Finite Group Databasebeta

← #38769388

Subgroups — #38769388

Order 25472.

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

normal subgroups only

This group has 314 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
1#27164496C211yesmaximal, characteristic, complemented
2#27164035C212yesmaximal
3C2×#20263874C211yesmaximal, characteristic, complemented
4#27163774C212yesmaximal, complemented
5#536×C199C211yesmaximal, characteristic
6C2×#15691870C2212yes—
7Q8×Dic199—24no—
8#20263914—22no—
9#20263914C2212yes—
10#20263882—24no—
11#20263874—22no—
12C2×#41×C199C2211yescharacteristic
13C22×Q8×C199C2211yescharacteristic
14C4⋊Q8×C199C412yescomplemented
15#124×C199C412yescomplemented
16#123×C199—22no—
17C22⋊Q8×C199—22no—
18#117×C199C2211yescharacteristic
19#15691870—22no—
20#15691870—22no—
21C2×C199⋊C8—22no—
22C4⋊Dic199—24no—
23C4⋊Dic199—22no—
24C22⋊Dic199—22no—
25C4×Dic199—24no—
26C4×Dic199D414yes—
27#41×C199—22no—
28#41×C199—22no—
29C2×Q8×C199—22no—
30C2×Q8×C199—22no—
31C2×Q8×C199C2×C411yescharacteristic
32C2×Q8×C199C2×C412yes—
33C2×Q8×C199D412yes—
34C2×D4×C199—21no—
35C2×D4×C199C2×C411yescharacteristic
36C22×C4×C199—21no—
37C22×C4×C199D412yes—
38C22×C4×C199C2311yescharacteristic
39C4⋊C4×C199—24no—
40C4⋊C4×C199—21no—
41C22⋊C4×C199—21no—
42C22⋊C4×C199—24no—
43C42×C199C2×C412yes—
44C2×Dic199—24no—
45C199⋊C8—42no—
46C23×C199—21no—
47C23×C199C2×D411yescharacteristic
48Q8×C199—24no—
49Q8×C199—24no—
50Q8×C199—22no—
51D4×C199—22no—
52D4×C199—41no—
53D4×C199—41no—
54C2×C4×C199—22no—
55C2×C4×C199—22no—
56C2×C4×C199—22no—
57C2×C4×C199—21no—
58C2×C4×C199—42no—
59C2×C4×C199C22⋊C414yes—
60C2×C4×C199C22×C411yescharacteristic
61C2×C4×C199C2×D412yes—
62C2×C4×C199C2×D411yescharacteristic
63C2×C4×C199#4112yes—
64C22×C199—41no—
65C22×C199—42no—
66C22×C199—41no—
67C22×C199C2×C22⋊C411yescharacteristic
68C22×C199#12011yescharacteristic
69C22×C199#12311yescharacteristic
70C4×C199—24no—
71C4×C199—42no—
72C4×C199—41no—
73C4×C199C4×D412yes—
74C4×C199#12112yes—
75Dic199—44no—
76C2×C199—21no—
77C2×C199—21no—
78C2×C199—21no—
79C2×C199—81no—
80C2×C199#38611yescharacteristic
81C199#191211yescharacteristic, complemented, Sylow
82#1912—1991nomaximal, Sylow
83#536D19911yescharacteristic
84#519—1991no—
85#421—1992no—
86C2×#101—1991no—
87H(3,ℤ/4)—1992no—
88C2×#41C2×D19911yescharacteristic
89C22×Q8C2×D19911yescharacteristic
90C2×M16—1992no—
91C4⋊Q8Dic19912yescomplemented
92#124Dic19912yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—3984no—
96#117—3982no—
97#117—1992no—
98#117C2×D19911yescharacteristic
99#104—3984no—
100#101—3982no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic19911yescharacteristic
106C2×Q8C2×Dic19912yes—
107C2×Q8C22⋊D19912yes—
108C2×D4—21no—
109C2×D4C2×Dic19911yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D19912yes—
112C22×C4C22×D19911yescharacteristic
113M16—3982no—
114M16—3982no—
115C2×C8—3982no—
116C4⋊C4—24no—
117C4⋊C4—21no—
118C4⋊C4—3984no—
119C4⋊C4—3982no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—3982no—
123C42—3984no—
124C42—1994no—
125C42C2×Dic19912yes—
126C23—21no—
127C23C2×C22⋊D19911yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—22no—
131D4—22no—
132D4—41no—
133D4—41no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3984no—
140C2×C4C22⋊Dic19914yes—
141C2×C4C22×Dic19911yescharacteristic
142C2×C4D4×D19912yes—
143C2×C4C2×C22⋊D19911yescharacteristic
144C2×C4#1569188512yes—
145C8—7962no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊Dic19911yescharacteristic
150C22#2026392311yescharacteristic
151C22#2026393611yescharacteristic
152C4—24no—
153C4—42no—
154C4—41no—
155C4—7964no—
156C4D4×Dic19912yes—
157C4#2026392712yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#2716389811yescharacteristic
subgroupG/H
centerC2#27163898
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#536×C199C2
socleC2×C199#386
prime radicalC2×#41×C199C22
cosocleC22×C4×C199C23
hypercenter#536D199
hypocenterC199#1912
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #27164496C2semidirect
3. C2×#20263874C2semidirect
4. #27163774C2semidirect
14. C4⋊Q8×C199C4semidirect
15. #124×C199C4semidirect
81. C199#1912semidirect
91. C4⋊Q8Dic199semidirect
92. #124Dic199semidirect

Maximal subgroups

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

subgroupindex
#271644962
#271640352
C2×#202638742
#271637742
#536×C1992
#1912199