Finite Group Databasebeta

← #38742705

Subgroups — #38742705

Order 25216.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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×C4×Dic394C211yesmaximal, characteristic
2#27161103C211yesmaximal, characteristic
3C2×#20263244C212yesmaximal
4#27160847C212yesmaximal
5#423×C197C211yesmaximal, characteristic
6C22×Dic394C2211yescharacteristic
7C2×#15691342C2212yes—
8C4×Dic394—21no—
9C4×Dic394—21no—
10C4×Dic394—21no—
11C4×Dic394—21no—
12C2×C4⋊Dic197C2211yescharacteristic
13C2×#15691337C412yes—
14C2×C4×Dic197C412yes—
15C4⋊#13096802—21no—
16C4⋊#13096802—21no—
17#20263244—24no—
18C2×M16×C197C2212yes—
19C2×C42×C197C2211yescharacteristic
20C4⋊C8×C197—21no—
21C4⋊C8×C197—21no—
22C2×Dic394—21no—
23C2×Dic394—21no—
24C2×Dic394—21no—
25C2×Dic394—21no—
26C2×Dic394D412yes—
27C2×Dic394Q812yes—
28C22×Dic197C2×C412yes—
29C22×Dic197C2×C412yes—
30#15691342—22no—
31#15691342—22no—
32C2×#13096802—22no—
33C4⋊Dic197—21no—
34C4⋊Dic197—21no—
35#15691337—22no—
36#15691337—22no—
37C4×Dic197—22no—
38C4×Dic197—22no—
39C22×C4×C197C2×C411yescharacteristic
40C22×C4×C197C2×C411yescharacteristic
41C22×C4×C197C2311yescharacteristic
42M16×C197—22no—
43M16×C197—22no—
44C2×C8×C197—22no—
45C42×C197D411yescharacteristic
46C42×C197D411yescharacteristic
47C42×C197D411yescharacteristic
48C42×C197D411yescharacteristic
49Dic394—22no—
50Dic394—22no—
51Dic394—22no—
52Dic394—22no—
53C2×Dic197—22no—
54C2×Dic197—22no—
55C2×Dic197—22no—
56C2×Dic197—42no—
57C2×Dic197C4⋊C414yes—
58#13096802—42no—
59C23×C197C22×C411yescharacteristic
60C2×C4×C197—22no—
61C2×C4×C197—41no—
62C2×C4×C197C22⋊C411yescharacteristic
63C2×C4×C197C22⋊C411yescharacteristic
64C2×C4×C197C22⋊C411yescharacteristic
65C2×C4×C197C22⋊C411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C2×C4×C197C2×D411yescharacteristic
68C2×C4×C197C2×D411yescharacteristic
69C2×C4×C197C2×Q811yescharacteristic
70C2×C4×C197#4111yescharacteristic
71C2×C4×C197#4111yescharacteristic
72C8×C197—42no—
73C22×C197—21no—
74C22×C197—21no—
75C22×C197C2×C22⋊C411yescharacteristic
76C22×C197C2×C4⋊C411yescharacteristic
77C22×C197#11711yescharacteristic
78C4×C197—21no—
79C4×C197—21no—
80C4×C197—21no—
81C4×C197#10112yescomplemented
82C4×C197#12111yescharacteristic
83C4×C197#12111yescharacteristic
84C4×C197C22⋊Q811yescharacteristic
85C4×C197C22⋊Q811yescharacteristic
86Dic197—24no—
87Dic197—42no—
88C2×C197—21no—
89C2×C197—21no—
90C2×C197#38011yescharacteristic
91C2×C197C2×#10111yescharacteristic
92C2×C197#41311yescharacteristic
93C197#180111yescharacteristic, complemented, Sylow
94#1801—1971nomaximal, Sylow
95C2×C4×Q8—1971no—
96#423—1971no—
97#423D19711yescharacteristic
98C2×#101—1972no—
99#343—1972no—
100C22×Q8—1971no—
101C2×M16—1972no—
102C2×M16C2×D19712yes—
103C4×Q8—3941no—
104C4×Q8—3941no—
105C4×Q8—3941no—
106C4×Q8—3941no—
107C2×C4⋊C4—1972no—
108C2×C4⋊C4—1971no—
109C2×C42—1972no—
110C2×C42C2×D19711yescharacteristic
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—3941no—
114C4⋊C8—3941no—
115#101—3944no—
116C2×Q8—3941no—
117C2×Q8—3941no—
118C2×Q8—3941no—
119C2×Q8—3941no—
120C2×Q8—1972no—
121C2×Q8—1972no—
122C22×C4—1972no—
123C22×C4—1972no—
124C22×C4C4×D19711yescharacteristic
125C22×C4C4×D19711yescharacteristic
126C22×C4C22×D19711yescharacteristic
127M16—22no—
128M16—22no—
129M16—3942no—
130M16—3942no—
131C2×C8—22no—
132C2×C8—3942no—
133C4⋊C4—3942no—
134C4⋊C4—3942no—
135C4⋊C4—3941no—
136C4⋊C4—3941no—
137C42—3942no—
138C42—3942no—
139C42C22⋊D19711yescharacteristic
140C42C22⋊D19711yescharacteristic
141C42D78811yescharacteristic
142C42D78811yescharacteristic
143C23C2×C4×D19711yescharacteristic
144Q8—3942no—
145Q8—3942no—
146Q8—3942no—
147Q8—3942no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—7882no—
151C2×C4—3942no—
152C2×C4—3942no—
153C2×C4—3942no—
154C2×C4—1974no—
155C2×C4#1569133511yescharacteristic
156C2×C4#1569133511yescharacteristic
157C2×C4#1569133511yescharacteristic
158C2×C4#1569133511yescharacteristic
159C2×C4D4×D19711yescharacteristic
160C2×C4C2×C22⋊D19711yescharacteristic
161C2×C4C2×D78811yescharacteristic
162C2×C4Q8×D19711yescharacteristic
163C2×C4#1569135711yescharacteristic
164C2×C4#1569135911yescharacteristic
165C8—42no—
166C8—7882no—
167C22—21no—
168C22—21no—
169C22C2×#1569133511yescharacteristic
170C22C4⋊C4×D19711yescharacteristic
171C22#2026328411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—7882no—
176C4—3944no—
177C4#2026324412yescomplemented
178C4#2026329711yescharacteristic
179C4#2026330111yescharacteristic
180C4#2026330211yescharacteristic
181C4#2026330611yescharacteristic
182C2—21no—
183C2—21no—
184C2#2716093611yescharacteristic
185C2C2×#2026324411yescharacteristic
186C2#2716106111yescharacteristic
subgroupG/H
centerC22C2×#15691335
commutator, prime radicalC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#423×C197C2
socleC22×C197C2×C22⋊C4
cosocleC22×C4×C197C23
hypercenter#423D197
hypocenterC197#1801
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
81. C4×C197#101semidirect
93. C197#1801semidirect
177. C4#20263244semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic3942
#271611032
C2×#202632442
#271608472
#423×C1972
#1801197