Finite Group Databasebeta

← #38769886

Subgroups — #38769886

Order 25472.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263986C211yesmaximal, characteristic
2#27164592C211yesmaximal, characteristic
3C2×#20263883C211yesmaximal, characteristic
4#27164027C211yesmaximal, characteristic, complemented
5C2×#20263874C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9#20263986—22no—
10#20263986—22no—
11C2×#15691876—21no—
12C2×#15691873—21no—
13C2×#15691870C2211yescharacteristic
14C2×#15691870C2211yescharacteristic
15#20263953—21no—
16#20263944—21no—
17#20263938—22no—
18#20263931—22no—
19#20263914C2211yescharacteristic
20#20263883—22no—
21#20263879—22no—
22#20263874—22no—
23C22×Q8×C199C2211yescharacteristic
24C22×Q8×C199C2211yescharacteristic
25C4⋊Q8×C199—22no—
26C4⋊Q8×C199—22no—
27C2×C4⋊C4×C199—22no—
28C2×C42×C199C2211yescharacteristic
29#15691886—21no—
30#15691886—21no—
31#15691886—21no—
32#15691886—21no—
33C2×Dic398D411yescharacteristic
34C2×D796D411yescharacteristic
35C2×C22⋊D199—21no—
36C2×C4×D199—21no—
37#15691876—24no—
38#15691873—24no—
39#15691870—21no—
40#15691870—21no—
41#15691870—21no—
42#15691870D412yes—
43C2×C199⋊C8—21no—
44C2×C199⋊C8—21no—
45C4⋊Dic199D412yes—
46Dic199⋊C4—24no—
47C22⋊Dic199—21no—
48(C2×D199)⋊C4—24no—
49C4×Dic199—21no—
50C2×Q8×C199—21no—
51C2×Q8×C199—21no—
52C2×Q8×C199—21no—
53C2×Q8×C199—21no—
54C2×Q8×C199—21no—
55C2×Q8×C199—21no—
56C2×Q8×C199—21no—
57C2×Q8×C199—21no—
58C2×Q8×C199—21no—
59C2×Q8×C199—21no—
60C2×Q8×C199—21no—
61C2×Q8×C199D412yes—
62C22×C4×C199—21no—
63C22×C4×C199—21no—
64C22×C4×C199D412yes—
65C22×C4×C199C2311yescharacteristic
66C4⋊C4×C199—28no—
67C42×C199—21no—
68C42×C199D412yes—
69C22×D199—21no—
70Dic398—21no—
71Dic398—21no—
72D796—21no—
73D796—21no—
74C22⋊D199—41no—
75C22⋊D199—41no—
76C2×Dic199—22no—
77C2×Dic199—21no—
78C4×D199—41no—
79C4×D199—41no—
80C199⋊C8—22no—
81C199⋊C8—41no—
82C23×C199C2×D411yescharacteristic
83Q8×C199—22no—
84Q8×C199—22no—
85Q8×C199—41no—
86Q8×C199—41no—
87Q8×C199—41no—
88Q8×C199—41no—
89Q8×C199—41no—
90Q8×C199—41no—
91C2×C4×C199—24no—
92C2×C4×C199—22no—
93C2×C4×C199—22no—
94C2×C4×C199—22no—
95C2×C4×C199—22no—
96C2×C4×C199—22no—
97C2×C4×C199—41no—
98C2×C4×C199—41no—
99C2×C4×C199C2×D411yescharacteristic
100C2×C4×C199C2×D411yescharacteristic
101C2×C4×C199C2×D411yescharacteristic
102C2×C4×C199C2×D411yescharacteristic
103C2×C4×C199C2×D411yescharacteristic
104C2×C4×C199#4111yescharacteristic
105C22×C199—22no—
106C22×C199#12011yescharacteristic
107C22×C199#12011yescharacteristic
108C22×C199#12111yescharacteristic
109C2×D199—41no—
110C2×D199—42no—
111C4×C199—24no—
112C4×C199—42no—
113C4×C199—42no—
114C4×C199#12011yescharacteristic
115C4×C199#12111yescharacteristic
116C4×C199#12111yescharacteristic
117C4×C199#12711yescharacteristic
118Dic199—41no—
119Dic199—42no—
