Finite Group Databasebeta

← #38741739

Subgroups — #38741739

Order 25216.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#27161415C212yesmaximal, complemented
3#27161015C211yesmaximal, characteristic
4#27160817C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic
6#20263305—21no—
7#20263305C2212yes—
8#20263303—22no—
9#20263303—22no—
10C2×C4⋊Dic197C2211yescharacteristic
11C2×#15691337—21no—
12C2×C22⋊Dic197—21no—
13C2×C22⋊Dic197—21no—
14#20263260—22no—
15#20263251—22no—
16#20263233C2211yescharacteristic
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C2×Dic394—21no—
20C2×Dic394D412yes—
21C22×Dic197—21no—
22C22×Dic197—21no—
23C22×Dic197—21no—
24C22×Dic197—21no—
25C4⋊Dic197—21no—
26C4⋊Dic197—22no—
27C4⋊Dic197D412yes—
28#15691337—22no—
29#15691337—22no—
30#15691337—42no—
31C22⋊Dic197—22no—
32C22⋊Dic197—22no—
33C22⋊Dic197—42no—
34C24×C197D411yescharacteristic
35C22×C4×C197D411yescharacteristic
36C22×C4×C197C2311yescharacteristic
37C2×C8×C197—22no—
38C22⋊C4×C197—22no—
39Dic394—42no—
40Dic394—81no—
41C2×Dic197—22no—
42C2×Dic197—22no—
43C2×Dic197—22no—
44C2×Dic197—41no—
45C2×Dic197—41no—
46C2×Dic197—41no—
47C2×Dic197—41no—
48C2×Dic197—41no—
49C2×Dic197—81no—
50C2×Dic197—81no—
51C23×C197—21no—
52C23×C197—21no—
53C23×C197—41no—
54C23×C197—41no—
55C23×C197Q1612yes—
56C23×C197C2×D411yescharacteristic
57C2×C4×C197—21no—
58C2×C4×C197—41no—
59C2×C4×C197—41no—
60C2×C4×C197C2×D412yes—
61C8×C197—42no—
62C22×C197—21no—
63C22×C197—21no—
64C22×C197—22no—
65C22×C197—42no—
66C22×C197—41no—
67C22×C197—41no—
68C22×C197—81no—
69C22×C197#12011yescharacteristic
70C22×C197C2×Q1611yescharacteristic
71C22×C197#13711yescharacteristic
72C4×C197—22no—
73C4×C197—41no—
74Dic197—42no—
75Dic197—42no—
76Dic197—42no—
77Dic197—81no—
78C2×C197—21no—
79C2×C197—21no—
80C2×C197—42no—
81C2×C197#45111yescharacteristic
82C2×C197#45311yescharacteristic
83C2×C197#45811yescharacteristic
84C197#161411yescharacteristic, complemented, Sylow
85#1614—1971nomaximal, Sylow
86#543—1971no—
87#484—1972no—
88#399—1971no—
89#328—1972no—
90#323D19711yescharacteristic
91C22⋊Q8—3942no—
92C22⋊Q8—3942no—
93C22⋊Q8—3941no—
94C22⋊Q8—1972no—
95C2×C4⋊C4—3941no—
96C2×C4⋊C4—1971no—
97C2×C22⋊C4—3941no—
98C2×C22⋊C4—3941no—
99C2×C22⋊C4C2×D19711yescharacteristic
100#107—3942no—
101#103—3942no—
102C22⋊C8C2×D19712yes—
103#95—1971no—
104C24D78811yescharacteristic
105C2×Q8—3941no—
106C2×Q8—1972no—
107C22×C4—3941no—
108C22×C4—3941no—
109C22×C4—3941no—
110C22×C4—3941no—
111C22×C4D78811yescharacteristic
112C22×C4C22×D19711yescharacteristic
113C2×C8—22no—
114C4⋊C4—7882no—
115C4⋊C4—3942no—
116C4⋊C4—3941no—
117C4⋊C4—3942no—
118C4⋊C4—3942no—
119C4⋊C4—1972no—
120C22⋊C4—22no—
121C22⋊C4—7882no—
122C22⋊C4—3942no—
123C22⋊C4—3942no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic78812yes—
129C23C2×D78811yescharacteristic
130Q8—7882no—
131Q8—15761no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—7881no—
136C2×C4—7881no—
137C2×C4—7881no—
138C2×C4—7881no—
139C2×C4—7881no—
140C2×C4—15761no—
141C2×C4—15761no—
142C2×C4—3942no—
143C2×C4—3942no—
144C2×C4—3942no—
145C2×C4D4×D19712yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#2026329611yescharacteristic
155C22C2×Dic78811yescharacteristic
156C22#2026335911yescharacteristic
157C4—22no—
158C4—41no—
159C4—7882no—
160C4—7882no—
161C4—7882no—
162C4—15761no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2716123711yescharacteristic
167C2#2716124911yescharacteristic
168C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
prime radicalC24×C197D4
hypercenter#323D197
hypocenterC197#1614
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27161415C2semidirect
4. #27160817C2semidirect
84. C197#1614semidirect

Maximal subgroups

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

subgroupindex
#271617052
#271614152
#271610152
#271608172
#323×C1972
#1614197