Finite Group Databasebeta

← #38741774

Subgroups — #38741774

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#27161276C211yesmaximal, characteristic, complemented
3#27161271C211yesmaximal, characteristic, complemented
4#27160823C211yesmaximal, characteristic, complemented
5#27160811C211yesmaximal, characteristic, complemented
6#27160808C211yesmaximal, characteristic, complemented
7#493×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C2×#15691346—21no—
10C2×#15691343—21no—
11#20263322—21no—
12#20263298—22no—
13#20263297—21no—
14C4×Dic394C2211yescharacteristic
15C4×D788C2211yescharacteristic
16D4×Dic197—21no—
17C4⋊#13096802C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19#20263247—21no—
20#20263247—21no—
21C2×D8×C197—22no—
22#127×C197C2211yescharacteristic
23#127×C197C2211yescharacteristic
24C4×C8×C197C2211yescharacteristic
25#15691357—42no—
26#15691357—42no—
27C2×Dic394D411yescharacteristic
28C2×D788D411yescharacteristic
29C2×C22⋊D197—22no—
30C2×C22⋊D197—22no—
31C22×Dic197—22no—
32C2×C4×D197—21no—
33#15691346—42no—
34#15691343—42no—
35C2×#13096802—21no—
36C2×#13096802—21no—
37C4⋊Dic197D411yescharacteristic
38C4⋊Dic197D411yescharacteristic
39#15691337—21no—
40C22⋊Dic197—22no—
41#15691335—21no—
42C4×Dic197—21no—
43C2×D4×C197—21no—
44C2×D4×C197—21no—
45C2×D4×C197—21no—
46C2×D4×C197—21no—
47C2×D4×C197—21no—
48C2×D4×C197D412yes—
49D8×C197—44no—
50C2×C8×C197—21no—
51C42×C197C2311yescharacteristic
52C22×D197—21no—
53Dic394—21no—
54Dic394D812yescomplemented
55D788—21no—
56D788—21no—
57C22⋊D197—44no—
58C22⋊D197—82no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—44no—
63C4×D197—41no—
64C4×D197—41no—
65#13096802—41no—
66#13096802—41no—
67C23×C197—22no—
68C23×C197—41no—
69D4×C197—42no—
70D4×C197—42no—
71D4×C197—41no—
72D4×C197—41no—
73D4×C197—41no—
74D4×C197—41no—
75D4×C197—41no—
76D4×C197—41no—
77D4×C197—41no—
78D4×C197—41no—
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C8×C197—22no—
83C22×C197—42no—
84C22×C197—42no—
85C22×C197—42no—
86C22×C197—81no—
87C22×C197—81no—
88C22×C197—81no—
89C22×C197#12011yescharacteristic
90C2×D197—41no—
91C2×D197—41no—
92C2×D197—41no—
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—161no—
103C2×C197#44711yescharacteristic
104C2×C197#44911yescharacteristic
105C2×C197#45311yescharacteristic
106D197—81no—
107C197#162111yescharacteristic, complemented, Sylow
108#1621—1971nomaximal, Sylow
109#550—1971no—
110#493D19711yescharacteristic, complemented
111#460—1971no—
112#459—1971no—
113#331—1971no—
114Q8⋊C8—1971no—
115D4⋊C8—1971no—
116C2×#41—3941no—
117C2×SD16—3941no—
118C2×D8—22no—
119C2×D8—3941no—
120#127—3941no—
121#127C2×D19711yescharacteristic
122#127C2×D19711yescharacteristic
123#121—3942no—
124#121—3941no—
125C4×Q8—1971no—
126C4×D4—3941no—
127C4×D4—1971no—
128C4⋊C8—1971no—
129C4⋊C8—1971no—
130#102—3941no—
131#102—3941no—
132C4×C8C2×D19711yescharacteristic
133#41—7882no—
134#41—7882no—
135C2×Q8—1971no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—3942no—
142C2×D4—3942no—
143C2×D4—1971no—
144C2×D4C22⋊D19712yes—
145C22×C4—3942no—
146C22×C4—3941no—
147SD16—7882no—
148D8—44no—
149D8—7882no—
150C2×C8—21no—
151C2×C8—3941no—
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—41no—
162C23—3941no—
163Q8—3941no—
164Q8—1972no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—7884no—
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×C22⋊D19711yescharacteristic
188C8—22no—
189C8—7881no—
190C8—7881no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—7881no—
198C22—7881no—
199C22—7881no—
200C22#2026329411yescharacteristic
201C4—21no—
202C4—7881no—
203C4—7881no—
204C4—3942no—
205C4D8×D19711yescharacteristic
206C4#2026334511yescharacteristic
207C4#2026335011yescharacteristic
208C4#2026335011yescharacteristic
209C2—82no—
210C2—161no—
211C2—15761no—
212C2#2716121111yescharacteristic
213C2#2716122111yescharacteristic
214C2#2716124611yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#493×C197C2
socleC22×C197#120
hypercenter#493D197
hypocenterC197#1621
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #27161749C2semidirect
2. #27161276C2semidirect
3. #27161271C2semidirect
4. #27160823C2semidirect
5. #27160811C2semidirect
6. #27160808C2semidirect
7. #493×C197C2semidirect
54. Dic394D8semidirect
107. C197#1621semidirect
110. #493D197semidirect

Maximal subgroups

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

subgroupindex
#271617492
#271612762
#271612712
#271608232
#271608112
#271608082
#493×C1972
#1621197