Finite Group Databasebeta

← #38767805

Subgroups — #38767805

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#27164174C211yesmaximal, characteristic, complemented
3#27163929C211yesmaximal, characteristic
4#27163756C211yesmaximal, characteristic, complemented
5#27163755C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic
7#451×C199C211yesmaximal, characteristic, complemented
8C2×#15691876—21no—
9#20263935—21no—
10#20263934C2211yescharacteristic
11#20263933—21no—
12#20263933—21no—
13#20263932—21no—
14#20263931—21no—
15C2×(C2×D199)⋊C4—21no—
16C2×(C2×D199)⋊C4C2211yescharacteristic
17C22⋊C4×D199—21no—
18#20263879—21no—
19C22⋊(C199⋊C8)C2211yescharacteristic
20C199⋊#95—21no—
21C199⋊#95C2211yescharacteristic
22C22×Q8×C199C2211yescharacteristic
23C2×Q16×C199—21no—
24C22⋊Q8×C199C2211yescharacteristic
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—
34#15691876—42no—
35C2×C199⋊C8—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40Dic199⋊C4—21no—
41C22⋊Dic199—21no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—42no—
47C2×Q8×C199—21no—
48C2×Q8×C199—21no—
49C2×Q8×C199—42no—
50C2×Q8×C199D411yescharacteristic
51C22×C4×C199—21no—
52C22×C4×C199C2311yescharacteristic
53Q16×C199—42no—
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—41no—
64Dic398—81no—
65C2×Dic199—21no—
66C2×Dic199—21no—
67C2×Dic199—21no—
68C2×Dic199—41no—
69C2×Dic199—41no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C4×D199—81no—
73C4×D199—81no—
74C199⋊C8—41no—
75C23×C199C2×D411yescharacteristic
76Q8×C199—22no—
77Q8×C199—22no—
78Q8×C199—41no—
79Q8×C199—41no—
80Q8×C199—41no—
81C2×C4×C199—22no—
82C2×C4×C199—21no—
83C2×C4×C199—21no—
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—42no—
104C4×C199—41no—
105Dic199—41no—
106Dic199—41no—
107Dic199—41no—
108Dic199—81no—
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#451D19911yescharacteristic, complemented
120#393—1991no—
121#328—1991no—
122#328—1991no—
123#323—1991no—
124C22×Q8C2×D19911yescharacteristic
125C2×Q16—21no—
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—21no—
138#103—3981no—
139C22⋊C8—1991no—
140C22⋊C8C2×D19911yescharacteristic
141#95—3981no—
142#95—1991no—
143C24—1991no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—3981no—
148C2×Q8—1991no—
149C2×Q8C22⋊D19911yescharacteristic
150C22×C4—21no—
151C22×C4—3981no—
152C22×C4—3981no—
153C22×C4—3981no—
154C22×C4—1991no—
155C22×C4C22×D19911yescharacteristic
156Q16—42no—
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⋊C4C22⋊D19911yescharacteristic
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—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—7961no—
186Q8—15921no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—7961no—
194C2×C4—7961no—
195C2×C4—7961no—
196C2×C4—7961no—
197C2×C4—15921no—
198C2×C4—15921no—
199C2×C4—3981no—
200C2×C4—3981no—
201C2×C4—3981no—
202C2×C4D4×D19911yescharacteristic
203C2×C4C2×C22⋊D19911yescharacteristic
204C8—41no—
205C8—7961no—
206C22—21no—
207C22—21no—
208C22—7962no—
209C22—7961no—
210C22—7961no—
211C22—15921no—
212C22—3982no—
213C22#2026392311yescharacteristic
214C22Q16×D19911yescharacteristic
215C22#2026398711yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—7961no—
222C4—7961no—
223C4—7961no—
224C4—15921no—
225C2—21no—
226C2—21no—
227C2—7962no—
228C2#2716417211yescharacteristic
229C2#2716419411yescharacteristic
230C2#2716420411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#451×C199C2
socleC22×C199#120
prime radicalC23×D199D4
hypercenter#451D199
hypocenterC199#1602
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #27164174C2semidirect
4. #27163756C2semidirect
5. #27163755C2semidirect
7. #451×C199C2semidirect
62. C22×D199Q16semidirect
115. C199#1602semidirect
119. #451D199semidirect

Maximal subgroups

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

subgroupindex
#271646432
#271641742
#271639292
#271637562
#271637552
#271637412
#451×C1992
#1602199