Finite Group Databasebeta

← #38741670

Subgroups — #38741670

Order 25216.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161704C211yesmaximal, characteristic
2#27161237C211yesmaximal, characteristic, complemented
3#27161234C211yesmaximal, characteristic, complemented
4#27160992C211yesmaximal, characteristic
5#27160815C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic
7#328×C197C211yesmaximal, characteristic, complemented
8C22×Dic394C2211yescharacteristic
9C2×Dic788—21no—
10C2×#15691348—21no—
11#20263306—21no—
12#20263305C2211yescharacteristic
13#20263304—21no—
14#20263304—21no—
15#20263303—21no—
16#20263302—21no—
17C2×#15691335—21no—
18C2×#15691335C2211yescharacteristic
19C22⋊C4×D197—21no—
20#20263251—21no—
21#20263249—21no—
22C22⋊#13096802C2211yescharacteristic
23#20263233—21no—
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C2×Dic394—21no—
29C2×Dic394—21no—
30C2×Dic394—21no—
31C2×Dic394—42no—
32C2×Dic394D411yescharacteristic
33C22×Dic197—21no—
34C22×Dic197D411yescharacteristic
35C2×C4×D197—21no—
36C2×C4×D197—21no—
37Dic788—42no—
38#15691348—42no—
39C2×#13096802—21no—
40C4⋊Dic197—21no—
41C4⋊Dic197D411yescharacteristic
42#15691337—21no—
43#15691337—21no—
44#15691337—21no—
45C22⋊Dic197—21no—
46#15691335—21no—
47#15691335—21no—
48#15691335—21no—
49#15691335—21no—
50#15691335—42no—
51C2×Q8×C197D411yescharacteristic
52C22×C4×C197—21no—
53C22×C4×C197C2311yescharacteristic
54C2×C8×C197—21no—
55C4⋊C4×C197—21no—
56C4⋊C4×C197D411yescharacteristic
57C22⋊C4×C197—21no—
58C22×D197—21no—
59C22×D197—21no—
60C22×D197—41no—
61C22×D197—41no—
62C22×D197Q1612yescomplemented
63Dic394—22no—
64Dic394—22no—
65Dic394—41no—
66Dic394—41no—
67Dic394—41no—
68Dic394—81no—
69C2×Dic197—22no—
70C2×Dic197—21no—
71C2×Dic197—21no—
72C2×Dic197—21no—
73C2×Dic197—41no—
74C2×Dic197—41no—
75C2×Dic197—41no—
76C4×D197—81no—
77C4×D197—81no—
78#13096802—41no—
79C23×C197C2×D411yescharacteristic
80Q8×C197—41no—
81C2×C4×C197—21no—
82C2×C4×C197—21no—
83C2×C4×C197—41no—
84C2×C4×C197—41no—
85C2×C4×C197—41no—
86C2×C4×C197—41no—
87C2×C4×C197C2×D411yescharacteristic
88C2×C4×C197C2×D411yescharacteristic
89C8×C197—41no—
90C22×C197—21no—
91C22×C197—21no—
92C22×C197#12011yescharacteristic
93C22×C197C2×Q1611yescharacteristic
94C22×C197#13711yescharacteristic
95C2×D197—22no—
96C2×D197—41no—
97C2×D197—41no—
98C2×D197—42no—
99C2×D197—81no—
100C4×C197—21no—
101C4×C197—21no—
102C4×C197—41no—
103C4×C197—41no—
104C4×C197—81no—
105Dic197—41no—
106Dic197—41no—
107Dic197—41no—
108Dic197—42no—
109C2×C197—21no—
110C2×C197—21no—
111C2×C197#45111yescharacteristic
112C2×C197#45511yescharacteristic
113C2×C197B(4,𝔽2)11yescharacteristic
114D197—42no—
115C197#160211yescharacteristic, complemented, Sylow
116#1602—1971nomaximal, Sylow
117#543—1971no—
118#451—1971no—
119#451—1971no—
120#393—1971no—
121#328—1971no—
122#328D19711yescharacteristic, complemented
123#323—1971no—
124C22×Q8—1971no—
125C2×Q16—3941no—
126C2×Q16—3941no—
127C22⋊Q8—3941no—
128C22⋊Q8—3941no—
129C22⋊Q8—3941no—
130C22⋊Q8—3941no—
131C22⋊Q8—3941no—
132C22⋊Q8—1971no—
133C22⋊Q8C2×D19711yescharacteristic
134C2×C22⋊C4—3941no—
135C2×C22⋊C4—3941no—
136C2×C22⋊C4—1971no—
137#103—3941no—
138#103—3941no—
139C22⋊C8—1971no—
140C22⋊C8C2×D19711yescharacteristic
141#95—3941no—
142#95C2×D19711yescharacteristic
143C24—1971no—
144C2×Q8—7882no—
145C2×Q8—3941no—
146C2×Q8—3941no—
147C2×Q8—3941no—
148C2×Q8—1971no—
149C2×Q8D78811yescharacteristic
150C22×C4—21no—
151C22×C4—3941no—
152C22×C4—3941no—
153C22×C4—3941no—
154C22×C4—1971no—
155C22×C4C22×D19711yescharacteristic
156Q16—7882no—
157Q16—7882no—
158C2×C8—21no—
159C2×C8—3941no—
160C4⋊C4—21no—
161C4⋊C4—3941no—
162C4⋊C4—3941no—
163C4⋊C4—3941no—
164C4⋊C4—3941no—
165C4⋊C4—1971no—
166C4⋊C4D78811yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—7882no—
169C22⋊C4—3941no—
170C22⋊C4—3941no—
171C22⋊C4—3941no—
172C22⋊C4—3941no—
173C22⋊C4—3941no—
174C23—7881no—
175C23—7881no—
176C23—3941no—
177C23—3941no—
178C23—1972no—
179C23D4×D19711yescharacteristic
180Q8—41no—
181Q8—7881no—
182Q8—7881no—
183Q8—7881no—
184Q8—15761no—
185Q8—3942no—
186Q8—3942no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—7881no—
194C2×C4—7881no—
195C2×C4—7881no—
196C2×C4—15761no—
197C2×C4—15761no—
198C2×C4—3942no—
199C2×C4—3941no—
200C2×C4—3941no—
201C2×C4—3941no—
202C2×C4D4×D19711yescharacteristic
203C2×C4C2×D78811yescharacteristic
204C8—41no—
205C8—7881no—
206C22—21no—
207C22—21no—
208C22—7882no—
209C22—7881no—
210C22—7881no—
211C22—15761no—
212C22—3942no—
213C22#2026329611yescharacteristic
214C22Q16×D19711yescharacteristic
215C22#2026335511yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—7881no—
222C4—7881no—
223C4—7882no—
224C4—7881no—
225C2—21no—
226C2—21no—
227C2—7882no—
228C2#2716123611yescharacteristic
229C2#2716125811yescharacteristic
230C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radicalC23×D197D4
hypercenter#328D197
hypocenterC197#1602
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27161237C2semidirect
3. #27161234C2semidirect
5. #27160815C2semidirect
7. #328×C197C2semidirect
62. C22×D197Q16semidirect
115. C197#1602semidirect
122. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271617042
#271612372
#271612342
#271609922
#271608152
#271608022
#328×C1972
#1602197