Finite Group Databasebeta

← D97⋊#354 #27158005

Subgroups — D97⋊#354 #27158005

Order 12416.

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

normal subgroups only

This group has 284 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
1D97⋊#99C212yesmaximal
2#20261642C212yesmaximal, complemented
3#20261641C212yesmaximal, complemented
4#124×D97C211yesmaximal, characteristic
5C2×C22⋊#87604—22no—
6#15690869C2214yes—
7#15690868—22no—
8C2×Q8×D97C411yescharacteristic, complemented
9C2×D4×D97C2211yescharacteristic
10#15690799C2211yescharacteristic
11#15690798C2211yescharacteristic
12#15690797C411yescharacteristic, complemented
13#15690796—22no—
14C22⋊C4×D97—22no—
15C42×D97C411yescharacteristic, complemented
16#124×C97C411yescharacteristic, complemented
17C22×#87604—42no—
18C22⋊#87604—24no—
19C22⋊#87604—42no—
20C23×D97D412yes—
21C2×Dic194—21no—
22C2×Dic194C2×C411yescharacteristic, complemented
23Q8×D97—41no—
24Q8×D97—41no—
25C2×D388C2311yescharacteristic
26C2×C22⋊D97D412yes—
27D4×D97—41no—
28D4×D97—41no—
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97C2×C411yescharacteristic
32C22⋊Dic97—22no—
33#13095488—22no—
34#13095488—22no—
35C4×Dic97—21no—
36C4×Dic97C2×C411yescharacteristic, complemented
37C2×Q8×C97C2×C411yescharacteristic, complemented
38C2×D4×C97C2×C411yescharacteristic
39C22⋊C4×C97—22no—
40C42×C97C2×C411yescharacteristic, complemented
41C23×C97C22⋊C412yes—
42C22×D97—22no—
43C22×D97—22no—
44C22×D97—44no—
45C22×D97C22⋊C411yescharacteristic
46C22×D97C2×D412yes—
47C2×#87604—44no—
48C2×#87604—84no—
49Q8×C97—41no—
50D4×C97—41no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97C22×C411yescharacteristic
54C22⋊D97—42no—
55C2×Dic97—21no—
56C2×Dic97—21no—
57C2×Dic97C22⋊C411yescharacteristic
58D388—41no—
59C4×D97—21no—
60C4×D97—21no—
61C4×D97—42no—
62C4×D97—41no—
63C4×D97—41no—
64Dic194—42no—
65Dic194—41no—
66C22×C97—22no—
67C22×C97—42no—
68C22×C97C2×C22⋊C411yescharacteristic
69C2×D97—22no—
70C2×D97—22no—
71C2×D97—44no—
72C2×D97—42no—
73C2×D97—82no—
74C2×D97#9911yescharacteristic
75C2×D97#9911yescharacteristic
76#87604—84no—
77C4×C97—21no—
78C4×C97—41no—
79C4×C97—41no—
80Dic97—21no—
81Dic97—41no—
82Dic97—41no—
83C2×C97—21no—
84C2×C97—42no—
85C2×C97C2×#9911yescharacteristic
86D97—21no—
87D97—42no—
88D97#35412yescomplemented
89C2×#354—971nomaximal, Sylow
90C97C2×#35411yescharacteristic, complemented, Sylow
91C2×#124—971no—
92C2×#99—972no—
93#354—972no—
94#354—972no—
95C22×Q8—971no—
96C22×D4—971no—
97#124—971no—
98#124—971no—
99#124—971no—
100#124—1942no—
101#124#8760411yescharacteristic, complemented
102C2×C22⋊C4—1942no—
103C2×C22⋊C4—1942no—
104C2×C42—971no—
105#99—974no—
106#99—1942no—
107C24—972no—
108C2×Q8—971no—
109C2×Q8—3881no—
110C2×Q8—3881no—
111C2×Q8—1941no—
112C2×Q8C2×#8760411yescharacteristic, complemented
113C2×D4—972no—
114C2×D4—971no—
115C2×D4—3881no—
116C2×D4—3881no—
117C2×D4C2×#8760411yescharacteristic
118C22×C4—971no—
119C22×C4—3882no—
120C22×C4—1941no—
121C22×C4—1941no—
122C22⋊C4—22no—
123C22⋊C4—3882no—
124C22⋊C4—1942no—
125C22⋊C4—1942no—
126C22⋊C4—1944no—
127C22⋊C4—1942no—
128C42—971no—
129C42—1941no—
130C42C2×#8760411yescharacteristic, complemented
131C23—971no—
132C23—972no—
133C23—3884no—
134C23—1942no—
135C23—1942no—
136C23C22⋊#8760412yes—
137Q8—41no—
138Q8—3881no—
139Q8—3882no—
140D4—41no—
141D4—3881no—
142D4—3882no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—7764no—
146C2×C4—971no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—3882no—
150C2×C4—3884no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4—1941no—
154C2×C4—1941no—
155C2×C4C22×#8760411yescharacteristic
156C22—22no—
157C22—42no—
158C22—7762no—
159C22—971no—
160C22—971no—
161C22—3882no—
162C22—3884no—
163C22—1942no—
164C22—1942no—
165C22C2×C22⋊#8760411yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—7764no—
170C4—3881no—
171C4—3881no—
172C4—1941no—
173C2—21no—
174C2—42no—
175C2—972no—
176C2—3882no—
177C2—1941no—
178C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC2×C97C2×#99
prime radicalC2×D4×D97C22
cosocleC2×D388C23
hypercenter#124#87604
hypocenterC97C2×#354
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20261642C2semidirect
3. #20261641C2semidirect
8. C2×Q8×D97C4semidirect
12. #15690797C4semidirect
15. C42×D97C4semidirect
16. #124×C97C4semidirect
22. C2×Dic194C2×C4semidirect
36. C4×Dic97C2×C4semidirect
37. C2×Q8×C97C2×C4semidirect
40. C42×C97C2×C4semidirect
88. D97#354semidirect
90. C97C2×#354semidirect
101. #124#87604semidirect
112. C2×Q8C2×#87604semidirect
130. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
D97⋊#992
#202616422
#202616412
#124×D972
C2×#35497