Finite Group Databasebeta

← C137×#7830 #46857216

Subgroups — C137×#7830 #46857216

Order 35072.

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

normal subgroups only

This group has 494 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#2733×C137C211yesmaximal, characteristic
2#2095×C137C211yesmaximal, characteristic, complemented
3#2094×C137C212yesmaximal, complemented
4#1272×C137C211yesmaximal, characteristic, complemented
5#1270×C137C212yesmaximal, complemented
6C2×#124×C137—21no—
7C2×#107×C137—21no—
8C2×#106×C137—22no—
9C2×#103×C137—21no—
10C2×#102×C137—22no—
11#394×C137—21no—
12#392×C137—22no—
13#391×C137C2211yescharacteristic
14#390×C137C2212yes—
15#388×C137—22no—
16#384×C137C2211yescharacteristic
17#336×C137C2211yescharacteristic
18#336×C137C2212yes—
19C22×Q8×C137D411yescharacteristic
20C22×D4×C137—21no—
21C22×D4×C137—21no—
22C22×D4×C137D412yes—
23C22×C8×C137—22no—
24C22×C8×C137—21no—
25#124×C137—42no—
26#124×C137—42no—
27C2×C4⋊C4×C137—22no—
28C2×C4⋊C4×C137—21no—
29C2×C4⋊C4×C137D412yes—
30C2×C4⋊C4×C137D411yescharacteristic
31C2×C22⋊C4×C137—22no—
32C2×C22⋊C4×C137—22no—
33C2×C22⋊C4×C137—22no—
34C2×C22⋊C4×C137—22no—
35C2×C42×C137—21no—
36C2×C42×C137C2311yescharacteristic
37#107×C137—42no—
38#106×C137—44no—
39#103×C137—42no—
40#102×C137—44no—
41#95×C137—21no—
42#95×C137—22no—
43C24×C137—22no—
44C2×Q8×C137—21no—
45C2×Q8×C137—41no—
46C2×Q8×C137—41no—
47C2×Q8×C137SD1612yes—
48C2×D4×C137—22no—
49C2×D4×C137—22no—
50C2×D4×C137—42no—
51C2×D4×C137—42no—
52C2×D4×C137—42no—
53C2×D4×C137—81no—
54C2×D4×C137—81no—
55C2×D4×C137—82no—
56C2×D4×C137—81no—
57C22×C4×C137—21no—
58C22×C4×C137—21no—
59C22×C4×C137—22no—
60C22×C4×C137—41no—
61C22×C4×C137C2×D411yescharacteristic
62C22×C4×C137C2×D412yes—
63C2×C8×C137—24no—
64C2×C8×C137—22no—
65C2×C8×C137—42no—
66C2×C8×C137—41no—
67C2×C8×C137—42no—
68C2×C8×C137—41no—
69C4⋊C4×C137—42no—
70C4⋊C4×C137—41no—
71C4⋊C4×C137—42no—
72C4⋊C4×C137—41no—
73C4⋊C4×C137—42no—
74C4⋊C4×C137—41no—
75C22⋊C4×C137—44no—
76C22⋊C4×C137—42no—
77C22⋊C4×C137—42no—
78C22⋊C4×C137—42no—
79C22⋊C4×C137—42no—
80C22⋊C4×C137—82no—
81C42×C137—41no—
82C42×C137—42no—
83C23×C137—42no—
84C23×C137—42no—
85C23×C137—42no—
86C23×C137—42no—
87C23×C137—42no—
88C23×C137—42no—
89C23×C137—42no—
90C23×C137#12011yescharacteristic
91Q8×C137—42no—
92Q8×C137—81no—
93D4×C137—44no—
94D4×C137—82no—
95D4×C137—84no—
96D4×C137—84no—
97C2×C4×C137—22no—
98C2×C4×C137—21no—
99C2×C4×C137—22no—
100C2×C4×C137—21no—
101C2×C4×C137—22no—
102C2×C4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137—42no—
105C2×C4×C137—42no—
106C2×C4×C137—42no—
107C2×C4×C137—42no—
108C2×C4×C137—41no—
109C2×C4×C137—81no—
110C2×C4×C137—82no—
111C2×C4×C137C2×SD1611yescharacteristic
112C2×C4×C137#13512yes—
113C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
114C2×C4×C137#13712yes—
115C8×C137—42no—
116C8×C137—44no—
117C22×C137—82no—
118C22×C137—82no—
119C22×C137—82no—
120C22×C137—82no—
121C22×C137—82no—
122C22×C137—82no—
123C22×C137—82no—
124C22×C137#45011yescharacteristic
125C22×C137#45212yes—
126C22×C137#45311yescharacteristic
