Finite Group Databasebeta

← C137×#13429 #46862815

Subgroups — C137×#13429 #46862815

Order 35072.

434 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 86 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#3199×C137C211yesmaximal, characteristic
2#2134×C137C212yesmaximal, complemented
3#1914×C137C212yesmaximal, complemented
4#1858×C137C211yesmaximal, characteristic, complemented
5#1392×C137C211yesmaximal, characteristic, complemented
6#568×C137C2211yescharacteristic
7#535×C137C2212yes—
8#482×C137—22no—
9#469×C137—21no—
10#466×C137—21no—
11#447×C137—22no—
12#430×C137—21no—
13#413×C137C2212yes—
14#413×C137C2211yescharacteristic
15C2×#100×C137—22no—
16#408×C137C2211yescharacteristic
17#355×C137—24no—
18#344×C137—22no—
19#341×C137—21no—
20C2×#41×C137C2311yescharacteristic
21C22×D4×C137D412yes—
22C2×D8×C137—21no—
23C2×D8×C137—21no—
24#131×C137—42no—
25C2×M16×C137—22no—
26C2×M16×C137—21no—
27C2×M16×C137—21no—
28C2×M16×C137—22no—
29C2×M16×C137D411yescharacteristic
30C22×C8×C137D411yescharacteristic
31#127×C137—22no—
32#124×C137—22no—
33#121×C137—22no—
34#121×C137—22no—
35#120×C137—22no—
36#120×C137—22no—
37#117×C137D412yes—
38#108×C137—42no—
39#107×C137—21no—
40#106×C137—21no—
41#102×C137—22no—
42#102×C137—22no—
43#101×C137—22no—
44#101×C137D412yes—
45#100×C137—21no—
46#100×C137—22no—
47#100×C137—44no—
48C22⋊C8×C137—22no—
49C24×C137—22no—
50#41×C137—41no—
51#41×C137—41no—
52C2×Q8×C137C2×D411yescharacteristic
53C2×D4×C137—22no—
54C2×D4×C137—22no—
55C2×D4×C137—42no—
56C2×D4×C137—42no—
57C2×D4×C137—42no—
58C2×D4×C137—42no—
59C2×D4×C137#4111yescharacteristic
60C2×D4×C137#4112yes—
61C22×C4×C137C2×D412yes—
62C22×C4×C137C2×D411yescharacteristic
63D8×C137—81no—
64D8×C137—81no—
65M16×C137—24no—
66M16×C137—22no—
67M16×C137—22no—
68M16×C137—41no—
69M16×C137—41no—
70M16×C137—41no—
71M16×C137—42no—
72M16×C137—42no—
73C2×C8×C137—21no—
74C2×C8×C137—21no—
75C2×C8×C137—41no—
76C2×C8×C137—41no—
77C2×C8×C137—42no—
78C2×C8×C137—41no—
79C2×C8×C137—42no—
80C4⋊C4×C137—22no—
81C22⋊C4×C137—22no—
82C22⋊C4×C137—42no—
83C22⋊C4×C137—42no—
84C42×C137—22no—
85C23×C137—44no—
86C23×C137—42no—
87C23×C137—162no—
88C23×C137#12312yes—
89C23×C137#12411yescharacteristic
90Q8×C137—41no—
91D4×C137—41no—
92D4×C137—42no—
93D4×C137—42no—
94D4×C137—42no—
95D4×C137—82no—
96D4×C137—162no—
97D4×C137—162no—
98C2×C4×C137—21no—
99C2×C4×C137—22no—
100C2×C4×C137—21no—
101C2×C4×C137—22no—
102C2×C4×C137—42no—
103C2×C4×C137#12011yescharacteristic
104C2×C4×C137#12112yes—
105C2×C4×C137#12111yescharacteristic
106C8×C137—42no—
107C8×C137—41no—
108C8×C137—42no—
109C8×C137—41no—
110C8×C137—82no—
111C22×C137—22no—
112C22×C137—21no—
113C22×C137—41no—
114C22×C137—42no—
115C22×C137—82no—
116C22×C137—82no—
117C22×C137—82no—
118C22×C137—164no—
119C22×C137#39411yescharacteristic
120C4×C137—21no—
121C4×C137—22no—
122C4×C137—82no—
123C4×C137B(4,𝔽2)11yescharacteristic
124C4×C137#45811yescharacteristic
125C2×C137—21no—
126C2×C137—41no—
127C2×C137—42no—
128C2×C137—162no—
