Finite Group Databasebeta

← #27140778

Subgroups — #27140778

Order 12416.

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

normal subgroups only

This group has 202 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#20261068C212yesmaximal
2#20260674C211yesmaximal, characteristic
3#20260472C212yesmaximal
4#20260461C211yesmaximal, characteristic
5#544×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic
7#15690743—22no—
8#15690734—22no—
9C22⋊#11715895C2212yes—
10#15690717—21no—
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
19C22×Dic97—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97—22no—
23C2×Q8×C97—21no—
24C2×Q8×C97D412yes—
25C22×C4×C97D411yescharacteristic
26C22×C4×C97D411yescharacteristic
27C22×C4×C97C2311yescharacteristic
28C4⋊C4×C97—21no—
29C4⋊C4×C97—21no—
30C4⋊C4×C97—21no—
31C4⋊C4×C97—22no—
32C4⋊C4×C97—22no—
33C4⋊C4×C97—22no—
34C4⋊C4×C97D412yes—
35C22⋊C4×C97—22no—
36C42×C97—22no—
37C23×C97C2×D411yescharacteristic
38Q8×C97—42no—
39Q8×C97—81no—
40C2×C4×C97—21no—
41C2×C4×C97—21no—
42C2×C4×C97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—21no—
45C2×C4×C97—41no—
46C2×C4×C97—41no—
47C2×C4×C97Q1612yes—
48C2×C4×C97C2×D412yes—
49C2×Dic97—41no—
50C2×Dic97—41no—
51C2×Dic97—42no—
52C2×Dic97—41no—
53C2×Dic97—41no—
54#11715895—42no—
55C22×C97—21no—
56C22×C97—21no—
57C22×C97#12011yescharacteristic
58C22×C97C2×Q1611yescharacteristic
59C22×C97#13711yescharacteristic
60C4×C97—22no—
61C4×C97—22no—
62C4×C97—42no—
63C4×C97—41no—
64C4×C97—42no—
65Dic97—81no—
66Dic97—81no—
67C2×C97—21no—
68C2×C97—21no—
69C2×C97#45111yescharacteristic
70C2×C97#45411yescharacteristic
71C2×C97#45811yescharacteristic
72#1617—971nomaximal, Sylow
73C97#161711yescharacteristic, complemented, Sylow
74#544D9711yescharacteristic
75#484—972no—
76#400—971no—
77#328—972no—
78#324—971no—
79C4⋊Q8—22no—
80#125—22no—
81C22⋊Q8—21no—
82C22⋊Q8C2×D9712yes—
83#117—21no—
84C2×C4⋊C4—971no—
85C2×C4⋊C4C2×D9711yescharacteristic
86#107—1942no—
87#103—1942no—
88C22⋊C8—972no—
89#95—971no—
90#95—1941no—
91#95—1941no—
92C2×Q8—21no—
93C2×Q8C22⋊D9712yes—
94C22×C4—1941no—
95C22×C4—1941no—
96C22×C4C22⋊D9711yescharacteristic
97C22×C4C22⋊D9711yescharacteristic
98C22×C4C22×D9711yescharacteristic
99C2×C8—1942no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—1942no—
107C4⋊C4C22⋊D9712yes—
108C22⋊C4—22no—
109C42—22no—
110C23C2×C22⋊D9711yescharacteristic
111Q8—42no—
112Q8—81no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—3881no—
121C2×C4—3881no—
122C2×C4—3882no—
123C2×C4—3881no—
124C2×C4—3881no—
125C2×C4#1309550112yes—
126C2×C4C2×C22⋊D9712yes—
127C8—3882no—
128C22—21no—
129C22—21no—
130C22#1569077911yescharacteristic
131C22C2×#1309550111yescharacteristic
132C22#1569084111yescharacteristic
133C4—22no—
134C4—22no—
135C4—42no—
136C4—41no—
137C4—42no—
138C4—7761no—
139C4—7761no—
140C2—21no—
141C2—21no—
142C2#2026089111yescharacteristic
143C2#2026090811yescharacteristic
144C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#544×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#544D97
hypocenterC97#1617
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
73. C97#1617semidirect

Maximal subgroups

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

subgroupindex
#202610682
#202606742
#202604722
#202604612
#544×C972
#161797