Finite Group Databasebeta

← #38769729

Subgroups — #38769729

Order 25472.

340 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#15691875C211yesmaximal, characteristic
2C2×#20263878C211yesmaximal, characteristic, complemented
3C2×#20263875C211yesmaximal, characteristic
4C2×#20263865C211yesmaximal, characteristic, complemented
5#27163923C211yesmaximal, characteristic, complemented
6#27163919C211yesmaximal, characteristic
7C2×#107×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C22×D796C2211yescharacteristic
10C2×#15691875—22no—
11C2×#15691875—22no—
12C2×#15691875C414yescomplemented
13C22×C199⋊C8C2211yescharacteristic
14C2×(C2×D199)⋊C4—22no—
15C2×C4×Dic199C2211yescharacteristic
16#20263878—22no—
17#20263875—22no—
18#20263865—22no—
19C199⋊#95—22no—
20C22×C8×C199C2211yescharacteristic
21C2×C4⋊C4×C199C2211yescharacteristic
22C2×C4⋊C4×C199C2211yescharacteristic
23#107×C199—22no—
24C23×D199—21no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398C2×C412yes—
31C2×D796—21no—
32C2×D796—21no—
33C2×D796—21no—
34C2×D796—21no—
35C2×D796—21no—
36C2×D796C2×C412yes—
37C22×Dic199—21no—
38C22×Dic199D412yes—
39#15691875—28no—
40C2×C199⋊C8—21no—
41C2×C199⋊C8—21no—
42C2×C199⋊C8D412yes—
43(C2×D199)⋊C4—44no—
44C4×Dic199—21no—
45C4×Dic199—21no—
46C22×C4×C199—21no—
47C22×C4×C199—21no—
48C22×C4×C199C2311yescharacteristic
49C2×C8×C199C2×C412yes—
50C2×C8×C199D412yes—
51C2×C8×C199D412yes—
52C4⋊C4×C199—21no—
53C4⋊C4×C199—21no—
54C4⋊C4×C199—21no—
55C4⋊C4×C199—21no—
56C22×D199—22no—
57C22×D199—41no—
58C22×D199—41no—
59C22×D199—42no—
60C22×D199—42no—
61Dic398—22no—
62Dic398—22no—
63Dic398—41no—
64Dic398—41no—
65D796—22no—
66D796—22no—
67D796—41no—
68D796—41no—
69C2×Dic199—22no—
70C2×Dic199—22no—
71C2×Dic199—22no—
72C2×Dic199—22no—
73C2×Dic199—41no—
74C2×Dic199—41no—
75C199⋊C8—22no—
76C23×C199C2×D411yescharacteristic
77C2×C4×C199—41no—
78C2×C4×C199—41no—
79C2×C4×C199—42no—
80C2×C4×C199—42no—
81C2×C4×C199C22×C411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×D411yescharacteristic
85C2×C4×C199#4111yescharacteristic
86C2×C4×C199#4111yescharacteristic
87C8×C199C22⋊C414yescomplemented
88C22×C199C4×D411yescharacteristic
89C22×C199#12111yescharacteristic
90C22×C199#12711yescharacteristic
91C22×C199AGL(1,ℤ/8)12yes—
92C22×C199#13712yes—
93C2×D199—42no—
94C2×D199—44no—
95C2×D199—82no—
96C2×D199—81no—
97C2×D199—81no—
98C4×C199—81no—
99C4×C199—81no—
100C4×C199C2×C22⋊C411yescharacteristic
101C4×C199C4×D411yescharacteristic
102C4×C199#12111yescharacteristic
103C4×C199#12411yescharacteristic
104Dic199—42no—
105Dic199—42no—
106C2×C199#39011yescharacteristic
107C2×C199#44012yes—
108C2×C199#46812yes—
109C2×C199#49611yescharacteristic
110C2×C199#49711yescharacteristic
111D199—82no—
112C199#197311yescharacteristic, complemented, Sylow
113#1973—1991nomaximal, Sylow
114C22×SD16—1991no—
115C2×#107D19911yescharacteristic, complemented
