Finite Group Databasebeta

← #38744398

Subgroups — #38744398

Order 25216.

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

normal subgroups only

This group has 230 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#27161649C211yesmaximal, characteristic
2#101×D197C212yesmaximal, complemented
3#27160877C212yesmaximal, complemented
4#27160876C211yesmaximal, characteristic, complemented
5#355×C197C211yesmaximal, characteristic, complemented
6C2×#15691357C411yescharacteristic
7C2×Q8×D197C2211yescharacteristic
8M16×D197—22no—
9#20263326C411yescharacteristic
10#20263324C2211yescharacteristic
11#20263313C411yescharacteristic
12#20263308—22no—
13#20263303—22no—
14#20263283C411yescharacteristic
15#20263245C2212yes—
16#20263244—22no—
17#124×C197C2211yescharacteristic
18#101×C197C2212yes—
19#15691357—42no—
20C2×Dic394C2×C411yescharacteristic
21C2×Dic394C2×C411yescharacteristic
22C2×Dic394D412yes—
23Q8×D197—42no—
24C2×C22⋊D197—21no—
25C22×Dic197—21no—
26C2×C4×D197C2×C411yescharacteristic
27C2×C4×D197D412yes—
28#15691342—22no—
29#15691341—42no—
30C8×D197—42no—
31C4⋊Dic197—22no—
32#15691337—21no—
33#15691337—22no—
34C22⋊Dic197—22no—
35#15691335—21no—
36C4×Dic197C2×C411yescharacteristic
37C2×Q8×C197C2311yescharacteristic
38C2×D4×C197C2×C411yescharacteristic
39M16×C197—22no—
40C22⋊C4×C197—22no—
41C42×C197C2×C411yescharacteristic
42C22×D197C22⋊C411yescharacteristic
43Dic394—41no—
44Dic394—42no—
45Dic394—41no—
46C22⋊D197—42no—
47C2×Dic197—21no—
48C2×Dic197—21no—
49C2×Dic197—42no—
50C2×Dic197C22⋊C412yes—
51C2×Dic197C22⋊C411yescharacteristic
52C4×D197—22no—
53C4×D197—22no—
54C4×D197—22no—
55#13096802—42no—
56C23×C197—21no—
57Q8×C197—41no—
58D4×C197—41no—
59C2×C4×C197—21no—
60C2×C4×C197C22×C411yescharacteristic
61C2×C4×C197C2×D412yes—
62C8×C197—42no—
63C22×C197—42no—
64C22×C197—41no—
65C22×C197C2×C22⋊C411yescharacteristic
66C2×D197—41no—
67C2×D197#9912yes—
68C4×C197—22no—
69C4×C197—21no—
70C4×C197—41no—
71Dic197—21no—
72Dic197—22no—
73Dic197—41no—
74Dic197—41no—
75C2×C197—21no—
76C2×C197—81no—
77C2×C197C2×#9911yescharacteristic
78D197—22no—
79C197#213311yescharacteristic, complemented, Sylow
80#2133—1971nomaximal, Sylow
81#536—1971no—
82C2×#101—1972no—
83#356—1972no—
84#355—1971no—
85#355D19711yescharacteristic, complemented
86C2×#41—1971no—
87C22×Q8—1971no—
88C2×M16—3942no—
89C4⋊Q8—1971no—
90C4⋊Q8—1971no—
91#124—1971no—
92#124C2×D19711yescharacteristic
93#123—3942no—
94C22⋊Q8—3942no—
95#117—1971no—
96#101—3942no—
97#101—1972no—
98#101C2×D19712yes—
99#41—7882no—
100C2×Q8—7882no—
101C2×Q8—1971no—
102C2×Q8—1971no—
103C2×Q8—1972no—
104C2×Q8C22×D19711yescharacteristic
105C2×D4—3941no—
106C2×D4C4×D19711yescharacteristic
107C22×C4—3941no—
108C22×C4—1971no—
109C22×C4—1972no—
110M16—22no—
111M16—7882no—
112M16—3942no—
113C2×C8—7882no—
114C4⋊C4—3941no—
115C4⋊C4—3942no—
116C4⋊C4—3942no—
117C22⋊C4—22no—
118C22⋊C4—3941no—
119C22⋊C4—3942no—
120C42—1971no—
121C42C4×D19711yescharacteristic
122C23—21no—
123C23—1971no—
124Q8—41no—
125Q8—7881no—
126Q8—7882no—
127Q8—7881no—
128D4—41no—
129D4—7882no—
130C2×C4—21no—
131C2×C4—7882no—
132C2×C4—3942no—
133C2×C4—3942no—
134C2×C4—3942no—
135C2×C4—3941no—
136C2×C4—3941no—
137C2×C4—1972no—
138C2×C4—1971no—
139C2×C4C2×C4×D19711yescharacteristic
140C2×C4D4×D19712yes—
141C8—42no—
142C8—7882no—
143C22—42no—
144C22—41no—
145C22—7881no—
146C22—1972no—
147C22C22⋊C4×D19711yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—7881no—
152C4—7881no—
153C4—3941no—
154C4—3942no—
155C2—21no—
156C2—81no—
157C2—3942no—
158C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×Q8C22×D197
Fitting#355×C197C2
socleC2×C197C2×#99
prime radicalC2×#15691357C4
cosocleC2×Q8×C197C23
hypercenter#355D197
hypocenterC197#2133
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. #101×D197C2semidirect
3. #27160877C2semidirect
4. #27160876C2semidirect
5. #355×C197C2semidirect
79. C197#2133semidirect
85. #355D197semidirect

Maximal subgroups

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

subgroupindex
#271616492
#101×D1972
#271608772
#271608762
#355×C1972
#2133197