Finite Group Databasebeta

← D97⋊#329 #27157182

Subgroups — D97⋊#329 #27157182

Order 12416.

284 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 64 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#20261694C212yesmaximal, complemented
2#20261591C212yesmaximal, complemented
3#20261590C212yesmaximal, complemented
4#124×D97C211yesmaximal, characteristic
5C2×#13095519—22no—
6#15690865—22no—
7#15690865C2214yescomplemented
8C2×Q8×D97C411yescharacteristic
9C2×D4×D97C411yescharacteristic
10#15690799C2211yescharacteristic
11#15690798C411yescharacteristic
12#15690797C2211yescharacteristic
13#15690796—22no—
14C22⋊C4×D97—22no—
15C42×D97C2211yescharacteristic
16#124×C97C411yescharacteristic
17#13095519—28no—
18C2×#11715897—24no—
19C23×D97—21no—
20C2×Dic194—21no—
21C2×Dic194C2×C411yescharacteristic
22Q8×D97—42no—
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25D4×D97—42no—
26C2×C4×D97—21no—
27C2×C4×D97C2×C411yescharacteristic
28C2×C4×D97D412yes—
29C22⋊Dic97—22no—
30#13095488—22no—
31#13095488—22no—
32C4×Dic97D412yes—
33C4×Dic97C2311yescharacteristic
34C2×Q8×C97C2×C411yescharacteristic
35C2×D4×C97C2×C411yescharacteristic
36C22⋊C4×C97—22no—
37C42×C97C2×C411yescharacteristic
38C23×C97—21no—
39C22×D97—21no—
40C22×D97—42no—
41C22×D97—44no—
42C22×D97C22⋊C411yescharacteristic
43Q8×C97—41no—
44D4×C97—41no—
45C2×C4×C97—21no—
46C2×C4×C97C22⋊C412yes—
47C2×C4×C97C22×C411yescharacteristic
48C22⋊D97—42no—
49C2×Dic97—21no—
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C2×D412yes—
52D388—41no—
53C4×D97—22no—
54C4×D97—24no—
55C4×D97—42no—
56Dic194—42no—
57Dic194—41no—
58#11715897—28no—
59C22×C97—41no—
60C22×C97—42no—
61C22×C97C2×C22⋊C411yescharacteristic
62C2×D97—41no—
63C2×D97—42no—
64C2×D97—84no—
65C2×D97#10012yes—
66C2×D97#10414yes—
67C4×C97—21no—
68C4×C97—22no—
69C4×C97—41no—
70Dic97—21no—
71Dic97—22no—
72Dic97—41no—
73C2×C97—81no—
74C2×C97C2×#10011yescharacteristic
75C2×C97C2×#10412yes—
76D97—81no—
77D97#32914yescomplemented
78C2×#329—971nomaximal, Sylow
79C97C2×#32911yescharacteristic, complemented, Sylow
80C2×#124—971no—
81C2×#97—972no—
82#329—972no—
83#329—972no—
84C22×Q8—971no—
85C22×D4—971no—
86C22×C8—1942no—
87#124—971no—
88#124—971no—
89#124—971no—
90#124—1942no—
91#124#8760411yescharacteristic
92C2×C22⋊C4—1942no—
93C2×C42—971no—
94#97—974no—
95#97—1942no—
96C24—1941no—
97C2×Q8—971no—
98C2×Q8—3882no—
99C2×Q8—1941no—
100C2×Q8C2×#8760411yescharacteristic
101C2×D4—971no—
102C2×D4—3882no—
103C2×D4—1941no—
104C2×D4C2×#8760411yescharacteristic
105C22×C4—972no—
106C22×C4—971no—
107C22×C4—1941no—
108C2×C8—1944no—
109C2×C8—1948no—
110C22⋊C4—22no—
111C22⋊C4—1942no—
112C22⋊C4—1942no—
113C22⋊C4—1942no—
114C42—972no—
115C42—971no—
116C42C2×#8760411yescharacteristic
117C23—21no—
118C23—971no—
119C23—3882no—
120C23—3884no—
121C23—1941no—
122Q8—41no—
123Q8—3881no—
124Q8—3882no—
125D4—41no—
126D4—3881no—
127D4—3882no—
128C2×C4—21no—
129C2×C4—972no—
130C2×C4—971no—
131C2×C4—3882no—
132C2×C4—1944no—
133C2×C4—1942no—
134C2×C4—1941no—
135C2×C4C22⋊#8760412yes—
136C2×C4C22×#8760411yescharacteristic
137C8—1948no—
138C22—41no—
139C22—42no—
140C22—7764no—
141C22—972no—
142C22—974no—
143C22—3881no—
144C22—3882no—
145C22C2×C22⋊#8760411yescharacteristic
146C4—21no—
147C4—22no—
148C4—41no—
149C4—3881no—
150C4—1941no—
151C4—1942no—
152C2—81no—
153C2—7761no—
154C2—974no—
155C2D97⋊#10011yescharacteristic
156C2D97⋊#10412yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97C2×#329
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261694C2semidirect
2. #20261591C2semidirect
3. #20261590C2semidirect
7. #15690865C22semidirect
77. D97#329semidirect
79. C97C2×#329semidirect

Maximal subgroups

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

subgroupindex
#202616942
#202615912
#202615902
#124×D972
C2×#32997