Finite Group Databasebeta

← C137×#13445 #46862831

Subgroups — C137×#13445 #46862831

Order 35072.

432 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C137C211yesmaximal, characteristic
2C2×#364×C137C211yesmaximal, characteristic, complemented
3C2×#361×C137C211yesmaximal, characteristic, complemented
4#2139×C137C211yesmaximal, characteristic, complemented
5#1974×C137C211yesmaximal, characteristic, complemented
6#1970×C137C211yesmaximal, characteristic, complemented
7#1387×C137C211yesmaximal, characteristic, complemented
8C2×#135×C137C2211yescharacteristic
9C22×D8×C137C2211yescharacteristic
10#509×C137—22no—
11#509×C137—22no—
12C2×SD32×C137—21no—
13C2×D16×C137—21no—
14C2×M32×C137C2211yescharacteristic
15C2×M32×C137C2211yescharacteristic
16C2×#108×C137C2211yescharacteristic
17C2×#107×C137C2211yescharacteristic
18C2×#104×C137—21no—
19C2×#102×C137—21no—
20#408×C137—21no—
21C2×C4×C8×C137C2211yescharacteristic
22#390×C137—21no—
23#364×C137—22no—
24#361×C137—22no—
25#358×C137—21no—
26#357×C137—21no—
27C2×#41×C137—21no—
28C22×D4×C137—21no—
29#135×C137—21no—
30#135×C137—21no—
31#135×C137—21no—
32#135×C137—21no—
33C2×Q16×C137D411yescharacteristic
34C2×SD16×C137—21no—
35C2×D8×C137—21no—
36C2×D8×C137—21no—
37C2×D8×C137—21no—
38C2×D8×C137—21no—
39C2×D8×C137—21no—
40C2×D8×C137D412yes—
41C2×D8×C137D411yescharacteristic
42C2×M16×C137—21no—
43C22×C8×C137D411yescharacteristic
44C22×C8×C137C2311yescharacteristic
45C2×C4⋊C4×C137—21no—
46C2×C22⋊C4×C137—42no—
47C2×C42×C137D411yescharacteristic
48SD32×C137—24no—
49D16×C137—24no—
50M32×C137—21no—
51M32×C137—21no—
52M32×C137—21no—
53M32×C137D412yes—
54C2×C16×C137—21no—
55C2×C16×C137—21no—
56#108×C137—21no—
57#108×C137—21no—
58#107×C137—21no—
59#107×C137—21no—
60#104×C137—42no—
61#102×C137—42no—
62C22⋊C8×C137—42no—
63C4×C8×C137—21no—
64C4×C8×C137—21no—
65C24×C137—41no—
66#41×C137—41no—
67#41×C137—41no—
68#41×C137—41no—
69#41×C137—41no—
70C2×Q8×C137—21no—
71C2×D4×C137—22no—
72C2×D4×C137—21no—
73C2×D4×C137—41no—
74C2×D4×C137—41no—
75C2×D4×C137—41no—
76C2×D4×C137—41no—
77C2×D4×C137—41no—
78C2×D4×C137—41no—
79C22×C4×C137—21no—
80C22×C4×C137—41no—
81C22×C4×C137—41no—
82C22×C4×C137C2×D411yescharacteristic
83Q16×C137—21no—
84Q16×C137—21no—
85SD16×C137—41no—
86SD16×C137—41no—
87D8×C137—21no—
88D8×C137—22no—
89D8×C137—22no—
90D8×C137—21no—
91D8×C137—41no—
92D8×C137—41no—
93M16×C137—41no—
94M16×C137—41no—
95C2×C8×C137—21no—
96C2×C8×C137—21no—
97C2×C8×C137—41no—
98C2×C8×C137D812yes—
99C2×C8×C137C2×D411yescharacteristic
100C2×C8×C137C2×D411yescharacteristic
101C2×C8×C137C2×D411yescharacteristic
102C2×C8×C137#4111yescharacteristic
103C2×C8×C137#4111yescharacteristic
104C2×C8×C137#4111yescharacteristic
105C4⋊C4×C137—41no—
106C4⋊C4×C137—41no—
107C22⋊C4×C137—84no—
108C42×C137—21no—
109C42×C137—21no—
110C16×C137—22no—
111C16×C137—41no—
112C23×C137—41no—
113C23×C137—42no—
114C23×C137—82no—
115C23×C137—82no—
116C23×C137—81no—
117C23×C137—81no—
118C23×C137AGL(1,ℤ/8)11yescharacteristic
119Q8×C137—41no—
120Q8×C137—41no—
121D4×C137—42no—
122D4×C137—41no—
123D4×C137—42no—
124D4×C137—41no—
125D4×C137—81no—
126D4×C137—81no—
127D4×C137—81no—
128D4×C137—81no—
129C2×C4×C137—22no—
130C2×C4×C137—41no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—81no—
134C2×C4×C137—81no—
135C2×C4×C137—82no—
136C2×C4×C137—81no—
137C2×C4×C137#12111yescharacteristic
138C2×C4×C137#12111yescharacteristic
139C2×C4×C137#12111yescharacteristic
140C2×C4×C137C2×D811yescharacteristic
141C2×C4×C137#13511yescharacteristic
142C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
143C8×C137—22no—
144C8×C137—81no—
145C8×C137#12011yescharacteristic
146C8×C137#12311yescharacteristic
147C8×C137#12311yescharacteristic
148C8×C137#12411yescharacteristic
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—84no—
152C22×C137—81no—
153C22×C137—82no—
154C22×C137—81no—
155C22×C137—81no—
156C22×C137—162no—
157C22×C137—161no—
158C22×C137—161no—
159C22×C137#45911yescharacteristic
160C22×C137#46311yescharacteristic
161C22×C137#46911yescharacteristic
162C4×C137—42no—
163C4×C137—81no—
164C4×C137—161no—
165C4×C137#39411yescharacteristic
166C4×C137#45911yescharacteristic
167C4×C137#46311yescharacteristic
168C4×C137#46611yescharacteristic
169C2×C137—21no—
170C2×C137—21no—
171C2×C137—161no—
172C2×C137—162no—
173C2×C137#205411yescharacteristic