129C2×C137#204411yescharacteristic
130#13429C13711yesmaximal, characteristic, direct product factor, Sylow
131C137#1342911yescharacteristic, direct product factor, Sylow
132#3199C2×C13711yescharacteristic
133#2134C2×C13712yescomplemented
134#1914C2×C13712yescomplemented
135#1858C2×C13711yescharacteristic, complemented
136#1392C2×C13711yescharacteristic, complemented
137#568C22×C13711yescharacteristic
138#535C22×C13712yes—
139#482—22no—
140#469—21no—
141#466—21no—
142#447—22no—
143#430—21no—
144#413C22×C13712yes—
145#413C22×C13711yescharacteristic
146C2×#100—22no—
147#408C22×C13711yescharacteristic
148#355—24no—
149#344—22no—
150#341—21no—
151C2×#41C23×C13711yescharacteristic
152C22×D4D4×C13712yes—
153C2×D8—21no—
154C2×D8—21no—
155#131—42no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—22no—
160C2×M16D4×C13711yescharacteristic
161C22×C8D4×C13711yescharacteristic
162#127—22no—
163#124—22no—
164#121—22no—
165#121—22no—
166#120—22no—
167#120—22no—
168#117D4×C13712yes—
169#108—42no—
170#107—21no—
171#106—21no—
172#102—22no—
173#102—22no—
174#101—22no—
175#101D4×C13712yes—
176#100—21no—
177#100—22no—
178#100—44no—
179C22⋊C8—22no—
180C24—22no—
181#41—41no—
182#41—41no—
183C2×Q8C2×D4×C13711yescharacteristic
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4#41×C13711yescharacteristic
191C2×D4#41×C13712yes—
192C22×C4C2×D4×C13712yes—
193C22×C4C2×D4×C13711yescharacteristic
194D8—81no—
195D8—81no—
196M16—24no—
197M16—22no—
198M16—22no—
199M16—41no—
200M16—41no—
201M16—41no—
202M16—42no—
203M16—42no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—42no—
211C4⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—22no—
216C23—44no—
217C23—42no—
218C23—162no—
219C23#123×C13712yes—
220C23#124×C13711yescharacteristic
221Q8—41no—
222D4—41no—
223D4—42no—
224D4—42no—
225D4—42no—
226D4—82no—
227D4—162no—
228D4—162no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4#120×C13711yescharacteristic
235C2×C4#121×C13712yes—
236C2×C4#121×C13711yescharacteristic
237C8—42no—
238C8—41no—
239C8—42no—
240C8—41no—
241C8—82no—
242C22—22no—
243C22—21no—
244C22—41no—
245C22—42no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—164no—
250C22#394×C13711yescharacteristic
251C4—21no—
252C4—22no—
253C4—82no—
254C4B(4,𝔽2)×C13711yescharacteristic
255C4#458×C13711yescharacteristic
256C2—21no—
257C2—42no—
258C2—41no—
259C2—162no—
260C2#2044×C13711yescharacteristic
subgroupG/H
center, socleC2×C137#2044
commutator, Frattini, cosocleC2×#41C23×C137
prime radical#3199×C137C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C2×C4×C137 ◁ C2×#41×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2134×C137C2semidirect
3. #1914×C137C2semidirect
4. #1858×C137C2semidirect
5. #1392×C137C2semidirect
130. #13429C137direct (N × H)
131. C137#13429direct (N × H)
133. #2134C2×C137semidirect
134. #1914C2×C137semidirect
135. #1858C2×C137semidirect
136. #1392C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3199×C1372
#2134×C1372
#1914×C1372
#1858×C1372
#1392×C1372
#13429137