Finite Group Databasebeta

← #38743582

Subgroups — #38743582

Order 25216.

304 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#15691348C211yesmaximal, characteristic
2C2×#20263257C211yesmaximal, characteristic
3C2×#20263250C211yesmaximal, characteristic
4C2×#20263249C211yesmaximal, characteristic
5C2×#20263237C211yesmaximal, characteristic
6#27160985C211yesmaximal, characteristic
7#391×C197C211yesmaximal, characteristic
8C22×Dic394C2211yescharacteristic
9C2×#15691348—22no—
10C2×#15691348—22no—
11C2×#15691348C414yescomplemented
12C22×#13096802C2211yescharacteristic
13C22×#13096802C2211yescharacteristic
14C2×C4⋊Dic197C2211yescharacteristic
15#20263257—22no—
16#20263250—22no—
17#20263249—22no—
18#20263237—22no—
19#20263233—22no—
20C22×Q8×C197C2211yescharacteristic
21C2×C4⋊C4×C197C2211yescharacteristic
22C2×C42×C197C2211yescharacteristic
23#95×C197—22no—
24C2×Dic394—21no—
25C2×Dic394—21no—
26C2×Dic394—21no—
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394C2×C412yes—
30C22×Dic197—21no—
31C22×Dic197—21no—
32#15691348—28no—
33C2×#13096802—21no—
34C2×#13096802—21no—
35C2×#13096802C2×C412yes—
36C2×#13096802D412yes—
37C2×#13096802D412yes—
38C2×#13096802D412yes—
39C4⋊Dic197—21no—
40C4⋊Dic197—21no—
41C2×Q8×C197—21no—
42C2×Q8×C197—21no—
43C2×Q8×C197—21no—
44C2×Q8×C197—21no—
45C2×Q8×C197—21no—
46C2×Q8×C197C2×C412yes—
47C22×C4×C197—21no—
48C22×C4×C197—21no—
49C22×C4×C197D412yes—
50C22×C4×C197C2311yescharacteristic
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C42×C197—21no—
54C42×C197—21no—
55Dic394—22no—
56Dic394—22no—
57Dic394—41no—
58Dic394—41no—
59C2×Dic197—22no—
60C2×Dic197—41no—
61C2×Dic197—41no—
62C2×Dic197—41no—
63C2×Dic197—42no—
64#13096802—22no—
65#13096802C22⋊C414yes—
66C23×C197C2×D411yescharacteristic
67Q8×C197—22no—
68Q8×C197—22no—
69Q8×C197—41no—
70Q8×C197—41no—
71C2×C4×C197—22no—
72C2×C4×C197—22no—
73C2×C4×C197—22no—
74C2×C4×C197—22no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—42no—
79C2×C4×C197C22×C411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197C2×D411yescharacteristic
83C2×C4×C197#4111yescharacteristic
84C2×C4×C197#4111yescharacteristic
85C22×C197C4×D411yescharacteristic
86C22×C197#12111yescharacteristic
87C22×C197#12711yescharacteristic
88C22×C197#13712yes—
89C22×C197#13712yes—
90C4×C197—42no—
91C4×C197—42no—
92C4×C197—81no—
93C4×C197C2×C22⋊C411yescharacteristic
94C4×C197C4×D411yescharacteristic
95C4×C197#12111yescharacteristic
96C4×C197#12411yescharacteristic
97Dic197—42no—
98Dic197—81no—
99C2×C197#39011yescharacteristic
100C2×C197#44112yes—
101C2×C197#47012yes—
102C2×C197#49711yescharacteristic
103C2×C197#49711yescharacteristic
104C197#197111yescharacteristic, complemented, Sylow
105#1971—1971nomaximal, Sylow
106C22×Q16—1971no—
107C2×#106—1971no—
108C2×#103—1971no—
109C2×#103—1971no—
110C2×#97—1971no—
111#391—1971no—
112#391D19711yescharacteristic
113C22×Q8—1971no—
114C22×Q8C2×D19711yescharacteristic
115C2×Q16—3942no—
116C2×Q16—3942no—
117C2×Q16—1974no—
118C22×C8—1971no—
119C22×C8—1971no—
120C2×C4⋊C4—1971no—
121C2×C4⋊C4C2×D19711yescharacteristic
122C2×C42C2×D19711yescharacteristic
123#106—3942no—
124#103—3942no—
125#103—3942no—
126#97—3942no—
127#95—22no—
128#95—3942no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—3941no—
135C2×Q8—3941no—
136C2×Q8—3941no—
137C2×Q8—3941no—
138C2×Q8—3941no—
139C2×Q8—1972no—
140C2×Q8C4×D19712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—3941no—
144C22×C4—3941no—
145C22×C4C22⋊D19712yes—
146C22×C4C22×D19711yescharacteristic
147Q16—3948no—
148C2×C8—3941no—
149C2×C8—3941no—
150C2×C8—1972no—
151C2×C8—1972no—
152C2×C8—1972no—
153C2×C8—1972no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—3941no—
157C4⋊C4—3941no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D19711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—7881no—
166Q8—7881no—
167Q8—3942no—
168Q8—3942no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—7881no—
178C2×C4—7881no—
179C2×C4—7881no—
180C2×C4—7882no—
181C2×C4—3942no—
182C2×C4C2×C4×D19711yescharacteristic
183C2×C4D4×D19711yescharacteristic
184C2×C4D4×D19711yescharacteristic
185C2×C4D4×D19711yescharacteristic
186C2×C4#1569135711yescharacteristic
187C2×C4#1569135811yescharacteristic
188C8—3942no—
189C8—1974no—
190C22C4×C22⋊D19711yescharacteristic
191C22#2026329811yescharacteristic
192C22#2026332211yescharacteristic
193C22#2026335612yes—
194C22#2026335712yes—
195C4—42no—
196C4—42no—
197C4—81no—
198C4—7882no—
199C4—15761no—
200C4C22⋊C4×D19711yescharacteristic
201C4#2026328611yescharacteristic
202C4#2026329911yescharacteristic
203C4#2026331211yescharacteristic
204C2#2716097911yescharacteristic
205C2#2716118412yes—
206C2#2716133512yes—
207C2#2716147111yescharacteristic
208C2#2716147411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#391×C197C2
socle, prime radicalC23×C197C2×D4
cosocleC22×C4×C197C23
hypercenter#391D197
hypocenterC197#1971
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#15691348C4semidirect
104. C197#1971semidirect

Maximal subgroups

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

subgroupindex
C22×#156913482
C2×#202632572
C2×#202632502
C2×#202632492
C2×#202632372
#271609852
#391×C1972
#1971197