Finite Group Databasebeta

← C137×#12111 #46861497

Subgroups — C137×#12111 #46861497

Order 35072.

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

normal subgroups only

This group has 422 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#3199×C137C211yesmaximal, characteristic
2#2185×C137C212yesmaximal, complemented
3#2020×C137C211yesmaximal, characteristic, complemented
4#1349×C137C212yesmaximal, complemented
5#1315×C137C211yesmaximal, characteristic, complemented
6C2×#137×C137—21no—
7C22×Q16×C137C2211yescharacteristic, complemented
8#535×C137—21no—
9C2×SD32×C137—22no—
10#482×C137—22no—
11#469×C137—21no—
12#466×C137C2212yes—
13#447×C137—22no—
14C2×#101×C137—21no—
15#408×C137C2211yescharacteristic
16#358×C137—22no—
17C22⋊C16×C137C2212yescomplemented
18#341×C137C2211yescharacteristic, complemented
19C2×#41×C137D411yescharacteristic
20C22×Q8×C137—21no—
21C22×D4×C137—21no—
22#137×C137—42no—
23#137×C137—42no—
24C2×Q16×C137—22no—
25C2×Q16×C137—42no—
26C2×Q16×C137—41no—
27C2×SD16×C137—41no—
28C2×D8×C137—21no—
29C2×D8×C137D412yes—
30C2×M16×C137—21no—
31C2×M16×C137D411yescharacteristic
32C22×C8×C137C2311yescharacteristic
33#127×C137—41no—
34#124×C137—41no—
35#121×C137—22no—
36#121×C137—22no—
37#120×C137—41no—
38#120×C137—41no—
39#117×C137—21no—
40SD32×C137—44no—
41C2×C16×C137—22no—
42#107×C137D412yes—
43#106×C137—21no—
44#102×C137—22no—
45#102×C137—22no—
46#101×C137—42no—
47C22⋊C8×C137—22no—
48C24×C137—41no—
49#41×C137—41no—
50#41×C137—41no—
51C2×Q8×C137—22no—
52C2×Q8×C137—41no—
53C2×Q8×C137—41no—
54C2×Q8×C137—41no—
55C2×Q8×C137—41no—
56C2×Q8×C137—41no—
57C2×Q8×C137D811yescharacteristic
58C2×D4×C137—21no—
59C2×D4×C137—22no—
60C2×D4×C137—41no—
61C2×D4×C137—42no—
62C2×D4×C137—42no—
63C2×D4×C137—81no—
64C2×D4×C137—81no—
65C2×D4×C137D811yescharacteristic
66C22×C4×C137—21no—
67C22×C4×C137—41no—
68C22×C4×C137C2×D411yescharacteristic
69Q16×C137—24no—
70Q16×C137—42no—
71Q16×C137—44no—
72SD16×C137—44no—
73D8×C137—42no—
74D8×C137—81no—
75M16×C137—22no—
76M16×C137—41no—
77M16×C137—41no—
78C2×C8×C137—21no—
79C2×C8×C137—41no—
80C2×C8×C137—41no—
81C2×C8×C137C2×D412yes—
82C4⋊C4×C137—22no—
83C22⋊C4×C137—42no—
84C22⋊C4×C137—41no—
85C22⋊C4×C137—42no—
86C42×C137—41no—
87C16×C137—42no—
88C23×C137—21no—
89C23×C137—42no—
90C23×C137—82no—
91C23×C137—162no—
92C23×C137AGL(1,ℤ/8)11yescharacteristic
93Q8×C137—41no—
94Q8×C137—42no—
95Q8×C137—42no—
96Q8×C137—81no—
97Q8×C137—81no—
98D4×C137—41no—
99D4×C137—42no—
100D4×C137—42no—
101D4×C137—81no—
102D4×C137—82no—
103D4×C137—162no—
104D4×C137—162no—
105C2×C4×C137—21no—
106C2×C4×C137—21no—
107C2×C4×C137—21no—
108C2×C4×C137—41no—
109C2×C4×C137—41no—
110C2×C4×C137—42no—
111C2×C4×C137—42no—
112C2×C4×C137—81no—
113C2×C4×C137—81no—
114C2×C4×C137#12011yescharacteristic
115C2×C4×C137C2×D811yescharacteristic
116C8×C137—22no—
117C8×C137—42no—
118C8×C137—41no—
119C22×C137—21no—
120C22×C137—22no—
121C22×C137—41no—
122C22×C137—42no—
123C22×C137—82no—
124C22×C137—82no—
125C22×C137—82no—
126C22×C137—164no—
127C22×C137#44711yescharacteristic
128C4×C137—21no—
129C4×C137—41no—
130C4×C137—82no—
131C4×C137—82no—
132C4×C137#45311yescharacteristic
133C4×C137B(4,𝔽2)11yescharacteristic
134C2×C137—41no—
135C2×C137—81no—
136C2×C137—162no—
137C2×C137#159511yescharacteristic
138C2×C137#219412yes—
