Finite Group Databasebeta

← #38769252

Subgroups — #38769252

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×#20263943C211yesmaximal, characteristic
2#27164495C211yesmaximal, characteristic, complemented
3#27164024C211yesmaximal, characteristic
4#27164022C211yesmaximal, characteristic
5C2×#20263873C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7#339×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C22×D796C2211yescharacteristic
10C2×#15691870C2211yescharacteristic
11#20263943—22no—
12#20263943C414yescomplemented
13C4×Dic398—22no—
14#20263920—22no—
15#20263914C2211yescharacteristic
16#20263911—22no—
17C2×(C2×D199)⋊C4—22no—
18#20263877—22no—
19#20263875—22no—
20#20263873—22no—
21C2×M16×C199C2211yescharacteristic
22#117×C199C2211yescharacteristic
23C2×C42×C199C2211yescharacteristic
24C23×D199—21no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398C2×C412yes—
30C2×Dic398D412yes—
31C2×D796—21no—
32C2×D796—21no—
33C2×D796—21no—
34C2×D796—21no—
35C2×D796—21no—
36C2×D796C2×C412yes—
37C22×Dic199D412yes—
38#15691870—21no—
39#15691870—21no—
40C2×C199⋊C8—21no—
41C4⋊Dic199D412yes—
42Dic199⋊C4—24no—
43C22⋊Dic199—21no—
44(C2×D199)⋊C4—28no—
45C4×Dic199—24no—
46C4×Dic199—21no—
47C22×C4×C199—21no—
48C22×C4×C199C2311yescharacteristic
49M16×C199—21no—
50M16×C199—21no—
51C2×C8×C199—21no—
52C4⋊C4×C199D412yes—
53C22⋊C4×C199—21no—
54C42×C199—21no—
55C42×C199—21no—
56C42×C199C2×C412yes—
57C22×D199—22no—
58C22×D199—44no—
59C22×D199—41no—
60C22×D199—41no—
61Dic398—22no—
62Dic398—22no—
63Dic398—22no—
64Dic398—22no—
65D796—22no—
66D796—22no—
67D796—41no—
68D796—41no—
69C2×Dic199—22no—
70C2×Dic199—22no—
71C2×Dic199—22no—
72C2×Dic199C22⋊C414yes—
73C199⋊C8—41no—
74C23×C199C2×D411yescharacteristic
75C2×C4×C199—22no—
76C2×C4×C199—22no—
77C2×C4×C199—42no—
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×C199C2×C22⋊C411yescharacteristic
87C22×C199#12011yescharacteristic
88C22×C199#12311yescharacteristic
89C2×D199—44no—
90C2×D199—42no—
91C2×D199—82no—
92C2×D199—82no—
93C4×C199—42no—
94C4×C199—42no—
95C4×C199C4×D411yescharacteristic
96C4×C199C4×D411yescharacteristic
97C4×C199#12111yescharacteristic
98C4×C199#12111yescharacteristic
99Dic199—24no—
100Dic199—42no—
101C2×C199—21no—
102C2×C199—21no—
103C2×C199#38611yescharacteristic
104C2×C199#45412yes—
105D199—82no—
106C199#188811yescharacteristic, complemented, Sylow
107#1888—1991nomaximal, Sylow
108C2×#124—1991no—
109#519—1991no—
110#418—1991no—
111#418—1991no—
112C2×#101—1991no—
113#339—1991no—
114#339D19911yescharacteristic, complemented
115C22×Q8—1991no—
116C22×D4—1991no—
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×C42C2×D19911yescharacteristic
128#102—3982no—
129#102—3982no—
130#101—3982no—
131C24—3981no—
132C2×Q8—3981no—
133C2×Q8—3981no—
134C2×Q8—3981no—
135C2×Q8—3981no—
136C2×Q8—1992no—
137C2×Q8—1992no—
138C2×D4—3981no—
139C2×D4—3981no—
140C2×D4—3981no—
141C2×D4—3981no—
142C2×D4—3981no—
143C2×D4—1992no—
144C22×C4—21no—
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⋊C4D79612yes—
156C22⋊C4—21no—
157C22⋊C4—3988no—
158C22⋊C4—3981no—
159C42—21no—
160C42—21no—
161C42—3984no—
162C42—3981no—
163C42C4×D19912yes—
164C23—7964no—
165C23—7961no—
166C23—7961no—
167C23—3982no—
168C23D4×D19911yescharacteristic
169Q8—3982no—
170Q8—3982no—
171Q8—3982no—
172Q8—3982no—
173D4—7961no—
174D4—7961no—
175D4—3982no—
176D4—3982no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—3982no—
181C2×C4—3982no—
182C2×C4—3982no—
183C2×C4—1994no—
184C2×C4C2×C4×D19911yescharacteristic
185C2×C4D4×D19911yescharacteristic
186C2×C4D4×D19911yescharacteristic
187C2×C4C2×D79611yescharacteristic
188C2×C4#1569188611yescharacteristic
189C2×C4#1569188711yescharacteristic
190C8—41no—
191C8—7961no—
192C22—22no—
193C22—7964no—
194C22—7962no—
195C22—15922no—
196C22—15922no—
197C22C22⋊C4×D19911yescharacteristic
198C22#2026392511yescharacteristic
199C22#2026394011yescharacteristic
200C4—42no—
201C4—42no—
202C4—7962no—
203C4—3984no—
204C4#2026391711yescharacteristic
205C4C4×D79611yescharacteristic
206C4#2026392811yescharacteristic
207C4#2026393011yescharacteristic
208C2—21no—
209C2—21no—
210C2—15922no—
211C2#2716389611yescharacteristic
212C2#2716419212yes—
subgroupG/H
centerC22#20263925
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#339×C199C2
socleC22×C199#120
prime radicalC22×D796C22
cosocleC22×C4×C199C23
hypercenter#339D199
hypocenterC199#1888
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #27164495C2semidirect
5. C2×#20263873C2semidirect
6. #27163778C2semidirect
7. #339×C199C2semidirect
12. #20263943C4semidirect
106. C199#1888semidirect
114. #339D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639432
#271644952
#271640242
#271640222
C2×#202638732
#271637782
#339×C1992
#1888199