Finite Group Databasebeta

← #38741687

Subgroups — #38741687

Order 25216.

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

normal subgroups only

This group has 238 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#27161708C211yesmaximal, characteristic
2#27161237C211yesmaximal, characteristic
3#27161234C211yesmaximal, characteristic
4#27161000C211yesmaximal, characteristic
5#27160815C211yesmaximal, characteristic
6#27160804C211yesmaximal, characteristic
7#328×C197C211yesmaximal, characteristic
8C22×Dic394C2211yescharacteristic
9C2×Dic788—21no—
10C2×#15691348—21no—
11#20263325—21no—
12#20263324—21no—
13#20263316—21no—
14#20263316—21no—
15#20263305C2211yescharacteristic
16#20263303—21no—
17#20263282—21no—
18C2×C4⋊Dic197—21no—
19C2×#15691337C2211yescharacteristic
20#20263251—21no—
21#20263249—21no—
22C22⋊#13096802C2211yescharacteristic
23#20263233—21no—
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394—21no—
30C2×Dic394—42no—
31C2×Dic394D411yescharacteristic
32C22×Dic197—21no—
33C22×Dic197D411yescharacteristic
34C22×Dic197D411yescharacteristic
35Dic788—42no—
36#15691348—42no—
37C2×#13096802—21no—
38C4⋊Dic197—21no—
39C4⋊Dic197—42no—
40C4⋊Dic197D411yescharacteristic
41#15691337—21no—
42#15691337—21no—
43#15691337—21no—
44#15691337—21no—
45#15691337—21no—
46#15691337—21no—
47#15691337—21no—
48C22⋊Dic197—21no—
49C4×Dic197—21no—
50C4×Dic197—21no—
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—
58Dic394—22no—
59Dic394—22no—
60Dic394—41no—
61Dic394—41no—
62Dic394—41no—
63Dic394—81no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—22no—
67C2×Dic197—21no—
68C2×Dic197—21no—
69C2×Dic197—21no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C2×Dic197—41no—
73C2×Dic197Q1612yescomplemented
74#13096802—41no—
75C23×C197C2×D411yescharacteristic
76Q8×C197—41no—
77C2×C4×C197—22no—
78C2×C4×C197—21no—
79C2×C4×C197—21no—
80C2×C4×C197—41no—
81C2×C4×C197—41no—
82C2×C4×C197—41no—
83C2×C4×C197C2×D411yescharacteristic
84C2×C4×C197C2×D411yescharacteristic
85C8×C197—41no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197#12011yescharacteristic
89C22×C197C2×Q1611yescharacteristic
90C22×C197#13711yescharacteristic
91C4×C197—21no—
92C4×C197—21no—
93C4×C197—42no—
94C4×C197—41no—
95C4×C197—41no—
96Dic197—22no—
97Dic197—41no—
98Dic197—42no—
99Dic197—41no—
100Dic197—41no—
101C2×C197—21no—
102C2×C197—21no—
103C2×C197#45111yescharacteristic
104C2×C197#45611yescharacteristic
105C2×C197B(4,𝔽2)11yescharacteristic
106C197#160511yescharacteristic, complemented, Sylow
107#1605—1971nomaximal, Sylow
108#544—1971no—
109#451—1971no—
110#451—1971no—
111#395—1971no—
112#328—1971no—
113#328D19711yescharacteristic
114#324—1971no—
115C22×Q8—1971no—
116C2×Q16—3941no—
117C2×Q16—3941no—
118C4⋊Q8—3941no—
119C4⋊Q8—3941no—
120#125—3941no—
121#125—3941no—
122C22⋊Q8—3941no—
123C22⋊Q8—1971no—
124C22⋊Q8C2×D19711yescharacteristic
125#117—3941no—
126C2×C4⋊C4—3941no—
127C2×C4⋊C4—1971no—
128#103—3941no—
129#103—3941no—
130C22⋊C8—1971no—
131C22⋊C8C2×D19711yescharacteristic
132#95—3941no—
133#95C2×D19711yescharacteristic
134C2×Q8—7882no—
135C2×Q8—3941no—
136C2×Q8—3941no—
137C2×Q8—3941no—
138C2×Q8—1971no—
139C2×Q8D78811yescharacteristic
140C22×C4—21no—
141C22×C4—3941no—
142C22×C4—1971no—
143C22×C4—1971no—
144C22×C4C22×D19711yescharacteristic
145Q16—7882no—
146Q16—7882no—
147C2×C8—21no—
148C2×C8—3941no—
149C4⋊C4—21no—
150C4⋊C4—7882no—
151C4⋊C4—3941no—
152C4⋊C4—3941no—
153C4⋊C4—3941no—
154C4⋊C4—3941no—
155C4⋊C4—3941no—
156C4⋊C4—3941no—
157C4⋊C4—3941no—
158C4⋊C4—3941no—
159C4⋊C4—1971no—
160C4⋊C4D78811yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3941no—
163C42—3941no—
164C42—3941no—
165C23D4×D19711yescharacteristic
166Q8—41no—
167Q8—7881no—
168Q8—7881no—
169Q8—7881no—
170Q8—15761no—
171Q8—3942no—
172Q8—3942no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—7881no—
180C2×C4—7881no—
181C2×C4—7881no—
182C2×C4—3942no—
183C2×C4—3941no—
184C2×C4—3941no—
185C2×C4—3941no—
186C2×C4—3941no—
187C2×C4—3941no—
188C2×C4—1972no—
189C2×C4D4×D19711yescharacteristic
190C2×C4C2×D78811yescharacteristic
191C8—41no—
192C8—7881no—
193C22—21no—
194C22—21no—
195C22#2026329611yescharacteristic
196C22Q16×D19711yescharacteristic
197C22#2026335511yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—7881no—
204C4—7882no—
205C4—7881no—
206C4—7881no—
207C4—3942no—
208C2—21no—
209C2—21no—
210C2#2716123611yescharacteristic
211C2#2716126311yescharacteristic
212C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
hypercenter#328D197
hypocenterC197#1605
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
73. C2×Dic197Q16semidirect
106. C197#1605semidirect

Maximal subgroups

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

subgroupindex
#271617082
#271612372
#271612342
#271610002
#271608152
#271608042
#328×C1972
#1605197