Finite Group Databasebeta

← #38768799

Subgroups — #38768799

Order 25472.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#27164854C211yesmaximal, characteristic
2#27164000C212yesmaximal, complemented
3#27163986C212yesmaximal, complemented
4#27163985C211yesmaximal, characteristic, complemented
5#413×C199C211yesmaximal, characteristic, complemented
6#20264003—21no—
7#20264002—22no—
8#20264001—22no—
9#20264000—21no—
10#20264000C414yes—
11C2×#15691886—22no—
12C2×#15691886C2212yes—
13#41×D199—24no—
14C2×#15691870C2211yescharacteristic, complemented
15C22×C199⋊C8C2211yescharacteristic, complemented
16#20263873—22no—
17#20263871—22no—
18C22⋊(C199⋊C8)—22no—
19(C2×D199)⋊C8—24no—
20C2×#41×C199C2211yescharacteristic
21C2×M16×C199C2212yescomplemented
22#101×C199—21no—
23#100×C199—21no—
24#15691887—24no—
25#15691886—24no—
26#15691886—24no—
27#15691886—24no—
28#15691886—22no—
29#15691886—22no—
30#15691886—22no—
31#15691886—22no—
32#15691885—24no—
33#15691885—28no—
34C2×Dic398—22no—
35C2×Dic398D412yes—
36Q8×D199—24no—
37C2×D796—22no—
38C2×D796D412yes—
39C2×C22⋊D199—24no—
40C2×C22⋊D199C2×C414yes—
41D4×D199—24no—
42D4×D199—28no—
43C2×C4×D199—24no—
44C2×C4×D199D414yes—
45#15691870—22no—
46C2×C199⋊C8—22no—
47C2×C199⋊C8—21no—
48C2×C199⋊C8—21no—
49#41×C199—22no—
50#41×C199—22no—
51C2×Q8×C199D411yescharacteristic
52C2×D4×C199C2×C412yes—
53C2×D4×C199D411yescharacteristic
54C22×C4×C199D412yes—
55C22×C4×C199C2311yescharacteristic
56M16×C199—22no—
57M16×C199—22no—
58C2×C8×C199—22no—
59C22×D199—24no—
60C22×D199C22⋊C414yes—
61Dic398—24no—
62Dic398—22no—
63Dic398—22no—
64D796—24no—
65D796—22no—
66D796—22no—
67C22⋊D199—28no—
68C22⋊D199—24no—
69C22⋊D199—24no—
70C2×Dic199—24no—
71C2×Dic199C22⋊C414yes—
72C4×D199—28no—
73C4×D199—24no—
74C4×D199—24no—
75C199⋊C8—41no—
76C199⋊C8—41no—
77C23×C199C22⋊C412yes—
78C23×C199C22×C411yescharacteristic
79Q8×C199—22no—
80D4×C199—24no—
81D4×C199—22no—
82C2×C4×C199—22no—
83C2×C4×C199—22no—
84C2×C4×C199C22⋊C412yes—
85C2×C4×C199C2×D412yes—
86C2×C4×C199C2×D412yes—
87C2×C4×C199C2×D411yescharacteristic
88C2×C4×C199C2×D411yescharacteristic
89C8×C199—42no—
90C22×C199—22no—
91C22×C199—41no—
92C22×C199—42no—
93C22×C199C2×C22⋊C412yes—
94C22×C199C2×C22⋊C411yescharacteristic
95C2×D199—24no—
96C2×D199—44no—
97C2×D199—84no—
98C4×C199—22no—
99C4×C199#12012yes—
100C4×C199#12011yescharacteristic
101C4×C199#12011yescharacteristic
102Dic199—24no—
103C2×C199—22no—
104C2×C199—21no—
105C2×C199—42no—
106C2×C199#37911yescharacteristic
107D199—44no—
108C199#179111yescharacteristic, complemented, Sylow
109#1791—1991nomaximal, Sylow
110#585—1991no—
111#413—1992no—
112#413D19911yescharacteristic, complemented
113#408—1991no—
114#408—1992no—
1152−1+4—3982no—
1162−1+4—3981no—
117H(5,𝔽2)—3982no—
118H(5,𝔽2)—3981no—
119H(5,𝔽2)—1994no—
120C2×#41—3984no—
121C2×#41—3982no—
122C2×#41—1992no—
123C2×#41C2×D19911yescharacteristic
124C2×M16—1991no—
125C2×M16C2×D19912yescomplemented
126C22×C8—1991no—
127#101—21no—
128#101—3982no—
129#100—21no—
130#100—3982no—
131C22⋊C8—3984no—
132C22⋊C8—3982no—
133#41—22no—
134#41—22no—
135#41—3984no—
136#41—3984no—
137#41—3984no—
138#41—3984no—
139#41—3988no—
140#41—3984no—
141#41—3982no—
142#41—3982no—
143#41—3982no—
144#41—3982no—
145C2×Q8—3982no—
146C2×Q8—3984no—
147C2×Q8—1992no—
148C2×Q8C22⋊D19911yescharacteristic
149C2×D4—3982no—
150C2×D4—3984no—
151C2×D4—3988no—
152C2×D4—3984no—
153C2×D4—1994no—
154C2×D4—1992no—
155C2×D4C4×D19912yes—
156C2×D4C22⋊D19911yescharacteristic
157C22×C4—3984no—
158C22×C4—1994no—
159C22×C4D79612yes—
160C22×C4C22×D19911yescharacteristic
161M16—22no—
162M16—22no—
163M16—3982no—
164C2×C8—22no—
165C2×C8—3982no—
166C2×C8—3981no—
167C2×C8—3981no—
168C23—3984no—
169C23—1994no—
170C23(C2×D199)⋊C412yes—
171C23C2×C4×D19911yescharacteristic
172Q8—22no—
173Q8—3984no—
174Q8—3982no—
175Q8—3982no—
176D4—24no—
177D4—22no—
178D4—3984no—
179D4—3988no—
180D4—3984no—
181D4—3984no—
182D4—3982no—
183D4—3982no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3988no—
187C2×C4—3984no—
188C2×C4—3984no—
189C2×C4—3984no—
190C2×C4—1994no—
191C2×C4(C2×D199)⋊C412yes—
192C2×C4D4×D19912yes—
193C2×C4D4×D19912yes—
194C2×C4C2×C22⋊D19911yescharacteristic
195C2×C4C2×D79611yescharacteristic
196C8—42no—
197C8—7961no—
198C8—7961no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—7964no—
203C22—15924no—
204C22—3984no—
205C22C22⋊C4×D19912yes—
206C22C2×(C2×D199)⋊C411yescharacteristic
207C4—22no—
208C4—3984no—
209C4#2026392312yes—
210C4#2026392511yescharacteristic
211C4#2026392511yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—7964no—
216C2#2716387311yescharacteristic
subgroupG/H
centerC4#20263925
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#413×C199C2
socleC2×C199#379
prime radical#27164854C2
cosocleC22×C4×C199C23
hypercenter#413D199
hypocenterC199#1791
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27164000C2semidirect
3. #27163986C2semidirect
4. #27163985C2semidirect
5. #413×C199C2semidirect
14. C2×#15691870C22semidirect
15. C22×C199⋊C8C22semidirect
21. C2×M16×C199C22semidirect
108. C199#1791semidirect
112. #413D199semidirect
125. C2×M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648542
#271640002
#271639862
#271639852
#413×C1992
#1791199