Finite Group Databasebeta

← #27142135

Subgroups — #27142135

Order 12416.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#20261210C211yesmaximal, characteristic, complemented
2#20260740C212yesmaximal, complemented
3C2×#15690727C211yesmaximal, characteristic, complemented
4#20260495C212yesmaximal, complemented
5C2×#127×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7D4×Dic97—24no—
8#15690769C2212yes—
9C2×C22⋊Dic97—22no—
10#15690731—24no—
11#15690727—22no—
12C22×D4×C97—22no—
13C22×D4×C97C2211yescharacteristic
14C22×D4×C97C2211yescharacteristic
15#127×C97—22no—
16#127×C97C414yescomplemented
17C2×C42×C97C2211yescharacteristic
18C22×Dic97—24no—
19#13095495—22no—
20#13095495—22no—
21C2×#11715895—22no—
22C4⋊Dic97D414yescomplemented
23C22⋊Dic97—24no—
24C22⋊Dic97—22no—
25C4×Dic97—22no—
26C24×C97—21no—
27C24×C97D412yes—
28C2×D4×C97—28no—
29C2×D4×C97—22no—
30C2×D4×C97—22no—
31C2×D4×C97—21no—
32C2×D4×C97—22no—
33C2×D4×C97—22no—
34C2×D4×C97—48no—
35C2×D4×C97C2×C412yes—
36C2×D4×C97C2×C412yescomplemented
37C2×D4×C97D412yes—
38C22×C4×C97—21no—
39C22×C4×C97C2311yescharacteristic
40C42×C97—21no—
41C42×C97C2×C412yescomplemented
42C23×C97—22no—
43C23×C97—22no—
44C23×C97—22no—
45C23×C97—44no—
46C23×C97—44no—
47C23×C97—41no—
48C23×C97—41no—
49C23×C97C22⋊C414yes—
50C23×C97C2×D411yescharacteristic
51D4×C97—24no—
52D4×C97—24no—
53D4×C97—24no—
54D4×C97—416no—
55D4×C97—42no—
56C2×C4×C97—24no—
57C2×C4×C97—22no—
58C2×C4×C97C22×C411yescharacteristic
59C2×C4×C97C2×D412yes—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97#4112yes—
62C2×Dic97—24no—
63C2×Dic97—84no—
64#11715895—42no—
65C22×C97—22no—
66C22×C97—24no—
67C22×C97—44no—
68C22×C97—44no—
69C22×C97—44no—
70C22×C97—42no—
71C22×C97—82no—
72C22×C97—82no—
73C22×C97—82no—
74C22×C97C2×C22⋊C411yescharacteristic
75C22×C97#12011yescharacteristic
76C22×C97#12311yescharacteristic
77C4×C97—24no—
78C4×C97C4×D412yes—
79C4×C97#12112yes—
80Dic97—44no—
81C2×C97—21no—
82C2×C97—21no—
83C2×C97—44no—
84C2×C97—82no—
85C2×C97#38611yescharacteristic
86C2×C97#45312yes—
87#1887—971nomaximal, Sylow
88C97#188711yescharacteristic, complemented, Sylow
89C2×#127D9711yescharacteristic
90#518—971no—
91#418—972no—
92C2×#100—971no—
93#339—972no—
94C22×D4—22no—
95C22×D4C2×D9711yescharacteristic
96C22×D4C2×D9711yescharacteristic
97C2×M16—972no—
98#127—22no—
99#127Dic9714yescomplemented
100C4×D4—1944no—
101#117—972no—
102C2×C22⋊C4—1942no—
103C2×C42C2×D9711yescharacteristic
104#102—1944no—
105#100—1942no—
106C24—21no—
107C24C22⋊D9712yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic9712yes—
116C2×D4C2×Dic9712yescomplemented
117C2×D4C22⋊D9712yes—
118C22×C4—21no—
119C22×C4—1944no—
120C22×C4C22×D9711yescharacteristic
121M16—1942no—
122M16—1942no—
123C2×C8—1942no—
124C4⋊C4—974no—
125C22⋊C4—1944no—
126C22⋊C4—1942no—
127C42—21no—
128C42—1942no—
129C42C2×Dic9712yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic9714yes—
138C23C2×C22⋊D9711yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—7764no—
147C2×C4—1944no—
148C2×C4C22×Dic9711yescharacteristic
149C2×C4D4×D9712yes—
150C2×C4C2×C22⋊D9711yescharacteristic
151C2×C4#1309551012yes—
152C8—3882no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic9711yescharacteristic
163C22#1569077811yescharacteristic
164C22#1569079111yescharacteristic
165C4—24no—
166C4—3884no—
167C4D4×Dic9712yes—
168C4#1569078212yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#2026061511yescharacteristic
174C2#2026090212yes—
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×#127×C97C2
socleC22×C97#120
cosocleC22×C4×C97C23
hypercenterC2×#127D97
hypocenterC97#1887
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20261210C2semidirect
2. #20260740C2semidirect
3. C2×#15690727C2semidirect
4. #20260495C2semidirect
16. #127×C97C4semidirect
22. C4⋊Dic97D4semidirect
36. C2×D4×C97C2×C4semidirect
41. C42×C97C2×C4semidirect
88. C97#1887semidirect
99. #127Dic97semidirect
116. C2×D4C2×Dic97semidirect
129. C42C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612102
#202607402
C2×#156907272
#202604952
C2×#127×C972
#188797