Finite Group Databasebeta

← #38769233

Subgroups — #38769233

Order 25472.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#27164564C211yesmaximal, characteristic
2#27164034C212yesmaximal, complemented
3#27163992C211yesmaximal, characteristic, complemented
4#27163779C212yesmaximal, complemented
5#568×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2211yescharacteristic
7#20263953C411yescharacteristic, complemented
8#20263951C411yescharacteristic, complemented
9#20263944C411yescharacteristic, complemented
10#20263942C411yescharacteristic, complemented
11#20263927—22no—
12C4×C22⋊D199—22no—
13#20263914C2212yes—
14C2×C4×Dic199C2211yescharacteristic
15#20263881—24no—
16#20263870—22no—
17C2×#41×C199C2211yescharacteristic
18#131×C199—24no—
19C2×M16×C199—22no—
20C2×M16×C199C2212yes—
21#15691886—22no—
22#15691886—22no—
23C2×Dic398C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—22no—
26C2×C22⋊D199—21no—
27C22×Dic199—21no—
28C2×C4×D199—21no—
29C4⋊Dic199—22no—
30Dic199⋊C4—22no—
31C22⋊Dic199—22no—
32C22⋊Dic199—22no—
33(C2×D199)⋊C4—22no—
34C4×Dic199—21no—
35C4×Dic199—22no—
36C4×Dic199C2×C411yescharacteristic
37C4×Dic199C2×C411yescharacteristic
38#41×C199—22no—
39#41×C199—22no—
40C2×Q8×C199C2×C411yescharacteristic
41C2×D4×C199C2×C411yescharacteristic
42C2×D4×C199D412yes—
43C22×C4×C199D412yes—
44C22×C4×C199C2311yescharacteristic
45M16×C199—24no—
46M16×C199—22no—
47M16×C199D414yescomplemented
48C2×C8×C199—24no—
49C2×C8×C199—22no—
50C22×D199—21no—
51Dic398—22no—
52D796—22no—
53C22⋊D199—44no—
54C22⋊D199—42no—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—42no—
59C2×Dic199—42no—
60C4×D199—41no—
61C4×D199—41no—
62C23×C199C22⋊C412yes—
63C23×C199C2×D411yescharacteristic
64Q8×C199—22no—
65D4×C199—24no—
66D4×C199—22no—
67C2×C4×C199—22no—
68C2×C4×C199—22no—
69C2×C4×C199C22⋊C412yes—
70C2×C4×C199C22×C411yescharacteristic
71C2×C4×C199C2×D412yes—
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199#4112yes—
74C8×C199—24no—
75C22×C199—22no—
76C22×C199—41no—
77C22×C199—42no—
78C22×C199C2×C22⋊C411yescharacteristic
79C22×C199#12112yes—
80C2×D199—42no—
81C2×D199—41no—
82C4×C199—22no—
83C4×C199C4×D412yes—
84C4×C199#12011yescharacteristic
85C4×C199#12311yescharacteristic
86Dic199—22no—
87Dic199—41no—
88Dic199—41no—
89Dic199—82no—
90C2×C199—21no—
91C2×C199—22no—
92C2×C199—42no—
93C2×C199#38611yescharacteristic
94D199—81no—
95C199#188411yescharacteristic, complemented, Sylow
96#1884—1991nomaximal, Sylow
97#568D19911yescharacteristic, complemented
98#532—1991no—
99#421—1992no—
100#410—1991no—
101#339—1992no—
102C2×#41—1991no—
103C2×#41C2×D19911yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D19912yes—
107C4⋊Q8—1991no—
108#127—1991no—
109#124—1991no—
110#124—1991no—
111#121—3982no—
112C4×D4—3982no—
113#117—1992no—
114C2×C42—1991no—
115#104—3984no—
116#99—3982no—
117#41—22no—
118#41—22no—
119#41—3982no—
120#41—3982no—
121C2×Q8—1991no—
122C2×Q8C4×D19911yescharacteristic
123C2×D4—3982no—
124C2×D4—3981no—
125C2×D4—1991no—
126C2×D4C4×D19911yescharacteristic
127C2×D4D79612yes—
128C22×C4—3981no—
129C22×C4—3981no—
130C22×C4C22⋊D19912yes—
131C22×C4C22×D19911yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D19914yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—3982no—
138C4⋊C4—3982no—
139C22⋊C4—3982no—
140C22⋊C4—3982no—
141C22⋊C4—3982no—
142C42—3982no—
143C42—3981no—
144C42—1991no—
145C42—1991no—
146C23—3981no—
147C23(C2×D199)⋊C412yes—
148C23C2×D79611yescharacteristic
149Q8—22no—
150Q8—3982no—
151D4—24no—
152D4—22no—
153D4—7962no—
154D4—7964no—
155D4—3982no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—7962no—
159C2×C4—7961no—
160C2×C4—7961no—
161C2×C4—7962no—
162C2×C4—3981no—
163C2×C4—3981no—
164C2×C4—3981no—
165C2×C4(C2×D199)⋊C412yes—
166C2×C4C2×C4×D19911yescharacteristic
167C2×C4C2×C22⋊D19911yescharacteristic
168C2×C4C2×C22⋊D19912yes—
169C2×C4#1569188612yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—7962no—
175C22—7961no—
176C22C2×(C2×D199)⋊C411yescharacteristic
177C22#2026392912yes—
178C4—22no—
179C4—7961no—
180C4—7961no—
181C4—15922no—
182C4—3982no—
183C4C4×C22⋊D19912yes—
184C4#2026392411yescharacteristic
185C4#2026393811yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—15921no—
190C2#2716389711yescharacteristic
subgroupG/H
centerC4#20263924
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#568×C199C2
socleC2×C199#386
prime radical#27164564C2
cosocleC22×C4×C199C23
hypercenter#568D199
hypocenterC199#1884
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27164034C2semidirect
3. #27163992C2semidirect
4. #27163779C2semidirect
5. #568×C199C2semidirect
7. #20263953C4semidirect
8. #20263951C4semidirect
9. #20263944C4semidirect
10. #20263942C4semidirect
47. M16×C199D4semidirect
95. C199#1884semidirect
97. #568D199semidirect
134. M16C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645642
#271640342
#271639922
#271637792
#568×C1992
#1884199