Finite Group Databasebeta

← #38769232

Subgroups — #38769232

Order 25472.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#27164784C211yesmaximal, characteristic, complemented
2#27164568C211yesmaximal, characteristic
3#27164036C211yesmaximal, characteristic, complemented
4#27164034C211yesmaximal, characteristic, complemented
5#27163991C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7#339×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9C2×#15691886C2211yescharacteristic
10#20263963—22no—
11#20263961—22no—
12C2×#15691870C2211yescharacteristic
13M16×D199—22no—
14#20263955C411yescharacteristic, complemented
15#20263952C411yescharacteristic, complemented
16#20263943C411yescharacteristic, complemented
17#20263943C411yescharacteristic, complemented
18#20263929—22no—
19C4×D796—22no—
20#20263914C2211yescharacteristic
21#20263883—22no—
22#20263881—22no—
23#20263869—22no—
24C2×M16×C199C2211yescharacteristic
25#117×C199C2211yescharacteristic
26C2×C42×C199C2211yescharacteristic
27#15691886—21no—
28#15691886—21no—
29#15691886—21no—
30#15691886—21no—
31#15691886—21no—
32#15691886—21no—
33#15691886—21no—
34#15691886—21no—
35C2×Dic398C2×C411yescharacteristic
36C2×Dic398C2×C411yescharacteristic
37C2×D796—22no—
38C2×D796C2×C411yescharacteristic
39C2×D796C2×C411yescharacteristic
40C2×C22⋊D199—21no—
41C2×C22⋊D199D412yes—
42C2×C4×D199—21no—
43C2×C4×D199D412yes—
44#15691870—21no—
45#15691870D412yescomplemented
46#15691869—24no—
47C2×C199⋊C8—21no—
48C8×D199—24no—
49C4⋊Dic199—22no—
50C4⋊Dic199—21no—
51C22⋊Dic199—21no—
52(C2×D199)⋊C4—22no—
53(C2×D199)⋊C4—22no—
54C4×Dic199—21no—
55C22×C4×C199—21no—
56C22×C4×C199C2311yescharacteristic
57M16×C199—21no—
58M16×C199D412yescomplemented
59C2×C8×C199—21no—
60C4⋊C4×C199—21no—
61C22⋊C4×C199—21no—
62C42×C199—21no—
63C42×C199—21no—
64C42×C199C2×C411yescharacteristic
65C42×C199C2×C411yescharacteristic
66C22×D199—21no—
67C22×D199C22⋊C412yes—
68Dic398—21no—
69Dic398—21no—
70Dic398—21no—
71Dic398—21no—
72D796—21no—
73D796—21no—
74D796—21no—
75D796—21no—
76D796—44no—
77C22⋊D199—22no—
78C22⋊D199—22no—
79C22⋊D199—41no—
80C22⋊D199—41no—
81C2×Dic199—21no—
82C2×Dic199—41no—
83C2×Dic199C22⋊C412yes—
84C4×D199—22no—
85C4×D199—22no—
86C4×D199—41no—
87C4×D199—41no—
88C199⋊C8—22no—
89C23×C199C2×D411yescharacteristic
90C2×C4×C199—21no—
91C2×C4×C199—21no—
92C2×C4×C199—42no—
93C2×C4×C199—41no—
94C2×C4×C199C22×C411yescharacteristic
95C2×C4×C199C2×D411yescharacteristic
96C2×C4×C199C2×D411yescharacteristic
97C2×C4×C199C2×D411yescharacteristic
98C2×C4×C199#4111yescharacteristic
99C2×C4×C199#4111yescharacteristic
100C8×C199—22no—
101C22×C199—41no—
102C22×C199C2×C22⋊C411yescharacteristic
103C22×C199#12111yescharacteristic
104C22×C199#12111yescharacteristic
105C2×D199—22no—
106C2×D199—42no—
107C2×D199—42no—
108C2×D199—41no—
109C4×C199—22no—
110C4×C199—41no—
111C4×C199—81no—
112C4×C199C4×D411yescharacteristic
113C4×C199C4×D411yescharacteristic
114C4×C199#12011yescharacteristic
115C4×C199#12311yescharacteristic
116Dic199—22no—
117Dic199—41no—
118Dic199—81no—
119C2×C199—21no—
120C2×C199—21no—
121C2×C199—21no—
122C2×C199#38611yescharacteristic
123D199—42no—
124D199—81no—
125C199#188411yescharacteristic, complemented, Sylow
