Finite Group Databasebeta

← C137×#11838 #46861224

Subgroups — C137×#11838 #46861224

Order 35072.

280 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 90 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#3185×C137C211yesmaximal, characteristic
2#2165×C137C211yesmaximal, characteristic
3#1841×C137C211yesmaximal, characteristic
4#1342×C137C212yesmaximal
5#1340×C137C212yesmaximal
6#533×C137—21no—
7#498×C137C411yescharacteristic, complemented
8#494×C137C411yescharacteristic, complemented
9#487×C137—21no—
10#470×C137—21no—
11#470×C137C2212yes—
12#441×C137—21no—
13#441×C137C412yescomplemented
14C4×M16×C137C2211yescharacteristic
15#364×C137—21no—
16#349×C137C2212yes—
17#347×C137—21no—
18#344×C137—21no—
19#339×C137—21no—
20#338×C137C2211yescharacteristic
21H(3,ℤ/4)×C137C2211yescharacteristic
22C2×Q16×C137—21no—
23C2×Q16×C137C2×C412yes—
24C2×M16×C137—21no—
25C2×M16×C137D411yescharacteristic
26C2×M16×C137C2311yescharacteristic
27C4⋊Q8×C137—21no—
28C4⋊Q8×C137—21no—
29#125×C137—41no—
30C22⋊Q8×C137—22no—
31C22⋊Q8×C137—41no—
32C4×Q8×C137—22no—
33C4×Q8×C137—41no—
34#117×C137—21no—
35#117×C137—41no—
36C2×C42×C137D411yescharacteristic
37M32×C137—22no—
38#106×C137—21no—
39#106×C137C2×C412yes—
40#103×C137—22no—
41#103×C137—22no—
42#97×C137C2×C411yescharacteristic
43#97×C137D411yescharacteristic
44C4×C8×C137C2×C411yescharacteristic
45C4×C8×C137D411yescharacteristic
46C2×Q8×C137—22no—
47C2×Q8×C137—41no—
48C22×C4×C137—21no—
49C22×C4×C137C2×D411yescharacteristic
50Q16×C137—42no—
51Q16×C137—42no—
52M16×C137—41no—
53M16×C137—41no—
54M16×C137—41no—
55M16×C137—41no—
56C2×C8×C137—41no—
57C2×C8×C137C22⋊C411yescharacteristic
58C2×C8×C137C22⋊C411yescharacteristic
59C2×C8×C137C22×C411yescharacteristic
60C2×C8×C137C2×D411yescharacteristic
61C4⋊C4×C137—22no—
62C4⋊C4×C137—41no—
63C4⋊C4×C137—41no—
64C4⋊C4×C137—42no—
65C4⋊C4×C137—41no—
66C4⋊C4×C137—41no—
67C4⋊C4×C137—42no—
68C22⋊C4×C137—42no—
69C22⋊C4×C137—41no—
70C42×C137—21no—
71C42×C137—42no—
72C42×C137—41no—
73C42×C137C22⋊C411yescharacteristic
74C42×C137C22⋊C411yescharacteristic
75C16×C137—42no—
76C23×C137AGL(1,ℤ/8)11yescharacteristic
77Q8×C137—42no—
78Q8×C137—42no—
79Q8×C137—82no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137—21no—
84C2×C4×C137—42no—
85C2×C4×C137—42no—
86C2×C4×C137—42no—
87C2×C4×C137—81no—
88C2×C4×C137C2×C22⋊C411yescharacteristic
89C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
90C8×C137—21no—
91C8×C137—21no—
92C8×C137—21no—
93C8×C137—21no—
94C8×C137—81no—
95C22×C137—21no—
96C22×C137—41no—
97C22×C137#41811yescharacteristic
98C4×C137—22no—
99C4×C137—21no—
100C4×C137—41no—
101C4×C137—82no—
102C4×C137—82no—
103C4×C137—161no—
104C4×C137#41011yescharacteristic
105C4×C137#42111yescharacteristic
106C2×C137—21no—
107C2×C137—41no—
108C2×C137#151411yescharacteristic
109#11838C13711yesmaximal, characteristic, direct product factor, Sylow
110C137#1183811yescharacteristic, direct product factor, Sylow
111#3185C2×C13711yescharacteristic
112#2165C2×C13711yescharacteristic
113#1841C2×C13711yescharacteristic
114#1342C2×C13712yes—
115#1340C2×C13712yes—
116#533—21no—
117#498C4×C13711yescharacteristic, complemented
118#494C4×C13711yescharacteristic, complemented
119#487—21no—
120#470—21no—
121#470C22×C13712yes—
122#441—21no—
123#441C4×C13712yescomplemented
124C4×M16C22×C13711yescharacteristic
125#364—21no—
126#349C22×C13712yes—
127#347—21no—
128#344—21no—
129#339—21no—
130#338C22×C13711yescharacteristic
131H(3,ℤ/4)C22×C13711yescharacteristic
132C2×Q16—21no—
133C2×Q16C2×C4×C13712yes—
134C2×M16—21no—
135C2×M16D4×C13711yescharacteristic
136C2×M16C23×C13711yescharacteristic
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—41no—
144#117—21no—
145#117—41no—
146C2×C42D4×C13711yescharacteristic
147M32—22no—
148#106—21no—
149#106C2×C4×C13712yes—
150#103—22no—
151#103—22no—
152#97C2×C4×C13711yescharacteristic
153#97D4×C13711yescharacteristic
154C4×C8C2×C4×C13711yescharacteristic
155C4×C8D4×C13711yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C22×C4—21no—
159C22×C4C2×D4×C13711yescharacteristic
160Q16—42no—
161Q16—42no—
162M16—41no—
163M16—41no—
164M16—41no—
165M16—41no—
166C2×C8—41no—
167C2×C8C22⋊C4×C13711yescharacteristic
168C2×C8C22⋊C4×C13711yescharacteristic
169C2×C8C22×C4×C13711yescharacteristic
170C2×C8C2×D4×C13711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—41no—
180C42—21no—
181C42—42no—
182C42—41no—
183C42C22⋊C4×C13711yescharacteristic
184C42C22⋊C4×C13711yescharacteristic
185C16—42no—
186C23AGL(1,ℤ/8)×C13711yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4C2×C22⋊C4×C13711yescharacteristic
199C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—81no—
205C22—21no—
206C22—41no—
207C22#418×C13711yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—82no—
212C4—82no—
213C4—161no—
214C4#410×C13711yescharacteristic
215C4#421×C13711yescharacteristic
216C2—21no—
217C2—41no—
218C2#1514×C13711yescharacteristic
subgroupG/H
center, socleC2×C137#1514
commutatorC2×C8C22×C4×C137
Frattini, cosocleC2×M16C23×C137
prime radicalC23×C137AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C2×C4×C137 ◁ C4×M16×C137 ◁ G

Complements

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

NH ≅ G/Ntype
7. #498×C137C4semidirect
8. #494×C137C4semidirect
13. #441×C137C4semidirect
109. #11838C137direct (N × H)
110. C137#11838direct (N × H)
117. #498C4×C137semidirect
118. #494C4×C137semidirect
123. #441C4×C137semidirect

Maximal subgroups

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

subgroupindex
#3185×C1372
#2165×C1372
#1841×C1372
#1342×C1372
#1340×C1372
#11838137