Finite Group Databasebeta

← #38767888

Subgroups — #38767888

Order 25472.

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

normal subgroups only

This group has 246 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#27164595C211yesmaximal, characteristic
2#27164337C211yesmaximal, characteristic, complemented
3#27164334C211yesmaximal, characteristic, complemented
4#27163958C211yesmaximal, characteristic, complemented
5#27163751C211yesmaximal, characteristic, complemented
6#27163743C211yesmaximal, characteristic, complemented
7#327×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9#20263933—21no—
10#20263932—21no—
11#20263929C2211yescharacteristic
12#20263928—21no—
13#20263927—21no—
14#20263927—21no—
15C2×C4⋊Dic199C2211yescharacteristic
16C2×Dic199⋊C4C2211yescharacteristic
17#20263887—21no—
18#20263886—21no—
19#20263877—21no—
20#20263876—21no—
21C22⋊(C199⋊C8)C2211yescharacteristic
22C199⋊#95—21no—
23C199⋊#95—21no—
24#121×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27#15691885—42no—
28#15691885—42no—
29C2×Dic398—21no—
30C2×D796D411yescharacteristic
31C2×C22⋊D199—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199—21no—
34C22×Dic199—21no—
35C22×Dic199—21no—
36C22×Dic199D411yescharacteristic
37C22×Dic199D411yescharacteristic
38C2×C4×D199—21no—
39C2×C199⋊C8—21no—
40C4⋊Dic199—21no—
41C4⋊Dic199—21no—
42C4⋊Dic199D411yescharacteristic
43Dic199⋊C4—21no—
44Dic199⋊C4—21no—
45Dic199⋊C4—21no—
46C22⋊Dic199—21no—
47(C2×D199)⋊C4—21no—
48(C2×D199)⋊C4—21no—
49C2×D4×C199—21no—
50C2×D4×C199D411yescharacteristic
51C22×C4×C199—21no—
52C22×C4×C199C2311yescharacteristic
53C2×C8×C199—21no—
54C4⋊C4×C199D411yescharacteristic
55C22⋊C4×C199—21no—
56C22×D199—21no—
57Dic398—42no—
58D796—41no—
59C22⋊D199—42no—
60C22⋊D199—41no—
61C22⋊D199—41no—
62C22⋊D199—81no—
63C2×Dic199—21no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—21no—
67C2×Dic199—42no—
68C2×Dic199—41no—
69C2×Dic199—42no—
70C2×Dic199—41no—
71C2×Dic199SD1612yes—
72C4×D199—42no—
73C199⋊C8—41no—
74C23×C199—21no—
75C23×C199C2×D411yescharacteristic
76D4×C199—41no—
77D4×C199—41no—
78D4×C199—41no—
79C2×C4×C199—21no—
80C2×C4×C199—41no—
81C2×C4×C199—41no—
82C2×C4×C199—41no—
83C2×C4×C199—41no—
84C2×C4×C199C2×D411yescharacteristic
85C2×C4×C199C2×D411yescharacteristic
86C8×C199—41no—
87C22×C199—21no—
88C22×C199—21no—
89C22×C199—41no—
90C22×C199—41no—
91C22×C199—41no—
92C22×C199#12011yescharacteristic
93C22×C199C2×SD1611yescharacteristic
94C22×C199AGL(1,ℤ/8)11yescharacteristic
95C2×D199—41no—
96C2×D199—41no—
97C2×D199—41no—
98C4×C199—21no—
99C4×C199—21no—
100C4×C199—41no—
101C4×C199—81no—
102Dic199—22no—
103Dic199—41no—
104Dic199—41no—
105Dic199—81no—
106C2×C199—21no—
107C2×C199—21no—
108C2×C199—81no—
109C2×C199#45011yescharacteristic
110C2×C199#45411yescharacteristic
111C2×C199#45811yescharacteristic
112D199—81no—
113C199#161611yescharacteristic, complemented, Sylow
114#1616—1991nomaximal, Sylow
115#537—1991no—
116#481—1991no—
117#481—1991no—
118#400—1991no—
119#327—1991no—
120#327D19911yescharacteristic, complemented
121#324—1991no—
122C2×#41—3981no—
123C22⋊Q8—3981no—
124C22⋊Q8—3981no—
125#121—3981no—
126#121—3981no—
127#121—3981no—
128#121—1991no—
129#121C2×D19911yescharacteristic
130C2×C4⋊C4—1991no—
131C2×C4⋊C4—1991no—
132#106—3981no—
133#106—3981no—
134#102—3981no—
135#102—3981no—
136C22⋊C8—1991no—
137C22⋊C8C2×D19911yescharacteristic
138#95—3981no—
139#95—3981no—
140#95C2×D19911yescharacteristic
141#41—7962no—
142#41—7962no—
143C2×Q8—3981no—
144C2×D4—21no—
145C2×D4—3981no—
146C2×D4—3981no—
147C2×D4—3981no—
148C2×D4—1991no—
149C2×D4D79611yescharacteristic
150C22×C4—21no—
151C22×C4—3981no—
152C22×C4—3981no—
153C22×C4—3981no—
154C22×C4—1991no—
155C22×C4—1991no—
156C22×C4C22×D19911yescharacteristic
157C2×C8—21no—
158C2×C8—3981no—
159C4⋊C4—3981no—
160C4⋊C4—3981no—
161C4⋊C4—3981no—
162C4⋊C4—3981no—
163C4⋊C4—3981no—
164C4⋊C4—1991no—
165C4⋊C4D79611yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—3981no—
168C22⋊C4—3981no—
169C22⋊C4—3981no—
170C23—21no—
171C23—3981no—
172C23D4×D19911yescharacteristic
173Q8—7962no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—7961no—
178D4—7961no—
179D4—7961no—
180D4—7962no—
181D4—15921no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—7962no—
188C2×C4—7962no—
189C2×C4—7961no—
190C2×C4—7962no—
191C2×C4—7961no—
192C2×C4—3981no—
193C2×C4—3981no—
194C2×C4—3981no—
195C2×C4—3981no—
196C2×C4—1992no—
197C2×C4D4×D19911yescharacteristic
198C2×C4C2×D79611yescharacteristic
199C8—41no—
200C8—7961no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—7961no—
207C22—7961no—
208C22—7961no—
209C22#2026392511yescharacteristic
210C22SD16×D19911yescharacteristic
211C22#2026397911yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—7961no—
217C4—7961no—
218C4—15921no—
219C4—3982no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—15921no—
224C2#2716417011yescharacteristic
225C2#2716419211yescharacteristic
226C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#327×C199C2
socleC22×C199#120
prime radical#27164595C2
hypercenter#327D199
hypocenterC199#1616
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27164337C2semidirect
3. #27164334C2semidirect
4. #27163958C2semidirect
5. #27163751C2semidirect
6. #27163743C2semidirect
7. #327×C199C2semidirect
113. C199#1616semidirect
120. #327D199semidirect

Maximal subgroups

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

subgroupindex
#271645952
#271643372
#271643342
#271639582
#271637512
#271637432
#327×C1992
#1616199