Finite Group Databasebeta

← C137×#12101 #46861487

Subgroups — C137×#12101 #46861487

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#2734×C137C211yesmaximal, characteristic, complemented
2#2007×C137C211yesmaximal, characteristic
3#1610×C137C212yesmaximal
4#1346×C137C212yesmaximal, complemented
5#1319×C137C211yesmaximal, characteristic, complemented
6C2×#137×C137—21no—
7#537×C137C2211yescharacteristic
8C2×C4⋊Q8×C137—21no—
9#480×C137—22no—
10C2×#104×C137—21no—
11#398×C137C2212yes—
12#395×C137—22no—
13#391×C137C2211yescharacteristic
14#389×C137—21no—
15#387×C137—22no—
16#343×C137C2211yescharacteristic
17#327×C137—22no—
18#324×C137C2212yes—
19C2×#41×C137—21no—
20C22×Q8×C137—21no—
21C22×Q8×C137D411yescharacteristic
22#137×C137—42no—
23#137×C137—42no—
24C2×Q16×C137—41no—
25C2×SD16×C137—41no—
26C2×M16×C137—21no—
27C4⋊Q8×C137—42no—
28C4⋊Q8×C137—42no—
29C22⋊Q8×C137—41no—
30#121×C137—22no—
31#121×C137—41no—
32C2×C4⋊C4×C137—21no—
33C2×C4⋊C4×C137—22no—
34C2×C4⋊C4×C137—22no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137D412yes—
37C2×C4⋊C4×C137C2311yescharacteristic
38C2×C42×C137—21no—
39C2×C42×C137D411yescharacteristic
40#107×C137—42no—
41#104×C137—42no—
42#102×C137—42no—
43C22⋊C8×C137—22no—
44#95×C137—22no—
45#95×C137—22no—
46#95×C137—22no—
47#95×C137D412yes—
48#41×C137—41no—
49#41×C137—41no—
50#41×C137—41no—
51#41×C137—41no—
52C2×Q8×C137—21no—
53C2×Q8×C137—21no—
54C2×Q8×C137—42no—
55C2×Q8×C137—41no—
56C2×Q8×C137—41no—
57C2×Q8×C137—81no—
58C2×Q8×C137—81no—
59C2×Q8×C137D812yes—
60C2×D4×C137—21no—
61C2×D4×C137—22no—
62C2×D4×C137—41no—
63C22×C4×C137—21no—
64C22×C4×C137—22no—
65C22×C4×C137—21no—
66C22×C4×C137—22no—
67C22×C4×C137—41no—
68C22×C4×C137C2×D411yescharacteristic
69C22×C4×C137C2×D412yes—
70Q16×C137—44no—
71SD16×C137—44no—
72M16×C137—22no—
73M16×C137—41no—
74C2×C8×C137—41no—
75C2×C8×C137—42no—
76C4⋊C4×C137—24no—
77C4⋊C4×C137—22no—
78C4⋊C4×C137—41no—
79C4⋊C4×C137—42no—
80C4⋊C4×C137—44no—
81C4⋊C4×C137—42no—
82C4⋊C4×C137—42no—
83C4⋊C4×C137—81no—
84C22⋊C4×C137—42no—
85C42×C137—41no—
86C42×C137—42no—
87C23×C137—41no—
88C23×C137#12011yescharacteristic
89Q8×C137—42no—
90Q8×C137—41no—
91Q8×C137—41no—
92Q8×C137—81no—
93Q8×C137—84no—
94D4×C137—41no—
95D4×C137—41no—
96D4×C137—82no—
97D4×C137—81no—
98D4×C137—81no—
99C2×C4×C137—21no—
100C2×C4×C137—21no—
101C2×C4×C137—22no—
102C2×C4×C137—24no—
103C2×C4×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137—41no—
106C2×C4×C137—42no—
107C2×C4×C137—42no—
108C2×C4×C137—41no—
109C2×C4×C137—41no—
110C2×C4×C137—41no—
111C2×C4×C137—42no—
112C2×C4×C137—42no—
113C2×C4×C137—42no—
114C2×C4×C137—41no—
115C2×C4×C137—42no—
116C2×C4×C137—81no—
117C2×C4×C137—81no—
118C2×C4×C137C2×D811yescharacteristic
119C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
120C8×C137—42no—
121C8×C137—82no—
122C22×C137—21no—
123C22×C137—21no—
124C22×C137—81no—
125C22×C137—81no—
126C22×C137—81no—
127C22×C137#44711yescharacteristic
128C22×C137#45311yescharacteristic
