Finite Group Databasebeta

← #1597×C137 #31776608

Subgroups — #1597×C137 #31776608

Order 17536.

272 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 64 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#539×C137C211yesmaximal, characteristic, complemented
2#450×C137C211yesmaximal, characteristic, complemented
3#447×C137C211yesmaximal, characteristic
4#394×C137C211yesmaximal, characteristic, complemented
5#328×C137C211yesmaximal, characteristic, complemented
6#327×C137C211yesmaximal, characteristic, complemented
7#324×C137C211yesmaximal, characteristic, complemented
8C22×D4×C137C2211yescharacteristic
9C2×SD16×C137—21no—
10C2×D8×C137—21no—
11#126×C137—21no—
12#125×C137—21no—
13#123×C137—21no—
14#123×C137—21no—
15C22⋊Q8×C137—21no—
16C22⋊Q8×C137C2211yescharacteristic
17#121×C137C2211yescharacteristic
18C4×D4×C137—21no—
19C2×C4⋊C4×C137C2211yescharacteristic
20C2×C22⋊C4×C137—21no—
21C2×C22⋊C4×C137—21no—
22#102×C137—21no—
23#102×C137—21no—
24C22⋊C8×C137C2211yescharacteristic
25C22⋊C8×C137C2211yescharacteristic
26#95×C137C2211yescharacteristic
27C24×C137—21no—
28C2×Q8×C137D411yescharacteristic
29C2×D4×C137—21no—
30C2×D4×C137—21no—
31C2×D4×C137—21no—
32C2×D4×C137—42no—
33C2×D4×C137D411yescharacteristic
34C22×C4×C137—21no—
35C22×C4×C137—21no—
36C22×C4×C137D411yescharacteristic
37C22×C4×C137D411yescharacteristic
38C22×C4×C137C2311yescharacteristic
39SD16×C137—42no—
40D8×C137—42no—
41C2×C8×C137—21no—
42C2×C8×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137—21no—
48C4⋊C4×C137—21no—
49C4⋊C4×C137D411yescharacteristic
50C4⋊C4×C137D411yescharacteristic
51C22⋊C4×C137—21no—
52C22⋊C4×C137—21no—
53C22⋊C4×C137—21no—
54C22⋊C4×C137—21no—
55C22⋊C4×C137—21no—
56C22⋊C4×C137—42no—
57C22⋊C4×C137—42no—
58C42×C137—21no—
59C23×C137—22no—
60C23×C137—21no—
61C23×C137—41no—
62C23×C137—41no—
63C23×C137—41no—
64C23×C137—41no—
65C23×C137—41no—
66C23×C137—41no—
67C23×C137C2×D411yescharacteristic
68Q8×C137—41no—
69D4×C137—22no—
70D4×C137—22no—
71D4×C137—41no—
72D4×C137—41no—
73D4×C137—41no—
74D4×C137—41no—
75D4×C137—41no—
76C2×C4×C137—21no—
77C2×C4×C137—21no—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137—41no—
84C2×C4×C137—41no—
85C2×C4×C137—41no—
86C2×C4×C137—41no—
87C2×C4×C137—41no—
88C2×C4×C137—81no—
89C2×C4×C137C2×D411yescharacteristic
90C2×C4×C137C2×D411yescharacteristic
91C8×C137—41no—
92C8×C137—41no—
93C22×C137—21no—
94C22×C137—21no—
95C22×C137—42no—
96C22×C137—42no—
97C22×C137—41no—
98C22×C137—41no—
99C22×C137—41no—
100C22×C137—42no—
101C22×C137—81no—
102C22×C137—81no—
103C22×C137—81no—
104C22×C137—81no—
105C22×C137#12011yescharacteristic
106C22×C137#13511yescharacteristic
107C22×C137AGL(1,ℤ/8)11yescharacteristic
108C4×C137—21no—
109C4×C137—21no—
110C4×C137—41no—
111C4×C137—41no—
112C4×C137—41no—
113C4×C137—41no—
114C4×C137—41no—
115C4×C137—81no—
116C2×C137—21no—
117C2×C137—21no—
118C2×C137—81no—
119C2×C137—82no—
120C2×C137#44911yescharacteristic
121C2×C137#45411yescharacteristic
122C2×C137B(4,𝔽2)11yescharacteristic
123C137#159711yescharacteristic, direct product factor, Sylow
124#1597C13711yesmaximal, characteristic, direct product factor, Sylow
125#539C2×C13711yescharacteristic, complemented
126#450C2×C13711yescharacteristic, complemented
127#447C2×C13711yescharacteristic
128#394C2×C13711yescharacteristic, complemented
129#328C2×C13711yescharacteristic, complemented
130#327C2×C13711yescharacteristic, complemented
131#324C2×C13711yescharacteristic, complemented
132C22×D4C22×C13711yescharacteristic
133C2×SD16—21no—
134C2×D8—21no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C22×C13711yescharacteristic
141#121C22×C13711yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C22×C13711yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146#102—21no—
147#102—21no—
148C22⋊C8C22×C13711yescharacteristic
149C22⋊C8C22×C13711yescharacteristic
150#95C22×C13711yescharacteristic
151C24—21no—
152C2×Q8D4×C13711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4D4×C13711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4D4×C13711yescharacteristic
161C22×C4D4×C13711yescharacteristic
162C22×C4C23×C13711yescharacteristic
163SD16—42no—
164D8—42no—
165C2×C8—21no—
166C2×C8—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4D4×C13711yescharacteristic
174C4⋊C4D4×C13711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C42—21no—
183C23—22no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23C2×D4×C13711yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—41no—
199D4—41no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4C2×D4×C13711yescharacteristic
214C2×C4C2×D4×C13711yescharacteristic
215C8—41no—
216C8—41no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22#120×C13711yescharacteristic
230C22#135×C13711yescharacteristic
231C22AGL(1,ℤ/8)×C13711yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—81no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—81no—
244C2#449×C13711yescharacteristic
245C2#454×C13711yescharacteristic
246C2B(4,𝔽2)×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#120
commutator, Frattini, cosocleC22×C4C23×C137
prime radical#447×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #539×C137C2semidirect
2. #450×C137C2semidirect
4. #394×C137C2semidirect
5. #328×C137C2semidirect
6. #327×C137C2semidirect
7. #324×C137C2semidirect
123. C137#1597direct (N × H)
124. #1597C137direct (N × H)
125. #539C2×C137semidirect
126. #450C2×C137semidirect
128. #394C2×C137semidirect
129. #328C2×C137semidirect
130. #327C2×C137semidirect
131. #324C2×C137semidirect

Maximal subgroups

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

subgroupindex
#539×C1372
#450×C1372
#447×C1372
#394×C1372
#328×C1372
#327×C1372
#324×C1372
#1597137