Finite Group Databasebeta

← #38769725

Subgroups — #38769725

Order 25472.

376 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15691871C211yesmaximal, characteristic
2C2×#20263886C211yesmaximal, characteristic, complemented
3C2×#20263876C211yesmaximal, characteristic, complemented
4C2×#20263875C211yesmaximal, characteristic, complemented
5C2×#20263866C211yesmaximal, characteristic, complemented
6#27163920C211yesmaximal, characteristic, complemented
7#390×C199C211yesmaximal, characteristic, complemented
8C22×D796C2211yescharacteristic
9C2×#15691871—22no—
10C2×#15691871—22no—
11C2×#15691871C414yescomplemented
12C22×C199⋊C8C2211yescharacteristic
13C22×C199⋊C8C2211yescharacteristic
14C2×C4⋊Dic199C2211yescharacteristic
15C2×(C2×D199)⋊C4—22no—
16#20263886—22no—
17#20263876—22no—
18#20263875—22no—
19#20263866—22no—
20C22×D4×C199C2211yescharacteristic
21C2×C4⋊C4×C199C2211yescharacteristic
22C2×C22⋊C4×C199—22no—
23C2×C42×C199C2211yescharacteristic
24C23×D199—21no—
25C2×D796—21no—
26C2×D796—21no—
27C2×D796—21no—
28C2×D796—21no—
29C2×D796—21no—
30C2×D796C2×C412yes—
31C22×Dic199—21no—
32#15691871—28no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8C2×C412yes—
36C2×C199⋊C8D412yes—
37C2×C199⋊C8D412yes—
38C2×C199⋊C8D412yes—
39C4⋊Dic199—21no—
40C4⋊Dic199—21no—
41(C2×D199)⋊C4—44no—
42C24×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199—21no—
45C2×D4×C199—21no—
46C2×D4×C199—21no—
47C2×D4×C199—21no—
48C2×D4×C199C2×C412yes—
49C22×C4×C199—21no—
50C22×C4×C199D412yes—
51C22×C4×C199C2311yescharacteristic
52C4⋊C4×C199—21no—
53C4⋊C4×C199—21no—
54C22⋊C4×C199—44no—
55C42×C199—21no—
56C42×C199—21no—
57C22×D199—22no—
58C22×D199—41no—
59C22×D199—41no—
60C22×D199—42no—
61C22×D199—42no—
62D796—22no—
63D796—22no—
64D796—41no—
65D796—41no—
66C2×Dic199—41no—
67C2×Dic199—42no—
68C199⋊C8—22no—
69C199⋊C8C22⋊C414yescomplemented
70C23×C199—22no—
71C23×C199—41no—
72C23×C199—41no—
73C23×C199—42no—
74C23×C199—42no—
75C23×C199C2×D411yescharacteristic
76D4×C199—22no—
77D4×C199—22no—
78D4×C199—41no—
79D4×C199—41no—
80C2×C4×C199—22no—
81C2×C4×C199—22no—
82C2×C4×C199—22no—
83C2×C4×C199—41no—
84C2×C4×C199—42no—
85C2×C4×C199C22×C411yescharacteristic
86C2×C4×C199C2×D411yescharacteristic
87C2×C4×C199C2×D411yescharacteristic
88C2×C4×C199C2×D411yescharacteristic
89C2×C4×C199#4111yescharacteristic
90C2×C4×C199#4111yescharacteristic
91C22×C199—42no—
92C22×C199—44no—
93C22×C199—82no—
94C22×C199—81no—
95C22×C199—81no—
96C22×C199C4×D411yescharacteristic
97C22×C199#12111yescharacteristic
98C22×C199#12711yescharacteristic
99C22×C199AGL(1,ℤ/8)12yes—
100C22×C199AGL(1,ℤ/8)12yes—
101C2×D199—42no—
102C2×D199—44no—
103C2×D199—82no—
104C2×D199—81no—
105C2×D199—81no—
106C4×C199—42no—
107C4×C199—81no—
108C4×C199C2×C22⋊C411yescharacteristic
109C4×C199C4×D411yescharacteristic
110C4×C199#12111yescharacteristic
111C4×C199#12411yescharacteristic
112Dic199—81no—
113C2×C199—82no—
114C2×C199#39011yescharacteristic
115C2×C199#44212yes—
116C2×C199#46912yes—
117C2×C199#49611yescharacteristic
118C2×C199#49611yescharacteristic
119D199—82no—
