Finite Group Databasebeta

← C137×#28735 #46878121

Subgroups — C137×#28735 #46878121

Order 35072.

674 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501×C137C211yesmaximal, characteristic, complemented
2#3142×C137C211yesmaximal, characteristic
3#3141×C137C211yesmaximal, characteristic
4#2932×C137C211yesmaximal, characteristic
5#2079×C137C212yesmaximal, complemented
6#2078×C137C212yesmaximal, complemented
7#1998×C137C211yesmaximal, characteristic, complemented
8#1997×C137C211yesmaximal, characteristic, complemented
9#1816×C137C212yesmaximal, complemented
10#1815×C137C212yesmaximal, complemented
11#1778×C137C211yesmaximal, characteristic, complemented
12C2×#131×C137C2212yes—
13#532×C137C2211yescharacteristic
14C2×C4⋊Q8×C137C2211yescharacteristic
15C2×#125×C137C2211yescharacteristic
16#501×C137—22no—
17#501×C137—22no—
18#491×C137—21no—
19#490×C137—21no—
20#489×C137—22no—
21#488×C137—22no—
22#487×C137—21no—
23#486×C137—21no—
24#435×C137—22no—
25#435×C137—22no—
26C2×#108×C137C2212yes—
27C2×#107×C137C2211yescharacteristic
28C2×#107×C137C2211yescharacteristic
29C2×#106×C137C2211yescharacteristic
30C2×#106×C137C2211yescharacteristic
31#423×C137C2211yescharacteristic
32#417×C137C2212yes—
33#417×C137C2212yes—
34#417×C137C2212yes—
35#417×C137C2212yes—
36#408×C137C2212yes—
37C4×M16×C137C2211yescharacteristic
38C4×M16×C137C2211yescharacteristic
39C2×#97×C137C2211yescharacteristic
40#343×C137C2212yes—
41#343×C137C2212yes—
42#343×C137C2212yes—
43#343×C137C2212yes—
44#341×C137C2212yes—
45C2×#41×C137C2312yes—
46C22×Q8×C137—21no—
47#131×C137—28no—
48C2×M16×C137—22no—
49C2×M16×C137C2312yes—
50C2×M16×C137C2312yes—
51C2×M16×C137C2312yes—
52C22×C8×C137—22no—
53C22×C8×C137C2311yescharacteristic
54C22×C8×C137C2311yescharacteristic
55C4⋊Q8×C137—21no—
56C4⋊Q8×C137—21no—
57C4⋊Q8×C137—21no—
58C4⋊Q8×C137—21no—
59C4⋊Q8×C137D411yescharacteristic
60#127×C137D411yescharacteristic
61#125×C137—21no—
62#125×C137—21no—
63#125×C137—21no—
64#125×C137—21no—
65#124×C137D412yes—
66#121×C137—21no—
67#121×C137—21no—
68C4×D4×C137—21no—
69C4×D4×C137—21no—
70C2×C4⋊C4×C137—21no—
71C2×C4⋊C4×C137—21no—
72C2×C4⋊C4×C137—21no—
73C2×C4⋊C4×C137C2311yescharacteristic
74C2×C4⋊C4×C137C2311yescharacteristic
75C2×C4⋊C4×C137C2311yescharacteristic
76C2×C4⋊C4×C137C2311yescharacteristic
77C2×C42×C137C2311yescharacteristic
78#108×C137—24no—
79#107×C137—22no—
80#107×C137—22no—
81#106×C137—22no—
82#106×C137—22no—
83C4⋊C8×C137—21no—
84C4⋊C8×C137—21no—
85#103×C137—22no—
86#103×C137—22no—
87#103×C137—22no—
88#103×C137—22no—
89#102×C137—22no—
90#102×C137—22no—
91#102×C137—22no—
92#102×C137—22no—
93C22⋊C8×C137—24no—
94#97×C137—21no—
95#97×C137—21no—
96#97×C137D412yes—
97#97×C137D412yes—
98C4×C8×C137—21no—
99C4×C8×C137—21no—
100#41×C137—24no—
101#41×C137—24no—
102C2×Q8×C137—41no—
103C2×Q8×C137—41no—
104C2×Q8×C137—82no—
105C2×Q8×C137C2×D412yes—
106C2×D4×C137—22no—
107C2×D4×C137—21no—
108C2×D4×C137—21no—
109C2×D4×C137C2×D412yes—
110C22×C4×C137—22no—
111C22×C4×C137—21no—
112C22×C4×C137—21no—
113C22×C4×C137—21no—
114C22×C4×C137—21no—
115C22×C4×C137#4111yescharacteristic
116C22×C4×C137#4111yescharacteristic
117C22×C4×C137C2411yescharacteristic
118M16×C137—22no—
119M16×C137—22no—
120M16×C137—24no—
121M16×C137—22no—
122M16×C137—22no—
123M16×C137—44no—
124C2×C8×C137—22no—
125C2×C8×C137—22no—
126C2×C8×C137—22no—
127C2×C8×C137—22no—
128C2×C8×C137—44no—
129C2×C8×C137C2×D412yes—
130C2×C8×C137C2×D412yes—
