Finite Group Databasebeta

← #38767867

Subgroups — #38767867

Order 25472.

232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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#27164608C211yesmaximal, characteristic
2#27164346C211yesmaximal, characteristic, complemented
3#27164332C211yesmaximal, characteristic, complemented
4#27163945C211yesmaximal, characteristic, complemented
5#27163751C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic, complemented
7#328×C199C211yesmaximal, characteristic, complemented
8#20263950—21no—
9#20263944—21no—
10#20263940—21no—
11#20263930—21no—
12#20263929C2211yescharacteristic
13#20263925—21no—
14(C22⋊D199)⋊C4—21no—
15C2×C4⋊Dic199C2211yescharacteristic
16C2×(C2×D199)⋊C4—21no—
17C2×(C2×D199)⋊C4C2211yescharacteristic
18#20263889—21no—
19#20263886—21no—
20#20263879—21no—
21#20263877—21no—
22C22⋊(C199⋊C8)C2211yescharacteristic
23C199⋊#95—21no—
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28C2×D796—21no—
29C2×D796—21no—
30C2×D796D411yescharacteristic
31C2×C22⋊D199—21no—
32C22×Dic199—21no—
33C22×Dic199D411yescharacteristic
34C2×C4×D199—21no—
35C2×C199⋊C8—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199D411yescharacteristic
39Dic199⋊C4—21no—
40(C2×D199)⋊C4—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—21no—
46(C2×D199)⋊C4—42no—
47C4×Dic199—21no—
48C2×Q8×C199D411yescharacteristic
49C22×C4×C199—21no—
50C22×C4×C199C2311yescharacteristic
51C2×C8×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199D411yescharacteristic
54C22⋊C4×C199—21no—
55C22×D199—21no—
56C22×D199—21no—
57C22×D199—21no—
58C22×D199—21no—
59C22×D199—41no—
60C22×D199—41no—
61D796—41no—
62D796—81no—
63D796—81no—
64C22⋊D199—41no—
65C22⋊D199—41no—
66C2×Dic199—21no—
67C2×Dic199—21no—
68C2×Dic199—41no—
69C2×Dic199—41no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C4×D199—81no—
73C199⋊C8—41no—
74C23×C199C2×D411yescharacteristic
75Q8×C199—41no—
76C2×C4×C199—21no—
77C2×C4×C199—21no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199—41no—
81C2×C4×C199—41no—
82C2×C4×C199C2×D411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C8×C199—41no—
85C22×C199—21no—
86C22×C199—21no—
87C22×C199#12011yescharacteristic
88C22×C199#13511yescharacteristic
89C22×C199AGL(1,ℤ/8)11yescharacteristic
90C2×D199—42no—
91C2×D199—41no—
92C2×D199—41no—
93C2×D199—41no—
94C2×D199—41no—
95C2×D199—41no—
96C2×D199—41no—
97C2×D199—81no—
98C4×C199—21no—
99C4×C199—21no—
100C4×C199—41no—
101C4×C199—41no—
102C4×C199—81no—
103Dic199—41no—
104Dic199—41no—
105Dic199—81no—
106C2×C199—21no—
107C2×C199—21no—
108C2×C199#44911yescharacteristic
109C2×C199#45511yescharacteristic
110C2×C199#45811yescharacteristic
111D199—81no—
112D199—81no—
113C199#161211yescharacteristic, complemented, Sylow
114#1612—1991nomaximal, Sylow
115#538—1991no—
116#483—1991no—
117#480—1991no—
118#397—1991no—
119#328D19911yescharacteristic, complemented
120#327—1991no—
121#323—1991no—
122#126—3981no—
123#124—3981no—
124#123—3981no—
125C22⋊Q8C2×D19911yescharacteristic
126#121—3981no—
127#121—1991no—
128#120—3981no—
129C4×D4—3981no—
130C2×C4⋊C4—1991no—
131C2×C22⋊C4—3981no—
132C2×C22⋊C4—1991no—
133#107—3981no—
134#106—3981no—
135#103—3981no—
136#102—3981no—
137C22⋊C8—1991no—
138C22⋊C8C2×D19911yescharacteristic
139#95—3981no—
140#95C2×D19911yescharacteristic
141C24—1991no—
142C2×Q8D79611yescharacteristic
143C2×D4—3981no—
144C2×D4—3981no—
145C2×D4—3981no—
146C2×D4—1991no—
147C22×C4—21no—
148C22×C4—3981no—
149C22×C4—3981no—
150C22×C4—1991no—
151C22×C4C22×D19911yescharacteristic
152C2×C8—21no—
153C2×C8—3981no—
154C4⋊C4—21no—
155C4⋊C4—3981no—
156C4⋊C4—3981no—
157C4⋊C4—3981no—
158C4⋊C4—1991no—
159C4⋊C4D79611yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—7962no—
162C22⋊C4—3981no—
163C22⋊C4—3981no—
164C22⋊C4—3981no—
165C22⋊C4—3981no—
166C22⋊C4—3981no—
167C22⋊C4—3981no—
168C42—3981no—
169C23—7961no—
170C23—7961no—
171C23—3981no—
172C23—3981no—
173C23—3981no—
174C23—3981no—
175C23D4×D19911yescharacteristic
176Q8—41no—
177D4—7961no—
178D4—7961no—
179D4—7961no—
180D4—15921no—
181D4—15921no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—7961no—
189C2×C4—7961no—
190C2×C4—7961no—
191C2×C4—7961no—
192C2×C4—15921no—
193C2×C4—3981no—
194C2×C4—3981no—
195C2×C4D4×D19911yescharacteristic
196C2×C4C2×D79611yescharacteristic
197C8—41no—
198C8—7961no—
199C22—21no—
200C22—21no—
201C22—7962no—
202C22—7961no—
203C22—7961no—
204C22—7961no—
205C22—7961no—
206C22—7961no—
207C22—7961no—
208C22—15921no—
209C22#2026392511yescharacteristic
210C22#2026397811yescharacteristic
211C22#2026398211yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—81no—
217C4—7961no—
218C4—7961no—
219C4—15921no—
220C2—21no—
221C2—21no—
222C2—15921no—
223C2—15921no—
224C2#2716416611yescharacteristic
225C2#2716419711yescharacteristic
226C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199#120
prime radical#27164608C2
hypercenter#328D199
hypocenterC199#1612
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. #27164346C2semidirect
3. #27164332C2semidirect
4. #27163945C2semidirect
5. #27163751C2semidirect
6. #27163741C2semidirect
7. #328×C199C2semidirect
113. C199#1612semidirect
119. #328D199semidirect

Maximal subgroups

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

subgroupindex
#271646082
#271643462
#271643322
#271639452
#271637512
#271637412
#328×C1992
#1612199