)×C13711yescharacteristic
209C8—22no—
210C8—22no—
211C8—22no—
212C8—44no—
213C22—44no—
214C22—44no—
215C22—44no—
216C22—84no—
217C22—82no—
218C22—168no—
219C22D42×C13711yescharacteristic
220C4—21no—
221C4—44no—
222C4—82no—
223C4#575×C13712yes—
224C4#576×C13712yes—
225C2—84no—
226C2—162no—
227C2#3273×C13712yes—
228C2#3292×C13711yescharacteristic
subgroupG/H
center, socleC22×C137D42
commutator, Frattini, cosocleC42C24×C137
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3325×C137C2semidirect
2. #1709×C137C2semidirect
3. #1697×C137C2semidirect
4. #1634×C137C2semidirect
5. #1619×C137C2semidirect
6. #1589×C137C2semidirect
8. D42×C137C22semidirect
9. #494×C137C22semidirect
11. #486×C137C22semidirect
13. #476×C137C22semidirect
18. #436×C137C22semidirect
20. #335×C137C22semidirect
21. #333×C137C22semidirect
22. #331×C137C22semidirect
23. D4⋊C8×C137C22semidirect
24. C82×C137C22semidirect
114. #20302C137direct (N × H)
115. C137#20302direct (N × H)
116. #3325C2×C137semidirect
117. #1709C2×C137semidirect
118. #1697C2×C137semidirect
119. #1634C2×C137semidirect
120. #1619C2×C137semidirect
121. #1589C2×C137semidirect
123. D42C22×C137semidirect
124. #494C22×C137semidirect
126. #486C22×C137semidirect
128. #476C22×C137semidirect
133. #436C22×C137semidirect
135. #335C22×C137semidirect
136. #333C22×C137semidirect
137. #331C22×C137semidirect
138. D4⋊C8C22×C137semidirect
139. C82C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3325×C1372
#1709×C1372
#1697×C1372
#1634×C1372
#1619×C1372
#1589×C1372
#20302137