Finite Group Databasebeta

← C137×#12112 #46861498

Subgroups — C137×#12112 #46861498

Order 35072.

390 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 74 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#3202×C137C211yesmaximal, characteristic
2#2188×C137C212yesmaximal, complemented
3#2020×C137C211yesmaximal, characteristic
4#1350×C137C212yesmaximal, complemented
5#1315×C137C211yesmaximal, characteristic
6C2×#137×C137—21no—
7C22×Q16×C137C2211yescharacteristic
8#536×C137—21no—
9C2×Q32×C137—22no—
10#485×C137—22no—
11#470×C137—21no—
12#467×C137C2212yes—
13#451×C137—22no—
14C2×#101×C137—21no—
15#408×C137C2211yescharacteristic
16#358×C137—22no—
17C22⋊C16×C137C2212yes—
18#341×C137C2211yescharacteristic
19C2×#41×C137D411yescharacteristic
20C22×Q8×C137—21no—
21C22×Q8×C137—21no—
22#137×C137—42no—
23#137×C137—42no—
24C2×Q16×C137—21no—
25C2×Q16×C137—22no—
26C2×Q16×C137—42no—
27C2×Q16×C137—41no—
28C2×Q16×C137D412yes—
29C2×SD16×C137—41no—
30C2×M16×C137—21no—
31C2×M16×C137D411yescharacteristic
32C22×C8×C137C2311yescharacteristic
33C4⋊Q8×C137—41no—
34#124×C137—41no—
35#123×C137—41no—
36#123×C137—41no—
37C22⋊Q8×C137—22no—
38C22⋊Q8×C137—22no—
39#117×C137—21no—
40Q32×C137—44no—
41C2×C16×C137—22no—
42#107×C137D412yes—
43#106×C137—21no—
44#103×C137—22no—
45#103×C137—22no—
46#101×C137—42no—
47C22⋊C8×C137—22no—
48#41×C137—41no—
49#41×C137—41no—
50C2×Q8×C137—22no—
51C2×Q8×C137—22no—
52C2×Q8×C137—41no—
53C2×Q8×C137—41no—
54C2×Q8×C137—41no—
55C2×Q8×C137—41no—
56C2×Q8×C137—41no—
57C2×Q8×C137—41no—
58C2×Q8×C137—81no—
59C2×Q8×C137—81no—
60C2×Q8×C137D811yescharacteristic
61C2×D4×C137—21no—
62C2×D4×C137D811yescharacteristic
63C22×C4×C137—21no—
64C22×C4×C137—41no—
65C22×C4×C137—41no—
66C22×C4×C137C2×D411yescharacteristic
67Q16×C137—24no—
68Q16×C137—42no—
69Q16×C137—42no—
70Q16×C137—44no—
71Q16×C137—81no—
72SD16×C137—44no—
73M16×C137—22no—
74M16×C137—41no—
75M16×C137—41no—
76C2×C8×C137—21no—
77C2×C8×C137—41no—
78C2×C8×C137—41no—
79C2×C8×C137C2×D412yes—
80C4⋊C4×C137—22no—
81C4⋊C4×C137—42no—
82C4⋊C4×C137—42no—
83C22⋊C4×C137—41no—
84C22⋊C4×C137—42no—
85C22⋊C4×C137—42no—
86C42×C137—41no—
87C16×C137—42no—
88C23×C137—21no—
89C23×C137AGL(1,ℤ/8)11yescharacteristic
90Q8×C137—41no—
91Q8×C137—42no—
92Q8×C137—42no—
93Q8×C137—42no—
94Q8×C137—42no—
95Q8×C137—81no—
96Q8×C137—82no—
97Q8×C137—81no—
98D4×C137—41no—
99D4×C137—81no—
100C2×C4×C137—21no—
101C2×C4×C137—21no—
102C2×C4×C137—21no—
103C2×C4×C137—41no—
104C2×C4×C137—41no—
105C2×C4×C137—42no—
106C2×C4×C137—42no—
107C2×C4×C137—42no—
108C2×C4×C137—81no—
109C2×C4×C137—82no—
110C2×C4×C137—81no—
111C2×C4×C137#12011yescharacteristic
112C2×C4×C137C2×D811yescharacteristic
113C8×C137—22no—
114C8×C137—42no—
115C8×C137—41no—
116C22×C137—21no—
117C22×C137—22no—
118C22×C137—41no—
119C22×C137—42no—
120C22×C137#44711yescharacteristic
121C4×C137—21no—
122C4×C137—41no—
123C4×C137—82no—
124C4×C137—82no—
125C4×C137—82no—