131C2×C8×C137#4114yes—
132C4⋊C4×C137—21no—
133C4⋊C4×C137—21no—
134C4⋊C4×C137—21no—
135C4⋊C4×C137—21no—
136C4⋊C4×C137—21no—
137C4⋊C4×C137—21no—
138C4⋊C4×C137—21no—
139C4⋊C4×C137—21no—
140C4⋊C4×C137—21no—
141C4⋊C4×C137—21no—
142C4⋊C4×C137—41no—
143C4⋊C4×C137—41no—
144C4⋊C4×C137—41no—
145C4⋊C4×C137—41no—
146C4⋊C4×C137—41no—
147C4⋊C4×C137—41no—
148C22⋊C4×C137—22no—
149C22⋊C4×C137—22no—
150C42×C137C2×D411yescharacteristic
151C42×C137C2×D411yescharacteristic
152C42×C137C2×D411yescharacteristic
153C42×C137C2×D411yescharacteristic
154C23×C137—22no—
155C23×C137C2×#4111yescharacteristic
156Q8×C137—24no—
157Q8×C137—82no—
158Q8×C137—82no—
159D4×C137—24no—
160D4×C137—44no—
161D4×C137—81no—
162D4×C137—81no—
163C2×C4×C137—22no—
164C2×C4×C137—41no—
165C2×C4×C137—41no—
166C2×C4×C137—42no—
167C2×C4×C137—42no—
168C2×C4×C137—41no—
169C2×C4×C137—41no—
170C2×C4×C137—41no—
171C2×C4×C137—41no—
172C2×C4×C137—81no—
173C2×C4×C137—81no—
174C2×C4×C137—81no—
175C2×C4×C137—81no—
176C2×C4×C137#12111yescharacteristic
177C2×C4×C137#12111yescharacteristic
178C2×C4×C137#12111yescharacteristic
179C2×C4×C137#12111yescharacteristic
180C2×C4×C137C22×D411yescharacteristic
181C2×C4×C137C22×D411yescharacteristic
182C2×C4×C137C2×#4112yes—
183C2×C4×C1372−1+412yes—
184C8×C137—22no—
185C8×C137—42no—
186C8×C137—42no—
187C8×C137—42no—
188C8×C137#12414yes—
189C22×C137—21no—
190C22×C137—21no—
191C22×C137—42no—
192C22×C137—42no—
193C22×C137—42no—
194C22×C137C2×#12111yescharacteristic
195C22×C137#55612yes—
196C4×C137—21no—
197C4×C137—21no—
198C4×C137—21no—
199C4×C137—21no—
200C4×C137—42no—
201C4×C137—81no—
202C4×C137—81no—
203C4×C137—81no—
204C4×C137—81no—
205C4×C137C2×#12411yescharacteristic
206C4×C137#53611yescharacteristic
207C4×C137#54712yes—
208C2×C137—22no—
209C2×C137—82no—
210C2×C137#254711yescharacteristic
211C2×C137#306811yescharacteristic
212C2×C137#306911yescharacteristic
213#28735C13711yesmaximal, characteristic, direct product factor, Sylow
214C137#2873511yescharacteristic, direct product factor, Sylow
215C2×#501C2×C13711yescharacteristic, complemented
216#3142C2×C13711yescharacteristic
217#3141C2×C13711yescharacteristic
218#2932C2×C13711yescharacteristic
219#2079C2×C13712yescomplemented
220#2078C2×C13712yescomplemented
221#1998C2×C13711yescharacteristic, complemented
222#1997C2×C13711yescharacteristic, complemented
223#1816C2×C13712yescomplemented
224#1815C2×C13712yescomplemented
225#1778C2×C13711yescharacteristic, complemented
226C2×#131C22×C13712yes—
227#532C22×C13711yescharacteristic
228C2×C4⋊Q8C22×C13711yescharacteristic
229C2×#125C22×C13711yescharacteristic
230#501—22no—
231#501—22no—
232#491—21no—
233#490—21no—
234#489—22no—
235#488—22no—
236#487—21no—
237#486—21no—
238#435—22no—
239#435—22no—
240C2×#108C22×C13712yes—
241C2×#107C22×C13711yescharacteristic
242C2×#107C22×C13711yescharacteristic
243C2×#106C22×C13711yescharacteristic
244C2×#106C22×C13711yescharacteristic
245#423C22×C13711yescharacteristic
246#417C22×C13712yes—
247#417C22×C13712yes—
248#417C22×C13712yes—
249#417C22×C13712yes—
250#408C22×C13712yes—
251C4×M16C22×C13711yescharacteristic
252C4×M16C22×C13711yescharacteristic
253C2×#97C22×C13711yescharacteristic
254#343C22×C13712yes—
255#343C22×C13712yes—
256#343C22×C13712yes—
257#343C22×C13712yes—
258#341C22×C13712yes—
259C2×#41C23×C13712yes—
260C22×Q8—21no—
261#131—28no—
262C2×M16—22no—
263C2×M16C23×C13712yes—
264C2×M16C23×C13712yes—
