Finite Group Databasebeta

← #27140743

Subgroups — #27140743

Order 12416.

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

normal subgroups only

This group has 210 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#20261065C212yesmaximal
3#20260471C212yesmaximal
4#20260461C211yesmaximal, characteristic
5#398×C97C211yesmaximal, characteristic
6#15690809—22no—
7#15690801—22no—
8#15690789—21no—
9#15690789C2212yes—
10#15690769—21no—
11#15690741—22no—
12#15690733—22no—
13C22⋊#11715895C2212yes—
14C2×C4⋊C4×C97—21no—
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C4⋊C4×C97C2211yescharacteristic
17#95×C97—21no—
18#95×C97C2211yescharacteristic
19C2×Dic194—21no—
20C2×Dic194D412yes—
21C2×#11715895—22no—
22C4⋊Dic97—21no—
23C4⋊Dic97—21no—
24C4⋊Dic97—21no—
25C4⋊Dic97D412yes—
26#13095490—22no—
27#13095490—22no—
28C22⋊Dic97—22no—
29C4×Dic97—22no—
30C22×C4×C97—21no—
31C22×C4×C97—21no—
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97C2311yescharacteristic
35C4⋊C4×C97—22no—
36C4⋊C4×C97—22no—
37C4⋊C4×C97—42no—
38C23×C97C2×D411yescharacteristic
39C2×C4×C97—21no—
40C2×C4×C97—21no—
41C2×C4×C97—22no—
42C2×C4×C97—21no—
43C2×C4×C97—41no—
44C2×C4×C97—41no—
45C2×C4×C97—41no—
46C2×C4×C97—41no—
47C2×C4×C97—41no—
48C2×C4×C97—41no—
49C2×C4×C97—41no—
50C2×C4×C97SD1612yes—
51C2×C4×C97C2×D412yes—
52C2×Dic97—22no—
53C2×Dic97—22no—
54Dic194—42no—
55Dic194—81no—
56#11715895—42no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97#12011yescharacteristic
60C22×C97C2×SD1611yescharacteristic
61C22×C97#13711yescharacteristic
62C4×C97—22no—
63C4×C97—22no—
64C4×C97—42no—
65C4×C97—41no—
66C4×C97—81no—
67Dic97—42no—
68Dic97—42no—
69C2×C97—21no—
70C2×C97—21no—
71C2×C97#44811yescharacteristic
72C2×C97#45611yescharacteristic
73C2×C97#45811yescharacteristic
74#1611—971nomaximal, Sylow
75C97#161111yescharacteristic, complemented, Sylow
76#544—971no—
77#483—972no—
78#398D9711yescharacteristic
79#328—972no—
80#324—971no—
81C4⋊Q8—1942no—
82#125—1942no—
83C22⋊Q8—972no—
84C22⋊Q8—1941no—
85#117—1941no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D9711yescharacteristic
88C2×C4⋊C4C2×D9711yescharacteristic
89#106—1942no—
90#103—1942no—
91C22⋊C8—972no—
92#95—21no—
93#95C2×D9711yescharacteristic
94C2×Q8—972no—
95C2×Q8—1941no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D9711yescharacteristic
99C22×C4C22⋊D9711yescharacteristic
100C22×C4C22×D9711yescharacteristic
101C2×C8—1942no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—972no—
106C4⋊C4—1942no—
107C4⋊C4—1941no—
108C4⋊C4—1941no—
109C4⋊C4—1941no—
110C4⋊C4—1942no—
111C22⋊C4—1942no—
112C42—1942no—
113C23C2×C22⋊D9711yescharacteristic
114Q8—7761no—
115Q8—3882no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—1942no—
128C2×C4—1942no—
129C2×C4#1309549912yes—
130C2×C4D4×D9712yes—
131C8—3882no—
132C22—21no—
133C22—21no—
134C22#1569077811yescharacteristic
135C22C2×#1309549911yescharacteristic
136C22#1569084011yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—81no—
142C4—3882no—
143C4—3882no—
144C2—21no—
145C2—21no—
146C2#2026087311yescharacteristic
147C2#2026091711yescharacteristic
148C2#2026092411yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#398×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#398D97
hypocenterC97#1611
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
75. C97#1611semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202610652
#202604712
#202604612
#398×C972
#161197