Finite Group Databasebeta

← #38769752

Subgroups — #38769752

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×#15691873C211yesmaximal, characteristic
2C2×#20263890C211yesmaximal, characteristic, complemented
3#27164022C211yesmaximal, characteristic
4C2×#20263874C211yesmaximal, characteristic, complemented
5C2×#20263871C211yesmaximal, characteristic
6#27163974C211yesmaximal, characteristic, complemented
7#419×C199C211yesmaximal, characteristic, complemented
8C22×D796C2211yescharacteristic
9C2×#15691873—21no—
10C2×#15691873—21no—
11C2×#15691873—22no—
12C2×#15691873C414yescomplemented
13C2×#15691870C2211yescharacteristic
14C2×#15691870C2211yescharacteristic
15C22×C199⋊C8C2211yescharacteristic
16#20263890—22no—
17#20263875—22no—
18#20263874—22no—
19#20263871—22no—
20#20263866—21no—
21C4×C199⋊C8—21no—
22C22×Q8×C199C2211yescharacteristic
23C2×M16×C199C2211yescharacteristic
24#117×C199C2211yescharacteristic
25#103×C199—22no—
26C23×D199—21no—
27C2×D796—21no—
28C2×D796—21no—
29C2×D796—21no—
30C2×D796—21no—
31C2×D796—21no—
32C2×D796C2×C412yes—
33#15691873—24no—
34#15691873—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—
45C2×Q8×C199—21no—
46C2×Q8×C199—21no—
47C2×Q8×C199—21no—
48C2×Q8×C199—21no—
49C2×Q8×C199—21no—
50C2×Q8×C199C2×C412yes—
51C22×C4×C199—21no—
52C22×C4×C199C2311yescharacteristic
53M16×C199—21no—
54M16×C199—21no—
55C2×C8×C199—21no—
56C4⋊C4×C199D412yes—
57C22⋊C4×C199—21no—
58C42×C199—21no—
59C22×D199—22no—
60C22×D199—41no—
61C22×D199—41no—
62C22×D199—82no—
63D796—22no—
64D796—22no—
65D796—41no—
66D796—41no—
67C199⋊C8—22no—
68C199⋊C8—41no—
69C199⋊C8C22⋊C412yes—
70C199⋊C8C22⋊C412yes—
71C23×C199#4111yescharacteristic
72Q8×C199—22no—
73Q8×C199—22no—
74Q8×C199—41no—
75Q8×C199—41no—
76C2×C4×C199—22no—
77C2×C4×C199—22no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
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×C199C4×D411yescharacteristic
89C22×C199#12111yescharacteristic
90C22×C199#12411yescharacteristic
91C2×D199—42no—
92C2×D199—82no—
93C2×D199—82no—
94C2×D199—82no—
95C4×C199—42no—
96C4×C199—42no—
97C4×C199C2×C22⋊C411yescharacteristic
98C4×C199C4×D411yescharacteristic
99C4×C199#12111yescharacteristic
100C4×C199#12711yescharacteristic
101C2×C199—21no—
102C2×C199—21no—
103C2×C199#39011yescharacteristic
104C2×C199#47112yes—
105D199—82no—
106C199#197711yescharacteristic, complemented, Sylow
107#1977—1991nomaximal, Sylow
108C22×SD16—1991no—
109C2×#108—1991no—
110#419D19911yescharacteristic, complemented
111#418—1991no—
112C2×#101—1991no—
113C2×#100—1991no—
114#405—1991no—
115C22×Q8C2×D19911yescharacteristic
116C22×D4—1991no—
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—22no—
128#102—3982no—
129#101—3982no—
130#100—3982no—
131#97—3981no—
132C4×C8—3981no—
133C24—3981no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C4×D19912yes—
140C2×D4—3981no—
141C2×D4—3981no—
142C2×D4—3981no—
143C2×D4—3981no—
144C2×D4—3981no—
145C2×D4—1992no—
146C22×C4—21no—
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—7961no—
167C23—7961no—
168C23—15922no—
169C23—3982no—
170C23#1569188511yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—7961no—
176D4—7961no—
177D4—3982no—
178D4—3982no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
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—7962no—
196C22—15922no—
197C22—15922no—
198C22—15922no—
199C22(C22⋊D199)⋊C411yescharacteristic
200C22#2026392711yescharacteristic
201C22#2026394111yescharacteristic
202C4—42no—
203C4—42no—
204C4C22⋊C4×D19911yescharacteristic
205C4C4×C22⋊D19911yescharacteristic
206C4#2026392811yescharacteristic
207C4#2026395111yescharacteristic
208C2—21no—
209C2—21no—
210C2—15922no—
211C2#2716391811yescharacteristic
212C2#2716428312yes—
subgroupG/H
centerC22#20263927
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#419×C199C2
socleC22×C199#121
prime radicalC22×D796C22
cosocleC22×C4×C199C23
hypercenter#419D199
hypocenterC199#1977
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C2×#20263890C2semidirect
4. C2×#20263874C2semidirect
6. #27163974C2semidirect
7. #419×C199C2semidirect
12. C2×#15691873C4semidirect
106. C199#1977semidirect
110. #419D199semidirect

Maximal subgroups

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

subgroupindex
C22×#156918732
C2×#202638902
#271640222
C2×#202638742
C2×#202638712
#271639742
#419×C1992
#1977199