Finite Group Databasebeta

← #34102164

Subgroups — #34102164

Order 20096.

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

normal subgroups only

This group has 236 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#23757989C211yesmaximal, characteristic, complemented
2#23757455C211yesmaximal, characteristic, complemented
3#23757448C212yesmaximal, complemented
4#23757069C212yesmaximal, complemented
5#356×C157C211yesmaximal, characteristic, complemented
6#19097819C2211yescharacteristic
7#19097802—21no—
8#19097799—22no—
9#19097760—21no—
10#19097760C2212yes—
11#19097746—22no—
12C4×D628—21no—
13#19097729—22no—
14#19097699—21no—
15#19097697—22no—
16#19097690C2211yescharacteristic
17C4⋊Q8×C157C2211yescharacteristic
18#101×C157C2212yes—
19#14578955—41no—
20#14578954—22no—
21#14578954—21no—
22Q8×D157—41no—
23C2×D628—21no—
24C2×D628—21no—
25C2×D628D412yes—
26C2×C4×D157—22no—
27#14578944—41no—
28#14578943—42no—
29#14578941—41no—
30D1256—42no—
31#14578938—21no—
32C4⋊Dic157—21no—
33#14578931—22no—
34#14578931—22no—
35#14578931—22no—
36C4×Dic157—21no—
37C4×Dic157D412yes—
38C2×Q8×C157D411yescharacteristic
39C2×Q8×C157C2311yescharacteristic
40M16×C157—22no—
41C4⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43C22×D157—22no—
44Dic314—21no—
45Dic314—22no—
46D628—22no—
47D628—21no—
48D628—42no—
49D628—42no—
50D628—81no—
51C22⋊D157—41no—
52C2×Dic157—22no—
53C4×D157—44no—
54C4×D157—42no—
55C4×D157—41no—
56#11929434—41no—
57Q8×C157—41no—
58Q8×C157—41no—
59C2×C4×C157—21no—
60C2×C4×C157—21no—
61C2×C4×C157C2×D412yes—
62C2×C4×C157C2×D411yescharacteristic
63C8×C157—42no—
64C22×C157#12011yescharacteristic
65C2×D157—42no—
66C2×D157—42no—
67C2×D157—41no—
68C2×D157—42no—
69C4×C157—22no—
70C4×C157—22no—
71C4×C157—21no—
72C4×C157—41no—
73Dic157—42no—
74Dic157—41no—
75C2×C157—21no—
76C2×C157B(4,𝔽2)11yescharacteristic
77D157—81no—
78D157—82no—
79C157#220411yescharacteristic, complemented, Sylow
80#2204—1571nomaximal, Sylow
81#555—1571no—
82#456—1571no—
83#454—1572no—
84#356D15711yescharacteristic, complemented
85#355—1572no—
862−1+4—1571no—
87#137—3141no—
88AGL(1,ℤ/8)—3142no—
89C4⋊Q8C2×D15711yescharacteristic
90#124—3141no—
91#124—1572no—
92#121—3142no—
93C4×D4—3141no—
94#117—3142no—
95#104—3142no—
96#104—3141no—
97#101—1571no—
98#101C2×D15712yes—
99#41—3142no—
100#41—3141no—
101#41—6281no—
102C2×Q8—6281no—
103C2×Q8D62811yescharacteristic
104C2×Q8C22×D15711yescharacteristic
105C2×D4—3141no—
106C2×D4—3141no—
107C2×D4—1572no—
108C22×C4—3142no—
109Q16—6281no—
110SD16—6282no—
111SD16—6281no—
112D8—6282no—
113M16—22no—
114M16—3141no—
115C4⋊C4—22no—
116C4⋊C4—3141no—
117C22⋊C4—3142no—
118C22⋊C4—3142no—
119C22⋊C4—3142no—
120C42—3141no—
121C42—1572no—
122C42D62811yescharacteristic
123C23—3142no—
124Q8—41no—
125Q8—41no—
126Q8—3142no—
127Q8—3141no—
128D4—3142no—
129D4—3141no—
130D4—6282no—
131D4—6282no—
132D4—6281no—
133D4—12561no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—3142no—
137C2×C4—6284no—
138C2×C4—6282no—
139C2×C4—6281no—
140C2×C4D4×D15712yes—
141C2×C4C2×D62811yescharacteristic
142C8—42no—
143C8—6281no—
144C22—6282no—
145C22—6282no—
146C22—6281no—
147C22—6282no—
148C22#1909774111yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—6282no—
154C4—6281no—
155C2—21no—
156C2—12562no—
157C2—12561no—
158C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#356×C157C2
socleC2×C157B(4,𝔽2)
hypercenter#356D157
hypocenterC157#2204
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. #23757989C2semidirect
2. #23757455C2semidirect
3. #23757448C2semidirect
4. #23757069C2semidirect
5. #356×C157C2semidirect
79. C157#2204semidirect
84. #356D157semidirect

Maximal subgroups

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

subgroupindex
#237579892
#237574552
#237574482
#237570692
#356×C1572
#2204157