Finite Group Databasebeta

← C137×#11834 #46861220

Subgroups — C137×#11834 #46861220

Order 35072.

400 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 98 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#2638×C137C211yesmaximal, characteristic
2#1800×C137C211yesmaximal, characteristic
3#1465×C137C211yesmaximal, characteristic
4#1354×C137C212yesmaximal, complemented
5#1346×C137C212yesmaximal, complemented
6C2×C22⋊Q8×C137C411yescharacteristic
7C2×C4×D4×C137C2211yescharacteristic
8#423×C137—21no—
9C2×#100×C137—21no—
10#407×C137—21no—
11#398×C137C2211yescharacteristic
12#397×C137—21no—
13#397×C137C411yescharacteristic
14#395×C137C2211yescharacteristic
15#394×C137—21no—
16#393×C137C411yescharacteristic
17#387×C137—21no—
18#385×C137—21no—
19#381×C137C411yescharacteristic
20#343×C137C2212yes—
21#324×C137C2212yes—
22#323×C137—21no—
23C22×Q8×C137C2×C411yescharacteristic
24C22×D4×C137D411yescharacteristic
25C23×C4×C137C2×C411yescharacteristic
26C23×C4×C137D411yescharacteristic
27C2×M16×C137—22no—
28C2×M16×C137—41no—
29C22⋊Q8×C137—41no—
30C22⋊Q8×C137—41no—
31C4×D4×C137—41no—
32C4×D4×C137—41no—
33C2×C4⋊C4×C137—21no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137C2×C411yescharacteristic
37C2×C4⋊C4×C137C2311yescharacteristic
38C2×C22⋊C4×C137—21no—
39C2×C22⋊C4×C137—21no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—21no—
42C2×C22⋊C4×C137C2×C411yescharacteristic
43C2×C22⋊C4×C137D411yescharacteristic
44C2×C42×C137D411yescharacteristic
45C4⋊C8×C137—41no—
46C4⋊C8×C137—41no—
47#100×C137—42no—
48C22⋊C8×C137—22no—
49C22⋊C8×C137—41no—
50#95×C137—21no—
51#95×C137—21no—
52#95×C137—21no—
53#95×C137—21no—
54#95×C137—21no—
55#95×C137C2×C411yescharacteristic
56#95×C137C2×C411yescharacteristic
57C24×C137C22⋊C411yescharacteristic
58C24×C137C22⋊C411yescharacteristic
59C2×Q8×C137—41no—
60C2×Q8×C137—81no—
61C2×D4×C137—21no—
62C2×D4×C137—21no—
63C2×D4×C137—41no—
64C2×D4×C137—41no—
65C22×C4×C137—21no—
66C22×C4×C137—21no—
67C22×C4×C137—21no—
68C22×C4×C137—21no—
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—21no—
72C22×C4×C137—21no—
73C22×C4×C137—81no—
74C22×C4×C137—81no—
75C22×C4×C137C22⋊C411yescharacteristic
76C22×C4×C137C22⋊C411yescharacteristic
77C22×C4×C137C22×C411yescharacteristic
78C22×C4×C137C2×D411yescharacteristic
79C22×C4×C137C2×D411yescharacteristic
80M16×C137—42no—
81M16×C137—42no—
82M16×C137—82no—
83C2×C8×C137—42no—
84C2×C8×C137—42no—
85C4⋊C4×C137—41no—
86C4⋊C4×C137—41no—
87C4⋊C4×C137—41no—
88C4⋊C4×C137—41no—
89C4⋊C4×C137—41no—
90C4⋊C4×C137—41no—
91C4⋊C4×C137—81no—
92C22⋊C4×C137—41no—
93C22⋊C4×C137—41no—
94C22⋊C4×C137—42no—
95C22⋊C4×C137—42no—
96C22⋊C4×C137—81no—
97C22⋊C4×C137—81no—
98C42×C137—21no—
99C42×C137—21no—
100C23×C137—21no—
101C23×C137—21no—
102C23×C137—21no—
103C23×C137—21no—
104C23×C137—21no—
105C23×C137—21no—
106C23×C137—41no—
107C23×C137—41no—
108C23×C137—41no—
109C23×C137—41no—
110C23×C137C2×C22⋊C411yescharacteristic
111Q8×C137—82no—
112D4×C137—42no—
113D4×C137—42no—
114C2×C4×C137—21no—
115C2×C4×C137—21no—
116C2×C4×C137—21no—
117C2×C4×C137—21no—
118C2×C4×C137—21no—
119C2×C4×C137—41no—
120C2×C4×C137—42no—
121C2×C4×C137—41no—
122C2×C4×C137—41no—
123C2×C4×C137—41no—
124C2×C4×C137—41no—
125C2×C4×C137—41no—
126C2×C4×C137—41no—
127C2×C4×C137—41no—
128C2×C4×C137—41no—
129C2×C4×C137—41no—
130C2×C4×C137—41no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—41no—
134C2×C4×C137—41no—
135C2×C4×C137—41no—
136C2×C4×C137—81no—
137C2×C4×C137—81no—
138C2×C4×C137—81no—
139C2×C4×C137#9911yescharacteristic
140C2×C4×C137#9911yescharacteristic
