Finite Group Databasebeta

← C137×#10216 #46859602

Subgroups — C137×#10216 #46859602

Order 35072.

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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#3255×C137C211yesmaximal, characteristic, complemented
2#3235×C137C212yesmaximal
3#1963×C137C211yesmaximal, characteristic
4#1506×C137C212yesmaximal, complemented
5#1307×C137C211yesmaximal, characteristic, complemented
6#533×C137C2212yes—
7#532×C137C2211yescharacteristic
8#501×C137—22no—
9#490×C137—22no—
10#482×C137—24no—
11#476×C137—22no—
12#475×C137—22no—
13#474×C137—22no—
14#470×C137—22no—
15#462×C137—22no—
16#432×C137C2212yes—
17#423×C137C2211yescharacteristic
18C2×#104×C137—22no—
19#376×C137C2211yescharacteristic
20#328×C137—24no—
21C2×#41×C137—21no—
22C2×Q16×C137—22no—
23C2×M16×C137D412yes—
24C4⋊Q8×C137D412yes—
25C4⋊Q8×C137D412yes—
26C4⋊Q8×C137D411yescharacteristic
27#127×C137D411yescharacteristic
28#125×C137—22no—
29#124×C137—21no—
30C22⋊Q8×C137—22no—
31C22⋊Q8×C137—22no—
32#121×C137—22no—
33C4×Q8×C137—22no—
34C4×Q8×C137—22no—
35C4×D4×C137—22no—
36#117×C137—22no—
37C2×C42×C137—21no—
38C2×C42×C137C2311yescharacteristic
39#107×C137—24no—
40#106×C137—22no—
41#106×C137—24no—
42C4⋊C8×C137—22no—
43C4⋊C8×C137—22no—
44#104×C137—44no—
45#103×C137—24no—
46#102×C137—24no—
47#102×C137—24no—
48C22⋊C8×C137D414yes—
49#97×C137—22no—
50#95×C137—22no—
51#41×C137—42no—
52#41×C137—42no—
53C2×Q8×C137—22no—
54C2×Q8×C137—21no—
55C2×Q8×C137—22no—
56C2×D4×C137—22no—
57C2×D4×C137—21no—
58C2×D4×C137—41no—
59C22×C4×C137—41no—
60C22×C4×C137—41no—
61C22×C4×C137C2×D411yescharacteristic
62C22×C4×C137C2×D412yes—
63Q16×C137—82no—
64M16×C137—42no—
65C2×C8×C137—24no—
66C2×C8×C137—22no—
67C4⋊C4×C137—22no—
68C4⋊C4×C137—22no—
69C4⋊C4×C137—22no—
70C4⋊C4×C137—22no—
71C4⋊C4×C137—22no—
72C4⋊C4×C137—42no—
73C4⋊C4×C137—42no—
74C22⋊C4×C137—42no—
75C22⋊C4×C137—42no—
76C42×C137—22no—
77C42×C137—41no—
78C42×C137—42no—
79C42×C137C2×D411yescharacteristic
80C42×C137C2×D412yes—
81C42×C137#4111yescharacteristic
82C23×C137—41no—
83C23×C137#12711yescharacteristic
84Q8×C137—42no—
85Q8×C137—42no—
86Q8×C137—42no—
87Q8×C137—82no—
88D4×C137—44no—
89D4×C137—42no—
90D4×C137—82no—
91C2×C4×C137—21no—
92C2×C4×C137—21no—
93C2×C4×C137—42no—
94C2×C4×C137—42no—
95C2×C4×C137—42no—
96C2×C4×C137—42no—
97C2×C4×C137—41no—
98C2×C4×C137—82no—
99C2×C4×C137—81no—
100C2×C4×C137—81no—
101C2×C4×C137#12012yes—
102C2×C4×C137#12011yescharacteristic
103C2×C4×C137#12112yes—
104C2×C4×C137#12111yescharacteristic
105C8×C137—44no—
106C8×C137—42no—
107C22×C137—21no—
108C22×C137—22no—
109C22×C137—81no—
110C22×C137—82no—
111C22×C137#39211yescharacteristic
112C4×C137—21no—
113C4×C137—22no—
114C4×C137—22no—
115C4×C137—82no—
116C4×C137—81no—
117C4×C137—82no—
118C4×C137—82no—
119C4×C137#45311yescharacteristic
120C4×C137#45411yescharacteristic
121C2×C137—41no—
122C2×C137—161no—
123C2×C137#200611yescharacteristic
124C2×C137#200912yes—
125#10216C13711yesmaximal, characteristic, direct product factor, Sylow
126C137#1021611yescharacteristic, direct product factor, Sylow
127#3255C2×C13711yescharacteristic, complemented
128#3235C2×C13712yes—
129#1963C2×C13711yescharacteristic
130#1506C2×C13712yescomplemented
131#1307C2×C13711yescharacteristic, complemented
132#533C22×C13712yes—
133#532C22×C13711yescharacteristic
134#501—22no—
135#490—22no—
136#482—24no—
137#476—22no—
138#475—22no—
139#474—22no—
140#470—22no—
141#462—22no—
142#432C22×C13712yes—
143#423C22×C13711yescharacteristic
144C2×#104—22no—
145#376C22×C13711yescharacteristic
146#328—24no—
147C2×#41—21no—
148C2×Q16—22no—
149C2×M16D4×C13712yes—
150C4⋊Q8D4×C13711yescharacteristic
151C4⋊Q8D4×C13712yes—
152C4⋊Q8D4×C13712yes—
153#127D4×C13711yescharacteristic
154#125—22no—
155#124—21no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158#121—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×D4—22no—
162#117—22no—
163C2×C42—21no—
164C2×C42C23×C13711yescharacteristic
165#107—24no—
166#106—22no—
167#106—24no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170#104—44no—
171#103—24no—
172#102—24no—
173#102—24no—
174C22⋊C8D4×C13714yes—
175#97—22no—
176#95—22no—
177#41—42no—
178#41—42no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×D4×C13711yescharacteristic
188C22×C4C2×D4×C13712yes—
189Q16—82no—
190M16—42no—
191C2×C8—24no—
192C2×C8—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—41no—
204C42—42no—
205C42C2×D4×C13711yescharacteristic
206C42C2×D4×C13712yes—
207C42#41×C13711yescharacteristic
208C23—41no—
209C23#127×C13711yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—82no—
214D4—42no—
215D4—44no—
216D4—82no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4#120×C13712yes—
228C2×C4#120×C13711yescharacteristic
229C2×C4#121×C13712yes—
230C2×C4#121×C13711yescharacteristic
231C8—44no—
232C8—42no—
233C22—21no—
234C22—22no—
235C22—81no—
236C22—82no—
237C22#392×C13711yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—82no—
245C4#453×C13711yescharacteristic
246C4#454×C13711yescharacteristic
247C2—41no—
248C2—161no—
249C2#2006×C13711yescharacteristic
250C2#2009×C13712yes—
subgroupG/H
center, socleC22×C137#392
commutator, Frattini, cosocleC2×C42C23×C137
prime radical#532×C137C22
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
1. #3255×C137C2semidirect
4. #1506×C137C2semidirect
5. #1307×C137C2semidirect
125. #10216C137direct (N × H)
126. C137#10216direct (N × H)
127. #3255C2×C137semidirect
130. #1506C2×C137semidirect
131. #1307C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3255×C1372
#3235×C1372
#1963×C1372
#1506×C1372
#1307×C1372
#10216137