139#12111C13711yesmaximal, characteristic, direct product factor, Sylow
140C137#1211111yescharacteristic, direct product factor, Sylow
141#3199C2×C13711yescharacteristic
142#2185C2×C13712yescomplemented
143#2020C2×C13711yescharacteristic, complemented
144#1349C2×C13712yescomplemented
145#1315C2×C13711yescharacteristic, complemented
146C2×#137—21no—
147C22×Q16C22×C13711yescharacteristic, complemented
148#535—21no—
149C2×SD32—22no—
150#482—22no—
151#469—21no—
152#466C22×C13712yes—
153#447—22no—
154C2×#101—21no—
155#408C22×C13711yescharacteristic
156#358—22no—
157C22⋊C16C22×C13712yescomplemented
158#341C22×C13711yescharacteristic, complemented
159C2×#41D4×C13711yescharacteristic
160C22×Q8—21no—
161C22×D4—21no—
162#137—42no—
163#137—42no—
164C2×Q16—22no—
165C2×Q16—42no—
166C2×Q16—41no—
167C2×SD16—41no—
168C2×D8—21no—
169C2×D8D4×C13712yes—
170C2×M16—21no—
171C2×M16D4×C13711yescharacteristic
172C22×C8C23×C13711yescharacteristic
173#127—41no—
174#124—41no—
175#121—22no—
176#121—22no—
177#120—41no—
178#120—41no—
179#117—21no—
180SD32—44no—
181C2×C16—22no—
182#107D4×C13712yes—
183#106—21no—
184#102—22no—
185#102—22no—
186#101—42no—
187C22⋊C8—22no—
188C24—41no—
189#41—41no—
190#41—41no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8D8×C13711yescharacteristic
198C2×D4—21no—
199C2×D4—22no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4D8×C13711yescharacteristic
206C22×C4—21no—
207C22×C4—41no—
208C22×C4C2×D4×C13711yescharacteristic
209Q16—24no—
210Q16—42no—
211Q16—44no—
212SD16—44no—
213D8—42no—
214D8—81no—
215M16—22no—
216M16—41no—
217M16—41no—
218C2×C8—21no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8C2×D4×C13712yes—
222C4⋊C4—22no—
223C22⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C42—41no—
227C16—42no—
228C23—21no—
229C23—42no—
230C23—82no—
231C23—162no—
232C23AGL(1,ℤ/8)×C13711yescharacteristic
233Q8—41no—
234Q8—42no—
235Q8—42no—
236Q8—81no—
237Q8—81no—
238D4—41no—
239D4—42no—
240D4—42no—
241D4—81no—
242D4—82no—
243D4—162no—
244D4—162no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4#120×C13711yescharacteristic
255C2×C4C2×D8×C13711yescharacteristic
256C8—22no—
257C8—42no—
258C8—41no—
259C22—21no—
260C22—22no—
261C22—41no—
262C22—42no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—164no—
267C22#447×C13711yescharacteristic
268C4—21no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4#453×C13711yescharacteristic
273C4B(4,𝔽2)×C13711yescharacteristic
274C2—41no—
275C2—81no—
276C2—162no—
277C2#1595×C13711yescharacteristic
278C2#2194×C13712yes—
subgroupG/H
center, socleC22×C137#447
commutator, Frattini, cosocleC22×C8C23×C137
prime radical#3199×C137C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ #408×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2185×C137C2semidirect
3. #2020×C137C2semidirect
4. #1349×C137C2semidirect
5. #1315×C137C2semidirect
7. C22×Q16×C137C22semidirect
17. C22⋊C16×C137C22semidirect
18. #341×C137C22semidirect
139. #12111C137direct (N × H)
140. C137#12111direct (N × H)
142. #2185C2×C137semidirect
143. #2020C2×C137semidirect
144. #1349C2×C137semidirect
145. #1315C2×C137semidirect
147. C22×Q16C22×C137semidirect
157. C22⋊C16C22×C137semidirect
158. #341C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3199×C1372
#2185×C1372
#2020×C1372
#1349×C1372
#1315×C1372
#12111137