Finite Group Databasebeta

← #27143811

Subgroups — #27143811

Order 12416.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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#20261173C211yesmaximal, characteristic, complemented
2#20261171C211yesmaximal, characteristic, complemented
3#20261110C211yesmaximal, characteristic, complemented
4#20260851C211yesmaximal, characteristic, complemented
5#20260555C211yesmaximal, characteristic, complemented
6#20260547C211yesmaximal, characteristic, complemented
7#361×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9C2×D776C2211yescharacteristic
10C2×#13095496—21no—
11#15690818—21no—
12#15690806—21no—
13#15690755—21no—
14#15690754—21no—
15D1552—21no—
16#15690751—21no—
17#15690750C2211yescharacteristic
18#15690736—21no—
19C8×Dic97C2211yescharacteristic
20C2×D8×C97C2211yescharacteristic
21M32×C97C2211yescharacteristic
22#108×C97C2211yescharacteristic
23#13095511—21no—
24C2×D388—21no—
25C2×C22⋊D97—42no—
26Dic388D411yescharacteristic, complemented
27#13095500—21no—
28D776—21no—
29D776—21no—
30D776D411yescharacteristic, complemented
31#13095496—41no—
32#13095496—41no—
33#13095494—41no—
34C2×#11715895D411yescharacteristic
35C8×D97—41no—
36C4×Dic97D411yescharacteristic
37#13095486—21no—
38C2×D4×C97—21no—
39D8×C97—21no—
40D8×C97—21no—
41M16×C97—21no—
42C2×C8×C97C2311yescharacteristic
43C16×C97—21no—
44C23×C97—41no—
45C22×D97—41no—
46D4×C97—41no—
47D4×C97—41no—
48C2×C4×C97C2×D411yescharacteristic
49C22⋊D97—41no—
50C22⋊D97—84no—
51C2×Dic97—21no—
52D388—21no—
53D388—41no—
54D388—41no—
55C4×D97—41no—
56Dic194—21no—
57C8×C97—41no—
58C8×C97C2×D411yescharacteristic
59C8×C97#4111yescharacteristic
60#11715895D811yescharacteristic
61#11715895D811yescharacteristic
62C22×C97—82no—
63C22×C97—81no—
64C22×C97AGL(1,ℤ/8)11yescharacteristic
65C2×D97—41no—
66C2×D97—82no—
67C2×D97—81no—
68C4×C97#12111yescharacteristic
69C4×C97C2×D811yescharacteristic
70Dic97—22no—
71Dic97—41no—
72C2×C97—21no—
73C2×C97—161no—
74C2×C97#45911yescharacteristic
75D97—81no—
76D97—161no—
77#2213—971nomaximal, Sylow
78C97#221311yescharacteristic, complemented, Sylow
79#509—971no—
80#509—971no—
81#493—971no—
82#443—971no—
83#364—971no—
84#361—971no—
85#361D9711yescharacteristic, complemented
86#135—971no—
87C2×D8—971no—
88C2×D8—1941no—
89C2×D8C2×D9711yescharacteristic
90#131—1941no—
91#127—1941no—
92SD32—1941no—
93SD32—1941no—
94D16—1941no—
95D16—1941no—
96M32—971no—
97M32C2×D9711yescharacteristic
98#108C2×D9711yescharacteristic
99#104—1941no—
100C4×C8—971no—
101#41—1941no—
102C2×D4—21no—
103C2×D4—3882no—
104C2×D4—1941no—
105Q16—971no—
106SD16—1941no—
107D8—21no—
108D8—21no—
109D8—971no—
110D8—3881no—
111D8—3881no—
112D8—1941no—
113D8—1941no—
114M16—21no—
115M16—3881no—
116C2×C8—971no—
117C2×C8—3881no—
118C2×C8C22×D9711yescharacteristic
119C42—971no—
120C16—21no—
121C16—1941no—
122C23—41no—
123C23—3881no—
124Q8—1941no—
125D4—41no—
126D4—41no—
127D4—7764no—
128D4—3881no—
129D4—3881no—
130D4—3881no—
131D4—1941no—
132C2×C4—3881no—
133C2×C4—1941no—
134C2×C4D4×D9711yescharacteristic
135C8—41no—
136C8—971no—
137C8—971no—
138C8D4×D9711yescharacteristic
139C8#1309551111yescharacteristic
140C22—82no—
141C22—81no—
142C22—7762no—
143C22—7761no—
144C22—3881no—
145C22#1569083711yescharacteristic
146C4—3881no—
147C4—1942no—
148C4#1569078311yescharacteristic
149C4D8×D9711yescharacteristic
150C2—21no—
151C2—7761no—
152C2—161no—
153C2—15521no—
154C2#2026092911yescharacteristic
subgroupG/H
centerC2#20260929
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#361×C97C2
socleC2×C97#459
hypercenter#361D97
hypocenterC97#2213
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #20261173C2semidirect
2. #20261171C2semidirect
3. #20261110C2semidirect
4. #20260851C2semidirect
5. #20260555C2semidirect
6. #20260547C2semidirect
7. #361×C97C2semidirect
26. Dic388D4semidirect
30. D776D4semidirect
78. C97#2213semidirect
85. #361D97semidirect

Maximal subgroups

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

subgroupindex
#202611732
#202611712
#202611102
#202608512
#202605552
#202605472
#361×C972
#221397