Finite Group Databasebeta

← C137×#10227 #46859613

Subgroups — C137×#10227 #46859613

Order 35072.

466 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 122 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#3229×C137C212yesmaximal
2#3162×C137C211yesmaximal, characteristic
3#1960×C137C211yesmaximal, characteristic
4#1518×C137C212yesmaximal
5#1281×C137C211yesmaximal, characteristic
6#533×C137C2212yes—
7C2×C4⋊Q8×C137C2211yescharacteristic
8#501×C137—22no—
9#501×C137—22no—
10#498×C137—22no—
11#487×C137—22no—
12#483×C137—24no—
13#475×C137—24no—
14#431×C137C2212yes—
15#427×C137—22no—
16C2×#103×C137—22no—
17C2×#97×C137C2211yescharacteristic
18#382×C137C2211yescharacteristic
19#328×C137—24no—
20C22×Q8×C137—21no—
21C22×C8×C137D412yes—
22C4⋊Q8×C137—22no—
23C4⋊Q8×C137—21no—
24C4⋊Q8×C137D412yes—
25C4⋊Q8×C137D412yes—
26C4⋊Q8×C137D412yes—
27#125×C137—22no—
28C22⋊Q8×C137—22no—
29C22⋊Q8×C137—22no—
30C4×Q8×C137—22no—
31C4×Q8×C137—22no—
32#117×C137—22no—
33C2×C4⋊C4×C137—22no—
34C2×C4⋊C4×C137—21no—
35C2×C42×C137C2311yescharacteristic
36#107×C137—24no—
37#106×C137—22no—
38#106×C137—22no—
39#106×C137—28no—
40C4⋊C8×C137—22no—
41#103×C137—28no—
42#103×C137—44no—
43C22⋊C8×C137D414yes—
44#97×C137—22no—
45C4×C8×C137—22no—
46#95×C137—22no—
47C2×Q8×C137—22no—
48C2×Q8×C137—22no—
49C2×Q8×C137—22no—
50C2×Q8×C137—41no—
51C2×Q8×C137—42no—
52C2×Q8×C137—42no—
53C22×C4×C137—41no—
54C22×C4×C137—41no—
55C22×C4×C137C2×D412yes—
56C22×C4×C137C2×D411yescharacteristic
57C2×C8×C137—22no—
58C2×C8×C137—22no—
59C2×C8×C137—24no—
60C2×C8×C137SD1614yes—
61C4⋊C4×C137—24no—
62C4⋊C4×C137—22no—
63C4⋊C4×C137—22no—
64C4⋊C4×C137—22no—
65C4⋊C4×C137—22no—
66C4⋊C4×C137—42no—
67C4⋊C4×C137—42no—
68C4⋊C4×C137—42no—
69C4⋊C4×C137—42no—
70C22⋊C4×C137—42no—
71C42×C137—42no—
72C42×C137C2×D412yes—
73C42×C137C2×D411yescharacteristic
74C42×C137#4111yescharacteristic
75C23×C137#12711yescharacteristic
76Q8×C137—44no—
77Q8×C137—44no—
78Q8×C137—82no—
79Q8×C137—82no—
80C2×C4×C137—42no—
81C2×C4×C137—42no—
82C2×C4×C137—42no—
83C2×C4×C137—42no—
84C2×C4×C137—81no—
85C2×C4×C137—81no—
86C2×C4×C137—82no—
87C2×C4×C137—81no—
88C2×C4×C137#12012yes—
89C2×C4×C137#12011yescharacteristic
90C2×C4×C137#12112yes—
91C2×C4×C137#12111yescharacteristic
92C2×C4×C137C2×SD1612yes—
93C2×C4×C137#13712yes—
94C8×C137—24no—
95C8×C137—44no—
96C22×C137—22no—
97C22×C137#39211yescharacteristic
98C22×C137#49211yescharacteristic
99C22×C137#49711yescharacteristic
100C4×C137—22no—
101C4×C137—22no—
102C4×C137—82no—
103C4×C137—82no—
104C4×C137—82no—
105C4×C137—161no—
106C4×C137#44812yes—
107C4×C137#46112yes—
108C2×C137—21no—
109C2×C137—21no—
110C2×C137#200912yes—
111C2×C137#201311yescharacteristic
112#10227C13711yesmaximal, characteristic, direct product factor, Sylow
113C137#1022711yescharacteristic, direct product factor, Sylow
114#3229C2×C13712yes—
115#3162C2×C13711yescharacteristic
116#1960C2×C13711yescharacteristic
117#1518C2×C13712yes—
118#1281C2×C13711yescharacteristic
119#533C22×C13712yes—
120C2×C4⋊Q8C22×C13711yescharacteristic
121#501—22no—
122#501—22no—
123#498—22no—
124#487—22no—
125#483—24no—
126#475—24no—
127#431C22×C13712yes—
128#427—22no—
129C2×#103—22no—
130C2×#97C22×C13711yescharacteristic
131#382C22×C13711yescharacteristic
132#328—24no—
133C22×Q8—21no—
134C22×C8D4×C13712yes—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8D4×C13712yes—
138C4⋊Q8D4×C13712yes—
139C4⋊Q8D4×C13712yes—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C42C23×C13711yescharacteristic
149#107—24no—
150#106—22no—
151#106—22no—
152#106—28no—
153C4⋊C8—22no—
154#103—28no—
155#103—44no—
156C22⋊C8D4×C13714yes—
157#97—22no—
158C4×C8—22no—
159#95—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4C2×D4×C13712yes—
169C22×C4C2×D4×C13711yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8SD16×C13714yes—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C22⋊C4—42no—
184C42—42no—
185C42C2×D4×C13712yes—
186C42C2×D4×C13711yescharacteristic
187C42#41×C13711yescharacteristic
188C23#127×C13711yescharacteristic
189Q8—44no—
190Q8—44no—
191Q8—82no—
192Q8—82no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4#120×C13712yes—
202C2×C4#120×C13711yescharacteristic
203C2×C4#121×C13712yes—
204C2×C4#121×C13711yescharacteristic
205C2×C4C2×SD16×C13712yes—
206C2×C4#137×C13712yes—
207C8—24no—
208C8—44no—
209C22—22no—
210C22#392×C13711yescharacteristic
211C22#492×C13711yescharacteristic
212C22#497×C13711yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4—161no—
219C4#448×C13712yes—
220C4#461×C13712yes—
221C2—21no—
222C2—21no—
223C2#2009×C13712yes—
224C2#2013×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#392
commutator, Frattini, cosocleC2×C42C23×C137
prime radicalC23×C137#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
112. #10227C137direct (N × H)
113. C137#10227direct (N × H)

Maximal subgroups

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

subgroupindex
#3229×C1372
#3162×C1372
#1960×C1372
#1518×C1372
#1281×C1372
#10227137