Finite Group Databasebeta

← C97×#1505 #27137463

Subgroups — C97×#1505 #27137463

Order 12416.

272 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 92 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#532×C97C211yesmaximal, characteristic
2#432×C97C211yesmaximal, characteristic, complemented
3#376×C97C211yesmaximal, characteristic, complemented
4#327×C97C214yesmaximal, complemented
5C2×#41×C97—21no—
6C4⋊Q8×C97C411yescharacteristic, complemented
7#127×C97C411yescharacteristic, complemented
8#124×C97—21no—
9#121×C97—21no—
10#121×C97C2212yes—
11C4×D4×C97—21no—
12C4×D4×C97C412yescomplemented
13C2×C42×C97—21no—
14C2×C42×C97C2211yescharacteristic
15C4⋊C8×C97—21no—
16C22⋊C8×C97C2212yes—
17#97×C97—21no—
18#95×C97—21no—
19#95×C97C2212yes—
20#41×C97—44no—
21C2×Q8×C97—21no—
22C2×D4×C97—22no—
23C2×D4×C97—21no—
24C2×D4×C97—21no—
25C2×D4×C97C2×C412yes—
26C22×C4×C97—22no—
27C22×C4×C97—22no—
28C22×C4×C97D411yescharacteristic
29C22×C4×C97D411yescharacteristic
30C22×C4×C97C2311yescharacteristic
31C2×C8×C97—22no—
32C4⋊C4×C97—21no—
33C4⋊C4×C97C2×C412yes—
34C22⋊C4×C97—22no—
35C22⋊C4×C97—22no—
36C42×C97—24no—
37C42×C97C2×C411yescharacteristic
38C42×C97C2×C411yescharacteristic
39C42×C97D411yescharacteristic
40C42×C97D411yescharacteristic
41C23×C97—22no—
42C23×C97C2×D411yescharacteristic
43Q8×C97—42no—
44D4×C97—42no—
45D4×C97—42no—
46D4×C97—44no—
47D4×C97—42no—
48C2×C4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97—24no—
51C2×C4×C97—22no—
52C2×C4×C97—41no—
53C2×C4×C97—44no—
54C2×C4×C97—42no—
55C2×C4×C97—42no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C8×C97—42no—
63C22×C97—21no—
64C22×C97—21no—
65C22×C97—42no—
66C22×C97—42no—
67C22×C97—42no—
68C22×C97C2×C22⋊C411yescharacteristic
69C22×C97AGL(1,ℤ/8)12yes—
70C4×C97—21no—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—21no—
74C4×C97—21no—
75C4×C97—44no—
76C4×C97—42no—
77C4×C97#10412yes—
78C2×C97—22no—
79C2×C97—82no—
80C2×C97#41011yescharacteristic
81C2×C97#41811yescharacteristic
82C2×C97C2×#10411yescharacteristic
83#1505C9711yesmaximal, characteristic, direct product factor, Sylow
84C97#150511yescharacteristic, direct product factor, Sylow
85#532C2×C9711yescharacteristic
86#432C2×C9711yescharacteristic, complemented
87#376C2×C9711yescharacteristic, complemented
88#327C2×C9714yescomplemented
89C2×#41—21no—
90C4⋊Q8C4×C9711yescharacteristic, complemented
91#127C4×C9711yescharacteristic, complemented
92#124—21no—
93#121—21no—
94#121C22×C9712yes—
95C4×D4—21no—
96C4×D4C4×C9712yescomplemented
97C2×C42—21no—
98C2×C42C22×C9711yescharacteristic
99C4⋊C8—21no—
100C22⋊C8C22×C9712yes—
101#97—21no—
102#95—21no—
103#95C22×C9712yes—
104#41—44no—
105C2×Q8—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4C2×C4×C9712yes—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4D4×C9711yescharacteristic
113C22×C4D4×C9711yescharacteristic
114C22×C4C23×C9711yescharacteristic
115C2×C8—22no—
116C4⋊C4—21no—
117C4⋊C4C2×C4×C9712yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—24no—
121C42C2×C4×C9711yescharacteristic
122C42C2×C4×C9711yescharacteristic
123C42D4×C9711yescharacteristic
124C42D4×C9711yescharacteristic
125C23—22no—
126C23C2×D4×C9711yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—44no—
131D4—42no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4C22⋊C4×C9711yescharacteristic
141C2×C4C22⋊C4×C9711yescharacteristic
142C2×C4C22⋊C4×C9711yescharacteristic
143C2×C4C22⋊C4×C9711yescharacteristic
144C2×C4C22×C4×C9711yescharacteristic
145C2×C4C2×D4×C9711yescharacteristic
146C8—42no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22C2×C22⋊C4×C9711yescharacteristic
153C22AGL(1,ℤ/8)×C9712yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—44no—
160C4—42no—
161C4#104×C9712yes—
162C2—22no—
163C2—82no—
164C2#410×C9711yescharacteristic
165C2#418×C9711yescharacteristic
166C2C2×#104×C9711yescharacteristic
subgroupG/H
center, socleC22×C97C2×C22⋊C4
commutatorC2×C4C22×C4×C97
Frattini, cosocleC22×C4C23×C97
prime radical#532×C97C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C97 ◁ C2×C42×C97 ◁ G

Complements

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

NH ≅ G/Ntype
2. #432×C97C2semidirect
3. #376×C97C2semidirect
4. #327×C97C2semidirect
6. C4⋊Q8×C97C4semidirect
7. #127×C97C4semidirect
12. C4×D4×C97C4semidirect
83. #1505C97direct (N × H)
84. C97#1505direct (N × H)
86. #432C2×C97semidirect
87. #376C2×C97semidirect
88. #327C2×C97semidirect
90. C4⋊Q8C4×C97semidirect
91. #127C4×C97semidirect
96. C4×D4C4×C97semidirect

Maximal subgroups

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

subgroupindex
#532×C972
#432×C972
#376×C972
#327×C972
#150597