Finite Group Databasebeta

← #38743877

Subgroups — #38743877

Order 25216.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#27161705C211yesmaximal, characteristic, complemented
2#27160934C212yesmaximal
3#27160934C211yesmaximal, characteristic
4#27160845C212yesmaximal, complemented
5#342×C197C211yesmaximal, characteristic, complemented
6C24×D197C2211yescharacteristic
7#20263305—22no—
8#20263303—24no—
9C2×C22⋊Dic197C2211yescharacteristic
10C2×C22⋊Dic197C2212yes—
11C2×#15691335—24no—
12C2×#15691335—22no—
13C2×C22⋊C4×C197C2212yes—
14C2×C22⋊C4×C197C2211yescharacteristic
15C23×D197—22no—
16C23×D197—21no—
17C23×D197—21no—
18C23×D197—22no—
19C23×D197—44no—
20C23×D197Q812yescomplemented
21C2×Dic394—22no—
22C22×Dic197—21no—
23C22×Dic197D414yes—
24C4⋊Dic197—22no—
25#15691337—24no—
26C22⋊Dic197—22no—
27C22⋊Dic197—22no—
28C22⋊Dic197—22no—
29#15691335—44no—
30#15691335—48no—
31C24×C197C2311yescharacteristic
32C22×C4×C197—22no—
33C22×C4×C197D412yes—
34C22⋊C4×C197—22no—
35C22⋊C4×C197—22no—
36C22⋊C4×C197—22no—
37C22×D197—28no—
38C22×D197—24no—
39C22×D197—24no—
40C22×D197—22no—
41C22×D197—44no—
42C22×D197—44no—
43C22×D197—48no—
44C22×D197—44no—
45C22×D197—42no—
46C22×D197—41no—
47C22×D197—42no—
48C22×D197—41no—
49Dic394—82no—
50C2×Dic197—24no—
51C2×Dic197—44no—
52C2×Dic197—42no—
53C2×Dic197—41no—
54C23×C197—42no—
55C23×C197C2×D412yes—
56C23×C197C2×D411yescharacteristic
57C23×C197C2×Q811yescharacteristic
58C23×C197#4111yescharacteristic
59C23×C197#4112yes—
60C2×C4×C197—22no—
61C2×C4×C197—42no—
62C2×C4×C197—44no—
63C2×C4×C197—42no—
64C22×C197—24no—
65C22×C197—22no—
66C22×C197—41no—
67C22×C197—41no—
68C22×C197—42no—
69C22×C197#12011yescharacteristic
70C22×C197C22⋊Q812yes—
71C22×C197C22⋊Q811yescharacteristic
72C22×C197#12412yes—
73C22×C197#12411yescharacteristic
74C2×D197—28no—
75C2×D197—24no—
76C2×D197—44no—
77C2×D197—48no—
78C2×D197—44no—
79C2×D197—42no—
80C2×D197—42no—
81C2×D197—44no—
82C4×C197—42no—
83C4×C197—82no—
84Dic197—44no—
85Dic197—81no—
86C2×C197—21no—
87C2×C197—22no—
88C2×C197—22no—
89C2×C197—21no—
90C2×C197#39311yescharacteristic
91C2×C197B(4,𝔽2)12yes—
92D197—44no—
93C197#202911yescharacteristic, complemented, Sylow
94#2029—1971nomaximal, Sylow
95#543—1971no—
96#379—1972no—
97#379—1971no—
98#342—1972no—
99#342D19711yescharacteristic, complemented
100C25—1971no—
101C22⋊Q8—3944no—
102C22⋊Q8—3942no—
103C2×C22⋊C4—3944no—
104C2×C22⋊C4—3942no—
105C2×C22⋊C4—1971no—
106C2×C22⋊C4—1972no—
107C2×C22⋊C4C2×D19712yes—
108C2×C22⋊C4C2×D19711yescharacteristic
109C24—7884no—
110C24—3941no—
111C24—3942no—
112C24—3942no—
113C24—3941no—
114C24—1972no—
115C24C22×D19711yescharacteristic
116C2×Q8—3942no—
117C22×C4—22no—
118C22×C4—3941no—
119C22×C4—1974no—
120C22×C4D78812yes—
121C4⋊C4—3944no—
122C4⋊C4—3942no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—7884no—
127C22⋊C4—7888no—
128C22⋊C4—3942no—
129C22⋊C4—3942no—
130C22⋊C4—3942no—
131C23—42no—
132C23—7884no—
133C23—7882no—
134C23—7884no—
135C23—7888no—
136C23—7884no—
137C23—7881no—
138C23—7882no—
139C23—7881no—
140C23—3948no—
141C23—3944no—
142C23—3944no—
143C23—3942no—
144C23D4×D19712yes—
145C23C2×D78811yescharacteristic
146C23Q8×D19711yescharacteristic
147C23#1569135711yescharacteristic
148C23#1569135812yes—
149Q8—15762no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—7884no—
155C2×C4—7882no—
156C2×C4—7881no—
157C2×C4—3944no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—7888no—
164C22—7884no—
165C22—7884no—
166C22—7882no—
167C22—7882no—
168C22—7884no—
169C22—3948no—
170C22—3944no—
171C22#2026329611yescharacteristic
172C22#2026330412yes—
173C22#2026330611yescharacteristic
174C22#2026331212yes—
175C22#2026331411yescharacteristic
176C4—42no—
177C4—82no—
178C4—7884no—
179C4—15761no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—7884no—
185C2#2716099211yescharacteristic
186C2#2716126712yes—
subgroupG/H
centerC22#20263296
commutator, cosocleC24×C197C23
FrattiniC24C22×D197
Fitting#342×C197C2
socleC22×C197#120
prime radicalC24×D197C22
hypercenter#342D197
hypocenterC197#2029
LCSG ▷ C24×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #27161705C2semidirect
4. #27160845C2semidirect
5. #342×C197C2semidirect
20. C23×D197Q8semidirect
93. C197#2029semidirect
99. #342D197semidirect

Maximal subgroups

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

subgroupindex
#271617052
#271609342
#271608452
#342×C1972
#2029197