Finite Group Databasebeta

← #38769381

Subgroups — #38769381

Order 25472.

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

normal subgroups only

This group has 362 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#27164587C211yesmaximal, characteristic
2#27164492C211yesmaximal, characteristic, complemented
3#27164036C211yesmaximal, characteristic, complemented
4#27164034C211yesmaximal, characteristic, complemented
5C2×#20263871C211yesmaximal, characteristic, complemented
6#27163774C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9C22×D796C2211yescharacteristic
10C2×#15691870C2211yescharacteristic
11#20263952C412yescomplemented
12#20263943C412yescomplemented
13#20263930—22no—
14#20263925—22no—
15C4×D796—22no—
16#20263917—22no—
17#20263914C2211yescharacteristic
18C22⋊C4×D199—22no—
19#20263883—22no—
20#20263881—22no—
21#20263871—22no—
22C2×M16×C199C2211yescharacteristic
23#117×C199C2211yescharacteristic
24#117×C199C2211yescharacteristic
25C23×D199D412yes—
26#15691886—21no—
27#15691886—21no—
28#15691886—21no—
29#15691886—21no—
30C2×Dic398C2×C411yescharacteristic, complemented
31C2×D796—24no—
32C2×D796—21no—
33C2×D796—21no—
34C2×D796—21no—
35C2×D796—21no—
36C2×D796C2×C412yes—
37C2×D796C2×C411yescharacteristic, complemented
38C2×D796D412yes—
39C2×C22⋊D199—21no—
40C2×C4×D199—24no—
41C2×C4×D199—21no—
42#15691870—21no—
43#15691870—21no—
44C2×C199⋊C8—21no—
45C4⋊Dic199—21no—
46C22⋊Dic199—21no—
47(C2×D199)⋊C4—24no—
48(C2×D199)⋊C4—24no—
49C4×Dic199D412yescomplemented
50C22×C4×C199C2311yescharacteristic
51M16×C199—21no—
52M16×C199—21no—
53C2×C8×C199—21no—
54C4⋊C4×C199—21no—
55C4⋊C4×C199—21no—
56C22⋊C4×C199—21no—
57C22⋊C4×C199—21no—
58C42×C199C2×C412yescomplemented
59C42×C199D412yescomplemented
60C22×D199—22no—
61C22×D199—22no—
62C22×D199—21no—
63C22×D199—82no—
64C22×D199C22⋊C414yes—
65Dic398—21no—
66Dic398—21no—
67D796—22no—
68D796—22no—
69D796—21no—
70D796—21no—
71D796—22no—
72D796—22no—
73D796—84no—
74C22⋊D199—41no—
75C22⋊D199—41no—
76C2×Dic199—21no—
77C2×Dic199—22no—
78C4×D199—41no—
79C4×D199—41no—
80C4×D199—84no—
81C199⋊C8—41no—
82C23×C199C2×D411yescharacteristic
83C2×C4×C199—22no—
84C2×C4×C199—22no—
85C2×C4×C199C22×C411yescharacteristic
86C2×C4×C199C2×D411yescharacteristic
87C2×C4×C199C2×D411yescharacteristic
88C2×C4×C199C2×D411yescharacteristic
89C2×C4×C199#4111yescharacteristic
90C2×C4×C199#4111yescharacteristic
91C8×C199—41no—
92C22×C199—41no—
93C22×C199C2×C22⋊C411yescharacteristic
94C22×C199#12011yescharacteristic
95C22×C199#12311yescharacteristic
96C2×D199—24no—
97C2×D199—44no—
98C2×D199—44no—
99C2×D199—42no—
100C2×D199—41no—
101C2×D199—82no—
102C4×C199—42no—
103C4×C199—42no—
104C4×C199C4×D411yescharacteristic
105C4×C199C4×D411yescharacteristic
106C4×C199#12111yescharacteristic
107C4×C199#12111yescharacteristic
108Dic199—42no—
109Dic199—41no—
110C2×C199—21no—
111C2×C199—21no—
112C2×C199—21no—
113C2×C199#38611yescharacteristic
114D199—44no—
115D199—81no—
116C199#191111yescharacteristic, complemented, Sylow
117#1911—1991nomaximal, Sylow
118#535—1991no—
119#518—1991no—
120#421—1991no—
121#421—1991no—
