Finite Group Databasebeta

← C137×#19058 #46868444

Subgroups — C137×#19058 #46868444

Order 35072.

840 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 222 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#3550×C137C211yesmaximal, characteristic
2#3237×C137C212yesmaximal, complemented
3#3135×C137C211yesmaximal, characteristic
4#1664×C137C218yesmaximal, complemented
5#1536×C137C211yesmaximal, characteristic
6#1531×C137C212yesmaximal, complemented
7#563×C137C2214yes—
8#559×C137—21no—
9Q82×C137—21no—
10#556×C137—22no—
11D4×Q8×C137—22no—
12#542×C137—24no—
13#536×C137—22no—
14#533×C137C2211yescharacteristic
15#533×C137C2212yes—
16#532×C137C2211yescharacteristic
17#501×C137C2214yes—
18#491×C137—21no—
19#491×C137C2214yes—
20#490×C137—21no—
21#489×C137C2214yes—
22#485×C137—24no—
23#477×C137—22no—
24#475×C137—22no—
25#432×C137C2212yes—
26#416×C137—22no—
27C2×#97×C137C2211yescharacteristic
28#330×C137C2218yes—
29#329×C137C2214yes—
30C2×#41×C137D412yes—
31C22×Q8×C137—22no—
32C22×C8×C137—21no—
33C4⋊Q8×C137—24no—
34C4⋊Q8×C137—24no—
35C4⋊Q8×C137D411yescharacteristic
36C4⋊Q8×C137D412yes—
37C4⋊Q8×C137D412yes—
38C4⋊Q8×C137C2312yes—
39#127×C137D411yescharacteristic
40#126×C137—28no—
41#125×C137—21no—
42#125×C137C2314yes—
43#124×C137—24no—
44#124×C137—24no—
45#124×C137C2312yes—
46#123×C137—24no—
47#123×C137—24no—
48C22⋊Q8×C137—24no—
49C22⋊Q8×C137—22no—
50C22⋊Q8×C137—22no—
51C22⋊Q8×C137—28no—
52C22⋊Q8×C137—22no—
53#121×C137—22no—
54C4×Q8×C137—24no—
55C4×Q8×C137—22no—
56C4×Q8×C137—22no—
57C4×Q8×C137—22no—
58C4×D4×C137—24no—
59C4×D4×C137—22no—
60#117×C137—22no—
61#117×C137—21no—
62#117×C137D414yes—
63C2×C42×C137C2311yescharacteristic
64#107×C137—24no—
65#106×C137—24no—
66C4⋊C8×C137—22no—
67#103×C137—24no—
68#103×C137—24no—
69#103×C137—24no—
70#102×C137—24no—
71C22⋊C8×C137—22no—
72#97×C137—21no—
73#97×C137C2312yes—
74#97×C137C2314yes—
75#41×C137—44no—
76C2×Q8×C137—24no—
77C2×Q8×C137—22no—
78C2×Q8×C137—22no—
79C2×Q8×C137—22no—
80C2×Q8×C137—84no—
81C2×Q8×C137C2×D412yes—
82C2×D4×C137—22no—
83C2×D4×C137—22no—
84C2×D4×C137C2×D412yes—
85C22×C4×C137—24no—
86C22×C4×C137—22no—
87C22×C4×C137C2×D411yescharacteristic
88C22×C4×C137C2×D412yes—
89C2×C8×C137—22no—
90C2×C8×C137—24no—
91C2×C8×C137—42no—
92C4⋊C4×C137—28no—
93C4⋊C4×C137—22no—
94C4⋊C4×C137—22no—
95C4⋊C4×C137—22no—
96C4⋊C4×C137—24no—
97C4⋊C4×C137—24no—
98C4⋊C4×C137—22no—
99C4⋊C4×C137—22no—
100C4⋊C4×C137—42no—
101C4⋊C4×C137C2×D414yes—
102C4⋊C4×C137C2×D414yes—
103C22⋊C4×C137—24no—
104C22⋊C4×C137—28no—
105C22⋊C4×C137—24no—
106C22⋊C4×C137—24no—
107C22⋊C4×C137—41no—
108C42×C137—24no—
109C42×C137—24no—
110C42×C137—41no—
111C42×C137C2×D411yescharacteristic
112C42×C137C2×D412yes—
113C42×C137C2411yescharacteristic
114C23×C137—22no—
115C23×C137#12011yescharacteristic
116Q8×C137—42no—
117Q8×C137—44no—
118Q8×C137—44no—
119Q8×C137—44no—
120Q8×C137—84no—
121D4×C137—42no—
122D4×C137—44no—
123D4×C137—82no—
124C2×C4×C137—22no—
125C2×C4×C137—24no—
126C2×C4×C137—22no—
127C2×C4×C137—24no—
128C2×C4×C137—42no—
129C2×C4×C137—44no—
130C2×C4×C137—42no—
131C2×C4×C137—84no—
132C2×C4×C137#12012yes—
133C2×C4×C137#12011yescharacteristic
134C2×C4×C137C22×D412yes—
135C2×C4×C137C22×D411yescharacteristic
136C8×C137—44no—
137C8×C137—42no—
138C22×C137—21no—
139C22×C137—22no—
140C22×C137—42no—
141C22×C137—44no—
142C22×C137C2×#12011yescharacteristic
143C4×C137—21no—
144C4×C137—22no—
145C4×C137—22no—
146C4×C137—42no—
147C4×C137—44no—
148C4×C137—44no—
149C4×C137—82no—
150C4×C137#45512yes—
