Finite Group Databasebeta

← C97×#1793 #27137751

Subgroups — C97×#1793 #27137751

Order 12416.

552 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C2×2−1+4×C97C211yesmaximal, characteristic
2C2×#104×C97C212yesmaximal, complemented
3#417×C97C212yesmaximal, complemented
4C2×#101×C97C211yesmaximal, characteristic, complemented
5#391×C97C211yesmaximal, characteristic, complemented
62−1+4×C97—24no—
72−1+4×C97—22no—
82−1+4×C97C414yescomplemented
9C2×#41×C97—22no—
10C2×#41×C97—22no—
11C2×#41×C97C2212yes—
12C22×Q8×C97—22no—
13C22×Q8×C97C2211yescharacteristic
14C2×M16×C97C2212yes—
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C42×C97C2211yescharacteristic
17#104×C97—24no—
18#103×C97—22no—
19#102×C97—22no—
20#101×C97—22no—
21#95×C97—22no—
22#41×C97—28no—
23#41×C97—28no—
24#41×C97—28no—
25#41×C97—28no—
26#41×C97—22no—
27#41×C97—22no—
28#41×C97C2×C414yes—
29#41×C97D414yes—
30C2×Q8×C97—22no—
31C2×Q8×C97—28no—
32C2×Q8×C97—28no—
33C2×Q8×C97—22no—
34C2×Q8×C97—22no—
35C2×Q8×C97C2×C412yes—
36C2×Q8×C97D412yes—
37C2×Q8×C97D412yes—
38C2×D4×C97—22no—
39C2×D4×C97—22no—
40C2×D4×C97D412yes—
41C22×C4×C97—24no—
42C22×C4×C97—22no—
43C22×C4×C97—21no—
44C22×C4×C97—21no—
45C22×C4×C97D412yes—
46C22×C4×C97C2311yescharacteristic
47M16×C97—22no—
48M16×C97—22no—
49C2×C8×C97—22no—
50C4⋊C4×C97—22no—
51C42×C97—22no—
52C23×C97—22no—
53C23×C97C2×D411yescharacteristic
54Q8×C97—28no—
55Q8×C97—24no—
56Q8×C97—22no—
57Q8×C97—24no—
58Q8×C97C22⋊C414yes—
59D4×C97—28no—
60D4×C97—24no—
61D4×C97—24no—
62D4×C97—22no—
63D4×C97C22⋊C414yes—
64C2×C4×C97—216no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—24no—
69C2×C4×C97—24no—
70C2×C4×C97—22no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—42no—
74C2×C4×C97—41no—
75C2×C4×C97C22⋊C414yes—
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C2×D412yes—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D411yescharacteristic
80C8×C97—42no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—24no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97C2×C22⋊C411yescharacteristic
87C22×C97#12011yescharacteristic
88C22×C97#12011yescharacteristic
89C4×C97—24no—
90C4×C97—24no—
91C4×C97—42no—
92C4×C97—81no—
93C4×C97C2×C22⋊C412yes—
94C4×C97#12012yes—
95C2×C97—21no—
96C2×C97—21no—
97C2×C97—44no—
98C2×C97#37911yescharacteristic
99C2×C97#45411yescharacteristic
100C2×C97#45611yescharacteristic
101#1793C9711yesmaximal, characteristic, direct product factor, Sylow
102C97#179311yescharacteristic, direct product factor, Sylow
103C2×2−1+4C2×C9711yescharacteristic
104C2×#104C2×C9712yescomplemented
105#417C2×C9712yescomplemented
106C2×#101C2×C9711yescharacteristic, complemented
107#391C2×C9711yescharacteristic, complemented
1082−1+4—24no—
1092−1+4—22no—
1102−1+4C4×C9714yescomplemented
111C2×#41—22no—
112C2×#41—22no—
113C2×#41C22×C9712yes—
114C22×Q8—22no—
115C22×Q8C22×C9711yescharacteristic
116C2×M16C22×C9712yes—
117C2×C4⋊C4C22×C9711yescharacteristic
118C2×C42C22×C9711yescharacteristic
119#104—24no—
120#103—22no—
121#102—22no—
122#101—22no—
123#95—22no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C2×C4×C9714yes—
131#41D4×C9714yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×C4×C9712yes—
138C2×Q8D4×C9712yes—
139C2×Q8D4×C9712yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4D4×C9712yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4D4×C9712yes—
148C22×C4C23×C9711yescharacteristic
149M16—22no—
150M16—22no—
151C2×C8—22no—
152C4⋊C4—22no—
153C42—22no—
154C23—22no—
155C23C2×D4×C9711yescharacteristic
156Q8—28no—
157Q8—24no—
158Q8—22no—
159Q8—24no—
160Q8C22⋊C4×C9714yes—
161D4—28no—
162D4—24no—
163D4—24no—
164D4—22no—
165D4C22⋊C4×C9714yes—
166C2×C4—216no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4C22⋊C4×C9714yes—
178C2×C4C22×C4×C9711yescharacteristic
179C2×C4C2×D4×C9712yes—
180C2×C4C2×D4×C9712yes—
181C2×C4C2×D4×C9711yescharacteristic
182C8—42no—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—42no—
187C22—42no—
188C22C2×C22⋊C4×C9711yescharacteristic
189C22#120×C9711yescharacteristic
190C22#120×C9711yescharacteristic
191C4—24no—
192C4—24no—
193C4—42no—
194C4—81no—
195C4C2×C22⋊C4×C9712yes—
196C4#120×C9712yes—
197C2—21no—
198C2—21no—
199C2—44no—
200C2#379×C9711yescharacteristic
201C2#454×C9711yescharacteristic
202C2#456×C9711yescharacteristic
subgroupG/H
center, socleC22×C97#120
commutatorC2×C4C22×C4×C97
Frattini, cosocleC22×C4C23×C97
prime radicalC2×2−1+4×C97C2
LCSG ▷ C2×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. C2×#101×C97C2semidirect
5. #391×C97C2semidirect
8. 2−1+4×C97C4semidirect
101. #1793C97direct (N × H)
102. C97#1793direct (N × H)
104. C2×#104C2×C97semidirect
105. #417C2×C97semidirect
106. C2×#101C2×C97semidirect
107. #391C2×C97semidirect
110. 2−1+4C4×C97semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4×C972
C2×#104×C972
#417×C972
C2×#101×C972
#391×C972
#179397