Finite Group Databasebeta

← #38767839

Subgroups — #38767839

Order 25472.

228 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 51 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#27164643C211yesmaximal, characteristic
2#27164348C211yesmaximal, characteristic, complemented
3#27163944C211yesmaximal, characteristic
4#27163756C211yesmaximal, characteristic, complemented
5#27163755C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic
7#483×C199C211yesmaximal, characteristic, complemented
8#20263935—21no—
9#20263934C2211yescharacteristic
10#20263933—21no—
11#20263933—21no—
12#20263932—21no—
13#20263931—21no—
14C2×(C2×D199)⋊C4—21no—
15C2×(C2×D199)⋊C4C2211yescharacteristic
16C22⋊C4×D199—21no—
17#20263886—21no—
18#20263878—21no—
19C22⋊(C199⋊C8)C2211yescharacteristic
20C199⋊#95—21no—
21C199⋊#95C2211yescharacteristic
22C22⋊Q8×C199C2211yescharacteristic
23C2×C4⋊C4×C199C2211yescharacteristic
24#106×C199—21no—
25#103×C199—21no—
26C22⋊C8×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28C2×Dic398—21no—
29C2×Dic398D411yescharacteristic
30C22×Dic199—21no—
31C22×Dic199D411yescharacteristic
32C2×C4×D199—21no—
33C2×C4×D199—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—21no—
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40C22⋊Dic199—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—42no—
46C2×Q8×C199D411yescharacteristic
47C22×C4×C199—21no—
48C22×C4×C199C2311yescharacteristic
49C2×C8×C199—21no—
50C4⋊C4×C199—21no—
51C4⋊C4×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199D411yescharacteristic
54C22⋊C4×C199—21no—
55C22×D199—21no—
56C22×D199—21no—
57C22×D199—41no—
58C22×D199—41no—
59C22×D199SD1612yes—
60Dic398—41no—
61Dic398—81no—
62C2×Dic199—21no—
63C2×Dic199—21no—
64C2×Dic199—21no—
65C2×Dic199—41no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C4×D199—81no—
70C4×D199—81no—
71C199⋊C8—41no—
72C23×C199C2×D411yescharacteristic
73Q8×C199—41no—
74C2×C4×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199—41no—
77C2×C4×C199—41no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199C2×D411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C8×C199—41no—
83C22×C199—21no—
84C22×C199—21no—
85C22×C199#12011yescharacteristic
86C22×C199C2×SD1611yescharacteristic
87C22×C199#13711yescharacteristic
88C2×D199—22no—
89C2×D199—41no—
90C2×D199—41no—
91C2×D199—42no—
92C2×D199—81no—
93C4×C199—21no—
94C4×C199—21no—
95C4×C199—41no—
96C4×C199—41no—
97C4×C199—81no—
98Dic199—41no—
99Dic199—41no—
100Dic199—41no—
101Dic199—81no—
102C2×C199—21no—
103C2×C199—21no—
104C2×C199#44811yescharacteristic
105C2×C199#45511yescharacteristic
106C2×C199#45811yescharacteristic
107D199—42no—
108C199#160811yescharacteristic, complemented, Sylow
109#1608—1991nomaximal, Sylow
110#543—1991no—
111#483—1991no—
112#483D19911yescharacteristic, complemented
113#397—1991no—
114#328—1991no—
115#328—1991no—
116#323—1991no—
117C22⋊Q8—3981no—
118C22⋊Q8—3981no—
119C22⋊Q8—3981no—
120C22⋊Q8—3981no—
121C22⋊Q8—3981no—
122C22⋊Q8—1991no—
123C22⋊Q8C2×D19911yescharacteristic
124C2×C4⋊C4C2×D19911yescharacteristic
125C2×C22⋊C4—3981no—
126C2×C22⋊C4—3981no—
127C2×C22⋊C4—1991no—
128#106—21no—
129#106—3981no—
130#103—21no—
131#103—3981no—
132C22⋊C8—1991no—
133C22⋊C8C2×D19911yescharacteristic
134#95—3981no—
135#95—1991no—
136C24—1991no—
137C2×Q8—3981no—
138C2×Q8—1991no—
139C2×Q8C22⋊D19911yescharacteristic
140C22×C4—21no—
141C22×C4—3981no—
142C22×C4—3981no—
143C22×C4—3981no—
144C22×C4—1991no—
145C22×C4C22×D19911yescharacteristic
146C2×C8—21no—
147C2×C8—3981no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—3981no—
152C4⋊C4—3981no—
153C4⋊C4—3981no—
154C4⋊C4—3981no—
155C4⋊C4—1991no—
156C4⋊C4C22⋊D19911yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—7962no—
159C22⋊C4—3981no—
160C22⋊C4—3981no—
161C22⋊C4—3981no—
162C22⋊C4—3981no—
163C22⋊C4—3981no—
164C23—7961no—
165C23—7961no—
166C23—3981no—
167C23—3981no—
168C23—1992no—
169C23D4×D19911yescharacteristic
170Q8—41no—
171Q8—7961no—
172Q8—15921no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—7961no—
180C2×C4—7961no—
181C2×C4—7961no—
182C2×C4—7961no—
183C2×C4—15921no—
184C2×C4—15921no—
185C2×C4—3981no—
186C2×C4—3981no—
187C2×C4—3981no—
188C2×C4D4×D19911yescharacteristic
189C2×C4C2×C22⋊D19911yescharacteristic
190C8—41no—
191C8—7961no—
192C22—21no—
193C22—21no—
194C22—7962no—
195C22—7961no—
196C22—7961no—
197C22—15921no—
198C22—3982no—
199C22#2026392311yescharacteristic
200C22SD16×D19911yescharacteristic
201C22#2026398411yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—7961no—
208C4—7961no—
209C4—7961no—
210C4—15921no—
211C2—21no—
212C2—21no—
213C2—7962no—
214C2#2716415511yescharacteristic
215C2#2716419411yescharacteristic
216C2#2716420711yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#483×C199C2
socleC22×C199#120
prime radicalC23×D199D4
hypercenter#483D199
hypocenterC199#1608
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. #27164348C2semidirect
4. #27163756C2semidirect
5. #27163755C2semidirect
7. #483×C199C2semidirect
108. C199#1608semidirect
112. #483D199semidirect

Maximal subgroups

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

subgroupindex
#271646432
#271643482
#271639442
#271637562
#271637552
#271637412
#483×C1992
#1608199