120C199#197211yescharacteristic, complemented, Sylow
121#1972—1991nomaximal, Sylow
122C22×D8—1991no—
123C2×#106—1991no—
124C2×#102—1991no—
125C2×#102—1991no—
126C2×#97—1991no—
127#390—1991no—
128#390D19911yescharacteristic, complemented
129C22×D4—1991no—
130C22×D4C2×D19911yescharacteristic
131C2×D8—3982no—
132C2×D8—3982no—
133C2×D8—1994no—
134C22×C8—1991no—
135C22×C8—1991no—
136C2×C4⋊C4—1991no—
137C2×C4⋊C4C2×D19911yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—3982no—
140C2×C42C2×D19911yescharacteristic
141#106—3982no—
142#102—3982no—
143#102—3982no—
144#97—3982no—
145C24—21no—
146C24—3981no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—3981no—
153C2×D4—3981no—
154C2×D4—3981no—
155C2×D4—3981no—
156C2×D4—3981no—
157C2×D4—1992no—
158C2×D4C4×D19912yes—
159C22×C4—21no—
160C22×C4—3981no—
161C22×C4C22⋊D19912yes—
162C22×C4C22×D19911yescharacteristic
163D8—3988no—
164C2×C8—3981no—
165C2×C8—3981no—
166C2×C8—1992no—
167C2×C8—1992no—
168C2×C8—1992no—
169C2×C8—1992no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—3981no—
173C4⋊C4—3981no—
174C22⋊C4—44no—
175C22⋊C4—7964no—
176C42—21no—
177C42—21no—
178C23—22no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—41no—
183C23—7961no—
184C23—7962no—
185C23—7962no—
186C23—7961no—
187C23—3982no—
188C23C2×C22⋊D19911yescharacteristic
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4—7961no—
194D4—7961no—
195D4—3982no—
196D4—3982no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—7961no—
203C2×C4—7962no—
204C2×C4C2×C4×D19911yescharacteristic
205C2×C4D4×D19911yescharacteristic
206C2×C4D4×D19911yescharacteristic
207C2×C4D4×D19911yescharacteristic
208C2×C4#1569188511yescharacteristic
209C2×C4#1569188611yescharacteristic
210C8—3982no—
211C8—1994no—
212C22—42no—
213C22—44no—
214C22—82no—
215C22—81no—
216C22—81no—
217C22—7962no—
218C22—7964no—
219C22—15922no—
220C22—15921no—
221C22—15921no—
222C22C4×C22⋊D19911yescharacteristic
223C22#2026392711yescharacteristic
224C22#2026395111yescharacteristic
225C22#2026398012yes—
226C22#2026398112yes—
227C4—42no—
228C4—81no—
229C4—15921no—
230C4C22⋊C4×D19911yescharacteristic
231C4(C22⋊D199)⋊C411yescharacteristic
232C4#2026392811yescharacteristic
233C4#2026394111yescharacteristic
234C2—82no—
235C2—15922no—
236C2#2716391811yescharacteristic
237C2#2716412812yes—
238C2#2716426912yes—
239C2#2716440511yescharacteristic
240C2#2716440811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#390×C199C2
socleC23×C199C2×D4
prime radicalC22×#15691871C2
cosocleC22×C4×C199C23
hypercenter#390D199
hypocenterC199#1972
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#20263886C2semidirect
3. C2×#20263876C2semidirect
4. C2×#20263875C2semidirect
5. C2×#20263866C2semidirect
6. #27163920C2semidirect
7. #390×C199C2semidirect
11. C2×#15691871C4semidirect
69. C199⋊C8C22⋊C4semidirect
120. C199#1972semidirect
128. #390D199semidirect

Maximal subgroups

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

subgroupindex
C22×#156918712
C2×#202638862
C2×#202638762
C2×#202638752
C2×#202638662
#271639202
#390×C1992
#1972199