Finite Group Databasebeta

← C137×#12096 #46861482

Subgroups — C137×#12096 #46861482

Order 35072.

726 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 726 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#2679×C137C211yesmaximal, characteristic, complemented
2#2006×C137C211yesmaximal, characteristic, complemented
3#1598×C137C212yesmaximal, complemented
4#1345×C137C212yesmaximal, complemented
5#1318×C137C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C137—21no—
7#537×C137C2211yescharacteristic, complemented
8C2×#120×C137—22no—
9C2×#120×C137—21no—
10#447×C137—22no—
11C2×#104×C137—21no—
12#394×C137—22no—
13#394×C137C2212yes—
14#390×C137—21no—
15#390×C137C2211yescharacteristic
16#343×C137C2211yescharacteristic, complemented
17#327×C137—22no—
18#324×C137C2212yescomplemented
19C25×C137—21no—
20C2×#41×C137—21no—
21C22×D4×C137—21no—
22C22×D4×C137—22no—
23C22×D4×C137D411yescharacteristic
24C22×D4×C137D412yes—
25AGL(1,ℤ/8)×C137—42no—
26AGL(1,ℤ/8)×C137—42no—
27C2×SD16×C137—41no—
28C2×D8×C137—41no—
29C2×D8×C137—42no—
30C2×M16×C137—21no—
31C22⋊Q8×C137—41no—
32#121×C137—22no—
33#121×C137—41no—
34#120×C137—48no—
35#120×C137—42no—
36#120×C137—42no—
37C2×C4⋊C4×C137—21no—
38C2×C4⋊C4×C137C2311yescharacteristic
39C2×C22⋊C4×C137—22no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—22no—
42C2×C22⋊C4×C137—22no—
43C2×C22⋊C4×C137—22no—
44C2×C42×C137D411yescharacteristic
45#104×C137—42no—
46#102×C137—42no—
47C22⋊C8×C137—22no—
48#95×C137D412yes—
49C24×C137—22no—
50C24×C137—21no—
51C24×C137—44no—
52C24×C137—42no—
53C24×C137—82no—
54C24×C137—81no—
55C24×C137—81no—
56#41×C137—41no—
57#41×C137—41no—
58#41×C137—41no—
59#41×C137—41no—
60C2×Q8×C137—21no—
61C2×D4×C137—21no—
62C2×D4×C137—24no—
63C2×D4×C137—22no—
64C2×D4×C137—21no—
65C2×D4×C137—42no—
66C2×D4×C137—41no—
67C2×D4×C137—41no—
68C2×D4×C137—44no—
69C2×D4×C137—44no—
70C2×D4×C137—44no—
71C2×D4×C137—41no—
72C2×D4×C137—81no—
73C2×D4×C137—81no—
74C2×D4×C137—82no—
75C2×D4×C137D812yescomplemented
76C22×C4×C137—21no—
77C22×C4×C137—22no—
78C22×C4×C137—41no—
79C22×C4×C137C2×D411yescharacteristic
80C22×C4×C137C2×D412yes—
81SD16×C137—44no—
82D8×C137—44no—
83D8×C137—84no—
84M16×C137—22no—
85M16×C137—41no—
86C2×C8×C137—41no—
87C2×C8×C137—42no—
88C4⋊C4×C137—22no—
89C4⋊C4×C137—41no—
90C4⋊C4×C137—81no—
91C22⋊C4×C137—44no—
92C22⋊C4×C137—44no—
93C22⋊C4×C137—42no—
94C22⋊C4×C137—44no—
95C22⋊C4×C137—42no—
96C22⋊C4×C137—42no—
97C22⋊C4×C137—42no—
98C42×C137—41no—
99C23×C137—24no—
100C23×C137—22no—
101C23×C137—42no—
102C23×C137—42no—
103C23×C137—42no—
104C23×C137—42no—
105C23×C137—42no—
106C23×C137—42no—
107C23×C137—41no—
108C23×C137—42no—
109C23×C137—41no—
110C23×C137—42no—
111C23×C137—41no—
112C23×C137—88no—
113C23×C137—84no—
114C23×C137—88no—
115C23×C137—84no—
116C23×C137—84no—
117C23×C137#12011yescharacteristic
118Q8×C137—41no—
119Q8×C137—41no—
120D4×C137—42no—
121D4×C137—41no—
122D4×C137—41no—
123D4×C137—44no—
124D4×C137—81no—
125D4×C137—84no—
126D4×C137—82no—
127D4×C137—82no—
128D4×C137—88no—
129D4×C137—81no—
130D4×C137—81no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—22no—
134C2×C4×C137—22no—
135C2×C4×C137—41no—
136C2×C4×C137—41no—
137C2×C4×C137—42no—
138C2×C4×C137—42no—
139C2×C4×C137—42no—
140C2×C4×C137—42no—
141C2×C4×C137—41no—
142C2×C4×C137—81no—
143C2×C4×C137—81no—
144C2×C4×C137C2×D811yescharacteristic
145C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
146C8×C137—42no—
147C8×C137—82no—
148C22×C137—21no—
149C22×C137—21no—
150C22×C137—44no—
151C22×C137—44no—
152C22×C137—44no—
153C22×C137—42no—
154C22×C137—44no—
155C22×C137—82no—
156C22×C137—82no—
157C22×C137—82no—
158C22×C137—816no—
159C22×C137—82no—
160C22×C137—81no—
161C22×C137—81no—
162C22×C137—82no—
163C22×C137—81no—
164C22×C137—81no—
165C22×C137—81no—
166C22×C137#44711yescharacteristic
167C22×C137#45311yescharacteristic
168C22×C137B(4,𝔽2)11yescharacteristic
169C4×C137—21no—