141C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C137#13711yescharacteristic
143C8×C137—82no—
144C8×C137—82no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—42no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137—81no—
155C22×C137—81no—
156C22×C137—81no—
157C22×C137C2×#9911yescharacteristic
158C22×C137#41711yescharacteristic
159C22×C137#42111yescharacteristic
160C4×C137—21no—
161C4×C137—21no—
162C4×C137—21no—
163C4×C137—21no—
164C4×C137—41no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137—81no—
168C4×C137—81no—
169C4×C137—81no—
170C4×C137—81no—
171C2×C137—21no—
172C2×C137—21no—
173C2×C137—81no—
174C2×C137—81no—
175C2×C137#151311yescharacteristic
176C2×C137#212111yescharacteristic
177C2×C137#213511yescharacteristic
178#11834C13711yesmaximal, characteristic, direct product factor, Sylow
179C137#1183411yescharacteristic, direct product factor, Sylow
180#2638C2×C13711yescharacteristic
181#1800C2×C13711yescharacteristic
182#1465C2×C13711yescharacteristic
183#1354C2×C13712yescomplemented
184#1346C2×C13712yescomplemented
185C2×C22⋊Q8C4×C13711yescharacteristic
186C2×C4×D4C22×C13711yescharacteristic
187#423—21no—
188C2×#100—21no—
189#407—21no—
190#398C22×C13711yescharacteristic
191#397—21no—
192#397C4×C13711yescharacteristic
193#395C22×C13711yescharacteristic
194#394—21no—
195#393C4×C13711yescharacteristic
196#387—21no—
197#385—21no—
198#381C4×C13711yescharacteristic
199#343C22×C13712yes—
200#324C22×C13712yes—
201#323—21no—
202C22×Q8C2×C4×C13711yescharacteristic
203C22×D4D4×C13711yescharacteristic
204C23×C4C2×C4×C13711yescharacteristic
205C23×C4D4×C13711yescharacteristic
206C2×M16—22no—
207C2×M16—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×C4×C13711yescharacteristic
216C2×C4⋊C4C23×C13711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4C2×C4×C13711yescharacteristic
222C2×C22⋊C4D4×C13711yescharacteristic
223C2×C42D4×C13711yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—41no—
226#100—42no—
227C22⋊C8—22no—
228C22⋊C8—41no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C2×C4×C13711yescharacteristic
235#95C2×C4×C13711yescharacteristic
236C24C22⋊C4×C13711yescharacteristic
237C24C22⋊C4×C13711yescharacteristic
238C2×Q8—41no—
239C2×Q8—81no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C22⋊C4×C13711yescharacteristic
255C22×C4C22⋊C4×C13711yescharacteristic
256C22×C4C22×C4×C13711yescharacteristic
257C22×C4C2×D4×C13711yescharacteristic
258C22×C4C2×D4×C13711yescharacteristic
259M16—42no—
260M16—42no—
261M16—82no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C42—21no—
278C42—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊C4×C13711yescharacteristic
290Q8—82no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#99×C13711yescharacteristic
319C2×C4#99×C13711yescharacteristic
320C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
321C2×C4#137×C13711yescharacteristic
322C8—82no—
323C8—82no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×#99×C13711yescharacteristic
337C22#417×C13711yescharacteristic
338C22#421×C13711yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—81no—
354C2#1513×C13711yescharacteristic
355C2#2121×C13711yescharacteristic
356C2#2135×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C4⋊C4C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1354×C137C2semidirect
5. #1346×C137C2semidirect
178. #11834C137direct (N × H)
179. C137#11834direct (N × H)
183. #1354C2×C137semidirect
184. #1346C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2638×C1372
#1800×C1372
#1465×C1372
#1354×C1372
#1346×C1372
#11834137