Finite Group Databasebeta

← #1596×C137 #31776607

Subgroups — #1596×C137 #31776607

Order 17536.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#543×C137C211yesmaximal, characteristic, complemented
2#450×C137C212yesmaximal, complemented
3#392×C137C211yesmaximal, characteristic
4#328×C137C212yesmaximal, complemented
5#323×C137C211yesmaximal, characteristic, complemented
6C22×D4×C137—21no—
7C22×D4×C137C2211yescharacteristic
8C2×SD16×C137—22no—
9C22⋊Q8×C137—22no—
10C22⋊Q8×C137—21no—
11C22⋊Q8×C137—22no—
12C22⋊Q8×C137C2212yes—
13C2×C22⋊C4×C137—21no—
14C2×C22⋊C4×C137—21no—
15C2×C22⋊C4×C137C2211yescharacteristic
16#102×C137—22no—
17C22⋊C8×C137C2212yes—
18#95×C137C2211yescharacteristic
19C24×C137—21no—
20C24×C137D411yescharacteristic
21C2×Q8×C137—21no—
22C2×Q8×C137D412yescomplemented
23C2×D4×C137—22no—
24C2×D4×C137—42no—
25C2×D4×C137—42no—
26C2×D4×C137—42no—
27C2×D4×C137—42no—
28C22×C4×C137—21no—
29C22×C4×C137—21no—
30C22×C4×C137—21no—
31C22×C4×C137D411yescharacteristic
32C22×C4×C137C2311yescharacteristic
33SD16×C137—44no—
34C2×C8×C137—22no—
35C4⋊C4×C137—21no—
36C4⋊C4×C137—22no—
37C4⋊C4×C137—22no—
38C4⋊C4×C137D412yescomplemented
39C22⋊C4×C137—22no—
40C22⋊C4×C137—22no—
41C22⋊C4×C137—22no—
42C22⋊C4×C137—42no—
43C23×C137—21no—
44C23×C137—21no—
45C23×C137—22no—
46C23×C137—41no—
47C23×C137—41no—
48C23×C137—41no—
49C23×C137—41no—
50C23×C137—41no—
51C23×C137—41no—
52C23×C137—41no—
53C23×C137—41no—
54C23×C137SD1612yescomplemented
55C23×C137C2×D411yescharacteristic
56Q8×C137—42no—
57Q8×C137—81no—
58D4×C137—24no—
59D4×C137—42no—
60D4×C137—48no—
61C2×C4×C137—22no—
62C2×C4×C137—22no—
63C2×C4×C137—21no—
64C2×C4×C137—22no—
65C2×C4×C137—41no—
66C2×C4×C137—41no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C2×C4×C137—81no—
70C2×C4×C137—81no—
71C2×C4×C137C2×D412yes—
72C8×C137—42no—
73C22×C137—21no—
74C22×C137—21no—
75C22×C137—22no—
76C22×C137—42no—
77C22×C137—42no—
78C22×C137—42no—
79C22×C137—41no—
80C22×C137—41no—
81C22×C137—42no—
82C22×C137—81no—
83C22×C137—81no—
84C22×C137—81no—
85C22×C137—81no—
86C22×C137—81no—
87C22×C137#12011yescharacteristic
88C22×C137C2×SD1611yescharacteristic
89C22×C137#13711yescharacteristic
90C4×C137—22no—
91C4×C137—42no—
92C4×C137—41no—
93C4×C137—42no—
94C4×C137—42no—
95C2×C137—21no—
96C2×C137—21no—
97C2×C137—42no—
98C2×C137—82no—
99C2×C137#44811yescharacteristic
100C2×C137#45311yescharacteristic
101C2×C137B(4,𝔽2)11yescharacteristic
102C137#159611yescharacteristic, direct product factor, Sylow
103#1596C13711yesmaximal, characteristic, direct product factor, Sylow
104#543C2×C13711yescharacteristic, complemented
105#450C2×C13712yescomplemented
106#392C2×C13711yescharacteristic
107#328C2×C13712yescomplemented
108#323C2×C13711yescharacteristic, complemented
109C22×D4—21no—
110C22×D4C22×C13711yescharacteristic
111C2×SD16—22no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C22×C13712yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4C22×C13711yescharacteristic
119#102—22no—
120C22⋊C8C22×C13712yes—
121#95C22×C13711yescharacteristic
122C24—21no—
123C24D4×C13711yescharacteristic
124C2×Q8—21no—
125C2×Q8D4×C13712yescomplemented
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4D4×C13711yescharacteristic
135C22×C4C23×C13711yescharacteristic
136SD16—44no—
137C2×C8—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4D4×C13712yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—42no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23SD16×C13712yescomplemented
158C23C2×D4×C13711yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—24no—
162D4—42no—
163D4—48no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×D4×C13712yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#120×C13711yescharacteristic
191C22C2×SD16×C13711yescharacteristic
192C22#137×C13711yescharacteristic
193C4—22no—
194C4—42no—
195C4—41no—
196C4—42no—
197C4—42no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#448×C13711yescharacteristic
203C2#453×C13711yescharacteristic
204C2B(4,𝔽2)×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#120
commutator, Frattini, cosocleC22×C4C23×C137
prime radical#392×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. #543×C137C2semidirect
2. #450×C137C2semidirect
4. #328×C137C2semidirect
5. #323×C137C2semidirect
22. C2×Q8×C137D4semidirect
38. C4⋊C4×C137D4semidirect
54. C23×C137SD16semidirect
102. C137#1596direct (N × H)
103. #1596C137direct (N × H)
104. #543C2×C137semidirect
105. #450C2×C137semidirect
107. #328C2×C137semidirect
108. #323C2×C137semidirect
125. C2×Q8D4×C137semidirect
141. C4⋊C4D4×C137semidirect
157. C23SD16×C137semidirect

Maximal subgroups

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

subgroupindex
#543×C1372
#450×C1372
#392×C1372
#328×C1372
#323×C1372
#1596137