Finite Group Databasebeta

← #34100536

Subgroups — #34100536

Order 20096.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#19097758C211yesmaximal, characteristic
2#23757750C211yesmaximal, characteristic, complemented
3#23757278C211yesmaximal, characteristic
4C2×#19097689C211yesmaximal, characteristic, complemented
5#23757034C211yesmaximal, characteristic, complemented
6#23757032C211yesmaximal, characteristic, complemented
7#418×C157C211yesmaximal, characteristic
8C22×Dic314C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097758—22no—
11#19097758C414yescomplemented
12C4×Dic314—22no—
13#19097736—22no—
14#19097730C2211yescharacteristic
15#19097727—22no—
16C2×C22⋊Dic157—22no—
17C2×C4×Dic157C2211yescharacteristic
18#19097692—22no—
19#19097689—22no—
20C22×D4×C157C2211yescharacteristic
21C2×M16×C157C2211yescharacteristic
22#117×C157C2211yescharacteristic
23#102×C157—22no—
24C2×Dic314—21no—
25C2×Dic314—21no—
26C2×Dic314—21no—
27C2×Dic314—21no—
28C2×Dic314C2×C412yes—
29C2×Dic314D412yes—
30C22×Dic157—21no—
31C22×Dic157D412yes—
32#14578938—21no—
33#14578938—21no—
34C2×#11929434—21no—
35C4⋊Dic157D412yes—
36#14578933—24no—
37C22⋊Dic157—28no—
38C22⋊Dic157—21no—
39C4×Dic157—24no—
40C4×Dic157—21no—
41C4×Dic157—21no—
42C4×Dic157C2×C412yes—
43C24×C157—21no—
44C2×D4×C157—21no—
45C2×D4×C157—21no—
46C2×D4×C157—21no—
47C2×D4×C157—21no—
48C2×D4×C157—21no—
49C2×D4×C157C2×C412yes—
50C22×C4×C157C2311yescharacteristic
51M16×C157—21no—
52M16×C157—21no—
53C2×C8×C157—21no—
54C4⋊C4×C157D412yes—
55C22⋊C4×C157—21no—
56C42×C157—21no—
57Dic314—22no—
58Dic314—22no—
59Dic314—22no—
60Dic314—22no—
61C2×Dic157—22no—
62C2×Dic157—22no—
63C2×Dic157—22no—
64C2×Dic157—22no—
65C2×Dic157—42no—
66C2×Dic157C22⋊C414yes—
67#11929434—41no—
68C23×C157—22no—
69C23×C157—44no—
70C23×C157—41no—
71C23×C157—41no—
72C23×C157C2×D411yescharacteristic
73D4×C157—22no—
74D4×C157—22no—
75D4×C157—41no—
76D4×C157—41no—
77C2×C4×C157—22no—
78C2×C4×C157C22×C411yescharacteristic
79C2×C4×C157C2×D411yescharacteristic
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157#4111yescharacteristic
83C2×C4×C157#4111yescharacteristic
84C8×C157—41no—
85C22×C157—22no—
86C22×C157—44no—
87C22×C157—42no—
88C22×C157—82no—
89C22×C157—82no—
90C22×C157C2×C22⋊C411yescharacteristic
91C22×C157#12011yescharacteristic
92C22×C157#12311yescharacteristic
93C4×C157—42no—
94C4×C157C4×D411yescharacteristic
95C4×C157C4×D411yescharacteristic
96C4×C157#12111yescharacteristic
97C4×C157#12111yescharacteristic
98Dic157—24no—
99Dic157—42no—
100Dic157—42no—
101C2×C157—21no—
102C2×C157—21no—
103C2×C157—82no—
104C2×C157#38611yescharacteristic
105C2×C157#45412yes—
106C157#188811yescharacteristic, complemented, Sylow
107#1888—1571nomaximal, Sylow
108C2×#124—1571no—
109#519—1571no—
110#418—1571no—
111#418D15711yescharacteristic
112C2×#101—1571no—
113#339—1571no—
114#339—1571no—
115C22×Q8—1571no—
116C22×D4C2×D15711yescharacteristic
117C2×M16—1571no—
118C2×M16C2×D15711yescharacteristic
119#124—3142no—
120#124—1574no—
121C4×Q8—3142no—
122C4×Q8—3142no—
123#117—3142no—
124#117—1571no—
125#117C2×D15711yescharacteristic
126C2×C22⋊C4—3142no—
127C2×C42—1571no—
128#102—22no—
129#102—3142no—
130#101—3142no—
131C24—21no—
132C2×Q8—3141no—
133C2×Q8—3141no—
134C2×Q8—3141no—
135C2×Q8—3141no—
136C2×Q8—1572no—
137C2×Q8—1572no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C4×D15712yes—
144C22×C4—3141no—
145C22×C4—1572no—
146C22×C4C22×D15711yescharacteristic
147M16—21no—
148M16—21no—
149M16—3141no—
150M16—3141no—
151C2×C8—21no—
152C2×C8—3141no—
153C4⋊C4—3144no—
154C4⋊C4—1572no—
155C4⋊C4C22⋊D15712yes—
156C22⋊C4—21no—
157C22⋊C4—3141no—
158C22⋊C4—3148no—
159C42—21no—
160C42—3144no—
161C42—3141no—
162C42—3141no—
163C42—1572no—
164C23—22no—
165C23—44no—
166C23—41no—
167C23—41no—
168C23D4×D15711yescharacteristic
169Q8—3142no—
170Q8—3142no—
171Q8—3142no—
172Q8—3142no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177C2×C4—22no—
178C2×C4—3142no—
179C2×C4—3142no—
180C2×C4—3142no—
181C2×C4—3142no—
182C2×C4—6282no—
183C2×C4—1574no—
184C2×C4C2×C4×D15711yescharacteristic
185C2×C4D4×D15711yescharacteristic
186C2×C4D4×D15711yescharacteristic
187C2×C4C2×C22⋊D15711yescharacteristic
188C2×C4#1457895311yescharacteristic
189C2×C4#1457895411yescharacteristic
190C8—41no—
191C8—6281no—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—82no—
197C22C22⋊C4×D15711yescharacteristic
198C22#1909773911yescharacteristic
199C22#1909775311yescharacteristic
200C4—42no—
201C4—3144no—
202C4—6282no—
203C4—6282no—
204C4#1909773111yescharacteristic
205C4C4×C22⋊D15711yescharacteristic
206C4#1909774211yescharacteristic
207C4#1909774411yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2375714911yescharacteristic
212C2#2375744412yes—
subgroupG/H
centerC22#19097739
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#418×C157C2
socleC22×C157#120
prime radicalC22×D4×C157C22
cosocleC22×C4×C157C23
hypercenter#418D157
hypocenterC157#1888
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #23757750C2semidirect
4. C2×#19097689C2semidirect
5. #23757034C2semidirect
6. #23757032C2semidirect
11. #19097758C4semidirect
106. C157#1888semidirect

Maximal subgroups

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

subgroupindex
C2×#190977582
#237577502
#237572782
C2×#190976892
#237570342
#237570322
#418×C1572
#1888157