127C22×C137#45312yes—
128C22×C137#45811yescharacteristic
129C4×C137—21no—
130C4×C137—21no—
131C4×C137—22no—
132C4×C137—22no—
133C4×C137—42no—
134C4×C137—82no—
135C4×C137—81no—
136C4×C137—161no—
137C2×C137—162no—
138C2×C137#161311yescharacteristic
139C2×C137#161812yes—
140C2×C137#163112yes—
141C2×C137#163411yescharacteristic
142C2×C137#165511yescharacteristic
143#7830C13711yesmaximal, characteristic, direct product factor, Sylow
144C137#783011yescharacteristic, direct product factor, Sylow
145#2733C2×C13711yescharacteristic
146#2095C2×C13711yescharacteristic, complemented
147#2094C2×C13712yescomplemented
148#1272C2×C13711yescharacteristic, complemented
149#1270C2×C13712yescomplemented
150C2×#124—21no—
151C2×#107—21no—
152C2×#106—22no—
153C2×#103—21no—
154C2×#102—22no—
155#394—21no—
156#392—22no—
157#391C22×C13711yescharacteristic
158#390C22×C13712yes—
159#388—22no—
160#384C22×C13711yescharacteristic
161#336C22×C13712yes—
162#336C22×C13711yescharacteristic
163C22×Q8D4×C13711yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4D4×C13712yes—
167C22×C8—22no—
168C22×C8—21no—
169#124—42no—
170#124—42no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4D4×C13712yes—
174C2×C4⋊C4D4×C13711yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—21no—
180C2×C42C23×C13711yescharacteristic
181#107—42no—
182#106—44no—
183#103—42no—
184#102—44no—
185#95—21no—
186#95—22no—
187C24—22no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8SD16×C13712yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—82no—
200C2×D4—81no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4C2×D4×C13711yescharacteristic
206C22×C4C2×D4×C13712yes—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C42—41no—
226C42—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#120×C13711yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—44no—
238D4—82no—
239D4—84no—
240D4—84no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4C2×SD16×C13711yescharacteristic
256C2×C4#135×C13712yes—
257C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
258C2×C4#137×C13712yes—
259C8—42no—
260C8—44no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#450×C13711yescharacteristic
269C22#452×C13712yes—
270C22#453×C13711yescharacteristic
271C22#453×C13712yes—
272C22#458×C13711yescharacteristic
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—82no—
279C4—81no—
280C4—161no—
281C2—162no—
282C2#1613×C13711yescharacteristic
283C2#1618×C13712yes—
284C2#1631×C13712yes—
285C2#1634×C13711yescharacteristic
286C2#1655×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
prime radical#2733×C137C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2095×C137C2semidirect
3. #2094×C137C2semidirect
4. #1272×C137C2semidirect
5. #1270×C137C2semidirect
143. #7830C137direct (N × H)
144. C137#7830direct (N × H)
146. #2095C2×C137semidirect
147. #2094C2×C137semidirect
148. #1272C2×C137semidirect
149. #1270C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2733×C1372
#2095×C1372
#2094×C1372
#1272×C1372
#1270×C1372
#7830137