Finite Group Databasebeta

← #27140692

Subgroups — #27140692

Order 12416.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261361C211yesmaximal, characteristic
2#20260891C212yesmaximal, complemented
3#20260471C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic
5#393×C97C211yesmaximal, characteristic
6C2×#13095501—22no—
7#15690789—21no—
8#15690789C2212yes—
9#15690787—22no—
10#15690787—22no—
11C2×C22⋊Dic97—21no—
12#15690734—22no—
13C22⋊#11715895C2212yes—
14C22×Q8×C97C2211yescharacteristic
15C2×C22⋊C4×C97—21no—
16C2×C22⋊C4×C97C2211yescharacteristic
17#95×C97—21no—
18#95×C97C2211yescharacteristic
19C2×Dic194—21no—
20C2×Dic194D412yes—
21C22×Dic97—21no—
22C22×Dic97—21no—
23#13095501—44no—
24C2×#11715895—22no—
25C4⋊Dic97—21no—
26C4⋊Dic97D412yes—
27#13095490—22no—
28#13095490—22no—
29C22⋊Dic97—22no—
30C22⋊Dic97—22no—
31C24×C97D411yescharacteristic
32C2×Q8×C97—22no—
33C2×Q8×C97—42no—
34C22×C4×C97—21no—
35C22×C4×C97—21no—
36C22×C4×C97D411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C22⋊C4×C97—22no—
39C22⋊C4×C97—42no—
40C23×C97—21no—
41C23×C97—21no—
42C23×C97—41no—
43C23×C97—41no—
44C23×C97Q1612yescomplemented
45C23×C97C2×D411yescharacteristic
46Q8×C97—24no—
47Q8×C97—42no—
48C2×C4×C97—22no—
49C2×C4×C97—21no—
50C2×C4×C97—41no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D412yes—
58C2×Dic97—22no—
59C2×Dic97—22no—
60C2×Dic97—81no—
61C2×Dic97—81no—
62Dic194—42no—
63Dic194—81no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97—22no—
68C22×C97—42no—
69C22×C97—41no—
70C22×C97—41no—
71C22×C97—81no—
72C22×C97#12011yescharacteristic
73C22×C97C2×Q1611yescharacteristic
74C22×C97#13711yescharacteristic
75C4×C97—22no—
76C4×C97—41no—
77C4×C97—42no—
78C4×C97—81no—
79Dic97—42no—
80Dic97—42no—
81C2×C97—21no—
82C2×C97—21no—
83C2×C97—42no—
84C2×C97#45111yescharacteristic
85C2×C97#45511yescharacteristic
86C2×C97B(4,𝔽2)11yescharacteristic
87#1602—971nomaximal, Sylow
88C97#160211yescharacteristic, complemented, Sylow
89#543—971no—
90#451—972no—
91#393D9711yescharacteristic
92#328—972no—
93#323—971no—
94C22×Q8C2×D9711yescharacteristic
95C2×Q16—1942no—
96C22⋊Q8—972no—
97C22⋊Q8—1942no—
98C22⋊Q8—1942no—
99C22⋊Q8—1941no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—1941no—
102C2×C22⋊C4C2×D9711yescharacteristic
103#103—1942no—
104C22⋊C8—972no—
105#95—21no—
106#95C2×D9711yescharacteristic
107C24C22⋊D9711yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—972no—
111C2×Q8—1941no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—1941no—
115C22×C4—1941no—
116C22×C4C22⋊D9711yescharacteristic
117C22×C4C22×D9711yescharacteristic
118Q16—3884no—
119C2×C8—1942no—
120C4⋊C4—972no—
121C4⋊C4—1942no—
122C4⋊C4—1941no—
123C4⋊C4—1942no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4—1942no—
127C22⋊C4—1942no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1309550112yescomplemented
133C23C2×C22⋊D9711yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—7761no—
137Q8—3882no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—7761no—
148C2×C4—7761no—
149C2×C4—1942no—
150C2×C4—1942no—
151C2×C4D4×D9712yes—
152C8—3882no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1569077811yescharacteristic
161C22C2×#1309550111yescharacteristic
162C22#1569084011yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4—3882no—
168C4—3882no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2026089011yescharacteristic
173C2#2026091211yescharacteristic
174C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#393×C97C2
socleC22×C97#120
prime radicalC24×C97D4
hypercenter#393D97
hypocenterC97#1602
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #20260891C2semidirect
3. #20260471C2semidirect
44. C23×C97Q16semidirect
88. C97#1602semidirect
132. C23#13095501semidirect

Maximal subgroups

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

subgroupindex
#202613612
#202608912
#202604712
#202604592
#393×C972
#160297