Finite Group Databasebeta

← #38770502

Subgroups — #38770502

Order 25472.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164590C211yesmaximal, characteristic
2#27163999C211yesmaximal, characteristic
3#27163990C211yesmaximal, characteristic
4#27163809C211yesmaximal, characteristic, complemented
5#27163808C211yesmaximal, characteristic, complemented
6#27163807C211yesmaximal, characteristic, complemented
7#352×C199C211yesmaximal, characteristic, complemented
8C2×#15691885C2211yescharacteristic
9C22×Dic398C411yescharacteristic, complemented
10M16×D199—21no—
11#20263954—21no—
12#20263941—21no—
13#20263939C411yescharacteristic, complemented
14#20263938C411yescharacteristic, complemented
15#20263936C2211yescharacteristic
16#20263933—21no—
17#20263932—21no—
18#20263911—21no—
19#20263911C411yescharacteristic, complemented
20#20263873—21no—
21#20263872C2211yescharacteristic
22#20263870C2211yescharacteristic
23#20263869—21no—
24#123×C199C2211yescharacteristic
25#100×C199C2211yescharacteristic
26#99×C199C2211yescharacteristic
27#15691885—41no—
28#15691885—41no—
29C2×Dic398—21no—
30C2×Dic398—21no—
31C2×Dic398—41no—
32C2×Dic398—41no—
33C2×Dic398D411yescharacteristic
34C2×C22⋊D199C2×C411yescharacteristic
35C2×C22⋊D199D411yescharacteristic
36C22×Dic199C2×C411yescharacteristic
37C22×Dic199C2×C411yescharacteristic
38C22×Dic199D411yescharacteristic
39C2×C4×D199D411yescharacteristic
40#15691870—21no—
41#15691869—41no—
42C8×D199—41no—
43C4⋊Dic199—21no—
44Dic199⋊C4—21no—
45Dic199⋊C4—21no—
46Dic199⋊C4—21no—
47Dic199⋊C4—41no—
48C22⋊Dic199—21no—
49C22⋊Dic199—21no—
50C22⋊Dic199C2×C411yescharacteristic
51(C2×D199)⋊C4—21no—
52(C2×D199)⋊C4—21no—
53C4×Dic199—21no—
54C4×Dic199—41no—
55C2×D4×C199C2311yescharacteristic
56C22×C4×C199C2×C411yescharacteristic
57M16×C199—21no—
58C4⋊C4×C199—21no—
59C22⋊C4×C199—21no—
60C22⋊C4×C199—21no—
61C22⋊C4×C199C2×C411yescharacteristic
62C22×D199C22⋊C411yescharacteristic
63Dic398—41no—
64Dic398—42no—
65Dic398—41no—
66Dic398—41no—
67C22⋊D199—41no—
68C22⋊D199—41no—
69C2×Dic199—21no—
70C2×Dic199—21no—
71C2×Dic199—21no—
72C2×Dic199—21no—
73C2×Dic199—21no—
74C2×Dic199—21no—
75C2×Dic199—41no—
76C2×Dic199—41no—
77C2×Dic199C22⋊C411yescharacteristic
78C2×Dic199C22⋊C411yescharacteristic
79C2×Dic199C22⋊C411yescharacteristic
80C4×D199—21no—
81C4×D199—21no—
82C199⋊C8—41no—
83C23×C199C22×C411yescharacteristic
84C23×C199C2×D411yescharacteristic
85D4×C199—41no—
86C2×C4×C199—21no—
87C2×C4×C199—21no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199C2×D411yescharacteristic
91C8×C199—41no—
92C22×C199—21no—
93C22×C199—21no—
94C22×C199—41no—
95C22×C199—41no—
96C22×C199C2×C22⋊C411yescharacteristic
97C2×D199—21no—
98C2×D199—41no—
99C4×C199—21no—
100C4×C199—41no—
101C4×C199—41no—
102C4×C199—81no—
103Dic199—21no—
104Dic199—21no—
105Dic199—41no—
106Dic199—41no—
107Dic199—81no—
108Dic199#9911yescharacteristic
109Dic199#9911yescharacteristic
