Finite Group Databasebeta

← #38768158

Subgroups — #38768158

Order 25472.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#27164760C211yesmaximal, characteristic
2#27164412C211yesmaximal, characteristic, complemented
3#27164378C211yesmaximal, characteristic, complemented
4#27164377C211yesmaximal, characteristic, complemented
5#27163757C211yesmaximal, characteristic, complemented
6#27163757C211yesmaximal, characteristic, complemented
7#329×C199C211yesmaximal, characteristic, complemented
8C2×Dic796—21no—
9C2×#15691875—21no—
10#20263955C2211yescharacteristic, complemented
11#20263944—21no—
12#20263943C2211yescharacteristic
13#20263943C2211yescharacteristic
14#20263941—21no—
15#20263941—21no—
16#20263925—21no—
17#20263925—21no—
18#20263923—21no—
19#20263880—21no—
20#20263879—21no—
21#20263877—21no—
22#20263876—21no—
23#20263866C2211yescharacteristic, complemented
24#124×C199C2211yescharacteristic
25#97×C199C2211yescharacteristic, complemented
26#97×C199C2211yescharacteristic, complemented
27C23×D199—21no—
28C2×Dic398—21no—
29C2×Dic398D411yescharacteristic
30C2×Dic398D411yescharacteristic
31C2×D796—21no—
32C2×D796D411yescharacteristic
33C2×D796D411yescharacteristic
34C2×C22⋊D199—21no—
35C2×C22⋊D199—21no—
36Dic796—42no—
37#15691875—42no—
38C2×C199⋊C8—21no—
39C4⋊Dic199—21no—
40C4⋊Dic199—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—21no—
47(C2×D199)⋊C4—21no—
48C4×Dic199—21no—
49C2×Q8×C199D411yescharacteristic
50C2×D4×C199D411yescharacteristic
51C2×C8×C199—21no—
52C2×C8×C199—21no—
53C22⋊C4×C199—21no—
54C22⋊C4×C199—21no—
55C42×C199C2311yescharacteristic
56C22×D199—21no—
57C22×D199—21no—
58C22×D199—81no—
59C22×D199—81no—
60C22×D199—81no—
61Dic398—41no—
62Dic398—41no—
63Dic398—41no—
64Dic398—41no—
65D796—41no—
66D796—41no—
67D796—81no—
68C22⋊D199—81no—
69C22⋊D199—81no—
70C2×Dic199—21no—
71C2×Dic199—21no—
72C2×Dic199—41no—
73C199⋊C8—41no—
74C23×C199—21no—
75Q8×C199—41no—
76D4×C199—41no—
77C2×C4×C199—21no—
78C2×C4×C199C2×D411yescharacteristic
79C2×C4×C199C2×D411yescharacteristic
80C2×C4×C199C2×D411yescharacteristic
81C8×C199—22no—
82C8×C199—41no—
83C22×C199—41no—
84C22×C199—41no—
85C22×C199—41no—
86C22×C199#12011yescharacteristic
87C2×D199—41no—
88C2×D199—41no—
89C2×D199—41no—
90C2×D199—41no—
91C2×D199—41no—
92C2×D199—41no—
93C2×D199—84no—
94C4×C199—21no—
95C4×C199—21no—
96C4×C199—21no—
97C4×C199—41no—
98Dic199—41no—
99Dic199—41no—
100Dic199—81no—
101C2×C199—81no—
102C2×C199#45311yescharacteristic
103C2×C199#45511yescharacteristic
104C2×C199#45511yescharacteristic
105D199—81no—
106D199—81no—
107C199#165911yescharacteristic, complemented, Sylow
108#1659—1991nomaximal, Sylow
109H(3,𝔽4)—1991no—
110#497—1991no—
111#489—1991no—
112#489—1991no—
113#329—1991no—
114#329—1991no—
115#329D19911yescharacteristic, complemented
116C2×Q16—3981no—
117C2×SD16—3981no—
118C4⋊Q8—1991no—
119#124—3981no—
120#124—3981no—
121#124—3981no—
122#124—1991no—
123#124—1991no—
124#124C2×D19911yescharacteristic
125#120—3981no—
126#120—3981no—
127#120—3981no—
128#103—3981no—
129#103—3981no—
130#102—3981no—
131#102—3981no—
132#97—1991no—
133#97C2×D19911yescharacteristic, complemented
134#97C2×D19911yescharacteristic, complemented
135C24—3981no—
136C2×Q8—3981no—
137C2×Q8—1991no—
138C2×Q8—1991no—
139C2×Q8D79611yescharacteristic
140C2×D4—3981no—
141C2×D4—3981no—
142C2×D4—3981no—
143C2×D4—1991no—
144C2×D4—1991no—
145C2×D4D79611yescharacteristic
146Q16—7962no—
147SD16—7962no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—3981no—
151C4⋊C4—3981no—
152C4⋊C4—3981no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—3981no—
156C22⋊C4—3981no—
157C22⋊C4—3981no—
158C22⋊C4—3981no—
159C22⋊C4—3981no—
160C22⋊C4—3981no—
161C22⋊C4—3981no—
162C42—3981no—
163C42C22×D19911yescharacteristic
164C23—21no—
165C23—15921no—
166C23—15921no—
167C23—15921no—
168C23—3981no—
169C23—3981no—
170Q8—41no—
171Q8—7961no—
172Q8—7961no—
173Q8—7961no—
174Q8—7961no—
175D4—41no—
176D4—7961no—
177D4—7961no—
178D4—15921no—
179D4—15921no—
180D4—15921no—
181C2×C4—21no—
182C2×C4—7961no—
183C2×C4—3981no—
184C2×C4—3981no—
185C2×C4D4×D19911yescharacteristic
186C2×C4D4×D19911yescharacteristic
187C2×C4C2×D79611yescharacteristic
188C8—22no—
189C8—41no—
190C8—7961no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—7961no—
195C22—7961no—
196C22—7961no—
197C22—7961no—
198C22—7961no—
199C22—7961no—
200C22—15924no—
201C22#2026392511yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—7961no—
207C4—7961no—
208C4—15921no—
209C2—81no—
210C2—15921no—
211C2—15921no—
212C2#2716418711yescharacteristic
213C2#2716419711yescharacteristic
214C2#2716419811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#329×C199C2
socleC22×C199#120
prime radical#27164760C2
hypercenter#329D199
hypocenterC199#1659
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27164412C2semidirect
3. #27164378C2semidirect
4. #27164377C2semidirect
5. #27163757C2semidirect
6. #27163757C2semidirect
7. #329×C199C2semidirect
10. #20263955C22semidirect
23. #20263866C22semidirect
25. #97×C199C22semidirect
26. #97×C199C22semidirect
107. C199#1659semidirect
115. #329D199semidirect
133. #97C2×D199semidirect
134. #97C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271647602
#271644122
#271643782
#271643772
#271637572
#329×C1992
#1659199