Finite Group Databasebeta

← C137×#12215 #46861601

Subgroups — C137×#12215 #46861601

Order 35072.

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

normal subgroups only

This group has 400 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#2607×C137C211yesmaximal, characteristic
2#2030×C137C211yesmaximal, characteristic
3#1602×C137C212yesmaximal
4#1351×C137C212yesmaximal
5#1317×C137C211yesmaximal, characteristic
6#543×C137C2211yescharacteristic
7#451×C137—22no—
8#399×C137—21no—
9#393×C137C2212yes—
10#391×C137—22no—
11#387×C137—22no—
12#381×C137—21no—
13#381×C137C2211yescharacteristic
14C4×C22⋊C4×C137—21no—
15#342×C137—21no—
16#342×C137C2211yescharacteristic
17#328×C137—22no—
18#323×C137C2212yes—
19C22×Q8×C137D412yes—
20C23×C4×C137D411yescharacteristic
21C2×Q16×C137—42no—
22C22⋊Q8×C137—22no—
23C22⋊Q8×C137—42no—
24C2×C4⋊C4×C137—22no—
25C2×C4⋊C4×C137—21no—
26C2×C22⋊C4×C137—22no—
27C2×C22⋊C4×C137—21no—
28C2×C22⋊C4×C137—21no—
29C2×C22⋊C4×C137D411yescharacteristic
30C2×C22⋊C4×C137C2311yescharacteristic
31C2×C42×C137—21no—
32C2×C42×C137—21no—
33#103×C137—42no—
34C22⋊C8×C137—22no—
35#95×C137—22no—
36#95×C137—21no—
37#95×C137—22no—
38#95×C137—22no—
39#95×C137—42no—
40#95×C137D412yes—
41C24×C137C2×D411yescharacteristic
42C2×Q8×C137—22no—
43C2×Q8×C137—44no—
44C2×Q8×C137—42no—
45C22×C4×C137—21no—
46C22×C4×C137—22no—
47C22×C4×C137—21no—
48C22×C4×C137—21no—
49C22×C4×C137—22no—
50C22×C4×C137—22no—
51C22×C4×C137—41no—
52C22×C4×C137—41no—
53C22×C4×C137—41no—
54C22×C4×C137Q1612yescomplemented
55C22×C4×C137C2×D412yes—
56Q16×C137—84no—
57C2×C8×C137—42no—
58C4⋊C4×C137—22no—
59C4⋊C4×C137—44no—
60C4⋊C4×C137—42no—
61C4⋊C4×C137—81no—
62C22⋊C4×C137—41no—
63C22⋊C4×C137—44no—
64C22⋊C4×C137—41no—
65C22⋊C4×C137—41no—
66C22⋊C4×C137—41no—
67C22⋊C4×C137—41no—
68C22⋊C4×C137—41no—
69C42×C137—81no—
70C42×C137—81no—
71C23×C137—21no—
72C23×C137—21no—
73C23×C137—42no—
74C23×C137#12011yescharacteristic
75C23×C137C2×Q1611yescharacteristic
76C23×C137#13711yescharacteristic
77Q8×C137—44no—
78Q8×C137—82no—
79Q8×C137—82no—
80C2×C4×C137—24no—
81C2×C4×C137—22no—
82C2×C4×C137—22no—
83C2×C4×C137—42no—
84C2×C4×C137—41no—
85C2×C4×C137—41no—
86C2×C4×C137—42no—
87C2×C4×C137—42no—
88C2×C4×C137—42no—
89C2×C4×C137—42no—
90C2×C4×C137—42no—
91C2×C4×C137—42no—
92C2×C4×C137—42no—
93C2×C4×C137—41no—
94C2×C4×C137—42no—
95C2×C4×C137—81no—
96C2×C4×C137—82no—
97C2×C4×C137—81no—
98C2×C4×C137—82no—
99C8×C137—82no—
100C22×C137—22no—
101C22×C137—21no—
102C22×C137—21no—
103C22×C137—41no—
104C22×C137—41no—
105C22×C137—42no—
106C22×C137—81no—
107C22×C137#45111yescharacteristic
108C22×C137#45511yescharacteristic
109C22×C137B(4,𝔽2)11yescharacteristic
110C4×C137—42no—
111C4×C137—42no—
112C4×C137—44no—
113C4×C137—82no—
114C4×C137—81no—
115C4×C137—82no—
116C4×C137—161no—
117C2×C137—21no—
118C2×C137—21no—
119C2×C137—41no—
120C2×C137—41no—
121C2×C137#160211yescharacteristic
122C2×C137#219712yes—
123#12215C13711yesmaximal, characteristic, direct product factor, Sylow
124C137#1221511yescharacteristic, direct product factor, Sylow
125#2607C2×C13711yescharacteristic
126#2030C2×C13711yescharacteristic
127#1602C2×C13712yes—
128#1351C2×C13712yes—
129#1317C2×C13711yescharacteristic
130#543C22×C13711yescharacteristic
131#451—22no—
132#399—21no—
133#393C22×C13712yes—
134#391—22no—
135#387—22no—
136#381—21no—
137#381C22×C13711yescharacteristic
138C4×C22⋊C4—21no—
139#342—21no—
140#342C22×C13711yescharacteristic
141#328—22no—
142#323C22×C13712yes—
143C22×Q8D4×C13712yes—
144C23×C4D4×C13711yescharacteristic
145C2×Q16—42no—
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4D4×C13711yescharacteristic
154C2×C22⋊C4C23×C13711yescharacteristic
155C2×C42—21no—
156C2×C42—21no—
157#103—42no—
158C22⋊C8—22no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163#95—42no—
164#95D4×C13712yes—
165C24C2×D4×C13711yescharacteristic
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—42no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4Q16×C13712yescomplemented
179C22×C4C2×D4×C13712yes—
180Q16—84no—
181C2×C8—42no—
182C4⋊C4—22no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C22⋊C4—44no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C42—81no—
194C42—81no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#120×C13711yescharacteristic
199C23C2×Q16×C13711yescharacteristic
200C23#137×C13711yescharacteristic
201Q8—44no—
202Q8—82no—
203Q8—82no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—82no—
223C8—82no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#451×C13711yescharacteristic
232C22#455×C13711yescharacteristic
233C22B(4,𝔽2)×C13711yescharacteristic
234C4—42no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—82no—
240C4—161no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#1602×C13711yescharacteristic
246C2#2197×C13712yes—
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C137
prime radicalC24×C137C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C2×C22⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
54. C22×C4×C137Q16semidirect
123. #12215C137direct (N × H)
124. C137#12215direct (N × H)
178. C22×C4Q16×C137semidirect

Maximal subgroups

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

subgroupindex
#2607×C1372
#2030×C1372
#1602×C1372
#1351×C1372
#1317×C1372
#12215137