Finite Group Databasebeta

← #38741778

Subgroups — #38741778

Order 25216.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161749C211yesmaximal, characteristic, complemented
2#27161455C211yesmaximal, characteristic, complemented
3#27161280C211yesmaximal, characteristic, complemented
4#27161275C211yesmaximal, characteristic, complemented
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#331×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C2×#15691347—21no—
10C2×D1576—21no—
11C2×#15691343—22no—
12#20263323C2211yescharacteristic
13#20263322—21no—
14#20263298—22no—
15#20263297—21no—
16C4×Dic394C2211yescharacteristic
17C4×D788C2211yescharacteristic
18D4×Dic197—21no—
19#20263248—21no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22#127×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25#15691357—42no—
26#15691357—42no—
27C2×Dic394D411yescharacteristic
28C2×D788—21no—
29C2×D788—21no—
30C2×D788—21no—
31C2×D788D411yescharacteristic
32C2×C22⋊D197—22no—
33C2×C22⋊D197—22no—
34C22×Dic197—22no—
35C2×C4×D197—21no—
36#15691347—42no—
37D1576—42no—
38#15691343—44no—
39C2×#13096802—21no—
40C4⋊Dic197D411yescharacteristic
41C4⋊Dic197D411yescharacteristic
42#15691337—21no—
43C22⋊Dic197—22no—
44#15691335—21no—
45C4×Dic197—21no—
46C2×D4×C197—21no—
47C2×D4×C197—21no—
48C2×D4×C197D412yes—
49C2×C8×C197—21no—
50C2×C8×C197—21no—
51C42×C197C2311yescharacteristic
52C22×D197—21no—
53C22×D197—41no—
54Dic394—21no—
55Dic394D812yescomplemented
56D788—21no—
57D788—21no—
58D788—42no—
59D788—41no—
60D788—41no—
61D788—41no—
62D788—41no—
63C22⋊D197—44no—
64C22⋊D197—82no—
65C2×Dic197—21no—
66C2×Dic197—21no—
67C2×Dic197—21no—
68C2×Dic197—44no—
69C4×D197—41no—
70C4×D197—41no—
71#13096802—22no—
72C23×C197—22no—
73D4×C197—42no—
74D4×C197—41no—
75D4×C197—41no—
76D4×C197—41no—
77D4×C197—41no—
78C2×C4×C197C2×D411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C8×C197—41no—
82C8×C197—41no—
83C22×C197—42no—
84C22×C197—42no—
85C22×C197—42no—
86C22×C197#12011yescharacteristic
87C2×D197—41no—
88C2×D197—41no—
89C2×D197—41no—
90C2×D197—81no—
91C2×D197—81no—
92C2×D197—81no—
93C4×C197—21no—
94C4×C197C2×D811yescharacteristic
95C4×C197#13511yescharacteristic
96C4×C197AGL(1,ℤ/8)11yescharacteristic
97C4×C197AGL(1,ℤ/8)11yescharacteristic
98Dic197—22no—
99Dic197—41no—
100Dic197—41no—
101C2×C197—82no—
102C2×C197#44711yescharacteristic
103C2×C197#44911yescharacteristic
104C2×C197#45311yescharacteristic
105D197—81no—
106D197—161no—
107C197#162111yescharacteristic, complemented, Sylow
108#1621—1971nomaximal, Sylow
109#550—1971no—
110#493—1971no—
111#460—1971no—
112#459—1971no—
113#331D19711yescharacteristic, complemented
114Q8⋊C8—1971no—
115D4⋊C8—1971no—
116C2×#41—3941no—
117C2×SD16—3941no—
118C2×D8—3941no—
119C2×D8—3942no—
120#127—3941no—
121#127—1971no—
122#127C2×D19711yescharacteristic
123#121—3942no—
124#121—3941no—
125C4×Q8—1971no—
126C4×D4—3941no—
127C4×D4—1971no—
128C4⋊C8C2×D19711yescharacteristic
129C4⋊C8C2×D19711yescharacteristic
130#102—3941no—
131#102—3941no—
132C4×C8—1971no—
133#41—7882no—
134#41—7882no—
135C2×Q8—1971no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—3942no—
139C2×D4—3942no—
140C2×D4—3941no—
141C2×D4—3941no—
142C2×D4—3941no—
143C2×D4—1971no—
144C2×D4D78812yes—
145C22×C4—3942no—
146C22×C4—3941no—
147SD16—7882no—
148D8—7882no—
149D8—7884no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—3941no—
153C4⋊C4—3941no—
154C4⋊C4—1971no—
155C4⋊C4—1971no—
156C22⋊C4—3942no—
157C22⋊C4—3941no—
158C42—3941no—
159C42C22×D19711yescharacteristic
160C23—22no—
161C23—7881no—
162C23—3941no—
163Q8—3941no—
164Q8—1972no—
165D4—42no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—7882no—
171D4—7884no—
172D4—7881no—
173D4—7881no—
174D4—7881no—
175D4—7881no—
176D4—15762no—
177D4—3941no—
178D4—3941no—
179C2×C4—7884no—
180C2×C4—7881no—
181C2×C4—7881no—
182C2×C4—3941no—
183C2×C4—3941no—
184C2×C4—3941no—
185C2×C4D4×D19711yescharacteristic
186C2×C4D4×D19711yescharacteristic
187C2×C4C2×D78811yescharacteristic
188C8—41no—
189C8—41no—
190C8—3942no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—7881no—
195C22—7881no—
196C22—7881no—
197C22—15761no—
198C22—15761no—
199C22—15761no—
200C22#2026329611yescharacteristic
201C4—21no—
202C4—7881no—
203C4—7881no—
204C4—3942no—
205C4D8×D19711yescharacteristic
206C4#2026334711yescharacteristic
207C4#2026335011yescharacteristic
208C4#2026335311yescharacteristic
209C2—82no—
210C2—15761no—
211C2—31521no—
212C2#2716121411yescharacteristic
213C2#2716122511yescharacteristic
214C2#2716124911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197#120
hypercenter#331D197
hypocenterC197#1621
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #27161749C2semidirect
2. #27161455C2semidirect
3. #27161280C2semidirect
4. #27161275C2semidirect
5. #27160809C2semidirect
6. #27160806C2semidirect
7. #331×C197C2semidirect
55. Dic394D8semidirect
107. C197#1621semidirect
113. #331D197semidirect

Maximal subgroups

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

subgroupindex
#271617492
#271614552
#271612802
#271612752
#271608092
#271608062
#331×C1972
#1621197