Finite Group Databasebeta

← C137×#7827 #46857213

Subgroups — C137×#7827 #46857213

Order 35072.

358 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 102 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#2726×C137C211yesmaximal, characteristic
2#2100×C137C211yesmaximal, characteristic
3#2100×C137C211yesmaximal, characteristic
4#2100×C137C211yesmaximal, characteristic
5#1271×C137C211yesmaximal, characteristic
6#1271×C137C211yesmaximal, characteristic
7#1271×C137C211yesmaximal, characteristic
8C2×#125×C137—21no—
9#398×C137—21no—
10#398×C137—21no—
11#398×C137—21no—
12#389×C137—21no—
13#389×C137C2211yescharacteristic
14#389×C137C2211yescharacteristic
15#389×C137C2211yescharacteristic
16#387×C137—21no—
17#384×C137C2211yescharacteristic
18#340×C137—21no—
19#340×C137—21no—
20#340×C137—21no—
21#340×C137—21no—
22#340×C137—21no—
23#340×C137—21no—
24#336×C137C2211yescharacteristic
25#336×C137C2211yescharacteristic
26#336×C137C2211yescharacteristic
27C22×C8×C137—21no—
28C22×C8×C137—21no—
29C22×C8×C137—21no—
30#125×C137—42no—
31#125×C137—42no—
32C2×C4⋊C4×C137—21no—
33C2×C4⋊C4×C137—21no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137—21no—
37C2×C4⋊C4×C137—21no—
38C2×C4⋊C4×C137—21no—
39C2×C4⋊C4×C137—21no—
40C2×C4⋊C4×C137—21no—
41C2×C4⋊C4×C137—21no—
42C2×C4⋊C4×C137D411yescharacteristic
43C2×C4⋊C4×C137D411yescharacteristic
44C2×C4⋊C4×C137D411yescharacteristic
45C2×C4⋊C4×C137D411yescharacteristic
46C2×C4⋊C4×C137D411yescharacteristic
47C2×C4⋊C4×C137D411yescharacteristic
48C2×C42×C137—21no—
49C2×C42×C137C2311yescharacteristic
50#95×C137—21no—
51#95×C137—21no—
52#95×C137—21no—
53#95×C137—21no—
54#95×C137—21no—
55#95×C137—21no—
56C22×C4×C137—21no—
57C22×C4×C137—21no—
58C22×C4×C137—21no—
59C22×C4×C137—21no—
60C22×C4×C137—21no—
61C22×C4×C137—21no—
62C22×C4×C137—41no—
63C22×C4×C137C2×D411yescharacteristic
64C22×C4×C137C2×D411yescharacteristic
65C22×C4×C137C2×D411yescharacteristic
66C2×C8×C137—41no—
67C2×C8×C137—41no—
68C2×C8×C137—41no—
69C2×C8×C137—41no—
70C2×C8×C137—41no—
71C2×C8×C137—41no—
72C2×C8×C137—41no—
73C2×C8×C137—41no—
74C2×C8×C137—41no—
75C4⋊C4×C137—21no—
76C4⋊C4×C137—21no—
77C4⋊C4×C137—21no—
78C4⋊C4×C137—21no—
79C4⋊C4×C137—41no—
80C4⋊C4×C137—41no—
81C4⋊C4×C137—41no—
82C4⋊C4×C137—42no—
83C4⋊C4×C137—42no—
84C4⋊C4×C137—42no—
85C4⋊C4×C137—42no—
86C4⋊C4×C137—42no—
87C4⋊C4×C137—42no—
88C4⋊C4×C137—41no—
89C4⋊C4×C137—41no—
90C4⋊C4×C137—41no—
91C4⋊C4×C137—41no—
92C4⋊C4×C137—41no—
93C4⋊C4×C137—41no—
94C4⋊C4×C137—81no—
95C4⋊C4×C137SD1612yes—
96C4⋊C4×C137Q1612yes—
97C42×C137—41no—
98C42×C137—42no—
99C23×C137#12011yescharacteristic
100C2×C4×C137—21no—
101C2×C4×C137—21no—
102C2×C4×C137—21no—
103C2×C4×C137—21no—
104C2×C4×C137—21no—
105C2×C4×C137—21no—
106C2×C4×C137—22no—
107C2×C4×C137—22no—
108C2×C4×C137—41no—
109C2×C4×C137—41no—
110C2×C4×C137—41no—
111C2×C4×C137—41no—
112C2×C4×C137—41no—
113C2×C4×C137—41no—
114C2×C4×C137—41no—
115C2×C4×C137—41no—
116C2×C4×C137—41no—
117C2×C4×C137—41no—
118C2×C4×C137—41no—
119C2×C4×C137—41no—
120C2×C4×C137—41no—
121C2×C4×C137—41no—
122C2×C4×C137—41no—
123C2×C4×C137—41no—
124C2×C4×C137—81no—
125C2×C4×C137—81no—
126C2×C4×C137—81no—
127C2×C4×C137C2×SD1611yescharacteristic
128C2×C4×C137C2×Q1611yescharacteristic
129C2×C4×C137#13511yescharacteristic
130C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C137#13711yescharacteristic
132C2×C4×C137#13711yescharacteristic
