Finite Group Databasebeta

← #34102167

Subgroups — #34102167

Order 20096.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#23758003C211yesmaximal, characteristic, complemented
2#23757458C212yesmaximal
3#23757455C211yesmaximal, characteristic
4#23757071C212yesmaximal, complemented
5#356×C157C211yesmaximal, characteristic, complemented
6#19097819C2211yescharacteristic
7#19097804—22no—
8#19097802—21no—
9#19097770—22no—
10#19097769—21no—
11#19097769C2212yes—
12C4×Dic314—21no—
13#19097736—22no—
14#19097699—21no—
15#19097697—22no—
16#19097690C2211yescharacteristic
17C4⋊Q8×C157C2211yescharacteristic
18#101×C157C2212yes—
19#14578955—41no—
20#14578954—22no—
21#14578954—21no—
22C2×Dic314—21no—
23C2×Dic314—21no—
24C2×Dic314D412yes—
25Q8×D157—41no—
26Dic628—42no—
27#14578944—41no—
28#14578943—42no—
29#14578941—41no—
30#14578938—21no—
31C4⋊Dic157—21no—
32#14578933—22no—
33#14578933—22no—
34#14578933—22no—
35C4×Dic157—21no—
36C4×Dic157—22no—
37C4×Dic157D412yes—
38C2×Q8×C157D411yescharacteristic
39C2×Q8×C157C2311yescharacteristic
40M16×C157—22no—
41C4⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43Dic314—24no—
44Dic314—21no—
45Dic314—22no—
46Dic314—42no—
47Dic314—81no—
48D628—21no—
49D628—22no—
50C22⋊D157—41no—
51C2×Dic157—22no—
52C2×Dic157—22no—
53C4×D157—42no—
54C4×D157—41no—
55#11929434—41no—
56Q8×C157—41no—
57Q8×C157—41no—
58C2×C4×C157—21no—
59C2×C4×C157—21no—
60C2×C4×C157C2×D412yes—
61C2×C4×C157C2×D411yescharacteristic
62C8×C157—42no—
63C22×C157#12011yescharacteristic
64C2×D157—41no—
65C4×C157—22no—
66C4×C157—22no—
67C4×C157—21no—
68C4×C157—41no—
69Dic157—24no—
70Dic157—41no—
71Dic157—42no—
72C2×C157—21no—
73C2×C157B(4,𝔽2)11yescharacteristic
74D157—81no—
75C157#220511yescharacteristic, complemented, Sylow
76#2205—1571nomaximal, Sylow
77Q82—1571no—
78#456—1571no—
79#456—1572no—
80#356—1572no—
81#356D15711yescharacteristic, complemented
822−1+4—1571no—
83#137—3142no—
84#137—3141no—
85C4⋊Q8—3142no—
86C4⋊Q8—3141no—
87C4⋊Q8—1572no—
88C4⋊Q8C2×D15711yescharacteristic
89C4×Q8—3142no—
90C4×Q8—3141no—
91#104—3142no—
92#104—3141no—
93#101—1571no—
94#101C2×D15712yes—
95#41—3142no—
96#41—3141no—
97#41—6281no—
98C2×Q8—3141no—
99C2×Q8—3141no—
100C2×Q8—6281no—
101C2×Q8—1572no—
102C2×Q8D62811yescharacteristic
103C2×Q8C22×D15711yescharacteristic
104Q16—6282no—
105Q16—6281no—
106SD16—6282no—
107SD16—6281no—
108M16—22no—
109M16—3141no—
110C4⋊C4—22no—
111C4⋊C4—3142no—
112C4⋊C4—3141no—
113C4⋊C4—3142no—
114C4⋊C4—3142no—
115C42—3142no—
116C42—3141no—
117C42—1572no—
118C42D62811yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—3144no—
122Q8—3142no—
123Q8—3141no—
124Q8—6282no—
125Q8—12561no—
126D4—3142no—
127D4—3141no—
128D4—6281no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—3142no—
132C2×C4—3142no—
133C2×C4—6282no—
134C2×C4—6281no—
135C2×C4D4×D15712yes—
136C2×C4C2×D62811yescharacteristic
137C8—42no—
138C8—6281no—
139C22—6281no—
140C22#1909774111yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—3144no—
146C4—6282no—
147C4—6281no—
148C2—21no—
149C2—12561no—
150C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#356×C157C2
socleC2×C157B(4,𝔽2)
prime radical#19097819C22
hypercenter#356D157
hypocenterC157#2205
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #23758003C2semidirect
4. #23757071C2semidirect
5. #356×C157C2semidirect
75. C157#2205semidirect
81. #356D157semidirect

Maximal subgroups

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

subgroupindex
#237580032
#237574582
#237574552
#237570712
#356×C1572
#2205157