Finite Group Databasebeta

← #38767849

Subgroups — #38767849

Order 25472.

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

normal subgroups only

This group has 216 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#27164615C211yesmaximal, characteristic
2#27164349C211yesmaximal, characteristic, complemented
3#27164332C211yesmaximal, characteristic
4#27163948C211yesmaximal, characteristic, complemented
5#27163756C211yesmaximal, characteristic, complemented
6#27163753C211yesmaximal, characteristic
7#324×C199C211yesmaximal, characteristic, complemented
8#20263950—21no—
9#20263945—21no—
10#20263940—21no—
11#20263938—21no—
12#20263934C2211yescharacteristic
13#20263933—21no—
14#20263929C2211yescharacteristic
15C4×C22⋊D199—21no—
16C2×C4⋊Dic199C2211yescharacteristic
17C2×Dic199⋊C4—21no—
18#20263889—21no—
19#20263887—21no—
20#20263880—21no—
21#20263877—21no—
22C199⋊#95—21no—
23C199⋊#95C2211yescharacteristic
24C2×C4⋊C4×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26C22⋊C8×C199C2211yescharacteristic
27C2×Dic398D411yescharacteristic
28C2×D796D411yescharacteristic
29C2×C22⋊D199—21no—
30C22×Dic199—21no—
31C22×Dic199—21no—
32C2×C4×D199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199D411yescharacteristic
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—21no—
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40Dic199⋊C4—21no—
41Dic199⋊C4—42no—
42C22⋊Dic199—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47C4×Dic199—21no—
48C22×C4×C199D411yescharacteristic
49C22×C4×C199D411yescharacteristic
50C22×C4×C199C2311yescharacteristic
51C2×C8×C199—21no—
52C2×C8×C199—21no—
53C4⋊C4×C199—21no—
54C4⋊C4×C199—21no—
55C22×D199—21no—
56Dic398—41no—
57D796—41no—
58C22⋊D199—41no—
59C22⋊D199—41no—
60C2×Dic199—22no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—21no—
64C2×Dic199—41no—
65C2×Dic199—41no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C4×D199—81no—
70C23×C199C2×D411yescharacteristic
71C2×C4×C199—21no—
72C2×C4×C199—21no—
73C2×C4×C199—21no—
74C2×C4×C199—21no—
75C2×C4×C199—41no—
76C2×C4×C199—41no—
77C2×C4×C199C2×D411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C8×C199—41no—
80C8×C199—41no—
81C22×C199—21no—
82C22×C199—21no—
83C22×C199#12011yescharacteristic
84C22×C199#13511yescharacteristic
85C22×C199AGL(1,ℤ/8)11yescharacteristic
86C2×D199—41no—
87C2×D199—41no—
88C2×D199—41no—
89C4×C199—21no—
90C4×C199—21no—
91C4×C199—41no—
92C4×C199—41no—
93Dic199—42no—
94Dic199—41no—
95Dic199—41no—
96Dic199—41no—
97Dic199—81no—
98C2×C199—21no—
99C2×C199—21no—
100C2×C199#44911yescharacteristic
101C2×C199#45611yescharacteristic
102C2×C199#45811yescharacteristic
103D199—81no—
104C199#160911yescharacteristic, complemented, Sylow
105#1609—1991nomaximal, Sylow
106#539—1991no—
107#483—1991no—
108#480—1991no—
109#398—1991no—
110#328—1991no—
111#327—1991no—
112#324D19911yescharacteristic, complemented
113#126—3981no—
114#125—3981no—
115#123—3981no—
116#123—3981no—
117C22⋊Q8—3981no—
118C22⋊Q8—1991no—
119#121—1991no—
120C4×D4—3981no—
121C2×C4⋊C4—3981no—
122C2×C4⋊C4—1991no—
123C2×C4⋊C4C2×D19911yescharacteristic
124#107—3981no—
125#106—3981no—
126#103—3981no—
127#102—3981no—
128C22⋊C8C2×D19911yescharacteristic
129C22⋊C8C2×D19911yescharacteristic
130#95—3981no—
131#95—1991no—
132C2×Q8—1991no—
133C2×D4—3981no—
134C2×D4—1991no—
135C22×C4—3981no—
136C22×C4—3981no—
137C22×C4—3981no—
138C22×C4D79611yescharacteristic
139C22×C4D79611yescharacteristic
140C22×C4C22×D19911yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—7962no—
146C4⋊C4—3981no—
147C4⋊C4—3981no—
148C4⋊C4—3981no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C4⋊C4—3981no—
152C4⋊C4—1991no—
153C4⋊C4—1991no—
154C22⋊C4—3981no—
155C22⋊C4—3981no—
156C22⋊C4—3981no—
157C22⋊C4—3981no—
158C22⋊C4—3981no—
159C42—3981no—
160C23—3981no—
161C23C2×D79611yescharacteristic
162Q8—7961no—
163D4—7961no—
164D4—7961no—
165D4—7961no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—7961no—
173C2×C4—7961no—
174C2×C4—7961no—
175C2×C4—7961no—
176C2×C4—7961no—
177C2×C4—15921no—
178C2×C4—3982no—
179C2×C4—3981no—
180C2×C4—3981no—
181C2×C4—3981no—
182C2×C4D4×D19911yescharacteristic
183C2×C4D4×D19911yescharacteristic
184C8—41no—
185C8—41no—
186C22—21no—
187C22—21no—
188C22—7961no—
189C22—7961no—
190C22—7961no—
191C22#2026392511yescharacteristic
192C22#2026397711yescharacteristic
193C22#2026398311yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—7962no—
199C4—7961no—
200C4—7961no—
201C4—7961no—
202C4—15921no—
203C2—21no—
204C2—21no—
205C2—15921no—
206C2#2716416511yescharacteristic
207C2#2716420311yescharacteristic
208C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
prime radical#20263929C22
hypercenter#324D199
hypocenterC199#1609
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27164349C2semidirect
4. #27163948C2semidirect
5. #27163756C2semidirect
7. #324×C199C2semidirect
104. C199#1609semidirect
112. #324D199semidirect

Maximal subgroups

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

subgroupindex
#271646152
#271643492
#271643322
#271639482
#271637562
#271637532
#324×C1992
#1609199