Finite Group Databasebeta

← #34100520

Subgroups — #34100520

Order 20096.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#23758040C211yesmaximal, characteristic, complemented
2#23757290C212yesmaximal, complemented
3#23757247C211yesmaximal, characteristic, complemented
4#23757033C212yesmaximal, complemented
5#532×C157C211yesmaximal, characteristic
6#19097778—24no—
7C2×#14578938—22no—
8C2×#14578938C2212yes—
9#19097730C2212yes—
10#19097698—24no—
11#19097686—22no—
12C2×#41×C157C2211yescharacteristic
13C2×#41×C157C2211yescharacteristic
14C4⋊Q8×C157C411yescharacteristic, complemented
15#127×C157C411yescharacteristic, complemented
16#124×C157C411yescharacteristic, complemented
17#124×C157C411yescharacteristic, complemented
18#121×C157—22no—
19C4×D4×C157—22no—
20C2×C42×C157C2211yescharacteristic
21#14578938—24no—
22#14578938—22no—
23#14578938D414yescomplemented
24C2×#11929434—24no—
25C2×#11929434—22no—
26C4⋊Dic157—22no—
27C22⋊Dic157—22no—
28C4×Dic157—22no—
29#41×C157—22no—
30#41×C157—22no—
31#41×C157—22no—
32#41×C157—22no—
33C2×Q8×C157C2×C411yescharacteristic
34C2×Q8×C157C2×C411yescharacteristic
35C2×D4×C157—22no—
36C2×D4×C157—21no—
37C2×D4×C157C2×C411yescharacteristic
38C2×D4×C157C2×C411yescharacteristic
39C2×D4×C157D412yes—
40C22×C4×C157—21no—
41C22×C4×C157—21no—
42C22×C4×C157D412yes—
43C22×C4×C157C2311yescharacteristic
44C4⋊C4×C157—22no—
45C22⋊C4×C157—22no—
46C22⋊C4×C157—22no—
47C42×C157—21no—
48C42×C157C2×C411yescharacteristic
49C42×C157C2×C411yescharacteristic
50C2×Dic157—42no—
51#11929434—24no—
52C23×C157—21no—
53C23×C157C22⋊C412yes—
54C23×C157C2×D411yescharacteristic
55Q8×C157—22no—
56Q8×C157—22no—
57D4×C157—24no—
58D4×C157—22no—
59D4×C157—22no—
60D4×C157—44no—
61D4×C157—42no—
62C2×C4×C157—21no—
63C2×C4×C157—22no—
64C2×C4×C157—22no—
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—42no—
68C2×C4×C157—41no—
69C2×C4×C157—41no—
70C2×C4×C157C22⋊C412yes—
71C2×C4×C157C22×C411yescharacteristic
72C2×C4×C157C2×D412yes—
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157#4112yes—
75C22×C157—22no—
76C22×C157—41no—
77C22×C157—42no—
78C22×C157—42no—
79C22×C157—41no—
80C22×C157C2×C22⋊C411yescharacteristic
81C22×C157#12112yes—
82C4×C157—22no—
83C4×C157—22no—
84C4×C157—41no—
85C4×C157—41no—
86C4×C157C4×D412yes—
87C4×C157#12011yescharacteristic
88C4×C157#12311yescharacteristic
89Dic157—82no—
90C2×C157—22no—
91C2×C157—21no—
92C2×C157—42no—
93C2×C157—81no—
94C2×C157#38611yescharacteristic
95C157#188411yescharacteristic, complemented, Sylow
96#1884—1571nomaximal, Sylow
97#568—1571no—
98#532D15711yescharacteristic
99#421—1572no—
100#410—1571no—
101#339—1572no—
102C2×#41C2×D15711yescharacteristic
103C2×#41C2×D15711yescharacteristic
104#131—3144no—
105C2×M16—3142no—
106C2×M16—1572no—
107C4⋊Q8Dic15711yescharacteristic, complemented
108#127Dic15711yescharacteristic, complemented
109#124Dic15711yescharacteristic, complemented
110#124Dic15711yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—1572no—
114C2×C42C2×D15711yescharacteristic
115#104—3144no—
116#99—3142no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic15711yescharacteristic
122C2×Q8C2×Dic15711yescharacteristic
123C2×D4—21no—
124C2×D4—22no—
125C2×D4C2×Dic15711yescharacteristic
126C2×D4C2×Dic15711yescharacteristic
127C2×D4C22⋊D15712yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D15712yes—
131C22×C4C22×D15711yescharacteristic
132M16—3144no—
133M16—3142no—
134M16—1574no—
135C2×C8—3144no—
136C2×C8—3142no—
137C4⋊C4—22no—
138C4⋊C4—3142no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—3142no—
142C42—21no—
143C42—3142no—
144C42C2×Dic15711yescharacteristic
145C42C2×Dic15711yescharacteristic
146C23—21no—
147C23C22⋊Dic15712yes—
148C23C2×C22⋊D15711yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—6282no—
165C2×C4C22⋊Dic15712yes—
166C2×C4C22×Dic15711yescharacteristic
167C2×C4D4×D15712yes—
168C2×C4C2×C22⋊D15711yescharacteristic
169C2×C4#1457895312yes—
170C8—3144no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic15711yescharacteristic
177C22#1909774312yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—12562no—
183C4D4×Dic15712yes—
184C4#1909773911yescharacteristic
185C4#1909775211yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#2375715311yescharacteristic
subgroupG/H
centerC4#19097739
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting, prime radical#532×C157C2
socleC2×C157#386
cosocleC22×C4×C157C23
hypercenter#532D157
hypocenterC157#1884
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #23758040C2semidirect
2. #23757290C2semidirect
3. #23757247C2semidirect
4. #23757033C2semidirect
14. C4⋊Q8×C157C4semidirect
15. #127×C157C4semidirect
16. #124×C157C4semidirect
17. #124×C157C4semidirect
23. #14578938D4semidirect
95. C157#1884semidirect
107. C4⋊Q8Dic157semidirect
108. #127Dic157semidirect
109. #124Dic157semidirect
110. #124Dic157semidirect

Maximal subgroups

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

subgroupindex
#237580402
#237572902
#237572472
#237570332
#532×C1572
#1884157