110C2×C199—21no—
111C2×C199—41no—
112C2×C199—41no—
113C2×C199C2×#9911yescharacteristic
114D199—41no—
115C199#212311yescharacteristic, complemented, Sylow
116#2123—1991nomaximal, Sylow
117#536—1991no—
118#413—1991no—
119#410—1991no—
120#352—1991no—
121#352—1991no—
122#352—1991no—
123#352D19911yescharacteristic, complemented
124C2×#41—1991no—
125C22×Q8—1991no—
126C2×M16—3981no—
127C4⋊Q8—3981no—
128#124—3981no—
129#123—1991no—
130#123—1991no—
131#123—1991no—
132#123C2×D19911yescharacteristic
133C22⋊Q8—3981no—
134C22⋊Q8—3981no—
135#117—3981no—
136#117—1991no—
137#101—3981no—
138#100—1991no—
139#100C2×D19911yescharacteristic
140#99—3981no—
141#99—1991no—
142#99C2×D19911yescharacteristic
143#41—7961no—
144#41—7961no—
145C2×Q8—7961no—
146C2×Q8—7961no—
147C2×Q8—3981no—
148C2×Q8—3981no—
149C2×Q8—1991no—
150C2×D4—1991no—
151C2×D4—1991no—
152C2×D4C22×D19911yescharacteristic
153C22×C4—1991no—
154C22×C4—1991no—
155C22×C4—1991no—
156C22×C4—1991no—
157C22×C4C4×D19911yescharacteristic
158M16—21no—
159M16—7961no—
160M16—3981no—
161C2×C8—7961no—
162C4⋊C4—21no—
163C4⋊C4—7961no—
164C4⋊C4—3981no—
165C4⋊C4—3981no—
166C4⋊C4—3981no—
167C4⋊C4—3981no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3981no—
171C22⋊C4—3981no—
172C22⋊C4—3981no—
173C22⋊C4—3981no—
174C22⋊C4—1991no—
175C22⋊C4C4×D19911yescharacteristic
176C42—7961no—
177C42—3981no—
178C23—1991no—
179C23C2×C4×D19911yescharacteristic
180C23D4×D19911yescharacteristic
181Q8—7961no—
182Q8—7962no—
183Q8—7961no—
184Q8—7961no—
185D4—41no—
186D4—7961no—
187D4—7961no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—7961no—
193C2×C4—7961no—
194C2×C4—3981no—
195C2×C4—3981no—
196C2×C4—3981no—
197C2×C4—3981no—
198C2×C4—3981no—
199C2×C4—3981no—
200C2×C4—3981no—
201C2×C4—3981no—
202C2×C4—1991no—
203C2×C4—1991no—
204C2×C4—1991no—
205C2×C4D4×D19911yescharacteristic
206C8—41no—
207C8—7961no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—7961no—
213C22—3981no—
214C22C22⋊C4×D19911yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—7961no—
220C4—7961no—
221C4—15921no—
222C4—3981no—
223C4—3981no—
224C4—1991no—
225C4—1991no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—7961no—
230C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC23×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#352×C199C2
socleC2×C199C2×#99
prime radicalC2×#15691885C22
cosocleC2×D4×C199C23
hypercenter#352D199
hypocenterC199#2123
LCSG ▷ C23×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #27163809C2semidirect
5. #27163808C2semidirect
6. #27163807C2semidirect
7. #352×C199C2semidirect
9. C22×Dic398C4semidirect
13. #20263939C4semidirect
14. #20263938C4semidirect
19. #20263911C4semidirect
115. C199#2123semidirect
123. #352D199semidirect

Maximal subgroups

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

subgroupindex
#271645902
#271639992
#271639902
#271638092
#271638082
#271638072
#352×C1992
#2123199