174C2×C137#221211yescharacteristic
175C2×C137#221311yescharacteristic
176#13445C13711yesmaximal, characteristic, direct product factor, Sylow
177C137#1344511yescharacteristic, direct product factor, Sylow
178C2×#509C2×C13711yescharacteristic
179C2×#364C2×C13711yescharacteristic, complemented
180C2×#361C2×C13711yescharacteristic, complemented
181#2139C2×C13711yescharacteristic, complemented
182#1974C2×C13711yescharacteristic, complemented
183#1970C2×C13711yescharacteristic, complemented
184#1387C2×C13711yescharacteristic, complemented
185C2×#135C22×C13711yescharacteristic
186C22×D8C22×C13711yescharacteristic
187#509—22no—
188#509—22no—
189C2×SD32—21no—
190C2×D16—21no—
191C2×M32C22×C13711yescharacteristic
192C2×M32C22×C13711yescharacteristic
193C2×#108C22×C13711yescharacteristic
194C2×#107C22×C13711yescharacteristic
195C2×#104—21no—
196C2×#102—21no—
197#408—21no—
198C2×C4×C8C22×C13711yescharacteristic
199#390—21no—
200#364—22no—
201#361—22no—
202#358—21no—
203#357—21no—
204C2×#41—21no—
205C22×D4—21no—
206#135—21no—
207#135—21no—
208#135—21no—
209#135—21no—
210C2×Q16D4×C13711yescharacteristic
211C2×SD16—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8D4×C13712yes—
218C2×D8D4×C13711yescharacteristic
219C2×M16—21no—
220C22×C8D4×C13711yescharacteristic
221C22×C8C23×C13711yescharacteristic
222C2×C4⋊C4—21no—
223C2×C22⋊C4—42no—
224C2×C42D4×C13711yescharacteristic
225SD32—24no—
226D16—24no—
227M32—21no—
228M32—21no—
229M32—21no—
230M32D4×C13712yes—
231C2×C16—21no—
232C2×C16—21no—
233#108—21no—
234#108—21no—
235#107—21no—
236#107—21no—
237#104—42no—
238#102—42no—
239C22⋊C8—42no—
240C4×C8—21no—
241C4×C8—21no—
242C24—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C2×D4×C13711yescharacteristic
260Q16—21no—
261Q16—21no—
262SD16—41no—
263SD16—41no—
264D8—21no—
265D8—22no—
266D8—22no—
267D8—21no—
268D8—41no—
269D8—41no—
270M16—41no—
271M16—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8D8×C13712yes—
276C2×C8C2×D4×C13711yescharacteristic
277C2×C8C2×D4×C13711yescharacteristic
278C2×C8C2×D4×C13711yescharacteristic
279C2×C8#41×C13711yescharacteristic
280C2×C8#41×C13711yescharacteristic
281C2×C8#41×C13711yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—41no—
284C22⋊C4—84no—
285C42—21no—
286C42—21no—
287C16—22no—
288C16—41no—
289C23—41no—
290C23—42no—
291C23—82no—
292C23—82no—
293C23—81no—
294C23—81no—
295C23AGL(1,ℤ/8)×C13711yescharacteristic
296Q8—41no—
297Q8—41no—
298D4—42no—
299D4—41no—
300D4—42no—
301D4—41no—
302D4—81no—
303D4—81no—
304D4—81no—
305D4—81no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4#121×C13711yescharacteristic
315C2×C4#121×C13711yescharacteristic
316C2×C4#121×C13711yescharacteristic
317C2×C4C2×D8×C13711yescharacteristic
318C2×C4#135×C13711yescharacteristic
319C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
320C8—22no—
321C8—81no—
322C8#120×C13711yescharacteristic
323C8#123×C13711yescharacteristic
324C8#123×C13711yescharacteristic
325C8#124×C13711yescharacteristic
326C22—21no—
327C22—21no—
328C22—84no—
329C22—81no—
330C22—82no—
331C22—81no—
332C22—81no—
333C22—162no—
334C22—161no—
335C22—161no—
336C22#459×C13711yescharacteristic
337C22#463×C13711yescharacteristic
338C22#469×C13711yescharacteristic
339C4—42no—
340C4—81no—
341C4—161no—
342C4#394×C13711yescharacteristic
343C4#459×C13711yescharacteristic
344C4#463×C13711yescharacteristic
345C4#466×C13711yescharacteristic
346C2—21no—
347C2—21no—
348C2—162no—
349C2—161no—
350C2#2054×C13711yescharacteristic
351C2#2212×C13711yescharacteristic
352C2#2213×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#459
commutator, Frattini, cosocleC22×C8C23×C137
prime radicalC2×#509×C137C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C4×C8×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#364×C137C2semidirect
3. C2×#361×C137C2semidirect
4. #2139×C137C2semidirect
5. #1974×C137C2semidirect
6. #1970×C137C2semidirect
7. #1387×C137C2semidirect
176. #13445C137direct (N × H)
177. C137#13445direct (N × H)
179. C2×#364C2×C137semidirect
180. C2×#361C2×C137semidirect
181. #2139C2×C137semidirect
182. #1974C2×C137semidirect
183. #1970C2×C137semidirect
184. #1387C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C1372
C2×#364×C1372
C2×#361×C1372
#2139×C1372
#1974×C1372
#1970×C1372
#1387×C1372
#13445137