Finite Group Databasebeta

← #38769326

Subgroups — #38769326

Order 25472.

304 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 79 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×#20263986C211yesmaximal, characteristic
2#27164784C211yesmaximal, characteristic
3#27164034C211yesmaximal, characteristic
4#27164027C211yesmaximal, characteristic, complemented
5#27163790C211yesmaximal, characteristic, complemented
6C199⋊#338C211yesmaximal, characteristic, complemented
7C2×#101×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9#20263986—22no—
10#20263986—22no—
11C2×#15691876C412yescomplemented
12C2×#15691873C412yescomplemented
13#20263963—22no—
14#20263961—22no—
15C2×#15691870C2211yescharacteristic
16C2×#15691870C2211yescharacteristic
17M16×D199—22no—
18#20263914C2211yescharacteristic
19#20263881—22no—
20#20263879—22no—
21C22×Q8×C199C2211yescharacteristic
22C2×M16×C199C2211yescharacteristic
23C2×M16×C199C2211yescharacteristic
24#101×C199—22no—
25#15691886—21no—
26#15691886—21no—
27#15691886D412yes—
28#15691886D412yes—
29C2×Dic398C2×C411yescharacteristic
30C2×D796C2×C411yescharacteristic
31C2×C22⋊D199—21no—
32C2×C4×D199—21no—
33#15691876—24no—
34#15691873—24no—
35#15691870—21no—
36#15691870—21no—
37#15691870—21no—
38#15691870—21no—
39#15691869—24no—
40C2×C199⋊C8C2×C412yes—
41C2×C199⋊C8D412yes—
42C8×D199—24no—
43C4⋊Dic199—21no—
44C22⋊Dic199—21no—
45C4×Dic199—21no—
46C2×Q8×C199—21no—
47C2×Q8×C199—21no—
48C2×Q8×C199—21no—
49C2×Q8×C199—21no—
50C2×Q8×C199—21no—
51C2×Q8×C199C2×C412yes—
52C22×C4×C199—21no—
53C22×C4×C199C2311yescharacteristic
54M16×C199—21no—
55M16×C199—21no—
56M16×C199—21no—
57M16×C199D412yes—
58C2×C8×C199—21no—
59C2×C8×C199—21no—
60C22×D199—21no—
61Dic398—21no—
62Dic398C22⋊C412yes—
63D796—21no—
64D796C22⋊C412yes—
65C22⋊D199—22no—
66C22⋊D199—22no—
67C2×Dic199—21no—
68C2×Dic199—41no—
69C4×D199—22no—
70C4×D199—22no—
71C199⋊C8—22no—
72C199⋊C8—22no—
73C23×C199#4111yescharacteristic
74Q8×C199—22no—
75Q8×C199—22no—
76Q8×C199—41no—
77Q8×C199—41no—
78C2×C4×C199—22no—
79C2×C4×C199—41no—
80C2×C4×C199—41no—
81C2×C4×C199C22×C411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×D411yescharacteristic
85C2×C4×C199C2×D411yescharacteristic
86C2×C4×C199#4111yescharacteristic
87C8×C199—22no—
88C8×C199—41no—
89C22×C199—41no—
90C22×C199C4×D411yescharacteristic
91C22×C199#12111yescharacteristic
92C22×C199#12311yescharacteristic
93C2×D199—22no—
94C2×D199—81no—
95C4×C199—42no—
96C4×C199C2×C22⋊C411yescharacteristic
97C4×C199C4×D411yescharacteristic
98C4×C199#12011yescharacteristic
99C4×C199#12111yescharacteristic
100Dic199—22no—
101Dic199—81no—
102C2×C199—21no—
103C2×C199—21no—
104C2×C199—21no—
105C2×C199#38611yescharacteristic
106D199—42no—
107C199#190211yescharacteristic, complemented, Sylow
108#1902—1991nomaximal, Sylow
109C2×#137—1991no—
110#568—1991no—
111#421—1991no—
112#419—1991no—
113C2×#101D19911yescharacteristic, complemented
114#344—1991no—
115#338—1991no—
116C2×#41—1991no—
117C22×Q8C2×D19911yescharacteristic
118#137—3982no—
119#137—3982no—
120C2×Q16—1992no—
121C2×SD16—1992no—
122#131—3982no—
123#131—3982no—
124C2×M16—3982no—
125C2×M16—1991no—
126C2×M16—1991no—
127C2×M16C2×D19911yescharacteristic
128C2×M16C2×D19911yescharacteristic
129#117—1991no—
130#104—3982no—
131#103—3982no—
132#101—22no—
133#41—3981no—
134#41—3981no—
135#41—1992no—
136#41—1992no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—1991no—
143C2×Q8C4×D19912yes—
144C2×D4—3981no—
145C2×D4—1991no—
146C22×C4—21no—
147C22×C4—3981no—
148C22×C4C22×D19911yescharacteristic
149Q16—3984no—
150SD16—3984no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—3984no—
155M16—3981no—
156M16—3981no—
157M16—3981no—
158M16—3981no—
159M16C22⋊D19912yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—3984no—
163C2×C8—1992no—
164C2×C8—1992no—
165C4⋊C4—3981no—
166C22⋊C4—3981no—
167C42—3981no—
168C23—3981no—
169C23#1569188611yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174Q8—3981no—
175Q8—1992no—
176D4—3982no—
177D4—3982no—
178D4—3981no—
179D4—1992no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—7961no—
184C2×C4—3981no—
185C2×C4—3982no—
186C2×C4—3982no—
187C2×C4C2×C4×D19911yescharacteristic
188C2×C4D4×D19911yescharacteristic
189C2×C4D4×D19911yescharacteristic
190C2×C4D4×D19911yescharacteristic
191C2×C4C2×C22⋊D19911yescharacteristic
192C2×C4#1569188511yescharacteristic
193C8—22no—
194C8—41no—
195C8—3982no—
196C8—3982no—
197C22—41no—
198C22—15921no—
199C22—3982no—
200C22C4×C22⋊D19911yescharacteristic
201C22#2026392811yescharacteristic
202C22#2026393711yescharacteristic
203C4—42no—
204C4—15921no—
205C4—3982no—
206C4C22⋊C4×D19911yescharacteristic
207C4(C22⋊D199)⋊C411yescharacteristic
208C4#2026392311yescharacteristic
209C4#2026392611yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—7962no—
214C2#2716389411yescharacteristic
subgroupG/H
centerC2#27163894
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#101×C199C2
socleC2×C199#386
prime radicalC2×#15691886C22
cosocleC22×C4×C199C23
hypercenterC2×#101D199
hypocenterC199#1902
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
4. #27164027C2semidirect
5. #27163790C2semidirect
6. C199⋊#338C2semidirect
7. C2×#101×C199C2semidirect
11. C2×#15691876C4semidirect
12. C2×#15691873C4semidirect
107. C199#1902semidirect
113. C2×#101D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639862
#271647842
#271640342
#271640272
#271637902
C199⋊#3382
C2×#101×C1992
#1902199