Finite Group Databasebeta

← C97×#1471 #27137429

Subgroups — C97×#1471 #27137429

Order 12416.

322 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 96 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×#121×C97C211yesmaximal, characteristic
2#407×C97C211yesmaximal, characteristic, complemented
3C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
4#323×C97C214yesmaximal, complemented
5C22×D4×C97C411yescharacteristic
6C22×D4×C97C412yes—
7C23×C4×C97C2211yescharacteristic
8C2×M16×C97—21no—
9C22×C8×C97—21no—
10#121×C97—21no—
11#121×C97—21no—
12#121×C97—21no—
13#121×C97—21no—
14C2×C4⋊C4×C97C411yescharacteristic
15C2×C22⋊C4×C97C2212yes—
16C22⋊C8×C97—21no—
17C22⋊C8×C97—21no—
18C22⋊C8×C97C2212yes—
19C22⋊C8×C97C2212yes—
20C24×C97C2×C411yescharacteristic
21C24×C97C2×C412yes—
22C2×D4×C97—22no—
23C2×D4×C97—22no—
24C2×D4×C97—21no—
25C2×D4×C97—21no—
26C2×D4×C97—42no—
27C2×D4×C97—44no—
28C22×C4×C97—42no—
29C22×C4×C97C2×C411yescharacteristic
30C22×C4×C97C2×C412yes—
31C22×C4×C97D411yescharacteristic
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97D411yescharacteristic
35C22×C4×C97C2311yescharacteristic
36M16×C97—42no—
37C2×C8×C97—22no—
38C2×C8×C97—22no—
39C2×C8×C97—42no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97—21no—
42C22⋊C4×C97—22no—
43C22⋊C4×C97—22no—
44C23×C97—22no—
45C23×C97—22no—
46C23×C97—22no—
47C23×C97—24no—
48C23×C97—41no—
49C23×C97—42no—
50C23×C97—42no—
51C23×C97C22⋊C411yescharacteristic
52C23×C97C22⋊C411yescharacteristic
53C23×C97C22×C411yescharacteristic
54D4×C97—42no—
55D4×C97—44no—
56D4×C97—44no—
57D4×C97—42no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—42no—
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C8×C97—42no—
71C8×C97—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97—44no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97—84no—
84C22×C97#10012yes—
85C22×C97C2×C22⋊C411yescharacteristic
86C22×C97#13112yes—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—42no—
91C4×C97#9912yes—
92C2×C97—22no—
93C2×C97—22no—
94C2×C97—41no—
95C2×C97—82no—
96C2×C97#40811yescharacteristic
97C2×C97C2×#9911yescharacteristic
98C2×C97C2×#10011yescharacteristic
99#1471C9711yesmaximal, characteristic, direct product factor, Sylow
100C97#147111yescharacteristic, direct product factor, Sylow
101C2×#121C2×C9711yescharacteristic
102#407C2×C9711yescharacteristic, complemented
103C2×C22⋊C8C2×C9711yescharacteristic, complemented
104#323C2×C9714yescomplemented
105C22×D4C4×C9711yescharacteristic
106C22×D4C4×C9712yes—
107C23×C4C22×C9711yescharacteristic
108C2×M16—21no—
109C22×C8—21no—
110#121—21no—
111#121—21no—
112#121—21no—
113#121—21no—
114C2×C4⋊C4C4×C9711yescharacteristic
115C2×C22⋊C4C22×C9712yes—
116C22⋊C8—21no—
117C22⋊C8—21no—
118C22⋊C8C22×C9712yes—
119C22⋊C8C22×C9712yes—
120C24C2×C4×C9711yescharacteristic
121C24C2×C4×C9712yes—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—42no—
127C2×D4—44no—
128C22×C4—42no—
129C22×C4C2×C4×C9711yescharacteristic
130C22×C4C2×C4×C9712yes—
131C22×C4D4×C9711yescharacteristic
132C22×C4D4×C9711yescharacteristic
133C22×C4D4×C9711yescharacteristic
134C22×C4D4×C9711yescharacteristic
135C22×C4C23×C9711yescharacteristic
136M16—42no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—42no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C23—22no—
145C23—22no—
146C23—22no—
147C23—24no—
148C23—41no—
149C23—42no—
150C23—42no—
151C23C22⋊C4×C9711yescharacteristic
152C23C22⋊C4×C9711yescharacteristic
153C23C22×C4×C9711yescharacteristic
154D4—42no—
155D4—44no—
156D4—44no—
157D4—42no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4C22⋊C4×C9711yescharacteristic
167C2×C4C22⋊C4×C9711yescharacteristic
168C2×C4C2×D4×C9711yescharacteristic
169C2×C4C2×D4×C9711yescharacteristic
170C8—42no—
171C8—42no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—44no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—84no—
184C22#100×C9712yes—
185C22C2×C22⋊C4×C9711yescharacteristic
186C22#131×C9712yes—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—42no—
191C4#99×C9712yes—
192C2—22no—
193C2—22no—
194C2—41no—
195C2—82no—
196C2#408×C9711yescharacteristic
197C2C2×#99×C9711yescharacteristic
198C2C2×#100×C9711yescharacteristic
subgroupG/H
center, socleC22×C97C2×C22⋊C4
commutatorC23C22×C4×C97
Frattini, cosocleC22×C4C23×C97
prime radicalC2×#121×C97C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C97 ◁ C23×C4×C97 ◁ G

Complements

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

NH ≅ G/Ntype
2. #407×C97C2semidirect
3. C2×C22⋊C8×C97C2semidirect
4. #323×C97C2semidirect
99. #1471C97direct (N × H)
100. C97#1471direct (N × H)
102. #407C2×C97semidirect
103. C2×C22⋊C8C2×C97semidirect
104. #323C2×C97semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C972
#407×C972
C2×C22⋊C8×C972
#323×C972
#147197