Finite Group Databasebeta

← #27158027

Subgroups — #27158027

Order 12416.

250 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261721C212yesmaximal, complemented
2#20261645C212yesmaximal, complemented
3#20261643C212yesmaximal, complemented
4#20261281C211yesmaximal, characteristic
5D97⋊M16—22no—
6#15690871C2214yes—
7#15690870—22no—
8C2×#13095511C2211yescharacteristic
9#15690808C2211yescharacteristic
10#15690806C411yescharacteristic
11#15690799C411yescharacteristic
12#15690797C2211yescharacteristic
13#15690782—22no—
14C4×C22⋊D97—22no—
15C2×C4×Dic97C411yescharacteristic
16C2×#41×C97C411yescharacteristic
17#13095521—42no—
18#13095520—24no—
19#13095519—42no—
20#13095511—41no—
21#13095511—41no—
22C2×Dic194C2311yescharacteristic
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—22no—
25C2×C22⋊D97D412yes—
26C22×Dic97—21no—
27C2×C4×D97D412yes—
28#13095490—22no—
29C22⋊Dic97—22no—
30#13095488—22no—
31C4×Dic97—21no—
32C4×Dic97C2×C411yescharacteristic
33C4×Dic97C2×C411yescharacteristic
34#41×C97—41no—
35#41×C97—41no—
36C2×Q8×C97C2×C411yescharacteristic
37C2×D4×C97—21no—
38C2×D4×C97C2×C411yescharacteristic
39C22×C4×C97—21no—
40C22×C4×C97C2×C411yescharacteristic
41C23×C97—21no—
42C23×C97C22⋊C411yescharacteristic
43C22×D97C22⋊C412yes—
44Q8×C97—41no—
45D4×C97—41no—
46D4×C97—42no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97—21no—
50C2×C4×C97—41no—
51C2×C4×C97—41no—
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22×C411yescharacteristic
54C22⋊D97—42no—
55C22⋊D97—44no—
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—42no—
59C2×Dic97C2×D412yes—
60D388—41no—
61C4×D97—22no—
62C4×D97—22no—
63Dic194—41no—
64#11715897—44no—
65C22×C97—21no—
66C22×C97—41no—
67C22×C97—42no—
68C22×C97—41no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—22no—
71C2×D97—42no—
72C4×C97—21no—
73C4×C97—41no—
74C4×C97#9911yescharacteristic
75C4×C97#9911yescharacteristic
76Dic97—22no—
77Dic97—22no—
78Dic97—41no—
79C2×C97—21no—
80C2×C97—41no—
81C2×C97—81no—
82C2×C97C2×#9911yescharacteristic
83D97—42no—
84#2132—971nomaximal, Sylow
85C97#213211yescharacteristic, complemented, Sylow
86#532—971no—
87#413—972no—
88#356—972no—
89#355—972no—
90C2×#41—971no—
91C2×#41#8760411yescharacteristic
92C2×M16—1942no—
93C4⋊Q8—971no—
94#127—971no—
95#124—971no—
96#124—971no—
97#121—1942no—
98C4×D4—1942no—
99C2×C42—971no—
100#101—974no—
101#100—1942no—
102#41—41no—
103#41—41no—
104#41—3881no—
105#41—3881no—
106C2×Q8—971no—
107C2×Q8C2×#8760411yescharacteristic
108C2×D4—21no—
109C2×D4—972no—
110C2×D4—971no—
111C2×D4—1942no—
112C2×D4C2×#8760411yescharacteristic
113C22×C4—21no—
114C22×C4—972no—
115C22×C4—1941no—
116C22×C4C2×#8760411yescharacteristic
117M16—3882no—
118M16—1944no—
119C2×C8—3882no—
120C4⋊C4—1942no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C42—971no—
124C42—971no—
125C42—1941no—
126C23—21no—
127C23—972no—
128C23C22⋊#8760411yescharacteristic
129Q8—41no—
130Q8—3881no—
131D4—41no—
132D4—42no—
133D4—3881no—
134D4—3882no—
135D4—3884no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—972no—
142C2×C4—3882no—
143C2×C4—1941no—
144C2×C4—1941no—
145C2×C4—1942no—
146C2×C4—1942no—
147C2×C4C22⋊#8760411yescharacteristic
148C2×C4C22×#8760411yescharacteristic
149C8—3884no—
150C22—21no—
151C22—41no—
152C22—42no—
153C22—41no—
154C22—3882no—
155C22—1942no—
156C22C2×C22⋊#8760411yescharacteristic
157C4—21no—
158C4—41no—
159C4—3881no—
160C4—1942no—
161C4—1942no—
162C4#1569086811yescharacteristic
163C4#1569086811yescharacteristic
164C2—21no—
165C2—41no—
166C2—81no—
167C2—3882no—
168C2C2×#1569086811yescharacteristic
subgroupG/H
centerC4#15690868
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×#41×C97C4
socleC2×C97C2×#99
prime radical#20261281C2
cosocleC2×Dic194C23
hypercenterC2×#41#87604
hypocenterC97#2132
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261721C2semidirect
2. #20261645C2semidirect
3. #20261643C2semidirect
85. C97#2132semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202616452
#202616432
#202612812
#213297