Finite Group Databasebeta

← C137×#12109 #46861495

Subgroups — C137×#12109 #46861495

Order 35072.

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

normal subgroups only

This group has 410 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#3202×C137C211yesmaximal, characteristic, complemented
2#2187×C137C212yesmaximal, complemented
3#2018×C137C211yesmaximal, characteristic
4#1350×C137C212yesmaximal, complemented
5#1315×C137C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C137—21no—
7C22×D8×C137C2211yescharacteristic
8#536×C137—21no—
9C2×SD32×C137—22no—
10#485×C137—22no—
11#470×C137—21no—
12#467×C137C2212yes—
13#451×C137—22no—
14C2×#100×C137—21no—
15#408×C137C2211yescharacteristic
16#357×C137—22no—
17C22⋊C16×C137C2212yes—
18#341×C137C2211yescharacteristic
19C2×#41×C137D411yescharacteristic
20C22×Q8×C137—21no—
21C22×D4×C137—21no—
22AGL(1,ℤ/8)×C137—42no—
23AGL(1,ℤ/8)×C137—42no—
24C2×Q16×C137—21no—
25C2×Q16×C137D412yes—
26C2×SD16×C137—41no—
27C2×D8×C137—22no—
28C2×D8×C137—42no—
29C2×D8×C137—41no—
30C2×M16×C137—21no—
31C2×M16×C137D411yescharacteristic
32C22×C8×C137C2311yescharacteristic
33C4⋊Q8×C137—41no—
34#124×C137—41no—
35#123×C137—41no—
36#123×C137—41no—
37C22⋊Q8×C137—22no—
38C22⋊Q8×C137—22no—
39#117×C137—21no—
40SD32×C137—44no—
41C2×C16×C137—22no—
42#107×C137D412yes—
43#106×C137—21no—
44#103×C137—22no—
45#103×C137—22no—
46#100×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—81no—
54C2×Q8×C137—81no—
55C2×Q8×C137D811yescharacteristic
56C2×D4×C137—22no—
57C2×D4×C137—21no—
58C2×D4×C137—41no—
59C2×D4×C137—41no—
60C2×D4×C137—41no—
61C2×D4×C137—41no—
62C2×D4×C137—41no—
63C2×D4×C137D811yescharacteristic
64C22×C4×C137—21no—
65C22×C4×C137—41no—
66C22×C4×C137C2×D411yescharacteristic
67Q16×C137—42no—
68Q16×C137—81no—
69SD16×C137—44no—
70D8×C137—24no—
71D8×C137—42no—
72D8×C137—44no—
73M16×C137—22no—
74M16×C137—41no—
75M16×C137—41no—
76C2×C8×C137—21no—
77C2×C8×C137—41no—
78C2×C8×C137—41no—
79C2×C8×C137C2×D412yes—
80C4⋊C4×C137—22no—
81C4⋊C4×C137—42no—
82C4⋊C4×C137—42no—
83C22⋊C4×C137—41no—
84C22⋊C4×C137—42no—
85C22⋊C4×C137—42no—
86C42×C137—41no—
87C16×C137—42no—
88C23×C137—21no—
89C23×C137—42no—
90C23×C137—81no—
91C23×C137—81no—
92C23×C137—162no—
93C23×C137AGL(1,ℤ/8)11yescharacteristic
94Q8×C137—41no—
95Q8×C137—42no—
96Q8×C137—42no—
97Q8×C137—82no—
98D4×C137—41no—
99D4×C137—42no—
100D4×C137—42no—
101D4×C137—81no—
102D4×C137—81no—
103D4×C137—81no—
104C2×C4×C137—21no—
105C2×C4×C137—21no—
106C2×C4×C137—21no—
107C2×C4×C137—41no—
108C2×C4×C137—41no—
109C2×C4×C137—42no—
110C2×C4×C137—42no—
111C2×C4×C137—82no—
112C2×C4×C137#12011yescharacteristic
113C2×C4×C137C2×D811yescharacteristic
114C8×C137—22no—
115C8×C137—42no—
116C8×C137—41no—
117C22×C137—21no—
118C22×C137—22no—
119C22×C137—41no—
120C22×C137—42no—
121C22×C137—82no—
122C22×C137—162no—
123C22×C137—162no—
124C22×C137—162no—
125C22×C137#44711yescharacteristic
126C4×C137—21no—
127C4×C137—41no—
128C4×C137—82no—
129C4×C137—82no—
130C4×C137#45311yescharacteristic
131C4×C137B(4,𝔽2)11yescharacteristic
132C2×C137—41no—
133C2×C137—81no—