151C2×C137—41no—
152C2×C137—82no—
153C2×C137C2×#45511yescharacteristic
154C2×C137#302012yes—
155#19058C13711yesmaximal, characteristic, direct product factor, Sylow
156C137#1905811yescharacteristic, direct product factor, Sylow
157#3550C2×C13711yescharacteristic
158#3237C2×C13712yescomplemented
159#3135C2×C13711yescharacteristic
160#1664C2×C13718yescomplemented
161#1536C2×C13711yescharacteristic
162#1531C2×C13712yescomplemented
163#563C22×C13714yes—
164#559—21no—
165Q82—21no—
166#556—22no—
167D4×Q8—22no—
168#542—24no—
169#536—22no—
170#533C22×C13711yescharacteristic
171#533C22×C13712yes—
172#532C22×C13711yescharacteristic
173#501C22×C13714yes—
174#491—21no—
175#491C22×C13714yes—
176#490—21no—
177#489C22×C13714yes—
178#485—24no—
179#477—22no—
180#475—22no—
181#432C22×C13712yes—
182#416—22no—
183C2×#97C22×C13711yescharacteristic
184#330C22×C13718yes—
185#329C22×C13714yes—
186C2×#41D4×C13712yes—
187C22×Q8—22no—
188C22×C8—21no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8D4×C13711yescharacteristic
192C4⋊Q8D4×C13712yes—
193C4⋊Q8D4×C13712yes—
194C4⋊Q8C23×C13712yes—
195#127D4×C13711yescharacteristic
196#126—28no—
197#125—21no—
198#125C23×C13714yes—
199#124—24no—
200#124—24no—
201#124C23×C13712yes—
202#123—24no—
203#123—24no—
204C22⋊Q8—24no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—28no—
208C22⋊Q8—22no—
209#121—22no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×D4—24no—
215C4×D4—22no—
216#117—22no—
217#117—21no—
218#117D4×C13714yes—
219C2×C42C23×C13711yescharacteristic
220#107—24no—
221#106—24no—
222C4⋊C8—22no—
223#103—24no—
224#103—24no—
225#103—24no—
226#102—24no—
227C22⋊C8—22no—
228#97—21no—
229#97C23×C13714yes—
230#97C23×C13712yes—
231#41—44no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—84no—
237C2×Q8C2×D4×C13712yes—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4C2×D4×C13712yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4C2×D4×C13711yescharacteristic
244C22×C4C2×D4×C13712yes—
245C2×C8—22no—
246C2×C8—24no—
247C2×C8—42no—
248C4⋊C4—28no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4C2×D4×C13714yes—
258C4⋊C4C2×D4×C13714yes—
259C22⋊C4—24no—
260C22⋊C4—28no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—41no—
264C42—24no—
265C42—24no—
266C42—41no—
267C42C2×D4×C13711yescharacteristic
268C42C2×D4×C13712yes—
269C42C24×C13711yescharacteristic
270C23—22no—
271C23#120×C13711yescharacteristic
272Q8—42no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276Q8—84no—
277D4—42no—
278D4—44no—
279D4—82no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—84no—
288C2×C4#120×C13712yes—
289C2×C4#120×C13711yescharacteristic
290C2×C4C22×D4×C13712yes—
291C2×C4C22×D4×C13711yescharacteristic
292C8—42no—
293C8—44no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—44no—
298C22C2×#120×C13711yescharacteristic
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—44no—
304C4—44no—
305C4—82no—
306C4#455×C13712yes—
307C2—41no—
308C2—82no—
309C2C2×#455×C13711yescharacteristic
310C2#3020×C13712yes—
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#532×C137C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3237×C137C2semidirect
4. #1664×C137C2semidirect
6. #1531×C137C2semidirect
155. #19058C137direct (N × H)
156. C137#19058direct (N × H)
158. #3237C2×C137semidirect
160. #1664C2×C137semidirect
162. #1531C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3550×C1372
#3237×C1372
#3135×C1372
#1664×C1372
#1536×C1372
#1531×C1372
#19058137