Finite Group Databasebeta

← #38744402

Subgroups — #38744402

Order 25216.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161647C211yesmaximal, characteristic
2#27161062C212yesmaximal, complemented
3#27160876C211yesmaximal, characteristic, complemented
4#27160875C212yesmaximal, complemented
5#355×C197C211yesmaximal, characteristic, complemented
6C2×#15691359C2211yescharacteristic
7C2×D4×D197C411yescharacteristic
8M16×D197—22no—
9#20263323C411yescharacteristic
10#20263322C411yescharacteristic
11#20263315C2211yescharacteristic
12#20263299—22no—
13#20263296—22no—
14#20263283C411yescharacteristic
15#20263245C2212yescomplemented
16#20263242—22no—
17#124×C197C2211yescharacteristic
18#101×C197C2212yescomplemented
19C23×D197—21no—
20#15691359—42no—
21C2×D788—22no—
22C2×D788C2×C411yescharacteristic
23C2×D788C2×C411yescharacteristic
24C2×D788D412yes—
25C2×C22⋊D197—22no—
26C2×C22⋊D197—21no—
27D4×D197—42no—
28C2×C4×D197C2×C411yescharacteristic
29C2×C4×D197D412yes—
30#15691342—22no—
31#15691341—42no—
32C8×D197—42no—
33#15691337—21no—
34#15691335—21no—
35#15691335—22no—
36C4×Dic197C2×C411yescharacteristic
37C2×Q8×C197C2311yescharacteristic
38C2×D4×C197C2×C411yescharacteristic
39M16×C197—22no—
40C22⋊C4×C197—22no—
41C42×C197C2×C411yescharacteristic
42C22×D197—21no—
43C22×D197—42no—
44C22×D197—82no—
45C22×D197C22⋊C411yescharacteristic
46C22×D197C22⋊C412yes—
47D788—41no—
48D788—42no—
49D788—41no—
50D788—82no—
51C22⋊D197—42no—
52C22⋊D197—82no—
53C2×Dic197—21no—
54C2×Dic197C22⋊C411yescharacteristic
55C4×D197—22no—
56C4×D197—22no—
57C4×D197—22no—
58#13096802—42no—
59C23×C197—21no—
60Q8×C197—41no—
61D4×C197—41no—
62C2×C4×C197—21no—
63C2×C4×C197C22×C411yescharacteristic
64C2×C4×C197C2×D412yes—
65C8×C197—42no—
66C22×C197—42no—
67C22×C197—41no—
68C22×C197C2×C22⋊C411yescharacteristic
69C2×D197—21no—
70C2×D197—22no—
71C2×D197—42no—
72C2×D197—41no—
73C2×D197—42no—
74C2×D197—41no—
75C2×D197—84no—
76C4×C197—22no—
77C4×C197—21no—
78C4×C197—41no—
79Dic197—41no—
80Dic197#9912yes—
81C2×C197—21no—
82C2×C197—81no—
83C2×C197C2×#9911yescharacteristic
84D197—42no—
85D197—41no—
86D197—81no—
87C197#213411yescharacteristic, complemented, Sylow
88#2134—1971nomaximal, Sylow
89#535—1971no—
90#413—1972no—
91#355—1971no—
92#355—1972no—
93#355D19711yescharacteristic, complemented
94C2×#41—1971no—
95C22×D4—1971no—
96C2×M16—3942no—
97#127—1971no—
98#127—1971no—
99#124—1971no—
100#124C2×D19711yescharacteristic
101#121—3942no—
102#120—3942no—
103#117—1971no—
104#101—1972no—
105#101C2×D19712yescomplemented
106#100—3942no—
107C24—3941no—
108#41—7882no—
109C2×Q8C22×D19711yescharacteristic
110C2×D4—7882no—
111C2×D4—3942no—
112C2×D4—3942no—
113C2×D4—3941no—
114C2×D4—1971no—
115C2×D4—1971no—
116C2×D4—1972no—
117C2×D4C4×D19711yescharacteristic
118C22×C4—1971no—
119C22×C4—1972no—
120M16—22no—
121M16—7882no—
122M16—3942no—
123C2×C8—7882no—
124C4⋊C4—3941no—
125C22⋊C4—22no—
126C22⋊C4—3941no—
127C22⋊C4—3942no—
128C42—1971no—
129C42C4×D19711yescharacteristic
130C23—21no—
131C23—7882no—
132C23—15762no—
133C23—3941no—
134C23—1971no—
135C23—1972no—
136Q8—41no—
137D4—41no—
138D4—7881no—
139D4—7882no—
140D4—7881no—
141D4—7882no—
142D4—15762no—
143D4—15762no—
144C2×C4—21no—
145C2×C4—3942no—
146C2×C4—3942no—
147C2×C4—3942no—
148C2×C4—3941no—
149C2×C4—1971no—
150C2×C4C2×C4×D19711yescharacteristic
151C2×C4D4×D19712yes—
152C8—42no—
153C8—7882no—
154C22—42no—
155C22—41no—
156C22—7882no—
157C22—7881no—
158C22—7882no—
159C22—7881no—
160C22—15764no—
161C22—3941no—
162C22—3942no—
163C22C22⋊C4×D19711yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—7881no—
168C4—1972no—
169C2—21no—
170C2—81no—
171C2—7881no—
172C2—7882no—
173C2—15761no—
174C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×Q8C22×D197
Fitting#355×C197C2
socleC2×C197C2×#99
prime radical#27161647C2
cosocleC2×Q8×C197C23
hypercenter#355D197
hypocenterC197#2134
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #27161062C2semidirect
3. #27160876C2semidirect
4. #27160875C2semidirect
5. #355×C197C2semidirect
15. #20263245C22semidirect
18. #101×C197C22semidirect
87. C197#2134semidirect
93. #355D197semidirect
105. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616472
#271610622
#271608762
#271608752
#355×C1972
#2134197