120C2×C199—21no—
121C2×C199—21no—
122C2×C199#39211yescharacteristic
123C2×C199#45612yes—
124D199—81no—
125C199#200511yescharacteristic, complemented, Sylow
126#2005—1991nomaximal, Sylow
127C2×#137—1991no—
128#536—1991no—
129C2×C4⋊Q8D19911yescharacteristic, complemented
130C2×#104—1991no—
131#419—1991no—
132C2×#101—1991no—
133#339—1991no—
134C2×#41—1991no—
135C22×Q8C2×D19911yescharacteristic
136C22×Q8C2×D19911yescharacteristic
137#137—3982no—
138#137—3982no—
139C2×Q16—3981no—
140C2×SD16—3981no—
141C2×M16—1991no—
142C2×M16—1991no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—3981no—
146#124—3981no—
147#123—3982no—
148C22⋊Q8—3982no—
149#117—1991no—
150C2×C4⋊C4—22no—
151C2×C42C2×D19911yescharacteristic
152#104—3982no—
153#103—3982no—
154#101—3982no—
155#41—3981no—
156#41—3981no—
157#41—3981no—
158#41—3981no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—1991no—
171C2×Q8C22⋊D19912yes—
172C2×D4—3981no—
173C2×D4—1991no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—3981no—
177C22×C4C22⋊D19912yes—
178C22×C4C22×D19911yescharacteristic
179Q16—3984no—
180SD16—3984no—
181M16—3981no—
182M16—3981no—
183M16—3981no—
184M16—1992no—
185C2×C8—3981no—
186C2×C8—3981no—
187C4⋊C4—28no—
188C4⋊C4—3984no—
189C4⋊C4—1992no—
190C22⋊C4—3984no—
191C22⋊C4—3981no—
192C42—21no—
193C42—3981no—
194C42C22⋊D19912yes—
195C23—3981no—
196C23C2×C22⋊D19911yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—3981no—
206Q8—3981no—
207D4—7961no—
208D4—7961no—
209D4—3981no—
210D4—3981no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—7961no—
220C2×C4—7961no—
221C2×C4—3982no—
222C2×C4—3981no—
223C2×C4D4×D19911yescharacteristic
224C2×C4D4×D19911yescharacteristic
225C2×C4D4×D19911yescharacteristic
226C2×C4C2×C22⋊D19911yescharacteristic
227C2×C4C2×C22⋊D19911yescharacteristic
228C2×C4#1569188511yescharacteristic
229C8—7961no—
230C8—3982no—
231C22—22no—
232C22—7961no—
233C22—7962no—
234C22#2026392311yescharacteristic
235C22#2026392411yescharacteristic
236C22#2026392711yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—7961no—
241C4—7962no—
242C4#2026392311yescharacteristic
243C4#2026392611yescharacteristic
244C4#2026392711yescharacteristic
245C4#2026395111yescharacteristic
246C2—21no—
247C2—21no—
248C2—15921no—
249C2#2716392611yescharacteristic
250C2#2716420012yes—
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×C4⋊Q8×C199C2
socleC22×C199#120
prime radicalC2×#15691886C22
hypercenterC2×C4⋊Q8D199
hypocenterC199#2005
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #27164027C2semidirect
5. C2×#20263874C2semidirect
6. #27163778C2semidirect
7. C2×C4⋊Q8×C199C2semidirect
125. C199#2005semidirect
129. C2×C4⋊Q8D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639862
#271645922
C2×#202638832
#271640272
C2×#202638742
#271637782
C2×C4⋊Q8×C1992
#2005199