Finite Group Databasebeta

← #38769250

Subgroups — #38769250

Order 25472.

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

normal subgroups only

This group has 344 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
1C2×#20263942C211yesmaximal, characteristic
2#27164495C211yesmaximal, characteristic, complemented
3#27164023C211yesmaximal, characteristic
4C2×#20263873C211yesmaximal, characteristic, complemented
5#27163779C211yesmaximal, characteristic, complemented
6#27163777C211yesmaximal, characteristic, complemented
7#418×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×#15691870C2211yescharacteristic
10#20263942—22no—
11#20263942C414yescomplemented
12C4×Dic398—22no—
13#20263920—22no—
14#20263914C2211yescharacteristic
15#20263911—22no—
16C2×C22⋊Dic199—22no—
17C2×C4×Dic199C2211yescharacteristic
18#20263876—22no—
19#20263873—22no—
20C22×D4×C199C2211yescharacteristic
21C2×M16×C199C2211yescharacteristic
22#117×C199C2211yescharacteristic
23#102×C199—22no—
24C2×Dic398—21no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398C2×C412yes—
29C2×Dic398D412yes—
30C22×Dic199—21no—
31C22×Dic199D412yes—
32#15691870—21no—
33#15691870—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199D412yes—
36Dic199⋊C4—24no—
37C22⋊Dic199—28no—
38C22⋊Dic199—21no—
39C4×Dic199—24no—
40C4×Dic199—21no—
41C4×Dic199—21no—
42C4×Dic199C2×C412yes—
43C24×C199—21no—
44C2×D4×C199—21no—
45C2×D4×C199—21no—
46C2×D4×C199—21no—
47C2×D4×C199—21no—
48C2×D4×C199—21no—
49C2×D4×C199C2×C412yes—
50C22×C4×C199C2311yescharacteristic
51M16×C199—21no—
52M16×C199—21no—
53C2×C8×C199—21no—
54C4⋊C4×C199D412yes—
55C22⋊C4×C199—21no—
56C42×C199—21no—
57Dic398—22no—
58Dic398—22no—
59Dic398—22no—
60Dic398—22no—
61C2×Dic199—22no—
62C2×Dic199—22no—
63C2×Dic199—22no—
64C2×Dic199—22no—
65C2×Dic199—42no—
66C2×Dic199C22⋊C414yes—
67C199⋊C8—41no—
68C23×C199—22no—
69C23×C199—44no—
70C23×C199—41no—
71C23×C199—41no—
72C23×C199C2×D411yescharacteristic
73D4×C199—22no—
74D4×C199—22no—
75D4×C199—41no—
76D4×C199—41no—
77C2×C4×C199—22no—
78C2×C4×C199C22×C411yescharacteristic
79C2×C4×C199C2×D411yescharacteristic
80C2×C4×C199C2×D411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C2×C4×C199#4111yescharacteristic
83C2×C4×C199#4111yescharacteristic
84C8×C199—41no—
85C22×C199—22no—
86C22×C199—44no—
87C22×C199—42no—
88C22×C199—82no—
89C22×C199—82no—
90C22×C199C2×C22⋊C411yescharacteristic
91C22×C199#12011yescharacteristic
92C22×C199#12311yescharacteristic
93C4×C199—42no—
94C4×C199C4×D411yescharacteristic
95C4×C199C4×D411yescharacteristic
96C4×C199#12111yescharacteristic
97C4×C199#12111yescharacteristic
98Dic199—24no—
99Dic199—42no—
100Dic199—42no—
101C2×C199—21no—
102C2×C199—21no—
103C2×C199—82no—
104C2×C199#38611yescharacteristic
105C2×C199#45412yes—
106C199#188811yescharacteristic, complemented, Sylow
107#1888—1991nomaximal, Sylow
108C2×#124—1991no—
109#519—1991no—
110#418—1991no—
111#418D19911yescharacteristic
112C2×#101—1991no—
113#339—1991no—
114#339—1991no—
115C22×Q8—1991no—
116C22×D4C2×D19911yescharacteristic
117C2×M16—1991no—
118C2×M16C2×D19911yescharacteristic
119#124—3982no—
120#124—1994no—
121C4×Q8—3982no—
122C4×Q8—3982no—
123#117—3982no—
124#117—1991no—
125#117C2×D19911yescharacteristic
126C2×C22⋊C4—3982no—
127C2×C42—1991no—
128#102—22no—
129#102—3982no—
130#101—3982no—
131C24—21no—
132C2×Q8—3981no—
133C2×Q8—3981no—
134C2×Q8—3981no—
135C2×Q8—3981no—
136C2×Q8—1992no—
137C2×Q8—1992no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C4×D19912yes—
144C22×C4—3981no—
145C22×C4—1992no—
146C22×C4C22×D19911yescharacteristic
147M16—21no—
148M16—21no—
149M16—3981no—
150M16—3981no—
151C2×C8—21no—
152C2×C8—3981no—
153C4⋊C4—3984no—
154C4⋊C4—1992no—
155C4⋊C4C22⋊D19912yes—
156C22⋊C4—21no—
157C22⋊C4—3981no—
158C22⋊C4—3988no—
159C42—21no—
160C42—3984no—
161C42—3981no—
162C42—3981no—
163C42—1992no—
164C23—22no—
165C23—44no—
166C23—41no—
167C23—41no—
168C23D4×D19911yescharacteristic
169Q8—3982no—
170Q8—3982no—
171Q8—3982no—
172Q8—3982no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177C2×C4—22no—
178C2×C4—7962no—
179C2×C4—3982no—
180C2×C4—3982no—
181C2×C4—3982no—
182C2×C4—3982no—
183C2×C4—1994no—
184C2×C4C2×C4×D19911yescharacteristic
185C2×C4D4×D19911yescharacteristic
186C2×C4D4×D19911yescharacteristic
187C2×C4C2×C22⋊D19911yescharacteristic
188C2×C4#1569188511yescharacteristic
189C2×C4#1569188611yescharacteristic
190C8—41no—
191C8—7961no—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—82no—
197C22C22⋊C4×D19911yescharacteristic
198C22#2026392311yescharacteristic
199C22#2026393711yescharacteristic
200C4—42no—
201C4—7962no—
202C4—7962no—
203C4—3984no—
204C4(C22⋊D199)⋊C411yescharacteristic
205C4C4×C22⋊D19911yescharacteristic
206C4#2026392611yescharacteristic
207C4#2026392811yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2716389411yescharacteristic
212C2#2716418912yes—
subgroupG/H
centerC22#20263923
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#418×C199C2
socleC22×C199#120
prime radicalC22×D4×C199C22
cosocleC22×C4×C199C23
hypercenter#418D199
hypocenterC199#1888
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. #27164495C2semidirect
4. C2×#20263873C2semidirect
5. #27163779C2semidirect
6. #27163777C2semidirect
11. #20263942C4semidirect
106. C199#1888semidirect

Maximal subgroups

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

subgroupindex
C2×#202639422
#271644952
#271640232
C2×#202638732
#271637792
#271637772
#418×C1992
#1888199