Finite Group Databasebeta

← #38770490

Subgroups — #38770490

Order 25472.

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

normal subgroups only

This group has 264 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#27164526C211yesmaximal, characteristic
2#27163999C211yesmaximal, characteristic, complemented
3#27163990C211yesmaximal, characteristic, complemented
4#27163808C211yesmaximal, characteristic, complemented
5#27163806C211yesmaximal, characteristic, complemented
6#27163804C211yesmaximal, characteristic, complemented
7#352×C199C211yesmaximal, characteristic, complemented
8C2×#15691885C2211yescharacteristic
9C22×C4×D199C411yescharacteristic, complemented
10M16×D199—21no—
11#20263938C411yescharacteristic, complemented
12#20263933—21no—
13#20263928—21no—
14#20263925C411yescharacteristic, complemented
15#20263923C2211yescharacteristic
16#20263917—21no—
17(C22⋊D199)⋊C4—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
27C23×D199C2×C411yescharacteristic
28#15691885—41no—
29#15691885—41no—
30C2×Dic398D411yescharacteristic
31C2×D796—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199C2×C411yescharacteristic
34C2×C22⋊D199D411yescharacteristic
35C22×Dic199C2×C411yescharacteristic
36C22×Dic199D411yescharacteristic
37C2×C4×D199—21no—
38C2×C4×D199—21no—
39C2×C4×D199—41no—
40C2×C4×D199—41no—
41C2×C4×D199D411yescharacteristic
42#15691870—21no—
43#15691869—41no—
44C8×D199—41no—
45Dic199⋊C4—21no—
46Dic199⋊C4—21no—
47Dic199⋊C4—41no—
48C22⋊Dic199—21no—
49C22⋊Dic199C2×C411yescharacteristic
50(C2×D199)⋊C4—21no—
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×D199—21no—
63C22×D199—42no—
64C22×D199—41no—
65C22×D199C22⋊C411yescharacteristic
66Dic398—41no—
67D796—42no—
68C22⋊D199—41no—
69C22⋊D199—41no—
70C22⋊D199—42no—
71C2×Dic199—21no—
72C2×Dic199—21no—
73C2×Dic199—21no—
74C2×Dic199—21no—
75C2×Dic199—21no—
76C2×Dic199—41no—
77C2×Dic199C22⋊C411yescharacteristic
78C2×Dic199C22⋊C411yescharacteristic
79C2×Dic199C22⋊C411yescharacteristic
80C4×D199—21no—
81C4×D199—21no—
82C4×D199—42no—
83C4×D199—41no—
84C4×D199—41no—
85C199⋊C8—41no—
86C23×C199C22×C411yescharacteristic
87C23×C199C2×D411yescharacteristic
88D4×C199—41no—
89C2×C4×C199—21no—
90C2×C4×C199—21no—
91C2×C4×C199—41no—
92C2×C4×C199—41no—
93C2×C4×C199C2×D411yescharacteristic
94C8×C199—41no—
95C22×C199—21no—
96C22×C199—21no—
97C22×C199—41no—
98C22×C199—41no—
99C22×C199C2×C22⋊C411yescharacteristic
100C2×D199—22no—
101C2×D199—21no—
102C2×D199—41no—
103C2×D199—42no—
104C2×D199—41no—
105C2×D199—41no—
106C2×D199—42no—
107C4×C199—21no—
108C4×C199—41no—
109C4×C199—41no—
110C4×C199—81no—
111Dic199—21no—
112Dic199—21no—
113Dic199—41no—
114Dic199—81no—
115Dic199#9911yescharacteristic
116Dic199#9911yescharacteristic
117C2×C199—21no—
118C2×C199—41no—
119C2×C199—41no—
120C2×C199C2×#9911yescharacteristic
121D199—41no—
122D199—42no—
123C199#212011yescharacteristic, complemented, Sylow
124#2120—1991nomaximal, Sylow
125#525—1991no—
126#413—1991no—
127#410—1991no—
128#352—1991no—
129#352D19911yescharacteristic, complemented
130AΣL(1,𝔽16)—1991no—
131AΣL(1,𝔽16)—1991no—
132C2×#41—1991no—
133C23×C4—1991no—
134C2×M16—3981no—
135#123—1991no—
136#123C2×D19911yescharacteristic
137C22⋊Q8—3981no—
138#121—3981no—
139#120—1991no—
140#120—1991no—
141C4×D4—3981no—
142C4×D4—3981no—
143#117—3981no—
144#117—1991no—
145#101—3981no—
146#100—1991no—
147#100C2×D19911yescharacteristic
148#99—3981no—
149#99—1991no—
150#99C2×D19911yescharacteristic
151C24—1991no—
152#41—7961no—
153#41—7961no—
154C2×Q8—1991no—
155C2×D4—3981no—
156C2×D4—3981no—
157C2×D4—1991no—
158C2×D4—1991no—
159C2×D4C22×D19911yescharacteristic
160C22×C4—7961no—
161C22×C4—7961no—
162C22×C4—3981no—
163C22×C4—3981no—
164C22×C4—1991no—
165C22×C4—1991no—
166C22×C4—1991no—
167C22×C4C4×D19911yescharacteristic
168M16—21no—
169M16—7961no—
170M16—3981no—
171C2×C8—7961no—
172C4⋊C4—21no—
173C4⋊C4—7961no—
174C4⋊C4—3981no—
175C4⋊C4—3981no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—3981no—
179C22⋊C4—3981no—
180C22⋊C4—3981no—
181C22⋊C4—3981no—
182C22⋊C4—1991no—
183C22⋊C4C4×D19911yescharacteristic
184C42—7961no—
185C42—3981no—
186C23—7961no—
187C23—7962no—
188C23—3981no—
189C23—1991no—
190C23C2×C4×D19911yescharacteristic
191C23D4×D19911yescharacteristic
192Q8—7961no—
193D4—41no—
194D4—7962no—
195D4—7962no—
196D4—7961no—
197D4—7961no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—7962no—
203C2×C4—7961no—
204C2×C4—7961no—
205C2×C4—7961no—
206C2×C4—3981no—
207C2×C4—3981no—
208C2×C4—3981no—
209C2×C4—3981no—
210C2×C4—3981no—
211C2×C4—3981no—
212C2×C4—3981no—
213C2×C4—1991no—
214C2×C4—1991no—
215C2×C4—1991no—
216C2×C4D4×D19911yescharacteristic
217C8—41no—
218C8—7961no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—7961no—
224C22—7961no—
225C22—7962no—
226C22—7962no—
227C22—7961no—
228C22—3982no—
229C22—3981no—
230C22C22⋊C4×D19911yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—7961no—
236C4—15921no—
237C4—3981no—
238C4—3981no—
239C4—1991no—
240C4—1991no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2—7962no—
245C2—7961no—
246C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC23×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#352×C199C2
socleC2×C199C2×#99
prime radical#27164526C2
cosocleC2×D4×C199C23
hypercenter#352D199
hypocenterC199#2120
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
2. #27163999C2semidirect
3. #27163990C2semidirect
4. #27163808C2semidirect
5. #27163806C2semidirect
6. #27163804C2semidirect
7. #352×C199C2semidirect
9. C22×C4×D199C4semidirect
11. #20263938C4semidirect
14. #20263925C4semidirect
19. #20263911C4semidirect
123. C199#2120semidirect
129. #352D199semidirect

Maximal subgroups

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

subgroupindex
#271645262
#271639992
#271639902
#271638082
#271638062
#271638042
#352×C1992
#2120199