Finite Group Databasebeta

← #27140658

Subgroups — #27140658

Order 12416.

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

normal subgroups only

This group has 316 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, complemented
2#20260886C212yesmaximal, complemented
3#20260471C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#392×C97C211yesmaximal, characteristic
6C2×#13095499—22no—
7#15690789—21no—
8#15690789C2212yes—
9#15690787—22no—
10#15690787—22no—
11C2×C22⋊Dic97—21no—
12#15690731—22no—
13C22⋊#11715895C2212yes—
14C22×D4×C97—21no—
15C22×D4×C97C2211yescharacteristic
16C2×C22⋊C4×C97—21no—
17C2×C22⋊C4×C97C2211yescharacteristic
18#95×C97C2211yescharacteristic
19C2×Dic194—21no—
20C2×Dic194D412yescomplemented
21C22×Dic97—21no—
22C22×Dic97—21no—
23#13095499—44no—
24C2×#11715895—22no—
25C4⋊Dic97—21no—
26C4⋊Dic97D412yescomplemented
27#13095490—22no—
28#13095490—22no—
29C22⋊Dic97—22no—
30C22⋊Dic97—22no—
31C24×C97—21no—
32C24×C97D411yescharacteristic
33C2×D4×C97—22no—
34C2×D4×C97—42no—
35C2×D4×C97—42no—
36C2×D4×C97—42no—
37C2×D4×C97—42no—
38C22×C4×C97—21no—
39C22×C4×C97D411yescharacteristic
40C22×C4×C97C2311yescharacteristic
41C22⋊C4×C97—22no—
42C22⋊C4×C97—42no—
43C23×C97—21no—
44C23×C97—21no—
45C23×C97—22no—
46C23×C97—41no—
47C23×C97—41no—
48C23×C97—41no—
49C23×C97—41no—
50C23×C97—41no—
51C23×C97—41no—
52C23×C97—41no—
53C23×C97—41no—
54C23×C97SD1612yescomplemented
55C23×C97C2×D411yescharacteristic
56D4×C97—24no—
57D4×C97—42no—
58D4×C97—48no—
59C2×C4×C97—22no—
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C2×D412yes—
66C2×Dic97—22no—
67C2×Dic97—22no—
68C2×Dic97—81no—
69C2×Dic97—81no—
70Dic194—42no—
71Dic194—81no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—22no—
76C22×C97—42no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—42no—
82C22×C97—81no—
83C22×C97—81no—
84C22×C97—81no—
85C22×C97—81no—
86C22×C97—81no—
87C22×C97#12011yescharacteristic
88C22×C97C2×SD1611yescharacteristic
89C22×C97#13711yescharacteristic
90C4×C97—22no—
91C4×C97—42no—
92C4×C97—41no—
93Dic97—42no—
94Dic97—42no—
95C2×C97—21no—
96C2×C97—21no—
97C2×C97—42no—
98C2×C97—82no—
99C2×C97#44811yescharacteristic
100C2×C97#45311yescharacteristic
101C2×C97B(4,𝔽2)11yescharacteristic
102#1596—971nomaximal, Sylow
103C97#159611yescharacteristic, complemented, Sylow
104#543—971no—
105#450—972no—
106#392D9711yescharacteristic
107#328—972no—
108#323—971no—
109C22×D4—21no—
110C22×D4C2×D9711yescharacteristic
111C2×SD16—1942no—
112C22⋊Q8—972no—
113C22⋊Q8—1942no—
114C22⋊Q8—1942no—
115C22⋊Q8—1941no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—1941no—
118C2×C22⋊C4C2×D9711yescharacteristic
119#102—1942no—
120C22⋊C8—972no—
121#95C2×D9711yescharacteristic
122C24—21no—
123C24C22⋊D9711yescharacteristic
124C2×Q8—972no—
125C2×Q8—1941no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—21no—
132C22×C4—1941no—
133C22×C4—1941no—
134C22×C4C22⋊D9711yescharacteristic
135C22×C4C22×D9711yescharacteristic
136SD16—3884no—
137C2×C8—1942no—
138C4⋊C4—972no—
139C4⋊C4—1942no—
140C4⋊C4—1941no—
141C4⋊C4—1942no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—1942no—
145C22⋊C4—1942no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23#1309549912yescomplemented
158C23C2×C22⋊D9711yescharacteristic
159Q8—7761no—
160Q8—3882no—
161D4—24no—
162D4—42no—
163D4—48no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—7761no—
171C2×C4—7761no—
172C2×C4—1942no—
173C2×C4—1942no—
174C2×C4D4×D9712yes—
175C8—3882no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#1569077811yescharacteristic
191C22C2×#1309549911yescharacteristic
192C22#1569084011yescharacteristic
193C4—22no—
194C4—42no—
195C4—41no—
196C4—3882no—
197C4—3882no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#2026087311yescharacteristic
203C2#2026090211yescharacteristic
204C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#392×C97C2
socleC22×C97#120
hypercenter#392D97
hypocenterC97#1596
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #20261361C2semidirect
2. #20260886C2semidirect
3. #20260471C2semidirect
4. #20260459C2semidirect
20. C2×Dic194D4semidirect
26. C4⋊Dic97D4semidirect
54. C23×C97SD16semidirect
103. C97#1596semidirect
157. C23#13095499semidirect

Maximal subgroups

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

subgroupindex
#202613612
#202608862
#202604712
#202604592
#392×C972
#159697