Finite Group Databasebeta

← #38743862

Subgroups — #38743862

Order 25216.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#20263317C211yesmaximal, characteristic
2C2×#20263251C211yesmaximal, characteristic
3C2×#20263249C211yesmaximal, characteristic
4#27160984C211yesmaximal, characteristic
5#27160841C211yesmaximal, characteristic
6#27160833C211yesmaximal, characteristic
7#340×C197C211yesmaximal, characteristic
8C22×Dic394C2211yescharacteristic
9C22×#13096802C2211yescharacteristic
10#20263317—22no—
11#20263317—22no—
12C2×C4⋊Dic197—21no—
13C2×C4⋊Dic197C2211yescharacteristic
14C2×#15691337—21no—
15C2×C4×Dic197C2211yescharacteristic
16#20263251—22no—
17#20263249—22no—
18#20263233—21no—
19#20263233—21no—
20C22×C8×C197C2211yescharacteristic
21C2×C4⋊C4×C197C2211yescharacteristic
22C2×C4⋊C4×C197C2211yescharacteristic
23C2×Dic394—21no—
24C2×Dic394—21no—
25C2×Dic394—21no—
26C2×Dic394—21no—
27C2×Dic394—21no—
28C2×Dic394Q812yes—
29C22×Dic197—21no—
30C22×Dic197—21no—
31C22×Dic197D411yescharacteristic
32C22×Dic197D411yescharacteristic
33C2×#13096802—21no—
34C2×#13096802—21no—
35C2×#13096802—21no—
36C4⋊Dic197—24no—
37C4⋊Dic197—21no—
38C4⋊Dic197D412yes—
39#15691337—24no—
40C4×Dic197—21no—
41C4×Dic197—21no—
42C22×C4×C197—21no—
43C22×C4×C197—21no—
44C22×C4×C197C2311yescharacteristic
45C2×C8×C197—21no—
46C2×C8×C197—21no—
47C2×C8×C197—21no—
48C4⋊C4×C197—21no—
49C4⋊C4×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197D412yes—
52Dic394—22no—
53Dic394—22no—
54Dic394—41no—
55Dic394—41no—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—22no—
59C2×Dic197—22no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C2×Dic197—41no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68#13096802—41no—
69C23×C197C2×D411yescharacteristic
70C2×C4×C197—22no—
71C2×C4×C197—41no—
72C2×C4×C197—41no—
73C2×C4×C197—41no—
74C2×C4×C197—41no—
75C2×C4×C197—41no—
76C2×C4×C197C2×D411yescharacteristic
77C2×C4×C197C2×D411yescharacteristic
78C2×C4×C197C2×Q811yescharacteristic
79C2×C4×C197#4111yescharacteristic
80C2×C4×C197#4111yescharacteristic
81C2×C4×C197#4111yescharacteristic
82C8×C197—41no—
83C22×C197#12011yescharacteristic
84C22×C197C22⋊Q811yescharacteristic
85C22×C197#12411yescharacteristic
86C22×C197#13511yescharacteristic
87C22×C197#13511yescharacteristic
88C22×C197#13711yescharacteristic
89C22×C197#13711yescharacteristic
90C4×C197—42no—
91C4×C197—81no—
92C4×C197C22⋊Q811yescharacteristic
93C4×C197C22⋊Q811yescharacteristic
94C4×C197#12411yescharacteristic
95C4×C197#12411yescharacteristic
96Dic197—42no—
97Dic197—41no—
98Dic197—41no—
99Dic197—42no—
100C2×C197#39311yescharacteristic
101C2×C197#45211yescharacteristic
102C2×C197#45211yescharacteristic
103C2×C197#47911yescharacteristic
104C2×C197#47911yescharacteristic
105C2×C197#48811yescharacteristic
106C2×C197#49111yescharacteristic
107C197#202611yescharacteristic, complemented, Sylow
108#2026—1971nomaximal, Sylow
109C2×#125—1971no—
110C2×#103—1971no—
111C2×#103—1971no—
112#391—1971no—
113#340—1971no—
114#340D19711yescharacteristic
115#336—1971no—
116C22×Q8—1971no—
117C22×C8—1971no—
118C22×C8C2×D19711yescharacteristic
119#125—3942no—
120#125—3942no—
121C2×C4⋊C4—3941no—
122C2×C4⋊C4—3941no—
123C2×C4⋊C4—1971no—
124C2×C4⋊C4C2×D19711yescharacteristic
125C2×C4⋊C4C2×D19711yescharacteristic
126C2×C42—1971no—
127#103—3942no—
128#103—3942no—
129#95—3941no—
130#95—3941no—
131C2×Q8—3941no—
132C2×Q8—3941no—
133C2×Q8—3941no—
134C2×Q8—3941no—
135C2×Q8—3941no—
136C2×Q8—1972no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—3941no—
140C22×C4—3941no—
141C22×C4—1971no—
142C22×C4—1971no—
143C22×C4C22×D19711yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—3941no—
148C2×C8—3941no—
149C2×C8—3941no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—3944no—
154C4⋊C4—3944no—
155C4⋊C4—3941no—
156C4⋊C4—1972no—
157C4⋊C4D78812yes—
158C42—3941no—
159C42—3941no—
160C23D4×D19711yescharacteristic
161Q8—7881no—
162Q8—7881no—
163Q8—3942no—
164Q8—3942no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—7881no—
172C2×C4—7881no—
173C2×C4—7881no—
174C2×C4—7881no—
175C2×C4—3941no—
176C2×C4—3941no—
177C2×C4—3942no—
178C2×C4—3942no—
179C2×C4—3941no—
180C2×C4—3941no—
181C2×C4—3941no—
182C2×C4—3941no—
183C2×C4D4×D19711yescharacteristic
184C2×C4C2×D78811yescharacteristic
185C2×C4Q8×D19711yescharacteristic
186C2×C4#1569135711yescharacteristic
187C2×C4#1569135811yescharacteristic
188C2×C4#1569135811yescharacteristic
189C8—41no—
190C8—7881no—
191C22#2026329611yescharacteristic
192C22#2026330411yescharacteristic
193C22#2026331211yescharacteristic
194C22#2026334511yescharacteristic
195C22#2026334711yescharacteristic
196C22#2026335511yescharacteristic
197C22#2026335811yescharacteristic
198C4—42no—
199C4—81no—
200C4—7881no—
201C4—7882no—
202C4—7881no—
203C4—7882no—
204C4#2026330411yescharacteristic
205C4#2026330611yescharacteristic
206C4#2026331211yescharacteristic
207C4#2026331411yescharacteristic
208C2#2716099211yescharacteristic
209C2#2716124211yescharacteristic
210C2#2716124411yescharacteristic
211C2#2716138511yescharacteristic
212C2#2716138911yescharacteristic
213C2#2716143211yescharacteristic
214C2#2716144611yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#340×C197C2
socle, prime radicalC23×C197C2×D4
hypercenter#340D197
hypocenterC197#2026
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
107. C197#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#202633172
C2×#202632512
C2×#202632492
#271609842
#271608412
#271608332
#340×C1972
#2026197