Finite Group Databasebeta

← #27141379

Subgroups — #27141379

Order 12416.

164 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261138C211yesmaximal, characteristic
2#20261131C211yesmaximal, characteristic
3#20261087C211yesmaximal, characteristic
4#20261086C211yesmaximal, characteristic
5#20261085C211yesmaximal, characteristic
6#20260456C211yesmaximal, characteristic
7#487×C97C211yesmaximal, characteristic
8#15690810C2211yescharacteristic
9#15690810C2211yescharacteristic
10#15690743—22no—
11#15690741—22no—
12#15690735—22no—
13#15690734—22no—
14#15690733—22no—
15C4×#11715895C2211yescharacteristic
16C4×#11715895C2211yescharacteristic
17C4⋊Q8×C97C2211yescharacteristic
18C4⋊Q8×C97C2211yescharacteristic
19#103×C97—22no—
20C4×C8×C97C2211yescharacteristic
21C2×Dic194—21no—
22C2×Dic194—21no—
23C2×#11715895D412yes—
24C2×#11715895D412yes—
25C4⋊Dic97—21no—
26C4⋊Dic97—21no—
27C4⋊Dic97—21no—
28C4⋊Dic97—21no—
29C2×Q8×C97—21no—
30C2×Q8×C97—21no—
31C2×C8×C97D412yes—
32C4⋊C4×C97—21no—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—21no—
35C4⋊C4×C97—21no—
36C42×C97C2311yescharacteristic
37Q8×C97—42no—
38Q8×C97—42no—
39C2×C4×C97—41no—
40C2×C4×C97—41no—
41C2×C4×C97C2×D411yescharacteristic
42C2×C4×C97C2×D411yescharacteristic
43C2×C4×C97C2×D411yescharacteristic
44C2×Dic97—41no—
45C2×Dic97—41no—
46Dic194—42no—
47Dic194—42no—
48C8×C97—22no—
49#11715895—22no—
50#11715895—22no—
51C22×C97#12711yescharacteristic
52C4×C97—81no—
53C4×C97—81no—
54C4×C97#13512yes—
55C4×C97#13712yes—
56C4×C97#13712yes—
57Dic97—81no—
58Dic97—81no—
59C2×C97#49511yescharacteristic
60C2×C97#49711yescharacteristic
61C2×C97#49711yescharacteristic
62#1718—971nomaximal, Sylow
63C97#171811yescharacteristic, complemented, Sylow
64#500—971no—
65#498—971no—
66#487—971no—
67#487—971no—
68#487—971no—
69#487D9711yescharacteristic
70#322—971no—
71C4⋊Q8—971no—
72C4⋊Q8—971no—
73C4⋊Q8C2×D9711yescharacteristic
74C4⋊Q8C2×D9711yescharacteristic
75#107—1942no—
76#106—1942no—
77#103—22no—
78#103—1942no—
79#103—1942no—
80#103—1942no—
81C4×C8—971no—
82C4×C8—971no—
83C4×C8C2×D9711yescharacteristic
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—1941no—
87C2×Q8—1941no—
88C2×C8—972no—
89C2×C8—972no—
90C2×C8C22⋊D9712yes—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—1941no—
96C4⋊C4—1941no—
97C4⋊C4—1941no—
98C4⋊C4—1941no—
99C42C22×D9711yescharacteristic
100Q8—42no—
101Q8—42no—
102Q8—3882no—
103Q8—3882no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—3881no—
107C2×C4—3881no—
108C2×C4D4×D9711yescharacteristic
109C2×C4D4×D9711yescharacteristic
110C2×C4C2×C22⋊D9711yescharacteristic
111C8—22no—
112C8—1942no—
113C8—1942no—
114C22#1569080611yescharacteristic
115C4—81no—
116C4—81no—
117C4—7761no—
118C4—7761no—
119C4#1569083012yes—
120C4#1569083912yes—
121C4#1569084212yes—
122C2#2026111711yescharacteristic
123C2#2026112611yescharacteristic
124C2#2026112811yescharacteristic
subgroupG/H
centerC22#15690806
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#487×C97C2
socle, prime radicalC22×C97#127
hypercenter#487D97
hypocenterC97#1718
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
63. C97#1718semidirect

Maximal subgroups

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

subgroupindex
#202611382
#202611312
#202610872
#202610862
#202610852
#202604562
#487×C972
#171897