126C4×C137#45311yescharacteristic
127C4×C137B(4,𝔽2)11yescharacteristic
128C2×C137—41no—
129C2×C137—81no—
130C2×C137#159511yescharacteristic
131C2×C137#219512yes—
132#12112C13711yesmaximal, characteristic, direct product factor, Sylow
133C137#1211211yescharacteristic, direct product factor, Sylow
134#3202C2×C13711yescharacteristic
135#2188C2×C13712yescomplemented
136#2020C2×C13711yescharacteristic
137#1350C2×C13712yescomplemented
138#1315C2×C13711yescharacteristic
139C2×#137—21no—
140C22×Q16C22×C13711yescharacteristic
141#536—21no—
142C2×Q32—22no—
143#485—22no—
144#470—21no—
145#467C22×C13712yes—
146#451—22no—
147C2×#101—21no—
148#408C22×C13711yescharacteristic
149#358—22no—
150C22⋊C16C22×C13712yes—
151#341C22×C13711yescharacteristic
152C2×#41D4×C13711yescharacteristic
153C22×Q8—21no—
154C22×Q8—21no—
155#137—42no—
156#137—42no—
157C2×Q16—21no—
158C2×Q16—22no—
159C2×Q16—42no—
160C2×Q16—41no—
161C2×Q16D4×C13712yes—
162C2×SD16—41no—
163C2×M16—21no—
164C2×M16D4×C13711yescharacteristic
165C22×C8C23×C13711yescharacteristic
166C4⋊Q8—41no—
167#124—41no—
168#123—41no—
169#123—41no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172#117—21no—
173Q32—44no—
174C2×C16—22no—
175#107D4×C13712yes—
176#106—21no—
177#103—22no—
178#103—22no—
179#101—42no—
180C22⋊C8—22no—
181#41—41no—
182#41—41no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8D8×C13711yescharacteristic
194C2×D4—21no—
195C2×D4D8×C13711yescharacteristic
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C2×D4×C13711yescharacteristic
200Q16—24no—
201Q16—42no—
202Q16—42no—
203Q16—44no—
204Q16—81no—
205SD16—44no—
206M16—22no—
207M16—41no—
208M16—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8C2×D4×C13712yes—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C42—41no—
220C16—42no—
221C23—21no—
222C23AGL(1,ℤ/8)×C13711yescharacteristic
223Q8—41no—
224Q8—42no—
225Q8—42no—
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—82no—
230Q8—81no—
231D4—41no—
232D4—81no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4#120×C13711yescharacteristic
245C2×C4C2×D8×C13711yescharacteristic
246C8—22no—
247C8—42no—
248C8—41no—
249C22—21no—
250C22—22no—
251C22—41no—
252C22—42no—
253C22#447×C13711yescharacteristic
254C4—21no—
255C4—41no—
256C4—82no—
257C4—82no—
258C4—82no—
259C4#453×C13711yescharacteristic
260C4B(4,𝔽2)×C13711yescharacteristic
261C2—41no—
262C2—81no—
263C2#1595×C13711yescharacteristic
264C2#2195×C13712yes—
subgroupG/H
center, socleC22×C137#447
commutator, Frattini, cosocleC22×C8C23×C137
prime radicalC2×#41×C137D4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ #408×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2188×C137C2semidirect
4. #1350×C137C2semidirect
132. #12112C137direct (N × H)
133. C137#12112direct (N × H)
135. #2188C2×C137semidirect
137. #1350C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3202×C1372
#2188×C1372
#2020×C1372
#1350×C1372
#1315×C1372
#12112137