Finite Group Databasebeta

← #27140659

Subgroups — #27140659

Order 12416.

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

normal subgroups only

This group has 316 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#20260885C212yesmaximal, complemented
2#20260644C211yesmaximal, characteristic
3#20260472C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#543×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C22×C22⋊D97—21no—
8C2×#13095498—22no—
9C2×#13095488—21no—
10#15690730—22no—
11C22⋊#11715895C2212yes—
12#15690717C2211yescharacteristic
13C22⋊Q8×C97—22no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97—22no—
16C22⋊Q8×C97C2212yes—
17C2×C22⋊C4×C97—21no—
18C2×C22⋊C4×C97C2211yescharacteristic
19C23×D97—21no—
20C2×D388—22no—
21C2×D388—42no—
22C2×C22⋊D97—42no—
23C2×C22⋊D97—42no—
24C2×C22⋊D97—42no—
25C22×Dic97—21no—
26#13095498—44no—
27C2×#11715895—22no—
28#13095488—42no—
29C24×C97D411yescharacteristic
30C2×Q8×C97—21no—
31C2×Q8×C97D412yescomplemented
32C22×C4×C97—21no—
33C22×C4×C97—21no—
34C22×C4×C97D411yescharacteristic
35C22×C4×C97C2311yescharacteristic
36C4⋊C4×C97—21no—
37C4⋊C4×C97—22no—
38C4⋊C4×C97—22no—
39C4⋊C4×C97D412yescomplemented
40C22⋊C4×C97—22no—
41C22⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C23×C97—21no—
44C23×C97—21no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97SD1612yescomplemented
48C23×C97C2×D411yescharacteristic
49C22×D97—22no—
50C22×D97—41no—
51C22×D97—41no—
52C22×D97—41no—
53C22×D97—41no—
54C22×D97—41no—
55C22×D97—41no—
56Q8×C97—42no—
57Q8×C97—81no—
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—81no—
64C2×C4×C97—81no—
65C2×C4×C97C2×D412yes—
66C22⋊D97—48no—
67C2×Dic97—22no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70D388—24no—
71D388—42no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—22no—
76C22×C97—42no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—81no—
80C22×C97#12011yescharacteristic
81C22×C97C2×SD1611yescharacteristic
82C22×C97#13711yescharacteristic
83C2×D97—42no—
84C2×D97—42no—
85C2×D97—42no—
86C2×D97—81no—
87C2×D97—81no—
88C2×D97—81no—
89C2×D97—81no—
90C4×C97—22no—
91C4×C97—41no—
92C4×C97—42no—
93C4×C97—42no—
94Dic97—42no—
95C2×C97—21no—
96C2×C97—21no—
97C2×C97—42no—
98C2×C97#44811yescharacteristic
99C2×C97#45311yescharacteristic
100C2×C97B(4,𝔽2)11yescharacteristic
101D97—82no—
102#1596—971nomaximal, Sylow
103C97#159611yescharacteristic, complemented, Sylow
104#543D9711yescharacteristic, complemented
105#450—972no—
106#392—971no—
107#328—972no—
108#323—971no—
109C22×D4—971no—
110C22×D4—1941no—
111C2×SD16—1942no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D9712yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—1941no—
118C2×C22⋊C4C2×D9711yescharacteristic
119#102—1942no—
120C22⋊C8—972no—
121#95—971no—
122C24—1941no—
123C24C22⋊D9711yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D9712yescomplemented
126C2×D4—3882no—
127C2×D4—3882no—
128C2×D4—3882no—
129C2×D4—3882no—
130C2×D4—1942no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—1941no—
134C22×C4C22⋊D9711yescharacteristic
135C22×C4C22×D9711yescharacteristic
136SD16—3884no—
137C2×C8—1942no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D9712yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—3882no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—3881no—
151C23—3881no—
152C23—3881no—
153C23—3881no—
154C23—3881no—
155C23—3881no—
156C23—1942no—
157C23#1309549812yescomplemented
158C23C2×C22⋊D9711yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—3882no—
162D4—3888no—
163D4—1944no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—1942no—
174C2×C4C2×C22⋊D9712yes—
175C8—3882no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—7761no—
184C22—7761no—
185C22—7761no—
186C22—7761no—
187C22—3882no—
188C22—3882no—
189C22—3882no—
190C22#1569077911yescharacteristic
191C22C2×#1309549811yescharacteristic
192C22#1569084111yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—3882no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—7762no—
202C2#2026087411yescharacteristic
203C2#2026090311yescharacteristic
204C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#543×C97C2
socleC22×C97#120
prime radical#20260644C2
hypercenter#543D97
hypocenterC97#1596
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20260885C2semidirect
3. #20260472C2semidirect
4. #20260459C2semidirect
5. #543×C97C2semidirect
31. C2×Q8×C97D4semidirect
39. C4⋊C4×C97D4semidirect
47. C23×C97SD16semidirect
103. C97#1596semidirect
104. #543D97semidirect
125. C2×Q8C22⋊D97semidirect
141. C4⋊C4C22⋊D97semidirect
157. C23#13095498semidirect

Maximal subgroups

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

subgroupindex
#202608852
#202606442
#202604722
#202604592
#543×C972
#159697