Finite Group Databasebeta

← #1853×C157 #34096211

Subgroups — #1853×C157 #34096211

Order 20096.

360 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C4⋊C4×C157C211yesmaximal, characteristic
2C2×#106×C157C212yesmaximal, complemented
3C2×C22⋊C8×C157C211yesmaximal, characteristic
4#380×C157C211yesmaximal, characteristic
5#340×C157C212yesmaximal, complemented
6C23×C4×C157—21no—
7C23×C4×C157C2211yescharacteristic
8C22×C8×C157C2212yes—
9C2×C4⋊C4×C157—21no—
10C2×C4⋊C4×C157—24no—
11C2×C4⋊C4×C157C2211yescharacteristic
12C2×C4⋊C4×C157C2211yescharacteristic
13C2×C4⋊C4×C157C2212yes—
14C2×C22⋊C4×C157—21no—
15#106×C157—24no—
16C22⋊C8×C157C414yescomplemented
17#95×C157—21no—
18C24×C157D411yescharacteristic
19C22×C4×C157—22no—
20C22×C4×C157—22no—
21C22×C4×C157—24no—
22C22×C4×C157—22no—
23C22×C4×C157—21no—
24C22×C4×C157—21no—
25C22×C4×C157—42no—
26C22×C4×C157—42no—
27C22×C4×C157C2×C412yes—
28C22×C4×C157D411yescharacteristic
29C22×C4×C157Q812yes—
30C22×C4×C157C2311yescharacteristic
31C2×C8×C157—22no—
32C2×C8×C157—22no—
33C2×C8×C157C2×C414yescomplemented
34C4⋊C4×C157—24no—
35C4⋊C4×C157—22no—
36C4⋊C4×C157—22no—
37C4⋊C4×C157—22no—
38C4⋊C4×C157D414yes—
39C22⋊C4×C157—42no—
40C23×C157—21no—
41C23×C157—21no—
42C23×C157—21no—
43C23×C157—21no—
44C23×C157C4⋊C412yes—
45C23×C157SD1612yes—
46C23×C157SD1612yes—
47C23×C157C2×D411yescharacteristic
48C2×C4×C157—21no—
49C2×C4×C157—21no—
50C2×C4×C157—24no—
51C2×C4×C157—24no—
52C2×C4×C157—24no—
53C2×C4×C157—48no—
54C2×C4×C157—42no—
55C2×C4×C157—42no—
56C2×C4×C157—41no—
57C2×C4×C157—41no—
58C2×C4×C157—41no—
59C2×C4×C157—41no—
60C2×C4×C157—41no—
61C2×C4×C157—41no—
62C2×C4×C157C4⋊C412yes—
63C2×C4×C157C22×C411yescharacteristic
64C2×C4×C157C2×D412yes—
65C2×C4×C157C2×Q811yescharacteristic
66C2×C4×C157#4112yes—
67C8×C157—24no—
68C22×C157—22no—
69C22×C157—22no—
70C22×C157—22no—
71C22×C157—22no—
72C22×C157—22no—
73C22×C157—22no—
74C22×C157#10614yescomplemented
75C22×C157C2×C4⋊C411yescharacteristic
76C22×C157#12011yescharacteristic
77C22×C157#12311yescharacteristic
78C22×C157C2×SD1611yescharacteristic
79C22×C157C2×SD1611yescharacteristic
80C22×C157AGL(1,ℤ/8)11yescharacteristic
81C22×C157#13711yescharacteristic
82C4×C157—22no—
83C4×C157—44no—
84C4×C157—81no—
85C4×C157—81no—
86C4×C157C4×D412yes—
87C4×C157C22⋊Q812yes—
88C2×C157—24no—
89C2×C157#38511yescharacteristic
90C2×C157C2×#10611yescharacteristic
91C2×C157#42811yescharacteristic
92C2×C157#44811yescharacteristic
93C2×C157#45011yescharacteristic
94C2×C157#48111yescharacteristic
95C2×C157#48311yescharacteristic
96C157#185311yescharacteristic, direct product factor, Sylow
97#1853C15711yesmaximal, characteristic, direct product factor, Sylow
98C22×C4⋊C4C2×C15711yescharacteristic
99C2×#106C2×C15712yescomplemented
100C2×C22⋊C8C2×C15711yescharacteristic
101#380C2×C15711yescharacteristic
102#340C2×C15712yescomplemented
103C23×C4—21no—
104C23×C4C22×C15711yescharacteristic
105C22×C8C22×C15712yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4C22×C15711yescharacteristic
109C2×C4⋊C4C22×C15711yescharacteristic
110C2×C4⋊C4C22×C15712yes—
111C2×C22⋊C4—21no—
112#106—24no—
113C22⋊C8C4×C15714yescomplemented
114#95—21no—
115C24D4×C15711yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×C4×C15712yes—
125C22×C4D4×C15711yescharacteristic
126C22×C4Q8×C15712yes—
127C22×C4C23×C15711yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×C4×C15714yescomplemented
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4D4×C15714yes—
136C22⋊C4—42no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊C4×C15712yes—
142C23SD16×C15712yes—
143C23SD16×C15712yes—
144C23C2×D4×C15711yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—48no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4C4⋊C4×C15712yes—
160C2×C4C22×C4×C15711yescharacteristic
161C2×C4C2×D4×C15712yes—
162C2×C4C2×Q8×C15711yescharacteristic
163C2×C4#41×C15712yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#106×C15714yescomplemented
172C22C2×C4⋊C4×C15711yescharacteristic
173C22#120×C15711yescharacteristic
174C22#123×C15711yescharacteristic
175C22C2×SD16×C15711yescharacteristic
176C22C2×SD16×C15711yescharacteristic
177C22AGL(1,ℤ/8)×C15711yescharacteristic
178C22#137×C15711yescharacteristic
179C4—22no—
180C4—44no—
181C4—81no—
182C4—81no—
183C4C4×D4×C15712yes—
184C4C22⋊Q8×C15712yes—
185C2—24no—
186C2#385×C15711yescharacteristic
187C2C2×#106×C15711yescharacteristic
188C2#428×C15711yescharacteristic
189C2#448×C15711yescharacteristic
190C2#450×C15711yescharacteristic
191C2#481×C15711yescharacteristic
192C2#483×C15711yescharacteristic
subgroupG/H
center, socleC23×C157C2×D4
commutatorC2×C4C22×C4×C157
Frattini, cosocleC22×C4C23×C157
prime radicalC24×C157D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C157 ◁ C23×C4×C157 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#106×C157C2semidirect
5. #340×C157C2semidirect
16. C22⋊C8×C157C4semidirect
33. C2×C8×C157C2×C4semidirect
74. C22×C157#106semidirect
96. C157#1853direct (N × H)
97. #1853C157direct (N × H)
99. C2×#106C2×C157semidirect
102. #340C2×C157semidirect
113. C22⋊C8C4×C157semidirect
130. C2×C8C2×C4×C157semidirect
171. C22#106×C157semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×C1572
C2×#106×C1572
C2×C22⋊C8×C1572
#380×C1572
#340×C1572
#1853157