116C2×#103—1991no—
117C2×#102—1991no—
118C2×#97—1991no—
119#391—1991no—
120#390—1991no—
121C22×Q8—1991no—
122C22×D4—1991no—
123C2×SD16—3982no—
124C2×SD16—3982no—
125C2×SD16—1994no—
126C22×C8—1991no—
127C22×C8C2×D19911yescharacteristic
128C2×C4⋊C4C2×D19911yescharacteristic
129C2×C4⋊C4C2×D19911yescharacteristic
130C2×C22⋊C4—3982no—
131C2×C42—1991no—
132#107—22no—
133#103—3982no—
134#102—3982no—
135#97—3982no—
136#95—3982no—
137C24—3981no—
138C2×Q8—3981no—
139C2×Q8—3981no—
140C2×Q8—3981no—
141C2×Q8—3981no—
142C2×Q8—3981no—
143C2×Q8—1992no—
144C2×D4—3981no—
145C2×D4—3981no—
146C2×D4—3981no—
147C2×D4—3981no—
148C2×D4—3981no—
149C2×D4—1992no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—3981no—
153C22×C4—1992no—
154C22×C4C22×D19911yescharacteristic
155SD16—3988no—
156C2×C8—3981no—
157C2×C8—3981no—
158C2×C8—1992no—
159C2×C8C4×D19912yes—
160C2×C8C22⋊D19912yes—
161C2×C8D79612yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C22⋊C4—7964no—
167C42—3981no—
168C42—3981no—
169C23—7961no—
170C23—7962no—
171C23—7962no—
172C23—7961no—
173C23—3982no—
174C23D4×D19911yescharacteristic
175Q8—7961no—
176Q8—7961no—
177Q8—3982no—
178Q8—3982no—
179D4—7961no—
180D4—7961no—
181D4—3982no—
182D4—3982no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—7961no—
188C2×C4—7961no—
189C2×C4—3982no—
190C2×C4—3982no—
191C2×C4—3982no—
192C2×C4—3982no—
193C2×C4C2×C4×D19911yescharacteristic
194C2×C4D4×D19911yescharacteristic
195C2×C4C2×C22⋊D19911yescharacteristic
196C2×C4C2×D79611yescharacteristic
197C2×C4#1569188711yescharacteristic
198C2×C4#1569188711yescharacteristic
199C8—3982no—
200C8(C2×D199)⋊C414yescomplemented
201C22—7962no—
202C22—7964no—
203C22—15922no—
204C22—15921no—
205C22—15921no—
206C22#2026391711yescharacteristic
207C22#2026393011yescharacteristic
208C22#2026395111yescharacteristic
209C22#2026397912yes—
210C22#2026398712yes—
211C4—81no—
212C4—81no—
213C4—7962no—
214C4—7962no—
215C4C2×(C2×D199)⋊C411yescharacteristic
216C4#2026391711yescharacteristic
217C4#2026393011yescharacteristic
218C4#2026394411yescharacteristic
219C2—15922no—
220C2#2716391911yescharacteristic
221C2#2716411712yes—
222C2#2716426512yes—
223C2#2716440611yescharacteristic
224C2#2716441111yescharacteristic
subgroupG/H
centerC23D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#107×C199C2
socleC23×C199C2×D4
prime radicalC22×D796C22
cosocleC22×C4×C199C23
hypercenterC2×#107D199
hypocenterC199#1973
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#20263878C2semidirect
4. C2×#20263865C2semidirect
5. #27163923C2semidirect
7. C2×#107×C199C2semidirect
12. C2×#15691875C4semidirect
87. C8×C199C22⋊C4semidirect
112. C199#1973semidirect
115. C2×#107D199semidirect
200. C8(C2×D199)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#156918752
C2×#202638782
C2×#202638752
C2×#202638652
#271639232
#271639192
C2×#107×C1992
#1973199