Finite Group Databasebeta

← C137×#13350 #46862736

Subgroups — C137×#13350 #46862736

Order 35072.

738 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 94 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#3199×C137C211yesmaximal, characteristic, complemented
2#3027×C137C212yesmaximal, complemented
3#2128×C137C212yesmaximal, complemented
4#1794×C137C211yesmaximal, characteristic, complemented
5#1391×C137C211yesmaximal, characteristic, complemented
6#585×C137C2211yescharacteristic
7#535×C137—22no—
8#535×C137C2212yes—
9#482×C137—22no—
10#469×C137—21no—
11#466×C137—21no—
12B(4,𝔽2)×C137—24no—
13B(4,𝔽2)×C137—24no—
14#447×C137—22no—
15#421×C137—21no—
16#416×C137—21no—
17#410×C137—22no—
18#410×C137C2211yescharacteristic
19#410×C137C2212yes—
20#408×C137C2211yescharacteristic
21#353×C137—24no—
222−1+4×C137—21no—
232−1+4×C137—41no—
24H(5,𝔽2)×C137—22no—
25H(5,𝔽2)×C137—41no—
26H(5,𝔽2)×C137D412yes—
27C2×#41×C137—22no—
28C2×#41×C137—21no—
29C2×#41×C137—42no—
30C2×#41×C137C2311yescharacteristic
31C22×D4×C137D412yescomplemented
32C2×D8×C137—21no—
33C2×D8×C137—21no—
34C2×M16×C137D411yescharacteristic, complemented
35C22×C8×C137D411yescharacteristic, complemented
36#127×C137—22no—
37#127×C137—42no—
38#124×C137—22no—
39#124×C137—42no—
40#121×C137—22no—
41#121×C137—22no—
42#121×C137—44no—
43#120×C137—28no—
44#120×C137—22no—
45#120×C137—22no—
46#117×C137—22no—
47#117×C137—21no—
48#117×C137D412yescomplemented
49#107×C137—21no—
50#106×C137—21no—
51#104×C137—42no—
52#103×C137—41no—
53#102×C137—22no—
54#102×C137—22no—
55#102×C137—41no—
56#99×C137—28no—
57#99×C137—22no—
58#99×C137—22no—
59#99×C137D414yescomplemented
60C22⋊C8×C137—22no—
61C24×C137—22no—
62#41×C137—22no—
63#41×C137—22no—
64#41×C137—41no—
65#41×C137—41no—
66#41×C137—42no—
67#41×C137—42no—
68#41×C137—42no—
69#41×C137—41no—
70#41×C137—44no—
71#41×C137—42no—
72#41×C137—41no—
73#41×C137—42no—
74C2×Q8×C137—22no—
75C2×Q8×C137—21no—
76C2×Q8×C137—42no—
77C2×Q8×C137#4111yescharacteristic
78C2×D4×C137—21no—
79C2×D4×C137—22no—
80C2×D4×C137—24no—
81C2×D4×C137—22no—
82C2×D4×C137—22no—
83C2×D4×C137—42no—
84C2×D4×C137—42no—
85C2×D4×C137—42no—
86C2×D4×C137—44no—
87C2×D4×C137—41no—
88C2×D4×C137—48no—
89C2×D4×C137—42no—
90C2×D4×C137—42no—
91C2×D4×C137C2×D411yescharacteristic
92C2×D4×C137C2×D412yes—
93C22×C4×C137—24no—
94C22×C4×C137—41no—
95C22×C4×C137C2×D412yes—
96C22×C4×C137C2×D411yescharacteristic
97D8×C137—81no—
98D8×C137—81no—
99M16×C137—41no—
100C2×C8×C137—21no—
101C2×C8×C137—21no—
102C2×C8×C137—41no—
103C4⋊C4×C137—22no—
104C4⋊C4×C137—42no—
105C4⋊C4×C137—41no—
106C22⋊C4×C137—22no—
107C22⋊C4×C137—24no—
108C22⋊C4×C137—48no—
109C22⋊C4×C137—42no—
110C22⋊C4×C137—42no—
111C22⋊C4×C137—41no—
112C42×C137—22no—
113C42×C137—42no—
114C42×C137—41no—
115C23×C137—24no—
116C23×C137—41no—
117C23×C137—44no—
118C23×C137—42no—
119C23×C137—162no—
120C23×C137#12012yes—
121C23×C137#12711yescharacteristic
122Q8×C137—22no—
123Q8×C137—41no—
124Q8×C137—42no—
125Q8×C137—41no—
126D4×C137—22no—
127D4×C137—41no—
128D4×C137—42no—
129D4×C137—42no—
130D4×C137—44no—
131D4×C137—42no—
132D4×C137—42no—
133D4×C137—41no—
134D4×C137—42no—
135D4×C137—42no—
136D4×C137—82no—
137D4×C137—162no—
138D4×C137—162no—
139D4×C137—168no—
140C2×C4×C137—21no—
141C2×C4×C137—22no—
142C2×C4×C137—21no—
143C2×C4×C137—22no—
144C2×C4×C137—24no—
145C2×C4×C137—44no—
146C2×C4×C137—42no—
147C2×C4×C137—41no—
148C2×C4×C137—42no—
149C2×C4×C137—44no—
150C2×C4×C137—42no—
151C2×C4×C137—81no—
152C2×C4×C137#12011yescharacteristic
153C2×C4×C137#12112yes—
154C2×C4×C137#12111yescharacteristic
155C8×C137—41no—
156C8×C137—41no—
157C22×C137—22no—
158C22×C137—21no—
159C22×C137—41no—
160C22×C137—42no—
161C22×C137—42no—
162C22×C137—82no—
163C22×C137—82no—
164C22×C137—84no—
165C22×C137—88no—
166C22×C137—82no—
167C22×C137—161no—
168C22×C137#39211yescharacteristic
169C4×C137—21no—
170C4×C137—22no—
171C4×C137—42no—
172C4×C137—82no—
173C4×C137—84no—
174C4×C137—161no—
175C4×C137B(4,𝔽2)11yescharacteristic
176C4×C137B(4,𝔽2)11yescharacteristic
177C2×C137—21no—
178C2×C137—42no—
179C2×C137—41no—
180C2×C137—82no—
181C2×C137—162no—
182C2×C137#202111yescharacteristic
