Finite Group Databasebeta

← #38770011

Subgroups — #38770011

Order 25472.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#27164649C211yesmaximal, characteristic
2#27164027C212yesmaximal
3#27163787C211yesmaximal, characteristic
4#27163778C212yesmaximal
5#391×C199C211yesmaximal, characteristic
6C2×#15691870C2212yes—
7#20263954—22no—
8#20263946—22no—
9#20263934—22no—
10#20263914C2212yes—
11#20263879—22no—
12#20263879—22no—
13C22×Q8×C199C2211yescharacteristic
14C2×C4⋊C4×C199C2211yescharacteristic
15C2×C42×C199C2211yescharacteristic
16#95×C199—22no—
17C2×Dic398—22no—
18#15691870—22no—
19#15691870—22no—
20C2×C199⋊C8—22no—
21C4⋊Dic199—22no—
22C4⋊Dic199D412yes—
23C4⋊Dic199D412yes—
24Dic199⋊C4—24no—
25C22⋊Dic199—22no—
26C4×Dic199—22no—
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×C199D412yes—
34C22×C4×C199C2311yescharacteristic
35C4⋊C4×C199—22no—
36C42×C199—22no—
37Dic398—82no—
38C2×Dic199—22no—
39C2×Dic199—22no—
40C199⋊C8—42no—
41C23×C199C2×D411yescharacteristic
42Q8×C199—24no—
43Q8×C199—42no—
44C2×C4×C199—22no—
45C2×C4×C199—22no—
46C2×C4×C199—22no—
47C2×C4×C199—41no—
48C2×C4×C199—41no—
49C2×C4×C199—41no—
50C2×C4×C199—41no—
51C2×C4×C199—42no—
52C2×C4×C199C2×D412yes—
53C2×C4×C199C2×Q811yescharacteristic
54C2×C4×C199#4112yes—
55C2×C4×C199#4111yescharacteristic
56C22×C199—21no—
57C22×C199—21no—
58C22×C199#12011yescharacteristic
59C22×C199C22⋊Q811yescharacteristic
60C22×C199#12411yescharacteristic
61C4×C199—42no—
62C4×C199—42no—
63C4×C199—42no—
64C4×C199C22⋊Q812yes—
65C4×C199#12412yes—
66Dic199—42no—
67Dic199—42no—
68C2×C199—21no—
69C2×C199—21no—
70C2×C199#39311yescharacteristic
71C2×C199#45511yescharacteristic
72C2×C199#45611yescharacteristic
73C199#202811yescharacteristic, complemented, Sylow
74#2028—1991nomaximal, Sylow
75#544—1991no—
76#419—1992no—
77#391D19911yescharacteristic
78#343—1991no—
79#339—1992no—
80C22×Q8C2×D19911yescharacteristic
81C2×M16—1992no—
82C4⋊Q8—3982no—
83#125—3982no—
84C22⋊Q8—3982no—
85#117—1992no—
86C2×C4⋊C4C2×D19911yescharacteristic
87C2×C42C2×D19911yescharacteristic
88#103—3982no—
89#103—3982no—
90#95—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—3982no—
95C2×Q8Dic39812yescomplemented
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D19912yes—
99C22×C4C22×D19911yescharacteristic
100M16—3982no—
101M16—3982no—
102C2×C8—3982no—
103C4⋊C4—22no—
104C4⋊C4—3984no—
105C4⋊C4—3982no—
106C4⋊C4—1992no—
107C4⋊C4—1992no—
108C22⋊C4—3982no—
109C42—22no—
110C42—3982no—
111C23C2×C22⋊D19911yescharacteristic
112Q8—24no—
113Q8—42no—
114Q8—15922no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—3982no—
124C2×C4—3982no—
125C2×C4D4×D19912yes—
126C2×C4C2×Dic39811yescharacteristic
127C2×C4#1569188512yes—
128C2×C4#1569188611yescharacteristic
129C8—7962no—
130C22—21no—
131C22—21no—
132C22#2026392311yescharacteristic
133C22#2026393411yescharacteristic
134C22#2026394211yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—7962no—
139C4—7962no—
140C4#2026393212yes—
141C4#2026394112yes—
142C2—21no—
143C2—21no—
144C2#2716393011yescharacteristic
145C2#2716419511yescharacteristic
146C2#2716420011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#391×C199C2
socleC22×C199#120
prime radicalC23×C199C2×D4
hypercenter#391D199
hypocenterC199#2028
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C199Q8semidirect
73. C199#2028semidirect
95. C2×Q8Dic398semidirect

Maximal subgroups

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

subgroupindex
#271646492
#271640272
#271637872
#271637782
#391×C1992
#2028199