Finite Group Databasebeta

← #38769756

Subgroups — #38769756

Order 25472.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#15691874C211yesmaximal, characteristic
2C2×#20263890C211yesmaximal, characteristic, complemented
3#27164026C211yesmaximal, characteristic, complemented
4C2×#20263873C211yesmaximal, characteristic, complemented
5C2×#20263872C211yesmaximal, characteristic
6#27163974C211yesmaximal, characteristic, complemented
7#418×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×#15691874—21no—
10C2×#15691874—21no—
11C2×#15691874—22no—
12C2×#15691874C414yescomplemented
13C2×#15691870C2211yescharacteristic
14C2×#15691870C2211yescharacteristic
15C22×C199⋊C8C2211yescharacteristic
16#20263890—22no—
17#20263878—22no—
18#20263873—22no—
19#20263872—22no—
20#20263866—21no—
21C4×C199⋊C8—21no—
22C22×D4×C199C2211yescharacteristic
23C2×M16×C199C2211yescharacteristic
24#117×C199C2211yescharacteristic
25#102×C199—22no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398—21no—
31C2×Dic398C2×C412yes—
32C22×Dic199—21no—
33#15691874—24no—
34#15691874—24no—
35#15691870—21no—
36#15691870—21no—
37#15691870—21no—
38#15691870D412yes—
39C2×C199⋊C8—21no—
40C2×C199⋊C8—21no—
41C2×C199⋊C8C2×C412yes—
42C2×C199⋊C8D411yescharacteristic
43C2×C199⋊C8D411yescharacteristic
44C2×C199⋊C8D412yes—
45C24×C199—21no—
46C2×D4×C199—21no—
47C2×D4×C199—21no—
48C2×D4×C199—21no—
49C2×D4×C199—21no—
50C2×D4×C199—21no—
51C2×D4×C199C2×C412yes—
52C22×C4×C199C2311yescharacteristic
53M16×C199—21no—
54M16×C199—21no—
55C2×C8×C199—21no—
56C4⋊C4×C199D412yes—
57C22⋊C4×C199—21no—
58C42×C199—21no—
59Dic398—22no—
60Dic398—22no—
61Dic398—41no—
62Dic398—41no—
63C2×Dic199—22no—
64C2×Dic199—41no—
65C2×Dic199—41no—
66C199⋊C8—22no—
67C199⋊C8—41no—
68C199⋊C8C22⋊C412yes—
69C199⋊C8C22⋊C412yes—
70C23×C199—22no—
71C23×C199—41no—
72C23×C199—41no—
73C23×C199—82no—
74C23×C199#4111yescharacteristic
75D4×C199—22no—
76D4×C199—22no—
77D4×C199—41no—
78D4×C199—41no—
79C2×C4×C199—22no—
80C2×C4×C199C22×C411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×D411yescharacteristic
85C2×C4×C199#4111yescharacteristic
86C8×C199—41no—
87C22×C199—22no—
88C22×C199—42no—
89C22×C199—82no—
90C22×C199—82no—
91C22×C199—82no—
92C22×C199C4×D411yescharacteristic
93C22×C199#12111yescharacteristic
94C22×C199#12411yescharacteristic
95C4×C199—42no—
96C4×C199C2×C22⋊C411yescharacteristic
97C4×C199C4×D411yescharacteristic
98C4×C199#12111yescharacteristic
99C4×C199#12711yescharacteristic
100Dic199—42no—
101C2×C199—21no—
102C2×C199—21no—
103C2×C199—82no—
104C2×C199#39011yescharacteristic
105C2×C199#47112yes—
106C199#197711yescharacteristic, complemented, Sylow
107#1977—1991nomaximal, Sylow
108C22×SD16—1991no—
109C2×#108—1991no—
110#419—1991no—
111#418D19911yescharacteristic
112C2×#101—1991no—
113C2×#100—1991no—
114#405—1991no—
115C22×Q8—1991no—
116C22×D4C2×D19911yescharacteristic
117C2×SD16—3981no—
118C2×SD16—3981no—
119C2×SD16—3982no—
120C2×SD16—1994no—
121C2×M16—1991no—
122C2×M16—1991no—
123C2×M16C2×D19911yescharacteristic
124C22×C8—1991no—
125#117C2×D19911yescharacteristic
126#108—3982no—
127#103—3982no—
128#102—22no—
129#101—3982no—
130#100—3982no—
131#97—3981no—
132C4×C8—3981no—
133C24—21no—
134C2×Q8—3981no—
135C2×Q8—3981no—
136C2×Q8—3981no—
137C2×Q8—3981no—
138C2×Q8—3981no—
139C2×Q8—1992no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C4×D19912yes—
146C22×C4—3981no—
147C22×C4C22×D19911yescharacteristic
148SD16—3984no—
149SD16—3984no—
150M16—21no—
151M16—21no—
152M16—3981no—
153M16—3981no—
154M16—3981no—
155M16—1992no—
156C2×C8—21no—
157C2×C8—3981no—
158C2×C8—3981no—
159C2×C8—1992no—
160C2×C8—1991no—
161C2×C8—1991no—
162C2×C8—1992no—
163C4⋊C4C22⋊D19912yes—
164C22⋊C4—21no—
165C42—21no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1569188511yescharacteristic
171Q8—7961no—
172Q8—7961no—
173Q8—3982no—
174Q8—3982no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—7961no—
181C2×C4—7961no—
182C2×C4—3982no—
183C2×C4C2×C4×D19911yescharacteristic
184C2×C4D4×D19911yescharacteristic
185C2×C4D4×D19911yescharacteristic
186C2×C4D4×D19911yescharacteristic
187C2×C4C2×C22⋊D19911yescharacteristic
188C2×C4#1569188611yescharacteristic
189C8—41no—
190C8—7961no—
191C8—3982no—
192C8—1992no—
193C8—1992no—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22(C22⋊D199)⋊C411yescharacteristic
200C22#2026392711yescharacteristic
201C22#2026394111yescharacteristic
202C4—42no—
203C4—7962no—
204C4C22⋊C4×D19911yescharacteristic
205C4C4×C22⋊D19911yescharacteristic
206C4#2026392811yescharacteristic
207C4#2026395111yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2716391811yescharacteristic
212C2#2716427912yes—
subgroupG/H
centerC22#20263927
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#418×C199C2
socleC22×C199#121
prime radicalC22×D4×C199C22
cosocleC22×C4×C199C23
hypercenter#418D199
hypocenterC199#1977
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#20263890C2semidirect
3. #27164026C2semidirect
4. C2×#20263873C2semidirect
6. #27163974C2semidirect
12. C2×#15691874C4semidirect
106. C199#1977semidirect

Maximal subgroups

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

subgroupindex
C22×#156918742
C2×#202638902
#271640262
C2×#202638732
C2×#202638722
#271639742
#418×C1992
#1977199