134C2×C137—162no—
135C2×C137#159511yescharacteristic
136C2×C137#219312yes—
137#12109C13711yesmaximal, characteristic, direct product factor, Sylow
138C137#1210911yescharacteristic, direct product factor, Sylow
139#3202C2×C13711yescharacteristic, complemented
140#2187C2×C13712yescomplemented
141#2018C2×C13711yescharacteristic
142#1350C2×C13712yescomplemented
143#1315C2×C13711yescharacteristic, complemented
144C2×AGL(1,ℤ/8)—21no—
145C22×D8C22×C13711yescharacteristic
146#536—21no—
147C2×SD32—22no—
148#485—22no—
149#470—21no—
150#467C22×C13712yes—
151#451—22no—
152C2×#100—21no—
153#408C22×C13711yescharacteristic
154#357—22no—
155C22⋊C16C22×C13712yes—
156#341C22×C13711yescharacteristic
157C2×#41D4×C13711yescharacteristic
158C22×Q8—21no—
159C22×D4—21no—
160AGL(1,ℤ/8)—42no—
161AGL(1,ℤ/8)—42no—
162C2×Q16—21no—
163C2×Q16D4×C13712yes—
164C2×SD16—41no—
165C2×D8—22no—
166C2×D8—42no—
167C2×D8—41no—
168C2×M16—21no—
169C2×M16D4×C13711yescharacteristic
170C22×C8C23×C13711yescharacteristic
171C4⋊Q8—41no—
172#124—41no—
173#123—41no—
174#123—41no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#117—21no—
178SD32—44no—
179C2×C16—22no—
180#107D4×C13712yes—
181#106—21no—
182#103—22no—
183#103—22no—
184#100—42no—
185C22⋊C8—22no—
186C24—41no—
187#41—41no—
188#41—41no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8D8×C13711yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4D8×C13711yescharacteristic
202C22×C4—21no—
203C22×C4—41no—
204C22×C4C2×D4×C13711yescharacteristic
205Q16—42no—
206Q16—81no—
207SD16—44no—
208D8—24no—
209D8—42no—
210D8—44no—
211M16—22no—
212M16—41no—
213M16—41no—
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8C2×D4×C13712yes—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—41no—
225C16—42no—
226C23—21no—
227C23—42no—
228C23—81no—
229C23—81no—
230C23—162no—
231C23AGL(1,ℤ/8)×C13711yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—42no—
235Q8—82no—
236D4—41no—
237D4—42no—
238D4—42no—
239D4—81no—
240D4—81no—
241D4—81no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4#120×C13711yescharacteristic
251C2×C4C2×D8×C13711yescharacteristic
252C8—22no—
253C8—42no—
254C8—41no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—42no—
259C22—82no—
260C22—162no—
261C22—162no—
262C22—162no—
263C22#447×C13711yescharacteristic
264C4—21no—
265C4—41no—
266C4—82no—
267C4—82no—
268C4#453×C13711yescharacteristic
269C4B(4,𝔽2)×C13711yescharacteristic
270C2—41no—
271C2—81no—
272C2—162no—
273C2#1595×C13711yescharacteristic
274C2#2193×C13712yes—
subgroupG/H
center, socleC22×C137#447
commutator, Frattini, cosocleC22×C8C23×C137
prime radical#2018×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
1. #3202×C137C2semidirect
2. #2187×C137C2semidirect
4. #1350×C137C2semidirect
5. #1315×C137C2semidirect
137. #12109C137direct (N × H)
138. C137#12109direct (N × H)
139. #3202C2×C137semidirect
140. #2187C2×C137semidirect
142. #1350C2×C137semidirect
143. #1315C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3202×C1372
#2187×C1372
#2018×C1372
#1350×C1372
#1315×C1372
#12109137