Finite Group Databasebeta

← D97⋊#356 #27158025

Subgroups — D97⋊#356 #27158025

Order 12416.

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

normal subgroups only

This group has 240 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⋊#101C212yesmaximal
2#20261646C212yesmaximal, complemented
3#20261645C212yesmaximal, complemented
4C4⋊Q8×D97C211yesmaximal, characteristic
5D97⋊M16—22no—
6#15690871—22no—
7#15690871C2214yes—
8C2×Q8×D97C411yescharacteristic
9C2×Q8×D97C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690809—22no—
12#15690808C411yescharacteristic
13#15690808C2211yescharacteristic
14C4⋊C4×D97—22no—
15C42×D97C411yescharacteristic
16C4⋊Q8×C97C411yescharacteristic
17#13095521—42no—
18#13095520—24no—
19#13095519—42no—
20C2×Dic194—21no—
21C2×Dic194C2×C411yescharacteristic
22C2×Dic194D412yes—
23C2×Dic194C2311yescharacteristic
24Q8×D97—41no—
25Q8×D97—41no—
26Q8×D97—41no—
27Q8×D97—41no—
28C2×C4×D97—21no—
29C2×C4×D97—21no—
30C2×C4×D97C2×C411yescharacteristic
31C2×C4×D97D412yes—
32C4⋊Dic97—22no—
33#13095490—22no—
34#13095490—22no—
35C4×Dic97—21no—
36C4×Dic97C2×C411yescharacteristic
37C2×Q8×C97C2×C411yescharacteristic
38C2×Q8×C97C2×C411yescharacteristic
39C4⋊C4×C97—22no—
40C42×C97C2×C411yescharacteristic
41C22×D97C22⋊C411yescharacteristic
42Q8×C97—41no—
43Q8×C97—41no—
44C2×C4×C97—21no—
45C2×C4×C97—21no—
46C2×C4×C97C22⋊C412yes—
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—21no—
49C2×Dic97—21no—
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C2×D412yes—
52C4×D97—24no—
53C4×D97—21no—
54C4×D97—21no—
55C4×D97—22no—
56C4×D97—22no—
57C4×D97—41no—
58C4×D97—41no—
59Dic194—42no—
60Dic194—41no—
61Dic194—42no—
62Dic194—41no—
63#11715897—44no—
64C22×C97C2×C22⋊C411yescharacteristic
65C2×D97—22no—
66C2×D97#9911yescharacteristic
67C2×D97#9911yescharacteristic
68C4×C97—22no—
69C4×C97—21no—
70C4×C97—22no—
71C4×C97—41no—
72Dic97—22no—
73Dic97—22no—
74Dic97—21no—
75Dic97—41no—
76C2×C97—21no—
77C2×C97C2×#9911yescharacteristic
78D97—21no—
79D97#35612yescomplemented
80C2×#356—971nomaximal, Sylow
81C97C2×#35611yescharacteristic, complemented, Sylow
82C2×C4⋊Q8—971no—
83C2×#101—972no—
84#356—972no—
85#356—972no—
86C22×Q8—971no—
87C22×Q8—971no—
88C2×M16—1942no—
89C4⋊Q8—971no—
90C4⋊Q8—971no—
91C4⋊Q8—971no—
92C4⋊Q8—1942no—
93C4⋊Q8#8760411yescharacteristic
94C2×C4⋊C4—1942no—
95C2×C42—971no—
96#101—974no—
97#101—1942no—
98C2×Q8—972no—
99C2×Q8—971no—
100C2×Q8—971no—
101C2×Q8—3881no—
102C2×Q8—3881no—
103C2×Q8—3881no—
104C2×Q8—3881no—
105C2×Q8—1941no—
106C2×Q8C2×#8760411yescharacteristic
107C2×Q8C2×#8760411yescharacteristic
108C22×C4—972no—
109C22×C4—971no—
110C22×C4—1941no—
111C22×C4—1941no—
112M16—3882no—
113M16—1944no—
114C2×C8—3882no—
115C4⋊C4—22no—
116C4⋊C4—1942no—
117C4⋊C4—1942no—
118C4⋊C4—1942no—
119C42—971no—
120C42—1941no—
121C42C2×#8760411yescharacteristic
122C23—971no—
123Q8—41no—
124Q8—41no—
125Q8—3881no—
126Q8—3882no—
127Q8—3881no—
128Q8—3882no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—971no—
132C2×C4—972no—
133C2×C4—3881no—
134C2×C4—3881no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4—1941no—
138C2×C4—1941no—
139C2×C4—1944no—
140C2×C4—1941no—
141C2×C4—1941no—
142C2×C4C22⋊#8760412yes—
143C2×C4C22×#8760411yescharacteristic
144C8—3884no—
145C22—971no—
146C22—971no—
147C22—1942no—
148C22C2×C22⋊#8760411yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—3881no—
154C4—1942no—
155C4—1942no—
156C4—1941no—
157C2—21no—
158C2—972no—
159C2—1941no—
160C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC2×C97C2×#99
prime radicalC22×D97C22⋊C4
cosocleC2×Dic194C23
hypercenterC4⋊Q8#87604
hypocenterC97C2×#356
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20261646C2semidirect
3. #20261645C2semidirect
79. D97#356semidirect
81. C97C2×#356semidirect

Maximal subgroups

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

subgroupindex
D97⋊#1012
#202616462
#202616452
C4⋊Q8×D972
C2×#35697