Finite Group Databasebeta

← D97⋊#353 #27158002

Subgroups — D97⋊#353 #27158002

Order 12416.

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

normal subgroups only

This group has 368 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, complemented
2#20261640C212yesmaximal, complemented
3#20261639C212yesmaximal, complemented
4#127×D97C211yesmaximal, characteristic
5C2×C22⋊#87604—22no—
6#15690869C2214yescomplemented
7#15690868—22no—
8C2×D4×D97—22no—
9C2×D4×D97C411yescharacteristic, complemented
10C2×D4×D97C2211yescharacteristic
11#15690807C2211yescharacteristic
12#15690806—22no—
13#15690806C411yescharacteristic, complemented
14#15690806C2211yescharacteristic
15C42×D97C411yescharacteristic, complemented
16#127×C97C411yescharacteristic, complemented
17C22×#87604—42no—
18C22⋊#87604—24no—
19C22⋊#87604—42no—
20C23×D97—21no—
21C23×D97D412yes—
22C2×D388—22no—
23C2×D388C2×C411yescharacteristic, complemented
24C2×D388C2311yescharacteristic
25C2×C22⋊D97—22no—
26C2×C22⋊D97—22no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97D412yes—
29D4×D97—41no—
30D4×D97—41no—
31D4×D97—41no—
32D4×D97—41no—
33D4×D97—48no—
34C2×C4×D97—21no—
35C2×C4×D97C2×C411yescharacteristic
36C4×Dic97—21no—
37C4×Dic97C2×C411yescharacteristic, complemented
38C2×D4×C97—22no—
39C2×D4×C97C2×C411yescharacteristic
40C2×D4×C97C2×C411yescharacteristic, complemented
41C42×C97C2×C411yescharacteristic, complemented
42C23×C97—21no—
43C23×C97C22⋊C412yes—
44C22×D97—22no—
45C22×D97—22no—
46C22×D97—21no—
47C22×D97—44no—
48C22×D97—44no—
49C22×D97—41no—
50C22×D97—41no—
51C22×D97C22⋊C411yescharacteristic
52C22×D97C2×D412yes—
53C2×#87604—44no—
54C2×#87604—84no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—44no—
58C2×C4×C97—21no—
59C2×C4×C97C22×C411yescharacteristic
60C22⋊D97—42no—
61C22⋊D97—44no—
62C22⋊D97—44no—
63C22⋊D97—42no—
64C2×Dic97—21no—
65C2×Dic97C22⋊C411yescharacteristic
66D388—41no—
67D388—41no—
68D388—44no—
69C4×D97—24no—
70C4×D97—21no—
71C4×D97—21no—
72C22×C97—22no—
73C22×C97—42no—
74C22×C97—42no—
75C22×C97—41no—
76C22×C97C2×C22⋊C411yescharacteristic
77C2×D97—22no—
78C2×D97—22no—
79C2×D97—44no—
80C2×D97—42no—
81C2×D97—42no—
82C2×D97—41no—
83C2×D97—82no—
84C2×D97—82no—
85C2×D97—82no—
86C2×D97#9911yescharacteristic
87C2×D97#9911yescharacteristic
88#87604—84no—
89C4×C97—22no—
90C4×C97—21no—
91Dic97—22no—
92Dic97—21no—
93C2×C97—21no—
94C2×C97—42no—
95C2×C97—81no—
96C2×C97C2×#9911yescharacteristic
97D97—21no—
98D97—42no—
99D97—81no—
100D97#35312yescomplemented
101C2×#353—971nomaximal, Sylow
102C97C2×#35311yescharacteristic, complemented, Sylow
103C2×#127—971no—
104C2×#99—972no—
105#353—972no—
106#353—972no—
107C22×D4—971no—
108C22×D4—971no—
109C22×D4—1942no—
110#127—971no—
111#127—971no—
112#127—971no—
113#127—1942no—
114#127#8760411yescharacteristic, complemented
115C2×C22⋊C4—1942no—
116C2×C42—971no—
117#99—974no—
118#99—1942no—
119C24—972no—
120C24—1941no—
121C2×D4—22no—
122C2×D4—972no—
123C2×D4—971no—
124C2×D4—971no—
125C2×D4—3881no—
126C2×D4—3881no—
127C2×D4—3888no—
128C2×D4—3881no—
129C2×D4—3881no—
130C2×D4—1942no—
131C2×D4—1941no—
132C2×D4—1942no—
133C2×D4—1942no—
134C2×D4C2×#8760411yescharacteristic
135C2×D4C2×#8760411yescharacteristic, complemented
136C22×C4—971no—
137C22×C4—3882no—
138C22×C4—1941no—
139C22⋊C4—3882no—
140C22⋊C4—1944no—
141C42—971no—
142C42—1941no—
143C42C2×#8760411yescharacteristic, complemented
144C23—21no—
145C23—971no—
146C23—972no—
147C23—3881no—
148C23—3881no—
149C23—3884no—
150C23—3884no—
151C23—1942no—
152C23—1942no—
153C23—1941no—
154C23C22⋊#8760412yes—
155D4—41no—
156D4—41no—
157D4—44no—
158D4—3881no—
159D4—3884no—
160D4—3881no—
161D4—3882no—
162D4—3884no—
163D4—3884no—
164D4—3882no—
165C2×C4—21no—
166C2×C4—7764no—
167C2×C4—971no—
168C2×C4—3884no—
169C2×C4—1941no—
170C2×C4—1941no—
171C2×C4—1944no—
172C2×C4—1941no—
173C2×C4C22×#8760411yescharacteristic
174C22—22no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—7762no—
179C22—7762no—
180C22—7762no—
181C22—971no—
182C22—971no—
183C22—3882no—
184C22—3882no—
185C22—3884no—
186C22—3881no—
187C22—1942no—
188C22—1942no—
189C22C2×C22⋊#8760411yescharacteristic
190C4—22no—
191C4—21no—
192C4—7764no—
193C4—1942no—
194C4—1941no—
195C2—21no—
196C2—42no—
197C2—81no—
198C2—7761no—
199C2—972no—
200C2—3882no—
201C2—1941no—
202C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC2×C97C2×#99
prime radical#127×D97C2
cosocleC2×D388C23
hypercenter#127#87604
hypocenterC97C2×#353
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. D97⋊#99C2semidirect
2. #20261640C2semidirect
3. #20261639C2semidirect
6. #15690869C22semidirect
9. C2×D4×D97C4semidirect
13. #15690806C4semidirect
15. C42×D97C4semidirect
16. #127×C97C4semidirect
23. C2×D388C2×C4semidirect
37. C4×Dic97C2×C4semidirect
40. C2×D4×C97C2×C4semidirect
41. C42×C97C2×C4semidirect
100. D97#353semidirect
102. C97C2×#353semidirect
114. #127#87604semidirect
135. C2×D4C2×#87604semidirect
143. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
D97⋊#992
#202616402
#202616392
#127×D972
C2×#35397