Finite Group Databasebeta

← #27140773

Subgroups — #27140773

Order 12416.

246 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261051C212yesmaximal, complemented
2#20260674C211yesmaximal, characteristic, complemented
3#20260469C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic
7#15690741—22no—
8#15690731—22no—
9C22⋊#11715895C2212yes—
10#15690717—21no—
11#15690717—21no—
12#15690717C2211yescharacteristic
13C2×#41×C97—21no—
14C22⋊Q8×C97—22no—
15#121×C97—22no—
16#121×C97—21no—
17#121×C97C2212yes—
18C2×C4⋊C4×C97C2211yescharacteristic
19C22×Dic97—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97—22no—
23#41×C97—42no—
24#41×C97—42no—
25C2×Q8×C97—21no—
26C2×D4×C97—21no—
27C2×D4×C97—22no—
28C2×D4×C97—21no—
29C2×D4×C97D412yes—
30C22×C4×C97—22no—
31C22×C4×C97D411yescharacteristic
32C22×C4×C97D411yescharacteristic
33C22×C4×C97C2311yescharacteristic
34C4⋊C4×C97—21no—
35C4⋊C4×C97—22no—
36C4⋊C4×C97D412yes—
37C22⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C23×C97—22no—
40C23×C97C2×D411yescharacteristic
41Q8×C97—42no—
42D4×C97—42no—
43D4×C97—42no—
44D4×C97—42no—
45D4×C97—42no—
46D4×C97—81no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97—21no—
50C2×C4×C97—22no—
51C2×C4×C97—41no—
52C2×C4×C97—44no—
53C2×C4×C97—41no—
54C2×C4×C97SD1612yes—
55C2×C4×C97C2×D412yes—
56C2×Dic97—41no—
57C2×Dic97—41no—
58C2×Dic97—42no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61#11715895—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97—42no—
65C22×C97—42no—
66C22×C97—42no—
67C22×C97#12011yescharacteristic
68C22×C97C2×SD1611yescharacteristic
69C22×C97AGL(1,ℤ/8)11yescharacteristic
70C4×C97—22no—
71C4×C97—22no—
72C4×C97—42no—
73C4×C97—41no—
74Dic97—81no—
75Dic97—81no—
76C2×C97—21no—
77C2×C97—21no—
78C2×C97—82no—
79C2×C97#45011yescharacteristic
80C2×C97#45411yescharacteristic
81C2×C97#45811yescharacteristic
82#1616—971nomaximal, Sylow
83C97#161611yescharacteristic, complemented, Sylow
84#537D9711yescharacteristic
85#481—972no—
86#400—971no—
87#327—972no—
88#324—971no—
89C2×#41—21no—
90C22⋊Q8—22no—
91#121—22no—
92#121—21no—
93#121C2×D9712yes—
94C2×C4⋊C4—971no—
95C2×C4⋊C4C2×D9711yescharacteristic
96#106—1942no—
97#102—1942no—
98C22⋊C8—972no—
99#95—971no—
100#95—1941no—
101#95—1941no—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4C22⋊D9712yes—
109C22×C4—22no—
110C22×C4—1941no—
111C22×C4—1941no—
112C22×C4C22⋊D9711yescharacteristic
113C22×C4C22⋊D9711yescharacteristic
114C22×C4C22×D9711yescharacteristic
115C2×C8—1942no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—1942no—
119C4⋊C4C22⋊D9712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C23—22no—
123C23C2×C22⋊D9711yescharacteristic
124Q8—42no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—81no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—3881no—
138C2×C4—3881no—
139C2×C4—3882no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4#1309549912yes—
143C2×C4C2×C22⋊D9712yes—
144C8—3882no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22#1569077911yescharacteristic
151C22C2×#1309549911yescharacteristic
152C22#1569083611yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—41no—
157C4—7761no—
158C4—7761no—
159C2—21no—
160C2—21no—
161C2—82no—
162C2#2026088611yescharacteristic
163C2#2026090811yescharacteristic
164C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#1616
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20261051C2semidirect
2. #20260674C2semidirect
3. #20260469C2semidirect
4. #20260461C2semidirect
83. C97#1616semidirect

Maximal subgroups

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

subgroupindex
#202610512
#202606742
#202604692
#202604612
#537×C972
#161697