265C2×M16C23×C13712yes—
266C22×C8—22no—
267C22×C8C23×C13711yescharacteristic
268C22×C8C23×C13711yescharacteristic
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8D4×C13711yescharacteristic
274#127D4×C13711yescharacteristic
275#125—21no—
276#125—21no—
277#125—21no—
278#125—21no—
279#124D4×C13712yes—
280#121—21no—
281#121—21no—
282C4×D4—21no—
283C4×D4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C23×C13711yescharacteristic
288C2×C4⋊C4C23×C13711yescharacteristic
289C2×C4⋊C4C23×C13711yescharacteristic
290C2×C4⋊C4C23×C13711yescharacteristic
291C2×C42C23×C13711yescharacteristic
292#108—24no—
293#107—22no—
294#107—22no—
295#106—22no—
296#106—22no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299#103—22no—
300#103—22no—
301#103—22no—
302#103—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307C22⋊C8—24no—
308#97—21no—
309#97—21no—
310#97D4×C13712yes—
311#97D4×C13712yes—
312C4×C8—21no—
313C4×C8—21no—
314#41—24no—
315#41—24no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—82no—
319C2×Q8C2×D4×C13712yes—
320C2×D4—22no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4C2×D4×C13712yes—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4#41×C13711yescharacteristic
330C22×C4#41×C13711yescharacteristic
331C22×C4C24×C13711yescharacteristic
332M16—22no—
333M16—22no—
334M16—24no—
335M16—22no—
336M16—22no—
337M16—44no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—44no—
343C2×C8C2×D4×C13712yes—
344C2×C8C2×D4×C13712yes—
345C2×C8#41×C13714yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C42C2×D4×C13711yescharacteristic
365C42C2×D4×C13711yescharacteristic
366C42C2×D4×C13711yescharacteristic
367C42C2×D4×C13711yescharacteristic
368C23—22no—
369C23C2×#41×C13711yescharacteristic
370Q8—24no—
371Q8—82no—
372Q8—82no—
373D4—24no—
374D4—44no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4#121×C13711yescharacteristic
391C2×C4#121×C13711yescharacteristic
392C2×C4#121×C13711yescharacteristic
393C2×C4#121×C13711yescharacteristic
394C2×C4C22×D4×C13711yescharacteristic
395C2×C4C22×D4×C13711yescharacteristic
396C2×C4C2×#41×C13712yes—
397C2×C42−1+4×C13712yes—
398C8—22no—
399C8—42no—
400C8—42no—
401C8—42no—
402C8#124×C13714yes—
403C22—21no—
404C22—21no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22C2×#121×C13711yescharacteristic
409C22#556×C13712yes—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4C2×#124×C13711yescharacteristic
420C4#536×C13711yescharacteristic
421C4#547×C13712yes—
422C2—22no—
423C2—82no—
424C2#2547×C13711yescharacteristic
425C2#3068×C13711yescharacteristic
426C2#3069×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#121
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#532×C137C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#501×C137C2semidirect
5. #2079×C137C2semidirect
6. #2078×C137C2semidirect
7. #1998×C137C2semidirect
8. #1997×C137C2semidirect
9. #1816×C137C2semidirect
10. #1815×C137C2semidirect
11. #1778×C137C2semidirect
213. #28735C137direct (N × H)
214. C137#28735direct (N × H)
215. C2×#501C2×C137semidirect
219. #2079C2×C137semidirect
220. #2078C2×C137semidirect
221. #1998C2×C137semidirect
222. #1997C2×C137semidirect
223. #1816C2×C137semidirect
224. #1815C2×C137semidirect
225. #1778C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#501×C1372
#3142×C1372
#3141×C1372
#2932×C1372
#2079×C1372
#2078×C1372
#1998×C1372
#1997×C1372
#1816×C1372
#1815×C1372
#1778×C1372
#28735137