126#1884—1991nomaximal, Sylow
127#568—1991no—
128#532—1991no—
129#421—1991no—
130#421—1991no—
131#410—1991no—
132#339—1991no—
133#339D19911yescharacteristic, complemented
134C2×#41—1991no—
135C2×#41—1991no—
136#131—3982no—
137#131—3982no—
138C2×M16—3982no—
139C2×M16—1991no—
140C2×M16C2×D19911yescharacteristic
141C4⋊Q8—1991no—
142#127—1991no—
143#124—1991no—
144#124—1991no—
145#121—3982no—
146C4×D4—3982no—
147#117—1991no—
148#117C2×D19911yescharacteristic
149C2×C42C2×D19911yescharacteristic
150#104—3982no—
151#104—3982no—
152#99—3982no—
153#41—3981no—
154#41—3981no—
155#41—3981no—
156#41—3981no—
157#41—3981no—
158#41—3981no—
159#41—3981no—
160#41—3981no—
161C2×Q8—1991no—
162C2×Q8—1991no—
163C2×D4—3982no—
164C2×D4—3981no—
165C2×D4—1992no—
166C2×D4—1991no—
167C2×D4—1991no—
168C22×C4—21no—
169C22×C4—3981no—
170C22×C4—1992no—
171C22×C4C22×D19911yescharacteristic
172M16—21no—
173M16—3984no—
174M16—3981no—
175M16—1992no—
176M16D79612yescomplemented
177C2×C8—21no—
178C2×C8—3984no—
179C2×C8—3981no—
180C4⋊C4—21no—
181C4⋊C4—3982no—
182C4⋊C4—3981no—
183C22⋊C4—21no—
184C22⋊C4—3982no—
185C22⋊C4—3982no—
186C22⋊C4—3981no—
187C42—21no—
188C42—21no—
189C42—3981no—
190C42C4×D19911yescharacteristic
191C42C4×D19911yescharacteristic
192C23—3981no—
193C23—1992no—
194C23D4×D19911yescharacteristic
195Q8—3981no—
196Q8—3981no—
197Q8—3981no—
198Q8—3981no—
199D4—7964no—
200D4—7961no—
201D4—7961no—
202D4—3982no—
203D4—3982no—
204D4—3981no—
205D4—3981no—
206D4—3981no—
207D4—3981no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—7961no—
213C2×C4—7961no—
214C2×C4—7961no—
215C2×C4—3981no—
216C2×C4—3982no—
217C2×C4—3982no—
218C2×C4—1992no—
219C2×C4C2×C4×D19911yescharacteristic
220C2×C4D4×D19911yescharacteristic
221C2×C4D4×D19911yescharacteristic
222C2×C4C2×D79611yescharacteristic
223C2×C4#1569188611yescharacteristic
224C2×C4#1569188711yescharacteristic
225C8—22no—
226C8—3982no—
227C22—41no—
228C22—7962no—
229C22—7962no—
230C22—7961no—
231C22—3982no—
232C22C22⋊C4×D19911yescharacteristic
233C22#2026392811yescharacteristic
234C22#2026393011yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—7961no—
239C4—15921no—
240C4—3982no—
241C4#2026391711yescharacteristic
242C4C4×D79611yescharacteristic
243C4#2026392511yescharacteristic
244C4#2026394011yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—7962no—
249C2—15921no—
250C2#2716389611yescharacteristic
subgroupG/H
centerC4#20263925
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#339×C199C2
socleC2×C199#386
prime radical#27164568C2
cosocleC22×C4×C199C23
hypercenter#339D199
hypocenterC199#1884
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #27164784C2semidirect
3. #27164036C2semidirect
4. #27164034C2semidirect
5. #27163991C2semidirect
6. #27163778C2semidirect
7. #339×C199C2semidirect
14. #20263955C4semidirect
15. #20263952C4semidirect
16. #20263943C4semidirect
17. #20263943C4semidirect
45. #15691870D4semidirect
58. M16×C199D4semidirect
125. C199#1884semidirect
133. #339D199semidirect
176. M16D796semidirect

Maximal subgroups

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

subgroupindex
#271647842
#271645682
#271640362
#271640342
#271639912
#271637782
#339×C1992
#1884199