Finite Group Databasebeta

← #38767808

Subgroups — #38767808

Order 25472.

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

normal subgroups only

This group has 256 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#27164176C211yesmaximal, characteristic, complemented
3#27164173C211yesmaximal, characteristic, complemented
4#27163931C211yesmaximal, characteristic
5#27163754C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic
7#328×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C2×Dic796—21no—
10C2×#15691876—21no—
11#20263935—21no—
12#20263934C2211yescharacteristic
13#20263933—21no—
14#20263933—21no—
15#20263932—21no—
16#20263931—21no—
17C2×(C2×D199)⋊C4—21no—
18C2×(C2×D199)⋊C4C2211yescharacteristic
19C22⋊C4×D199—21no—
20#20263880—21no—
21#20263878—21no—
22C22⋊(C199⋊C8)C2211yescharacteristic
23C199⋊#95—21no—
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398—21no—
31C2×Dic398—42no—
32C2×Dic398D411yescharacteristic
33C22×Dic199—21no—
34C22×Dic199D411yescharacteristic
35C2×C4×D199—21no—
36C2×C4×D199—21no—
37Dic796—42no—
38#15691876—42no—
39C2×C199⋊C8—21no—
40C4⋊Dic199—21no—
41C4⋊Dic199D411yescharacteristic
42Dic199⋊C4—21no—
43Dic199⋊C4—21no—
44Dic199⋊C4—21no—
45C22⋊Dic199—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—21no—
48(C2×D199)⋊C4—21no—
49(C2×D199)⋊C4—21no—
50(C2×D199)⋊C4—42no—
51C2×Q8×C199D411yescharacteristic
52C22×C4×C199—21no—
53C22×C4×C199C2311yescharacteristic
54C2×C8×C199—21no—
55C4⋊C4×C199—21no—
56C4⋊C4×C199D411yescharacteristic
57C22⋊C4×C199—21no—
58C22×D199—21no—
59C22×D199—21no—
60C22×D199—41no—
61C22×D199—41no—
62C22×D199Q1612yescomplemented
63Dic398—22no—
64Dic398—22no—
65Dic398—41no—
66Dic398—41no—
67Dic398—41no—
68Dic398—81no—
69C2×Dic199—22no—
70C2×Dic199—21no—
71C2×Dic199—21no—
72C2×Dic199—21no—
73C2×Dic199—41no—
74C2×Dic199—41no—
75C2×Dic199—41no—
76C4×D199—81no—
77C4×D199—81no—
78C199⋊C8—41no—
79C23×C199C2×D411yescharacteristic
80Q8×C199—41no—
81C2×C4×C199—21no—
82C2×C4×C199—21no—
83C2×C4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199—41no—
87C2×C4×C199C2×D411yescharacteristic
88C2×C4×C199C2×D411yescharacteristic
89C8×C199—41no—
90C22×C199—21no—
91C22×C199—21no—
92C22×C199#12011yescharacteristic
93C22×C199C2×Q1611yescharacteristic
94C22×C199#13711yescharacteristic
95C2×D199—22no—
96C2×D199—41no—
97C2×D199—41no—
98C2×D199—42no—
99C2×D199—81no—
100C4×C199—21no—
101C4×C199—21no—
102C4×C199—41no—
103C4×C199—41no—
104C4×C199—81no—
105Dic199—41no—
106Dic199—41no—
107Dic199—41no—
108Dic199—42no—
109C2×C199—21no—
110C2×C199—21no—
111C2×C199#45111yescharacteristic
112C2×C199#45511yescharacteristic
113C2×C199B(4,𝔽2)11yescharacteristic
114D199—42no—
115C199#160211yescharacteristic, complemented, Sylow
116#1602—1991nomaximal, Sylow
117#543—1991no—
118#451—1991no—
119#451—1991no—
120#393—1991no—
121#328—1991no—
122#328D19911yescharacteristic, complemented
123#323—1991no—
124C22×Q8—1991no—
125C2×Q16—3981no—
126C2×Q16—3981no—
127C22⋊Q8—3981no—
128C22⋊Q8—3981no—
129C22⋊Q8—3981no—
130C22⋊Q8—3981no—
131C22⋊Q8—3981no—
132C22⋊Q8—1991no—
133C22⋊Q8C2×D19911yescharacteristic
134C2×C22⋊C4—3981no—
135C2×C22⋊C4—3981no—
136C2×C22⋊C4—1991no—
137#103—3981no—
138#103—3981no—
139C22⋊C8—1991no—
140C22⋊C8C2×D19911yescharacteristic
141#95—3981no—
142#95C2×D19911yescharacteristic
143C24—1991no—
144C2×Q8—7962no—
145C2×Q8—3981no—
146C2×Q8—3981no—
147C2×Q8—3981no—
148C2×Q8—1991no—
149C2×Q8D79611yescharacteristic
150C22×C4—21no—
151C22×C4—3981no—
152C22×C4—3981no—
153C22×C4—3981no—
154C22×C4—1991no—
155C22×C4C22×D19911yescharacteristic
156Q16—7962no—
157Q16—7962no—
158C2×C8—21no—
159C2×C8—3981no—
160C4⋊C4—21no—
161C4⋊C4—3981no—
162C4⋊C4—3981no—
163C4⋊C4—3981no—
164C4⋊C4—3981no—
165C4⋊C4—1991no—
166C4⋊C4D79611yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—7962no—
169C22⋊C4—3981no—
170C22⋊C4—3981no—
171C22⋊C4—3981no—
172C22⋊C4—3981no—
173C22⋊C4—3981no—
174C23—7961no—
175C23—7961no—
176C23—3981no—
177C23—3981no—
178C23—1992no—
179C23D4×D19911yescharacteristic
180Q8—41no—
181Q8—7961no—
182Q8—7961no—
183Q8—7961no—
184Q8—15921no—
185Q8—3982no—
186Q8—3982no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—7961no—
194C2×C4—7961no—
195C2×C4—7961no—
196C2×C4—15921no—
197C2×C4—15921no—
198C2×C4—3982no—
199C2×C4—3981no—
200C2×C4—3981no—
201C2×C4—3981no—
202C2×C4D4×D19911yescharacteristic
203C2×C4C2×D79611yescharacteristic
204C8—41no—
205C8—7961no—
206C22—21no—
207C22—21no—
208C22—7962no—
209C22—7961no—
210C22—7961no—
211C22—15921no—
212C22—3982no—
213C22#2026392511yescharacteristic
214C22Q16×D19911yescharacteristic
215C22#2026398411yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—7961no—
222C4—7961no—
223C4—7962no—
224C4—7961no—
225C2—21no—
226C2—21no—
227C2—7962no—
228C2#2716417511yescharacteristic
229C2#2716419711yescharacteristic
230C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199#120
prime radicalC23×D199D4
hypercenter#328D199
hypocenterC199#1602
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27164176C2semidirect
3. #27164173C2semidirect
5. #27163754C2semidirect
7. #328×C199C2semidirect
62. C22×D199Q16semidirect
115. C199#1602semidirect
122. #328D199semidirect

Maximal subgroups

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

subgroupindex
#271646432
#271641762
#271641732
#271639312
#271637542
#271637412
#328×C1992
#1602199