Finite Group Databasebeta

← #38770075

Subgroups — #38770075

Order 25472.

306 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 67 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#27164599C211yesmaximal, characteristic
2C2×#20263881C212yesmaximal, complemented
3#27164020C212yesmaximal, complemented
4#27163887C211yesmaximal, characteristic, complemented
5#343×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2212yes—
7#20263935—22no—
8#20263930—22no—
9#20263929—22no—
10C2×C4⋊Dic199C2211yescharacteristic
11C2×Dic199⋊C4—22no—
12C2×C4×Dic199C2211yescharacteristic
13#20263881—24no—
14#20263880—22no—
15#20263877—22no—
16C2×M16×C199C2212yes—
17C2×C4⋊C4×C199C2211yescharacteristic
18#15691886—22no—
19#15691886—22no—
20#15691886—22no—
21#15691886—22no—
22C2×Dic398D412yes—
23C2×D796—22no—
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26C22×Dic199—21no—
27C22×Dic199—21no—
28C2×C4×D199—22no—
29C4⋊Dic199—22no—
30C4⋊Dic199—22no—
31Dic199⋊C4—44no—
32(C2×D199)⋊C4—24no—
33C4×Dic199—21no—
34C4×Dic199D412yescomplemented
35C22×C4×C199D412yes—
36C22×C4×C199C2311yescharacteristic
37M16×C199—22no—
38M16×C199—22no—
39C2×C8×C199—22no—
40C4⋊C4×C199—22no—
41C22×D199—22no—
42Dic398—22no—
43Dic398—22no—
44D796—22no—
45D796—22no—
46D796—82no—
47C22⋊D199—42no—
48C22⋊D199—42no—
49C2×Dic199—22no—
50C2×Dic199—22no—
51C2×Dic199—22no—
52C2×Dic199—22no—
53C2×Dic199—42no—
54C2×Dic199—41no—
55C4×D199—42no—
56C4×D199—42no—
57C23×C199C2×D411yescharacteristic
58C2×C4×C199—22no—
59C2×C4×C199—42no—
60C2×C4×C199C2×D411yescharacteristic
61C2×C4×C199C2×D412yes—
62C2×C4×C199#4112yes—
63C2×C4×C199#4111yescharacteristic
64C8×C199—42no—
65C22×C199—21no—
66C22×C199—21no—
67C22×C199#12011yescharacteristic
68C22×C199#12111yescharacteristic
69C22×C199#12411yescharacteristic
70C2×D199—42no—
71C2×D199—42no—
72C2×D199—42no—
73C4×C199—42no—
74C4×C199#12112yes—
75C4×C199#12312yes—
76Dic199—24no—
77Dic199—42no—
78Dic199—81no—
79C2×C199—21no—
80C2×C199—21no—
81C2×C199#39411yescharacteristic
82C2×C199#45311yescharacteristic
83C2×C199#45611yescharacteristic
84D199—82no—
85C199#203911yescharacteristic, complemented, Sylow
86#2039—1991nomaximal, Sylow
87#537—1991no—
88C2×#104—1992no—
89#417—1992no—
90#384—1991no—
91#343D19911yescharacteristic, complemented
92C2×#41—1992no—
93C2×M16C2×D19912yes—
94C22⋊Q8—3982no—
95#121—3982no—
96#121—3982no—
97C2×C4⋊C4—3982no—
98C2×C4⋊C4—1991no—
99C2×C4⋊C4C2×D19911yescharacteristic
100C2×C42—1991no—
101#104—3984no—
102#103—3982no—
103#102—3982no—
104#41—3982no—
105#41—3982no—
106#41—3982no—
107#41—3982no—
108C2×Q8—1992no—
109C2×D4—3982no—
110C2×D4—3982no—
111C2×D4—1992no—
112C22×C4—3981no—
113C22×C4—3982no—
114C22×C4—3981no—
115C22×C4D79612yes—
116C22×C4C22×D19911yescharacteristic
117M16—22no—
118M16—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—7964no—
122C4⋊C4—3982no—
123C4⋊C4—3982no—
124C22⋊C4—3984no—
125C42—3981no—
126C42—1992no—
127C23—3982no—
128C23C2×D79611yescharacteristic
129Q8—3982no—
130Q8—3982no—
131D4—7962no—
132D4—7962no—
133D4—15922no—
134D4—3982no—
135D4—3982no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—7962no—
139C2×C4—7962no—
140C2×C4—7962no—
141C2×C4—7961no—
142C2×C4—3982no—
143C2×C4—3982no—
144C2×C4—3982no—
145C2×C4—3982no—
146C2×C4D4×D19912yes—
147C2×C4D4×D19911yescharacteristic
148C2×C4#1569188612yes—
149C2×C4#1569188711yescharacteristic
150C8—42no—
151C22—21no—
152C22—21no—
153C22—7962no—
154C22—7962no—
155C22—7962no—
156C22#2026392511yescharacteristic
157C22#2026393011yescharacteristic
158C22#2026394311yescharacteristic
159C4—42no—
160C4—7962no—
161C4—15921no—
162C4—3984no—
163C4#2026392812yes—
164C4#2026394012yes—
165C2—21no—
166C2—21no—
167C2—15922no—
168C2#2716393611yescharacteristic
169C2#2716418811yescharacteristic
170C2#2716420311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#343×C199C2
socleC22×C199#120
prime radical#27164599C2
hypercenter#343D199
hypocenterC199#2039
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#20263881C2semidirect
3. #27164020C2semidirect
4. #27163887C2semidirect
5. #343×C199C2semidirect
34. C4×Dic199D4semidirect
85. C199#2039semidirect
91. #343D199semidirect

Maximal subgroups

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

subgroupindex
#271645992
C2×#202638812
#271640202
#271638872
#343×C1992
#2039199