Finite Group Databasebeta

← #27140710

Subgroups — #27140710

Order 12416.

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

normal subgroups only

This group has 238 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#20260890C212yesmaximal
2#20260655C211yesmaximal, characteristic
3#20260472C212yesmaximal
4#20260461C211yesmaximal, characteristic
5#544×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×#13095501—22no—
8C2×#13095490—21no—
9#15690733—22no—
10C22⋊#11715895C2212yes—
11#15690717—21no—
12#15690717C2211yescharacteristic
13C4⋊Q8×C97—22no—
14#125×C97—22no—
15C22⋊Q8×C97—21no—
16C22⋊Q8×C97C2212yes—
17#117×C97—21no—
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×Dic194—22no—
20C2×Dic194—42no—
21C22×Dic97—21no—
22C22×Dic97—21no—
23#13095501—44no—
24C2×#11715895—22no—
25#13095490—42no—
26C2×Q8×C97—21no—
27C2×Q8×C97D412yes—
28C22×C4×C97D411yescharacteristic
29C22×C4×C97D411yescharacteristic
30C22×C4×C97C2311yescharacteristic
31C4⋊C4×C97—21no—
32C4⋊C4×C97—21no—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97—22no—
36C4⋊C4×C97—22no—
37C4⋊C4×C97D412yes—
38C22⋊C4×C97—22no—
39C42×C97—22no—
40C23×C97C2×D411yescharacteristic
41Q8×C97—42no—
42Q8×C97—81no—
43C2×C4×C97—21no—
44C2×C4×C97—21no—
45C2×C4×C97—22no—
46C2×C4×C97—22no—
47C2×C4×C97—21no—
48C2×C4×C97—41no—
49C2×C4×C97—41no—
50C2×C4×C97Q1612yescomplemented
51C2×C4×C97C2×D412yes—
52C2×Dic97—22no—
53C2×Dic97—22no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56C2×Dic97—41no—
57C2×Dic97—41no—
58Dic194—24no—
59Dic194—42no—
60#11715895—42no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97#12011yescharacteristic
64C22×C97C2×Q1611yescharacteristic
65C22×C97#13711yescharacteristic
66C4×C97—22no—
67C4×C97—22no—
68C4×C97—42no—
69C4×C97—41no—
70C4×C97—42no—
71Dic97—42no—
72Dic97—42no—
73C2×C97—21no—
74C2×C97—21no—
75C2×C97#45111yescharacteristic
76C2×C97#45611yescharacteristic
77C2×C97B(4,𝔽2)11yescharacteristic
78#1605—971nomaximal, Sylow
79C97#160511yescharacteristic, complemented, Sylow
80#544D9711yescharacteristic
81#451—972no—
82#395—971no—
83#328—972no—
84#324—971no—
85C22×Q8—971no—
86C2×Q16—1942no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D9712yes—
91#117—21no—
92C2×C4⋊C4—1941no—
93C2×C4⋊C4C2×D9711yescharacteristic
94#103—1942no—
95C22⋊C8—972no—
96#95—971no—
97#95—1941no—
98C2×Q8—21no—
99C2×Q8—3882no—
100C2×Q8—1942no—
101C2×Q8C22⋊D9712yes—
102C22×C4—1941no—
103C22×C4—1941no—
104C22×C4C22⋊D9711yescharacteristic
105C22×C4C22⋊D9711yescharacteristic
106C22×C4C22×D9711yescharacteristic
107Q16—3884no—
108C2×C8—1942no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—3882no—
116C4⋊C4C22⋊D9712yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D9711yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—3882no—
123Q8—1944no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—3881no—
132C2×C4—3881no—
133C2×C4—3881no—
134C2×C4—3881no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4#1309550112yescomplemented
138C2×C4C2×C22⋊D9712yes—
139C8—3882no—
140C22—21no—
141C22—21no—
142C22#1569077911yescharacteristic
143C22C2×#1309550111yescharacteristic
144C22#1569084111yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—3882no—
151C4—3882no—
152C2—21no—
153C2—21no—
154C2#2026089111yescharacteristic
155C2#2026091811yescharacteristic
156C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#544×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#544D97
hypocenterC97#1605
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
50. C2×C4×C97Q16semidirect
79. C97#1605semidirect
137. C2×C4#13095501semidirect

Maximal subgroups

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

subgroupindex
#202608902
#202606552
#202604722
#202604612
#544×C972
#160597