Finite Group Databasebeta

← #38767770

Subgroups — #38767770

Order 25472.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164580C211yesmaximal, characteristic, complemented
2#27164154C212yesmaximal, complemented
3#27163927C211yesmaximal, characteristic, complemented
4#27163753C212yesmaximal, complemented
5#323×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic, complemented
7C22×C22⋊D199—21no—
8C22×C22⋊D199—21no—
9C2×D1592—22no—
10#20263939—22no—
11#20263929—21no—
12#20263929C2212yes—
13#20263925—22no—
14C2×(C2×D199)⋊C4—21no—
15#20263877—22no—
16C199⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yescomplemented
19C23×D199—21no—
20C23×D199—21no—
21C2×D796—21no—
22C2×D796—22no—
23C2×D796—42no—
24C2×D796D412yescomplemented
25C2×C22⋊D199—22no—
26C2×C22⋊D199—22no—
27C2×C22⋊D199—42no—
28C2×C22⋊D199—42no—
29C2×C22⋊D199—42no—
30C2×C22⋊D199—42no—
31C2×C22⋊D199—42no—
32C22×Dic199—21no—
33C22×Dic199—21no—
34D1592—44no—
35C4⋊Dic199—21no—
36C4⋊Dic199D412yescomplemented
37(C2×D199)⋊C4—22no—
38(C2×D199)⋊C4—22no—
39(C2×D199)⋊C4—42no—
40C24×C199D411yescharacteristic
41C22×C4×C199D411yescharacteristic
42C22×C4×C199C2311yescharacteristic
43C2×C8×C199—22no—
44C22⋊C4×C199—22no—
45C22×D199—22no—
46C22×D199—22no—
47C22×D199—41no—
48C22×D199—41no—
49C22×D199—42no—
50C22×D199—41no—
51C22×D199—41no—
52C22×D199—41no—
53C22×D199—41no—
54C22×D199—81no—
55C22×D199—81no—
56D796—24no—
57D796—42no—
58D796—42no—
59D796—81no—
60C22⋊D199—44no—
61C22⋊D199—48no—
62C22⋊D199—42no—
63C22⋊D199—42no—
64C2×Dic199—22no—
65C2×Dic199—22no—
66C2×Dic199—41no—
67C2×Dic199—42no—
68C2×Dic199—41no—
69C23×C199—21no—
70C23×C199—21no—
71C23×C199—41no—
72C23×C199—41no—
73C23×C199D812yescomplemented
74C23×C199C2×D411yescharacteristic
75C2×C4×C199—21no—
76C2×C4×C199—41no—
77C2×C4×C199—41no—
78C2×C4×C199C2×D412yes—
79C8×C199—42no—
80C22×C199—21no—
81C22×C199—21no—
82C22×C199—22no—
83C22×C199—42no—
84C22×C199—41no—
85C22×C199—41no—
86C22×C199—81no—
87C22×C199#12011yescharacteristic
88C22×C199C2×D811yescharacteristic
89C22×C199AGL(1,ℤ/8)11yescharacteristic
90C2×D199—42no—
91C2×D199—42no—
92C2×D199—42no—
93C2×D199—42no—
94C2×D199—42no—
95C2×D199—42no—
96C2×D199—81no—
97C2×D199—81no—
98C2×D199—84no—
99C2×D199—81no—
100C2×D199—81no—
101C4×C199—22no—
102C4×C199—41no—
103Dic199—42no—
104Dic199—42no—
105C2×C199—21no—
106C2×C199—21no—
107C2×C199—42no—
108C2×C199#44711yescharacteristic
109C2×C199#45311yescharacteristic
110C2×C199B(4,𝔽2)11yescharacteristic
111D199—82no—
112D199—82no—
113C199#159511yescharacteristic, complemented, Sylow
114#1595—1991nomaximal, Sylow
115#534—1991no—
116#447—1992no—
117#392—1991no—
118#327—1992no—
119#323D19911yescharacteristic, complemented
120C22×D4—3981no—
121C22×D4—3981no—
122C22×D4—1991no—
123C2×D8—3982no—
124#123—3982no—
125#121—3981no—
126#121—1992no—
127#120—3982no—
128C2×C22⋊C4—3981no—
129C2×C22⋊C4C2×D19911yescharacteristic
130#102—3982no—
131C22⋊C8C2×D19912yescomplemented
132#95—1991no—
133C24—3981no—
134C24—3981no—
135C24D79611yescharacteristic
136C2×D4—7962no—
137C2×D4—7962no—
138C2×D4—7962no—
139C2×D4—7962no—
140C2×D4—7962no—
141C2×D4—7962no—
142C2×D4—3981no—
143C2×D4—3982no—
144C2×D4—3982no—
145C2×D4—3982no—
146C2×D4—1992no—
147C22×C4—3981no—
148C22×C4—3981no—
149C22×C4D79611yescharacteristic
150C22×C4C22×D19911yescharacteristic
151D8—7964no—
152C2×C8—22no—
153C4⋊C4—3981no—
154C4⋊C4—1992no—
155C22⋊C4—22no—
156C22⋊C4—7962no—
157C22⋊C4—3982no—
158C22⋊C4—3982no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—7961no—
164C23—7961no—
165C23—7961no—
166C23—7962no—
167C23—7961no—
168C23—7961no—
169C23—7961no—
170C23—15921no—
171C23—15921no—
172C23—3982no—
173C23—3982no—
174C23D159212yescomplemented
175C23C2×D79611yescharacteristic
176D4—7962no—
177D4—7962no—
178D4—7964no—
179D4—7968no—
180D4—7962no—
181D4—7962no—
182D4—15921no—
183D4—3984no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—7961no—
188C2×C4—7962no—
189C2×C4—7961no—
190C2×C4—3982no—
191C2×C4—3982no—
192C2×C4D4×D19912yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—7962no—
202C22—7962no—
203C22—7962no—
204C22—7962no—
205C22—7962no—
206C22—7962no—
207C22—15921no—
208C22—15921no—
209C22—15924no—
210C22—15921no—
211C22—15921no—
212C22#2026392511yescharacteristic
213C22C2×D159211yescharacteristic
214C22#2026398311yescharacteristic
215C4—22no—
216C4—41no—
217C4—7962no—
218C4—7962no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—15922no—
223C2—15922no—
224C2#2716415411yescharacteristic
225C2#2716418811yescharacteristic
226C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#323×C199C2
socleC22×C199#120
hypercenter#323D199
hypocenterC199#1595
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #27164580C2semidirect
2. #27164154C2semidirect
3. #27163927C2semidirect
4. #27163753C2semidirect
5. #323×C199C2semidirect
6. C22×D796C22semidirect
16. C199⋊#95C22semidirect
18. C22⋊C8×C199C22semidirect
24. C2×D796D4semidirect
36. C4⋊Dic199D4semidirect
73. C23×C199D8semidirect
113. C199#1595semidirect
119. #323D199semidirect
131. C22⋊C8C2×D199semidirect
174. C23D1592semidirect

Maximal subgroups

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

subgroupindex
#271645802
#271641542
#271639272
#271637532
#323×C1992
#1595199