Finite Group Databasebeta

← C137×#12256 #46861642

Subgroups — C137×#12256 #46861642

Order 35072.

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

normal subgroups only

This group has 442 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#2551×C137C211yesmaximal, characteristic
2#2060×C137C211yesmaximal, characteristic
3#1605×C137C212yesmaximal
4#1354×C137C212yesmaximal
5#1325×C137C211yesmaximal, characteristic
6#544×C137C2211yescharacteristic
7C2×C4⋊Q8×C137—22no—
8#451×C137—22no—
9#428×C137—21no—
10#395×C137—21no—
11#395×C137C2212yes—
12#389×C137—22no—
13#384×C137C2211yescharacteristic
14C4×C4⋊C4×C137—21no—
15#343×C137C2211yescharacteristic
16#339×C137—21no—
17#328×C137—22no—
18#324×C137C2212yes—
19C22×Q8×C137—21no—
20C22×Q8×C137D412yes—
21C2×Q16×C137—42no—
22C2×M16×C137—21no—
23C4⋊Q8×C137—41no—
24C4⋊Q8×C137—48no—
25#125×C137—41no—
26C22⋊Q8×C137—22no—
27#117×C137—21no—
28C2×C4⋊C4×C137—21no—
29C2×C4⋊C4×C137—21no—
30C2×C4⋊C4×C137—22no—
31C2×C4⋊C4×C137—22no—
32C2×C4⋊C4×C137—22no—
33C2×C4⋊C4×C137D411yescharacteristic
34C2×C4⋊C4×C137C2311yescharacteristic
35C2×C42×C137—22no—
36C2×C42×C137D411yescharacteristic
37#107×C137—41no—
38#106×C137—41no—
39#103×C137—42no—
40C22⋊C8×C137—22no—
41#95×C137—21no—
42#95×C137—22no—
43#95×C137D412yes—
44C2×Q8×C137—22no—
45C2×Q8×C137—44no—
46C2×Q8×C137—44no—
47C2×Q8×C137—42no—
48C2×Q8×C137—81no—
49C22×C4×C137—21no—
50C22×C4×C137—22no—
51C22×C4×C137—22no—
52C22×C4×C137—21no—
53C22×C4×C137C2×D411yescharacteristic
54C22×C4×C137C2×D412yes—
55Q16×C137—84no—
56M16×C137—22no—
57M16×C137—41no—
58C2×C8×C137—41no—
59C2×C8×C137—42no—
60C4⋊C4×C137—24no—
61C4⋊C4×C137—24no—
62C4⋊C4×C137—22no—
63C4⋊C4×C137—21no—
64C4⋊C4×C137—21no—
65C4⋊C4×C137—41no—
66C4⋊C4×C137—41no—
67C4⋊C4×C137—44no—
68C4⋊C4×C137—44no—
69C4⋊C4×C137—44no—
70C4⋊C4×C137—42no—
71C22⋊C4×C137—41no—
72C42×C137—21no—
73C42×C137—44no—
74C42×C137—41no—
75C42×C137Q1612yescomplemented
76C23×C137#12011yescharacteristic
77Q8×C137—44no—
78Q8×C137—82no—
79Q8×C137—84no—
80Q8×C137—82no—
81C2×C4×C137—22no—
82C2×C4×C137—22no—
83C2×C4×C137—22no—
84C2×C4×C137—21no—
85C2×C4×C137—21no—
86C2×C4×C137—24no—
87C2×C4×C137—24no—
88C2×C4×C137—22no—
89C2×C4×C137—42no—
90C2×C4×C137—42no—
91C2×C4×C137—42no—
92C2×C4×C137—42no—
93C2×C4×C137—42no—
94C2×C4×C137—42no—
95C2×C4×C137—41no—
96C2×C4×C137—41no—
97C2×C4×C137—41no—
98C2×C4×C137C2×Q1611yescharacteristic
99C2×C4×C137#13711yescharacteristic
100C8×C137—42no—
101C8×C137—82no—
102C22×C137—21no—
103C22×C137—21no—
104C22×C137#45111yescharacteristic
105C22×C137#45611yescharacteristic
106C22×C137B(4,𝔽2)11yescharacteristic
107C4×C137—24no—
108C4×C137—21no—
109C4×C137—21no—
110C4×C137—44no—
111C4×C137—42no—
112C4×C137—44no—
113C4×C137—81no—
114C4×C137—81no—
115C4×C137—81no—
116C2×C137—21no—
117C2×C137—21no—
118C2×C137#160511yescharacteristic
119C2×C137#219611yescharacteristic
120C2×C137#220511yescharacteristic
121#12256C13711yesmaximal, characteristic, direct product factor, Sylow
122C137#1225611yescharacteristic, direct product factor, Sylow
123#2551C2×C13711yescharacteristic
124#2060C2×C13711yescharacteristic
125#1605C2×C13712yes—
126#1354C2×C13712yes—
127#1325C2×C13711yescharacteristic
128#544C22×C13711yescharacteristic
129C2×C4⋊Q8—22no—
130#451—22no—
131#428—21no—
132#395—21no—
133#395C22×C13712yes—
134#389—22no—
135#384C22×C13711yescharacteristic
136C4×C4⋊C4—21no—
137#343C22×C13711yescharacteristic
138#339—21no—
139#328—22no—
140#324C22×C13712yes—
141C22×Q8—21no—
142C22×Q8D4×C13712yes—
143C2×Q16—42no—
144C2×M16—21no—
145C4⋊Q8—41no—
146C4⋊Q8—48no—
147#125—41no—
148C22⋊Q8—22no—
149#117—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4D4×C13711yescharacteristic
156C2×C4⋊C4C23×C13711yescharacteristic
157C2×C42—22no—
158C2×C42D4×C13711yescharacteristic
159#107—41no—
160#106—41no—
161#103—42no—
162C22⋊C8—22no—
163#95—21no—
164#95—22no—
165#95D4×C13712yes—
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—42no—
170C2×Q8—81no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×D4×C13711yescharacteristic
176C22×C4C2×D4×C13712yes—
177Q16—84no—
178M16—22no—
179M16—41no—
180C2×C8—41no—
181C2×C8—42no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C22⋊C4—41no—
194C42—21no—
195C42—44no—
196C42—41no—
197C42Q16×C13712yescomplemented
198C23#120×C13711yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—82no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4C2×Q16×C13711yescharacteristic
221C2×C4#137×C13711yescharacteristic
222C8—42no—
223C8—82no—
224C22—21no—
225C22—21no—
226C22#451×C13711yescharacteristic
227C22#456×C13711yescharacteristic
228C22B(4,𝔽2)×C13711yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2#1605×C13711yescharacteristic
241C2#2196×C13711yescharacteristic
242C2#2205×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C137
prime radicalC23×C137#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C2×C4⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
75. C42×C137Q16semidirect
121. #12256C137direct (N × H)
122. C137#12256direct (N × H)
197. C42Q16×C137semidirect

Maximal subgroups

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

subgroupindex
#2551×C1372
#2060×C1372
#1605×C1372
#1354×C1372
#1325×C1372
#12256137