Finite Group Databasebeta

← #38770076

Subgroups — #38770076

Order 25472.

290 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 71 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×#20263882C212yesmaximal, complemented
2#27164019C212yesmaximal, complemented
3#27163883C211yesmaximal, characteristic, complemented
4#27163787C211yesmaximal, characteristic, complemented
5#537×C199C211yesmaximal, characteristic
6C2×#15691870C2212yes—
7C2×C4⋊Dic199C2211yescharacteristic
8C2×C4×Dic199C2211yescharacteristic
9#20263882—24no—
10#20263879—22no—
11#20263876—22no—
12C199⋊#95—22no—
13C2×#41×C199C2212yes—
14C22⋊Q8×C199—22no—
15#121×C199—22no—
16#121×C199—22no—
17C2×C4⋊C4×C199C2211yescharacteristic
18C22×Dic199—21no—
19C22×Dic199—21no—
20#15691870—22no—
21#15691870—22no—
22C2×C199⋊C8—22no—
23C4⋊Dic199—22no—
24C4×Dic199—21no—
25C4×Dic199D412yescomplemented
26#41×C199—22no—
27#41×C199—22no—
28#41×C199—22no—
29#41×C199—22no—
30C2×Q8×C199D412yes—
31C2×D4×C199—22no—
32C2×D4×C199—22no—
33C2×D4×C199D412yes—
34C22×C4×C199—22no—
35C22×C4×C199D412yes—
36C22×C4×C199C2311yescharacteristic
37C4⋊C4×C199—22no—
38C4⋊C4×C199—22no—
39C22⋊C4×C199—24no—
40C2×Dic199—22no—
41C2×Dic199—41no—
42C2×Dic199—41no—
43C2×Dic199—42no—
44C2×Dic199—41no—
45C199⋊C8—42no—
46C23×C199—22no—
47C23×C199C2×D411yescharacteristic
48Q8×C199—22no—
49Q8×C199—22no—
50D4×C199—22no—
51D4×C199—22no—
52D4×C199—42no—
53D4×C199—42no—
54D4×C199—82no—
55C2×C4×C199—22no—
56C2×C4×C199—22no—
57C2×C4×C199—42no—
58C2×C4×C199—42no—
59C2×C4×C199—42no—
60C2×C4×C199C2×D411yescharacteristic
61C2×C4×C199C2×D412yes—
62C2×C4×C199#4112yes—
63C2×C4×C199#4111yescharacteristic
64C22×C199—21no—
65C22×C199—21no—
66C22×C199—42no—
67C22×C199—42no—
68C22×C199—42no—
69C22×C199#12011yescharacteristic
70C22×C199#12111yescharacteristic
71C22×C199#12411yescharacteristic
72C4×C199—42no—
73C4×C199—42no—
74C4×C199#12112yes—
75C4×C199#12312yes—
76Dic199—42no—
77Dic199—81no—
78C2×C199—21no—
79C2×C199—21no—
80C2×C199—82no—
81C2×C199#39411yescharacteristic
82C2×C199#45411yescharacteristic
83C2×C199#45511yescharacteristic
84C199#204011yescharacteristic, complemented, Sylow
85#2040—1991nomaximal, Sylow
86#537D19911yescharacteristic
87C2×#104—1992no—
88#417—1992no—
89#382—1991no—
90#343—1991no—
91C2×#41C2×D19912yes—
92C2×M16—1992no—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
96C2×C4⋊C4—1991no—
97C2×C4⋊C4C2×D19911yescharacteristic
98C2×C42—1991no—
99#104—3984no—
100#103—3982no—
101#102—3982no—
102#95—3982no—
103#41—22no—
104#41—22no—
105#41—22no—
106#41—22no—
107C2×Q8C22⋊D19912yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4C22⋊D19912yes—
111C22×C4—22no—
112C22×C4—3981no—
113C22×C4—3981no—
114C22×C4C22⋊D19912yes—
115C22×C4C22×D19911yescharacteristic
116M16—3982no—
117M16—3982no—
118C2×C8—3982no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—3982no—
122C22⋊C4—24no—
123C42—3981no—
124C42—1992no—
125C23—22no—
126C23C2×C22⋊D19911yescharacteristic
127Q8—22no—
128Q8—22no—
129D4—22no—
130D4—22no—
131D4—42no—
132D4—42no—
133D4—82no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—7961no—
140C2×C4—7961no—
141C2×C4—7962no—
142C2×C4—7961no—
143C2×C4—3982no—
144C2×C4D4×D19911yescharacteristic
145C2×C4C2×C22⋊D19912yes—
146C2×C4#1569188511yescharacteristic
147C2×C4#1569188512yes—
148C8—7962no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#2026392411yescharacteristic
155C22#2026392611yescharacteristic
156C22#2026394211yescharacteristic
157C4—42no—
158C4—42no—
159C4—7962no—
160C4—15921no—
161C4#2026392712yes—
162C4#2026393612yes—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2716393211yescharacteristic
167C2#2716419111yescharacteristic
168C2#2716419611yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting, prime radical#537×C199C2
socleC22×C199#120
hypercenter#537D199
hypocenterC199#2040
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#20263882C2semidirect
2. #27164019C2semidirect
3. #27163883C2semidirect
4. #27163787C2semidirect
25. C4×Dic199D4semidirect
84. C199#2040semidirect

Maximal subgroups

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

subgroupindex
C2×#202638822
#271640192
#271638832
#271637872
#537×C1992
#2040199