122C2×#100—1991no—
123H(3,ℤ/4)—1991no—
124H(3,ℤ/4)D19911yescharacteristic, complemented
125C2×#41—1991no—
126C22×D4—1991no—
127C2×M16—1991no—
128C2×M16C2×D19911yescharacteristic
129#127—1992no—
130#124—1992no—
131#121—3982no—
132#120—3982no—
133C4×D4—3982no—
134C4×D4—3982no—
135#117—1991no—
136#117C2×D19911yescharacteristic
137#117C2×D19911yescharacteristic
138C2×C22⋊C4—3982no—
139#104—3982no—
140#104—3982no—
141#100—3982no—
142C24—1992no—
143#41—3981no—
144#41—3981no—
145#41—3981no—
146#41—3981no—
147C2×Q8—1991no—
148C2×D4—3984no—
149C2×D4—3981no—
150C2×D4—3981no—
151C2×D4—3981no—
152C2×D4—3981no—
153C2×D4—3981no—
154C2×D4—1991no—
155C2×D4—1992no—
156C2×D4—1992no—
157C22×C4—3984no—
158C22×C4—3981no—
159C22×C4C22×D19911yescharacteristic
160M16—21no—
161M16—21no—
162M16—3981no—
163M16—3981no—
164C2×C8—21no—
165C2×C8—3981no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3981no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—3984no—
172C22⋊C4—3984no—
173C22⋊C4—3981no—
174C42—1992no—
175C42C4×D19912yescomplemented
176C42D79612yescomplemented
177C23—15922no—
178C23—3981no—
179C23—3982no—
180C23—3982no—
181C23—1994no—
182C23D4×D19911yescharacteristic
183Q8—3981no—
184Q8—3981no—
185D4—7961no—
186D4—7961no—
187D4—15924no—
188D4—3981no—
189D4—3982no—
190D4—3981no—
191D4—3982no—
192D4—3982no—
193D4—3982no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—7961no—
197C2×C4—7961no—
198C2×C4—15924no—
199C2×C4—3981no—
200C2×C4—3982no—
201C2×C4C2×C4×D19911yescharacteristic
202C2×C4D4×D19911yescharacteristic
203C2×C4D4×D19911yescharacteristic
204C2×C4C2×D79611yescharacteristic
205C2×C4#1569188611yescharacteristic
206C2×C4#1569188711yescharacteristic
207C8—41no—
208C8—7961no—
209C22—41no—
210C22—7964no—
211C22—7964no—
212C22—7962no—
213C22—7961no—
214C22—15922no—
215C22—3984no—
216C22C22⋊C4×D19911yescharacteristic
217C22#2026392511yescharacteristic
218C22#2026394011yescharacteristic
219C4—42no—
220C4—42no—
221C4—7962no—
222C4—7961no—
223C4#2026391711yescharacteristic
224C4C4×D79611yescharacteristic
225C4#2026392811yescharacteristic
226C4#2026393011yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—7964no—
231C2—15921no—
232C2#2716389611yescharacteristic
subgroupG/H
centerC2#27163896
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingH(3,ℤ/4)×C199C2
socleC2×C199#386
prime radical#27164587C2
cosocleC22×C4×C199C23
hypercenterH(3,ℤ/4)D199
hypocenterC199#1911
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #27164492C2semidirect
3. #27164036C2semidirect
4. #27164034C2semidirect
5. C2×#20263871C2semidirect
6. #27163774C2semidirect
7. H(3,ℤ/4)×C199C2semidirect
11. #20263952C4semidirect
12. #20263943C4semidirect
30. C2×Dic398C2×C4semidirect
37. C2×D796C2×C4semidirect
49. C4×Dic199D4semidirect
58. C42×C199C2×C4semidirect
59. C42×C199D4semidirect
116. C199#1911semidirect
124. H(3,ℤ/4)D199semidirect
175. C42C4×D199semidirect
176. C42D796semidirect

Maximal subgroups

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

subgroupindex
#271645872
#271644922
#271640362
#271640342
C2×#202638712
#271637742
H(3,ℤ/4)×C1992
#1911199