Finite Group Databasebeta

← #38741755

Subgroups — #38741755

Order 25216.

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

normal subgroups only

This group has 202 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#27161415C211yesmaximal, characteristic
3#27161412C211yesmaximal, characteristic
4#27161019C211yesmaximal, characteristic
5#27160815C211yesmaximal, characteristic
6#27160804C211yesmaximal, characteristic
7#328×C197C211yesmaximal, characteristic
8#20263325—21no—
9#20263324—21no—
10#20263316—21no—
11#20263316—21no—
12#20263305C2211yescharacteristic
13#20263303—21no—
14#20263282—21no—
15C2×C4⋊Dic197C2211yescharacteristic
16C2×#15691337C2211yescharacteristic
17#20263260—21no—
18#20263259—21no—
19#20263251—21no—
20#20263250—21no—
21C22⋊#13096802C2211yescharacteristic
22#20263233—21no—
23#20263233—21no—
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394D411yescharacteristic
29C22×Dic197—21no—
30C22×Dic197D411yescharacteristic
31C22×Dic197D411yescharacteristic
32C2×#13096802—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197D411yescharacteristic
37#15691337—21no—
38#15691337—21no—
39#15691337—21no—
40#15691337—21no—
41#15691337—21no—
42#15691337—21no—
43#15691337—21no—
44C22⋊Dic197—21no—
45C4×Dic197—21no—
46C4×Dic197—21no—
47C2×Q8×C197D411yescharacteristic
48C22×C4×C197—21no—
49C22×C4×C197C2311yescharacteristic
50C2×C8×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197D411yescharacteristic
53C22⋊C4×C197—21no—
54Dic394—41no—
55Dic394—81no—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—21no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—41no—
62C2×Dic197—42no—
63C2×Dic197—41no—
64C2×Dic197Q1612yes—
65#13096802—41no—
66C23×C197C2×D411yescharacteristic
67Q8×C197—41no—
68C2×C4×C197—21no—
69C2×C4×C197—21no—
70C2×C4×C197—41no—
71C2×C4×C197—41no—
72C2×C4×C197—41no—
73C2×C4×C197—41no—
74C2×C4×C197C2×D411yescharacteristic
75C2×C4×C197C2×D411yescharacteristic
76C8×C197—41no—
77C22×C197—21no—
78C22×C197—21no—
79C22×C197#12011yescharacteristic
80C22×C197C2×Q1611yescharacteristic
81C22×C197#13711yescharacteristic
82C4×C197—21no—
83C4×C197—21no—
84C4×C197—41no—
85C4×C197—41no—
86C4×C197—81no—
87Dic197—22no—
88Dic197—41no—
89Dic197—41no—
90Dic197—41no—
91Dic197—81no—
92C2×C197—21no—
93C2×C197—21no—
94C2×C197#45111yescharacteristic
95C2×C197#45411yescharacteristic
96C2×C197#45811yescharacteristic
97C197#161711yescharacteristic, complemented, Sylow
98#1617—1971nomaximal, Sylow
99#544—1971no—
100#484—1971no—
101#484—1971no—
102#400—1971no—
103#328—1971no—
104#328D19711yescharacteristic
105#324—1971no—
106C4⋊Q8—3941no—
107C4⋊Q8—3941no—
108#125—3941no—
109#125—3941no—
110C22⋊Q8—3941no—
111C22⋊Q8—1971no—
112C22⋊Q8C2×D19711yescharacteristic
113#117—3941no—
114C2×C4⋊C4—1971no—
115C2×C4⋊C4—1971no—
116#107—3941no—
117#107—3941no—
118#103—3941no—
119#103—3941no—
120C22⋊C8—1971no—
121C22⋊C8C2×D19711yescharacteristic
122#95—3941no—
123#95—3941no—
124#95C2×D19711yescharacteristic
125C2×Q8—3941no—
126C2×Q8—1971no—
127C2×Q8D78811yescharacteristic
128C22×C4—21no—
129C22×C4—3941no—
130C22×C4—1971no—
131C22×C4—1971no—
132C22×C4C22×D19711yescharacteristic
133C2×C8—21no—
134C2×C8—3941no—
135C4⋊C4—21no—
136C4⋊C4—3941no—
137C4⋊C4—3941no—
138C4⋊C4—3941no—
139C4⋊C4—3941no—
140C4⋊C4—3941no—
141C4⋊C4—3941no—
142C4⋊C4—3941no—
143C4⋊C4—3941no—
144C4⋊C4—3941no—
145C4⋊C4—3941no—
146C4⋊C4—1971no—
147C4⋊C4D78811yescharacteristic
148C22⋊C4—21no—
149C22⋊C4—3941no—
150C42—3941no—
151C42—3941no—
152C23D4×D19711yescharacteristic
153Q8—41no—
154Q8—7881no—
155Q8—15761no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—7881no—
163C2×C4—7882no—
164C2×C4—7881no—
165C2×C4—3941no—
166C2×C4—3941no—
167C2×C4—3941no—
168C2×C4—3941no—
169C2×C4—3941no—
170C2×C4—1972no—
171C2×C4D4×D19711yescharacteristic
172C2×C4C2×D78811yescharacteristic
173C8—41no—
174C8—7881no—
175C22—21no—
176C22—21no—
177C22#2026329611yescharacteristic
178C22Q16×D19711yescharacteristic
179C22#2026335511yescharacteristic
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—81no—
185C4—7881no—
186C4—7881no—
187C4—7881no—
188C4—15761no—
189C4—3942no—
190C2—21no—
191C2—21no—
192C2#2716123611yescharacteristic
193C2#2716125311yescharacteristic
194C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
hypercenter#328D197
hypocenterC197#1617
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
97. C197#1617semidirect

Maximal subgroups

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

subgroupindex
#271617082
#271614152
#271614122
#271610192
#271608152
#271608042
#328×C1972
#1617197