133C8×C137—81no—
134C8×C137—81no—
135C8×C137—81no—
136C22×C137#45011yescharacteristic
137C22×C137#45111yescharacteristic
138C22×C137#45211yescharacteristic
139C22×C137#45411yescharacteristic
140C22×C137#45411yescharacteristic
141C22×C137#45411yescharacteristic
142C22×C137#45811yescharacteristic
143C4×C137—21no—
144C4×C137—21no—
145C4×C137—21no—
146C4×C137—21no—
147C4×C137—21no—
148C4×C137—21no—
149C4×C137—42no—
150C4×C137—42no—
151C4×C137—81no—
152C4×C137—81no—
153C4×C137—81no—
154C4×C137—81no—
155C4×C137—161no—
156C2×C137#161511yescharacteristic
157C2×C137#161611yescharacteristic
158C2×C137#161711yescharacteristic
159C2×C137#163511yescharacteristic
160C2×C137#163611yescharacteristic
161C2×C137#163711yescharacteristic
162C2×C137#165811yescharacteristic
163#7827C13711yesmaximal, characteristic, direct product factor, Sylow
164C137#782711yescharacteristic, direct product factor, Sylow
165#2726C2×C13711yescharacteristic
166#2100C2×C13711yescharacteristic
167#2100C2×C13711yescharacteristic
168#2100C2×C13711yescharacteristic
169#1271C2×C13711yescharacteristic
170#1271C2×C13711yescharacteristic
171#1271C2×C13711yescharacteristic
172C2×#125—21no—
173#398—21no—
174#398—21no—
175#398—21no—
176#389—21no—
177#389C22×C13711yescharacteristic
178#389C22×C13711yescharacteristic
179#389C22×C13711yescharacteristic
180#387—21no—
181#384C22×C13711yescharacteristic
182#340—21no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—21no—
187#340—21no—
188#336C22×C13711yescharacteristic
189#336C22×C13711yescharacteristic
190#336C22×C13711yescharacteristic
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194#125—42no—
195#125—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4D4×C13711yescharacteristic
207C2×C4⋊C4D4×C13711yescharacteristic
208C2×C4⋊C4D4×C13711yescharacteristic
209C2×C4⋊C4D4×C13711yescharacteristic
210C2×C4⋊C4D4×C13711yescharacteristic
211C2×C4⋊C4D4×C13711yescharacteristic
212C2×C42—21no—
213C2×C42C23×C13711yescharacteristic
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4C2×D4×C13711yescharacteristic
228C22×C4C2×D4×C13711yescharacteristic
229C22×C4C2×D4×C13711yescharacteristic
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—81no—
259C4⋊C4SD16×C13712yes—
260C4⋊C4Q16×C13712yes—
261C42—41no—
262C42—42no—
263C23#120×C13711yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4C2×SD16×C13711yescharacteristic
292C2×C4C2×Q16×C13711yescharacteristic
293C2×C4#135×C13711yescharacteristic
294C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
295C2×C4#137×C13711yescharacteristic
296C2×C4#137×C13711yescharacteristic
297C8—81no—
298C8—81no—
299C8—81no—
300C22#450×C13711yescharacteristic
301C22#451×C13711yescharacteristic
302C22#452×C13711yescharacteristic
303C22#454×C13711yescharacteristic
304C22#454×C13711yescharacteristic
305C22#454×C13711yescharacteristic
306C22#458×C13711yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—161no—
320C2#1615×C13711yescharacteristic
321C2#1616×C13711yescharacteristic
322C2#1617×C13711yescharacteristic
323C2#1635×C13711yescharacteristic
324C2#1636×C13711yescharacteristic
325C2#1637×C13711yescharacteristic
326C2#1658×C13711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
163. #7827C137direct (N × H)
164. C137#7827direct (N × H)

Maximal subgroups

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

subgroupindex
#2726×C1372
#2100×C1372
#1271×C1372
#7827137