170C4×C137—21no—
171C4×C137—42no—
172C4×C137—42no—
173C4×C137—82no—
174C4×C137—81no—
175C2×C137—21no—
176C2×C137—21no—
177C2×C137—82no—
178C2×C137—84no—
179C2×C137—161no—
180C2×C137#159511yescharacteristic
181C2×C137#219611yescharacteristic
182C2×C137#220211yescharacteristic
183#12096C13711yesmaximal, characteristic, direct product factor, Sylow
184C137#1209611yescharacteristic, direct product factor, Sylow
185#2679C2×C13711yescharacteristic, complemented
186#2006C2×C13711yescharacteristic, complemented
187#1598C2×C13712yescomplemented
188#1345C2×C13712yescomplemented
189#1318C2×C13711yescharacteristic, complemented
190C2×AGL(1,ℤ/8)—21no—
191#537C22×C13711yescharacteristic, complemented
192C2×#120—22no—
193C2×#120—21no—
194#447—22no—
195C2×#104—21no—
196#394—22no—
197#394C22×C13712yes—
198#390—21no—
199#390C22×C13711yescharacteristic
200#343C22×C13711yescharacteristic, complemented
201#327—22no—
202#324C22×C13712yescomplemented
203C25—21no—
204C2×#41—21no—
205C22×D4—21no—
206C22×D4—22no—
207C22×D4D4×C13711yescharacteristic
208C22×D4D4×C13712yes—
209AGL(1,ℤ/8)—42no—
210AGL(1,ℤ/8)—42no—
211C2×SD16—41no—
212C2×D8—41no—
213C2×D8—42no—
214C2×M16—21no—
215C22⋊Q8—41no—
216#121—22no—
217#121—41no—
218#120—48no—
219#120—42no—
220#120—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C23×C13711yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C42D4×C13711yescharacteristic
229#104—42no—
230#102—42no—
231C22⋊C8—22no—
232#95D4×C13712yes—
233C24—22no—
234C24—21no—
235C24—44no—
236C24—42no—
237C24—82no—
238C24—81no—
239C24—81no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—41no—
256C2×D4—81no—
257C2×D4—81no—
258C2×D4—82no—
259C2×D4D8×C13712yescomplemented
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4C2×D4×C13711yescharacteristic
264C22×C4C2×D4×C13712yes—
265SD16—44no—
266D8—44no—
267D8—84no—
268M16—22no—
269M16—41no—
270C2×C8—41no—
271C2×C8—42no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—41no—
283C23—24no—
284C23—22no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—42no—
293C23—41no—
294C23—42no—
295C23—41no—
296C23—88no—
297C23—84no—
298C23—88no—
299C23—84no—
300C23—84no—
301C23#120×C13711yescharacteristic
302Q8—41no—
303Q8—41no—
304D4—42no—
305D4—41no—
306D4—41no—
307D4—44no—
308D4—81no—
309D4—84no—
310D4—82no—
311D4—82no—
312D4—88no—
313D4—81no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C2×D8×C13711yescharacteristic
329C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
330C8—42no—
331C8—82no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—44no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22—816no—
343C22—82no—
344C22—81no—
345C22—81no—
346C22—82no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#447×C13711yescharacteristic
351C22#453×C13711yescharacteristic
352C22B(4,𝔽2)×C13711yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—42no—
357C4—82no—
358C4—81no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2—84no—
363C2—161no—
364C2#1595×C13711yescharacteristic
365C2#2196×C13711yescharacteristic
366C2#2202×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C137
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. #2679×C137C2semidirect
2. #2006×C137C2semidirect
3. #1598×C137C2semidirect
4. #1345×C137C2semidirect
5. #1318×C137C2semidirect
7. #537×C137C22semidirect
16. #343×C137C22semidirect
18. #324×C137C22semidirect
75. C2×D4×C137D8semidirect
183. #12096C137direct (N × H)
184. C137#12096direct (N × H)
185. #2679C2×C137semidirect
186. #2006C2×C137semidirect
187. #1598C2×C137semidirect
188. #1345C2×C137semidirect
189. #1318C2×C137semidirect
191. #537C22×C137semidirect
200. #343C22×C137semidirect
202. #324C22×C137semidirect
259. C2×D4D8×C137semidirect

Maximal subgroups

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

subgroupindex
#2679×C1372
#2006×C1372
#1598×C1372
#1345×C1372
#1318×C1372
#12096137