Finite Group Databasebeta

← #27141953

Subgroups — #27141953

Order 12416.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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#20261213C211yesmaximal, characteristic
2#20260777C212yesmaximal
3#20260642C211yesmaximal, characteristic
4#20260496C212yesmaximal
5C2×#101×C97C211yesmaximal, characteristic
6Q8×Dic97—22no—
7Q8×Dic97—22no—
8#15690769—22no—
9#15690769C2212yes—
10C2×C4⋊Dic97C2211yescharacteristic
11C2×C4×Dic97C2211yescharacteristic
12#15690744—22no—
13#15690742—22no—
14#15690717—22no—
15C22×Q8×C97C2211yescharacteristic
16C2×M16×C97C2212yes—
17#101×C97C414yescomplemented
18C22×Dic97—21no—
19C22×Dic97—21no—
20C4⋊Dic97—22no—
21C4⋊Dic97—22no—
22C4⋊Dic97—22no—
23C4⋊Dic97D412yes—
24C4⋊Dic97D412yes—
25C22⋊Dic97—22no—
26C4×Dic97—24no—
27C4×Dic97—22no—
28C4×Dic97—22no—
29C2×Q8×C97—22no—
30C2×Q8×C97—22no—
31C2×Q8×C97C2×C412yes—
32C2×Q8×C97Q812yes—
33C22×C4×C97D412yes—
34C22×C4×C97C2311yescharacteristic
35M16×C97—22no—
36M16×C97C2×C414yes—
37C2×C8×C97—22no—
38C23×C97C2×D411yescharacteristic
39Q8×C97—24no—
40Q8×C97—24no—
41C2×C4×C97—22no—
42C2×C4×C97—22no—
43C2×C4×C97C4⋊C414yes—
44C2×C4×C97C22×C411yescharacteristic
45C2×C4×C97C2×D412yes—
46C2×C4×C97C2×Q811yescharacteristic
47C2×C4×C97#4112yes—
48C2×Dic97—22no—
49C2×Dic97—22no—
50C2×Dic97—22no—
51C2×Dic97—41no—
52C2×Dic97—41no—
53C2×Dic97—42no—
54C2×Dic97—41no—
55C8×C97—24no—
56C22×C97—21no—
57C22×C97—21no—
58C22×C97C2×C4⋊C411yescharacteristic
59C22×C97#12011yescharacteristic
60C22×C97#12311yescharacteristic
61C4×C97—24no—
62C4×C97C4×D412yes—
63C4×C97C22⋊Q812yes—
64Dic97—42no—
65Dic97—42no—
66Dic97—42no—
67Dic97—81no—
68C2×C97—21no—
69C2×C97—21no—
70C2×C97#38511yescharacteristic
71C2×C97#45411yescharacteristic
72C2×C97#45611yescharacteristic
73#1857—971nomaximal, Sylow
74C97#185711yescharacteristic, complemented, Sylow
75#519—971no—
76#428—972no—
77C2×#101D9711yescharacteristic
78#391—971no—
79#339—972no—
80C22×Q8C2×D9711yescharacteristic
81C2×M16C2×D9712yes—
82C4×Q8—1942no—
83C4×Q8—1942no—
84#117—972no—
85#117—1942no—
86C2×C4⋊C4—971no—
87C2×C42—971no—
88#107—1942no—
89#106—1942no—
90#101Dic9714yescomplemented
91#95—1942no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8Dic19412yes—
95C2×Q8C2×Dic9712yes—
96C22×C4—1941no—
97C22×C4—1941no—
98C22×C4D38812yes—
99C22×C4C22×D9711yescharacteristic
100M16—22no—
101M16C2×Dic9714yes—
102C2×C8—22no—
103C4⋊C4—972no—
104C4⋊C4—972no—
105C4⋊C4—1942no—
106C4⋊C4—1942no—
107C4⋊C4—1942no—
108C22⋊C4—1942no—
109C42—1944no—
110C42—1942no—
111C42—1942no—
112C23C2×D38811yescharacteristic
113Q8—24no—
114Q8—24no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—3882no—
118C2×C4—3881no—
119C2×C4—3881no—
120C2×C4—3881no—
121C2×C4—1942no—
122C2×C4—1942no—
123C2×C4—1942no—
124C2×C4C4⋊Dic9714yes—
125C2×C4C22×Dic9711yescharacteristic
126C2×C4D4×D9712yes—
127C2×C4C2×Dic19411yescharacteristic
128C2×C4#1309551012yes—
129C8—24no—
130C22—21no—
131C22—21no—
132C22C2×C4⋊Dic9711yescharacteristic
133C22#1569078011yescharacteristic
134C22#1569079411yescharacteristic
135C4—24no—
136C4—7761no—
137C4—3882no—
138C4—3882no—
139C4—3882no—
140C4D4×Dic9712yes—
141C4#1569078712yes—
142C2—21no—
143C2—21no—
144C2#2026061011yescharacteristic
145C2#2026091011yescharacteristic
146C2#2026092011yescharacteristic
subgroupG/H
centerC22#15690780
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#101×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenterC2×#101D97
hypocenterC97#1857
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
17. #101×C97C4semidirect
74. C97#1857semidirect
90. #101Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612132
#202607772
#202606422
#202604962
C2×#101×C972
#185797