Finite Group Databasebeta

← #38769991

Subgroups — #38769991

Order 25472.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20263955C211yesmaximal, characteristic
2C2×#20263879C212yesmaximal
3#27163780C212yesmaximal
4#27163773C211yesmaximal, characteristic
5#391×C199C211yesmaximal, characteristic
6C22×Dic398—21no—
7C22×C199⋊C8C2212yes—
8#20263955—24no—
9C2×C4⋊Dic199—21no—
10C2×C4⋊Dic199C2212yes—
11#20263879—24no—
12C22×Q8×C199C2211yescharacteristic
13C2×C4⋊C4×C199C2211yescharacteristic
14C2×C42×C199C2211yescharacteristic
15#95×C199—21no—
16#95×C199—21no—
17C2×Dic398—24no—
18C2×Dic398—42no—
19C2×Dic398—42no—
20C22×Dic199—22no—
21C2×C199⋊C8—22no—
22C2×C199⋊C8—22no—
23C2×C199⋊C8—22no—
24C4⋊Dic199—24no—
25C4⋊Dic199—22no—
26C4⋊Dic199D414yes—
27C2×Q8×C199—21no—
28C2×Q8×C199—22no—
29C2×Q8×C199—22no—
30C2×Q8×C199Q812yescomplemented
31C22×C4×C199—21no—
32C22×C4×C199—21no—
33C22×C4×C199D411yescharacteristic
34C22×C4×C199D411yescharacteristic
35C22×C4×C199C2311yescharacteristic
36C4⋊C4×C199—22no—
37C42×C199—22no—
38Dic398—48no—
39C2×Dic199—24no—
40C2×Dic199—42no—
41C2×Dic199—42no—
42C199⋊C8—42no—
43C23×C199C2×D411yescharacteristic
44Q8×C199—24no—
45Q8×C199—42no—
46C2×C4×C199—21no—
47C2×C4×C199—21no—
48C2×C4×C199—21no—
49C2×C4×C199—21no—
50C2×C4×C199—22no—
51C2×C4×C199—41no—
52C2×C4×C199—41no—
53C2×C4×C199—41no—
54C2×C4×C199—41no—
55C2×C4×C199—41no—
56C2×C4×C199SD1612yes—
57C2×C4×C199Q1612yes—
58C2×C4×C199C2×D412yes—
59C2×C4×C199C2×Q811yescharacteristic
60C2×C4×C199#4112yes—
61C2×C4×C199#4111yescharacteristic
62C22×C199#12011yescharacteristic
63C22×C199C22⋊Q811yescharacteristic
64C22×C199#12411yescharacteristic
65C22×C199C2×SD1611yescharacteristic
66C22×C199C2×Q1611yescharacteristic
67C22×C199AGL(1,ℤ/8)11yescharacteristic
68C22×C199#13711yescharacteristic
69C4×C199—22no—
70C4×C199—22no—
71C4×C199—42no—
72C4×C199—81no—
73C4×C199C22⋊Q812yes—
74C4×C199#12412yes—
75Dic199—44no—
76C2×C199#39311yescharacteristic
77C2×C199#45011yescharacteristic
78C2×C199#45111yescharacteristic
79C2×C199#47511yescharacteristic
80C2×C199#47711yescharacteristic
81C2×C199#48711yescharacteristic
82C2×C199#48911yescharacteristic
83C199#202411yescharacteristic, complemented, Sylow
84#2024—1991nomaximal, Sylow
85C2×C4⋊Q8—1991no—
86C2×#103—1992no—
87#391D19911yescharacteristic
88#340—1992no—
89#336—1991no—
90C22×Q8—3981no—
91C22×Q8C2×D19911yescharacteristic
92C22×C8—1992no—
93C4⋊Q8—3984no—
94C2×C4⋊C4—3981no—
95C2×C4⋊C4—1992no—
96C2×C4⋊C4C2×D19911yescharacteristic
97C2×C42C2×D19911yescharacteristic
98#103—3984no—
99#95—21no—
100#95—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—7962no—
105C2×Q8—7962no—
106C2×Q8—3984no—
107C2×Q8Dic39812yescomplemented
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—3982no—
111C22×C4C22⋊D19911yescharacteristic
112C22×C4C22⋊D19911yescharacteristic
113C22×C4C22×D19911yescharacteristic
114C2×C8—3982no—
115C2×C8—3982no—
116C2×C8—3982no—
117C4⋊C4—22no—
118C4⋊C4—3984no—
119C4⋊C4—3982no—
120C4⋊C4—1994no—
121C42—22no—
122C23C2×C22⋊D19911yescharacteristic
123Q8—24no—
124Q8—42no—
125Q8—7968no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—7962no—
137C2×C4—7962no—
138C2×C4—3984no—
139C2×C4#1569187312yes—
140C2×C4#1569187612yes—
141C2×C4D4×D19912yes—
142C2×C4C2×Dic39811yescharacteristic
143C2×C4#1569188512yes—
144C2×C4#1569188611yescharacteristic
145C8—7962no—
146C22#2026392311yescharacteristic
147C22#2026393411yescharacteristic
148C22#2026394211yescharacteristic
149C22C2×#1569187311yescharacteristic
150C22C2×#1569187611yescharacteristic
151C22#2026398011yescharacteristic
152C22#2026398511yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—81no—
157C4—7964no—
158C4#2026393212yes—
159C4#2026394112yes—
160C2#2716393011yescharacteristic
161C2#2716416811yescharacteristic
162C2#2716417311yescharacteristic
163C2#2716430311yescharacteristic
164C2#2716431311yescharacteristic
165C2#2716436811yescharacteristic
166C2#2716437711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D199
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#391×C199C2
socle, prime radicalC23×C199C2×D4
hypercenter#391D199
hypocenterC199#2024
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C199Q8semidirect
83. C199#2024semidirect
107. C2×Q8Dic398semidirect

Maximal subgroups

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

subgroupindex
C2×#202639552
C2×#202638792
#271637802
#271637732
#391×C1992
#2024199