129C22×C137B(4,𝔽2)11yescharacteristic
130C4×C137—21no—
131C4×C137—21no—
132C4×C137—44no—
133C4×C137—42no—
134C4×C137—42no—
135C4×C137—42no—
136C4×C137—82no—
137C4×C137—81no—
138C2×C137—21no—
139C2×C137—21no—
140C2×C137—161no—
141C2×C137#159511yescharacteristic
142C2×C137#220111yescharacteristic
143C2×C137#220511yescharacteristic
144#12101C13711yesmaximal, characteristic, direct product factor, Sylow
145C137#1210111yescharacteristic, direct product factor, Sylow
146#2734C2×C13711yescharacteristic, complemented
147#2007C2×C13711yescharacteristic
148#1610C2×C13712yes—
149#1346C2×C13712yescomplemented
150#1319C2×C13711yescharacteristic, complemented
151C2×#137—21no—
152#537C22×C13711yescharacteristic
153C2×C4⋊Q8—21no—
154#480—22no—
155C2×#104—21no—
156#398C22×C13712yes—
157#395—22no—
158#391C22×C13711yescharacteristic
159#389—21no—
160#387—22no—
161#343C22×C13711yescharacteristic
162#327—22no—
163#324C22×C13712yes—
164C2×#41—21no—
165C22×Q8—21no—
166C22×Q8D4×C13711yescharacteristic
167#137—42no—
168#137—42no—
169C2×Q16—41no—
170C2×SD16—41no—
171C2×M16—21no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C22⋊Q8—41no—
175#121—22no—
176#121—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4D4×C13712yes—
182C2×C4⋊C4C23×C13711yescharacteristic
183C2×C42—21no—
184C2×C42D4×C13711yescharacteristic
185#107—42no—
186#104—42no—
187#102—42no—
188C22⋊C8—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95D4×C13712yes—
193#41—41no—
194#41—41no—
195#41—41no—
196#41—41no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×Q8D8×C13712yes—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4C2×D4×C13711yescharacteristic
214C22×C4C2×D4×C13712yes—
215Q16—44no—
216SD16—44no—
217M16—22no—
218M16—41no—
219C2×C8—41no—
220C2×C8—42no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—81no—
229C22⋊C4—42no—
230C42—41no—
231C42—42no—
232C23—41no—
233C23#120×C13711yescharacteristic
234Q8—42no—
235Q8—41no—
236Q8—41no—
237Q8—81no—
238Q8—84no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×D8×C13711yescharacteristic
264C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
265C8—42no—
266C8—82no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22#447×C13711yescharacteristic
273C22#453×C13711yescharacteristic
274C22B(4,𝔽2)×C13711yescharacteristic
275C4—21no—
276C4—21no—
277C4—44no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—81no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2#1595×C13711yescharacteristic
287C2#2201×C13711yescharacteristic
288C2#2205×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C137
prime radical#537×C137C22
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
1. #2734×C137C2semidirect
4. #1346×C137C2semidirect
5. #1319×C137C2semidirect
144. #12101C137direct (N × H)
145. C137#12101direct (N × H)
146. #2734C2×C137semidirect
149. #1346C2×C137semidirect
150. #1319C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2734×C1372
#2007×C1372
#1610×C1372
#1346×C1372
#1319×C1372
#12101137