Finite Group Databasebeta

← #27140780

Subgroups — #27140780

Order 12416.

202 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261366C211yesmaximal, characteristic
2#20261071C212yesmaximal
3#20260674C211yesmaximal, characteristic
4#20260473C212yesmaximal
5#324×C97C211yesmaximal, characteristic
6#15690809—22no—
7#15690801—22no—
8#15690789—21no—
9#15690789C2212yes—
10#15690769—21no—
11C2×C4⋊Dic97C2211yescharacteristic
12#15690744—22no—
13#15690735—22no—
14#15690717—21no—
15#15690717—21no—
16#15690717C2211yescharacteristic
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C2×Dic194—21no—
20C2×Dic194D412yes—
21C22×Dic97—21no—
22C22×Dic97—21no—
23C4⋊Dic97—21no—
24C4⋊Dic97—21no—
25C4⋊Dic97—21no—
26C4⋊Dic97—22no—
27C4⋊Dic97D412yes—
28#13095490—22no—
29#13095490—22no—
30C22⋊Dic97—22no—
31C4×Dic97—22no—
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97C2311yescharacteristic
35C2×C8×C97—22no—
36C4⋊C4×C97—22no—
37C23×C97C2×D411yescharacteristic
38C2×C4×C97—21no—
39C2×C4×C97—21no—
40C2×C4×C97—21no—
41C2×C4×C97—41no—
42C2×C4×C97—41no—
43C2×C4×C97Q1612yes—
44C2×C4×C97C2×D412yes—
45C2×Dic97—22no—
46C2×Dic97—22no—
47C2×Dic97—41no—
48C2×Dic97—41no—
49C2×Dic97—42no—
50C2×Dic97—41no—
51C2×Dic97—41no—
52Dic194—42no—
53Dic194—81no—
54C8×C97—42no—
55C22×C97—21no—
56C22×C97—21no—
57C22×C97#12011yescharacteristic
58C22×C97C2×Q1611yescharacteristic
59C22×C97#13711yescharacteristic
60C4×C97—22no—
61C4×C97—22no—
62C4×C97—41no—
63Dic97—42no—
64Dic97—42no—
65Dic97—81no—
66Dic97—81no—
67C2×C97—21no—
68C2×C97—21no—
69C2×C97#45111yescharacteristic
70C2×C97#45411yescharacteristic
71C2×C97#45811yescharacteristic
72#1617—971nomaximal, Sylow
73C97#161711yescharacteristic, complemented, Sylow
74#544—971no—
75#484—972no—
76#400—971no—
77#328—972no—
78#324D9711yescharacteristic
79C4⋊Q8—1942no—
80#125—1942no—
81C22⋊Q8—972no—
82C22⋊Q8—1941no—
83#117—1941no—
84C2×C4⋊C4—971no—
85C2×C4⋊C4C2×D9711yescharacteristic
86#107—1942no—
87#103—1942no—
88C22⋊C8C2×D9712yes—
89#95—971no—
90#95—1941no—
91#95—1941no—
92C2×Q8—972no—
93C2×Q8—1941no—
94C22×C4—1941no—
95C22×C4—1941no—
96C22×C4D38811yescharacteristic
97C22×C4D38811yescharacteristic
98C22×C4C22×D9711yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—972no—
102C4⋊C4—1942no—
103C4⋊C4—1941no—
104C4⋊C4—1941no—
105C4⋊C4—1941no—
106C4⋊C4—1942no—
107C4⋊C4—1942no—
108C22⋊C4—1942no—
109C42—1942no—
110C23C2×D38811yescharacteristic
111Q8—7761no—
112Q8—3882no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—3881no—
119C2×C4—3881no—
120C2×C4—3882no—
121C2×C4—3881no—
122C2×C4—3881no—
123C2×C4—1942no—
124C2×C4—1942no—
125C2×C4Dic38812yes—
126C2×C4D4×D9712yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#1569078011yescharacteristic
131C22C2×Dic38811yescharacteristic
132C22#1569084311yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—7761no—
137C4—7761no—
138C4—3882no—
139C4—3882no—
140C2—21no—
141C2—21no—
142C2#2026089311yescharacteristic
143C2#2026091011yescharacteristic
144C2#2026092611yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#324D97
hypocenterC97#1617
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
73. C97#1617semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202610712
#202606742
#202604732
#324×C972
#161797