Finite Group Databasebeta

← C97×#2040 #27137998

Subgroups — C97×#2040 #27137998

Order 12416.

290 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 84 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#537×C97C211yesmaximal, characteristic
2C2×#104×C97C212yesmaximal, complemented
3#417×C97C212yesmaximal, complemented
4#382×C97C211yesmaximal, characteristic, complemented
5#343×C97C211yesmaximal, characteristic, complemented
6C2×#41×C97C2212yes—
7C2×M16×C97C2212yes—
8C22⋊Q8×C97—22no—
9#121×C97—22no—
10#121×C97—22no—
11C2×C4⋊C4×C97C2211yescharacteristic
12C2×C4⋊C4×C97C2211yescharacteristic
13C2×C42×C97C2211yescharacteristic
14#104×C97—24no—
15#103×C97—22no—
16#102×C97—22no—
17#95×C97—22no—
18#41×C97—22no—
19#41×C97—22no—
20#41×C97—22no—
21#41×C97—22no—
22C2×Q8×C97D412yes—
23C2×D4×C97—22no—
24C2×D4×C97—22no—
25C2×D4×C97D412yes—
26C22×C4×C97—21no—
27C22×C4×C97—22no—
28C22×C4×C97—21no—
29C22×C4×C97D412yes—
30C22×C4×C97C2311yescharacteristic
31M16×C97—22no—
32M16×C97—22no—
33C2×C8×C97—22no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97—22no—
36C4⋊C4×C97—22no—
37C22⋊C4×C97—24no—
38C42×C97—21no—
39C42×C97D412yescomplemented
40C23×C97—22no—
41C23×C97C2×D411yescharacteristic
42Q8×C97—22no—
43Q8×C97—22no—
44D4×C97—22no—
45D4×C97—22no—
46D4×C97—42no—
47D4×C97—42no—
48D4×C97—82no—
49C2×C4×C97—22no—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—42no—
55C2×C4×C97—42no—
56C2×C4×C97—42no—
57C2×C4×C97—42no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D412yes—
61C2×C4×C97#4112yes—
62C2×C4×C97#4111yescharacteristic
63C8×C97—42no—
64C22×C97—21no—
65C22×C97—21no—
66C22×C97—42no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97#12011yescharacteristic
70C22×C97#12111yescharacteristic
71C22×C97#12411yescharacteristic
72C4×C97—42no—
73C4×C97—42no—
74C4×C97—42no—
75C4×C97—81no—
76C4×C97#12112yes—
77C4×C97#12312yes—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97—82no—
81C2×C97#39411yescharacteristic
82C2×C97#45411yescharacteristic
83C2×C97#45511yescharacteristic
84#2040C9711yesmaximal, characteristic, direct product factor, Sylow
85C97#204011yescharacteristic, direct product factor, Sylow
86#537C2×C9711yescharacteristic
87C2×#104C2×C9712yescomplemented
88#417C2×C9712yescomplemented
89#382C2×C9711yescharacteristic, complemented
90#343C2×C9711yescharacteristic, complemented
91C2×#41C22×C9712yes—
92C2×M16C22×C9712yes—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
96C2×C4⋊C4C22×C9711yescharacteristic
97C2×C4⋊C4C22×C9711yescharacteristic
98C2×C42C22×C9711yescharacteristic
99#104—24no—
100#103—22no—
101#102—22no—
102#95—22no—
103#41—22no—
104#41—22no—
105#41—22no—
106#41—22no—
107C2×Q8D4×C9712yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4D4×C9712yes—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4D4×C9712yes—
115C22×C4C23×C9711yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C22⋊C4—24no—
123C42—21no—
124C42D4×C9712yescomplemented
125C23—22no—
126C23C2×D4×C9711yescharacteristic
127Q8—22no—
128Q8—22no—
129D4—22no—
130D4—22no—
131D4—42no—
132D4—42no—
133D4—82no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4C2×D4×C9711yescharacteristic
145C2×C4C2×D4×C9712yes—
146C2×C4#41×C9712yes—
147C2×C4#41×C9711yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#120×C9711yescharacteristic
155C22#121×C9711yescharacteristic
156C22#124×C9711yescharacteristic
157C4—42no—
158C4—42no—
159C4—42no—
160C4—81no—
161C4#121×C9712yes—
162C4#123×C9712yes—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#394×C9711yescharacteristic
167C2#454×C9711yescharacteristic
168C2#455×C9711yescharacteristic
subgroupG/H
center, socleC22×C97#120
commutator, Frattini, cosocleC22×C4C23×C97
prime radical#537×C97C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C97 ◁ C22×C4×C97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#104×C97C2semidirect
3. #417×C97C2semidirect
4. #382×C97C2semidirect
5. #343×C97C2semidirect
39. C42×C97D4semidirect
84. #2040C97direct (N × H)
85. C97#2040direct (N × H)
87. C2×#104C2×C97semidirect
88. #417C2×C97semidirect
89. #382C2×C97semidirect
90. #343C2×C97semidirect
124. C42D4×C97semidirect

Maximal subgroups

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

subgroupindex
#537×C972
C2×#104×C972
#417×C972
#382×C972
#343×C972
#204097