183#13350C13711yesmaximal, characteristic, direct product factor, Sylow
184C137#1335011yescharacteristic, direct product factor, Sylow
185#3199C2×C13711yescharacteristic, complemented
186#3027C2×C13712yescomplemented
187#2128C2×C13712yescomplemented
188#1794C2×C13711yescharacteristic, complemented
189#1391C2×C13711yescharacteristic, complemented
190#585C22×C13711yescharacteristic
191#535—22no—
192#535C22×C13712yes—
193#482—22no—
194#469—21no—
195#466—21no—
196B(4,𝔽2)—24no—
197B(4,𝔽2)—24no—
198#447—22no—
199#421—21no—
200#416—21no—
201#410—22no—
202#410C22×C13711yescharacteristic
203#410C22×C13712yes—
204#408C22×C13711yescharacteristic
205#353—24no—
2062−1+4—21no—
2072−1+4—41no—
208H(5,𝔽2)—22no—
209H(5,𝔽2)—41no—
210H(5,𝔽2)D4×C13712yes—
211C2×#41—22no—
212C2×#41—21no—
213C2×#41—42no—
214C2×#41C23×C13711yescharacteristic
215C22×D4D4×C13712yescomplemented
216C2×D8—21no—
217C2×D8—21no—
218C2×M16D4×C13711yescharacteristic, complemented
219C22×C8D4×C13711yescharacteristic, complemented
220#127—22no—
221#127—42no—
222#124—22no—
223#124—42no—
224#121—22no—
225#121—22no—
226#121—44no—
227#120—28no—
228#120—22no—
229#120—22no—
230#117—22no—
231#117—21no—
232#117D4×C13712yescomplemented
233#107—21no—
234#106—21no—
235#104—42no—
236#103—41no—
237#102—22no—
238#102—22no—
239#102—41no—
240#99—28no—
241#99—22no—
242#99—22no—
243#99D4×C13714yescomplemented
244C22⋊C8—22no—
245C24—22no—
246#41—22no—
247#41—22no—
248#41—41no—
249#41—41no—
250#41—42no—
251#41—42no—
252#41—42no—
253#41—41no—
254#41—44no—
255#41—42no—
256#41—41no—
257#41—42no—
258C2×Q8—22no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×Q8#41×C13711yescharacteristic
262C2×D4—21no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—48no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4C2×D4×C13711yescharacteristic
276C2×D4C2×D4×C13712yes—
277C22×C4—24no—
278C22×C4—41no—
279C22×C4C2×D4×C13712yes—
280C22×C4C2×D4×C13711yescharacteristic
281D8—81no—
282D8—81no—
283M16—41no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—41no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—48no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C42—22no—
297C42—42no—
298C42—41no—
299C23—24no—
300C23—41no—
301C23—44no—
302C23—42no—
303C23—162no—
304C23#120×C13712yes—
305C23#127×C13711yescharacteristic
306Q8—22no—
307Q8—41no—
308Q8—42no—
309Q8—41no—
310D4—22no—
311D4—41no—
312D4—42no—
313D4—42no—
314D4—44no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—42no—
319D4—42no—
320D4—82no—
321D4—162no—
322D4—162no—
323D4—168no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4#120×C13711yescharacteristic
337C2×C4#121×C13712yes—
338C2×C4#121×C13711yescharacteristic
339C8—41no—
340C8—41no—
341C22—22no—
342C22—21no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—82no—
347C22—82no—
348C22—84no—
349C22—88no—
350C22—82no—
351C22—161no—
352C22#392×C13711yescharacteristic
353C4—21no—
354C4—22no—
355C4—42no—
356C4—82no—
357C4—84no—
358C4—161no—
359C4B(4,𝔽2)×C13711yescharacteristic
360C4B(4,𝔽2)×C13711yescharacteristic
361C2—21no—
362C2—42no—
363C2—41no—
364C2—82no—
365C2—162no—
366C2#2021×C13711yescharacteristic
subgroupG/H
center, socleC2×C137#2021
commutator, Frattini, cosocleC2×#41C23×C137
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C2×C4×C137 ◁ C2×#41×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3199×C137C2semidirect
2. #3027×C137C2semidirect
3. #2128×C137C2semidirect
4. #1794×C137C2semidirect
5. #1391×C137C2semidirect
31. C22×D4×C137D4semidirect
34. C2×M16×C137D4semidirect
35. C22×C8×C137D4semidirect
48. #117×C137D4semidirect
59. #99×C137D4semidirect
183. #13350C137direct (N × H)
184. C137#13350direct (N × H)
185. #3199C2×C137semidirect
186. #3027C2×C137semidirect
187. #2128C2×C137semidirect
188. #1794C2×C137semidirect
189. #1391C2×C137semidirect
215. C22×D4D4×C137semidirect
218. C2×M16D4×C137semidirect
219. C22×C8D4×C137semidirect
232. #117D4×C137semidirect
243. #99D4×C137semidirect

Maximal subgroups

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

subgroupindex
#3199×C1372
#3027×C1372
#2128×C1372
#1794×C1372
#1391×C1372
#13350137