Finite Group Databasebeta

← #38754167

Subgroups — #38754167

Order 25216.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#27161696C212yesmaximal, complemented
2#27161373C212yesmaximal, complemented
3#27161357C212yesmaximal, complemented
4#27161323C212yesmaximal, complemented
5#27161320C211yesmaximal, characteristic, complemented
6#27161300C212yesmaximal, complemented
7#27161292C212yesmaximal, complemented
8#27161105C211yesmaximal, characteristic, complemented
9#466×C197C211yesmaximal, characteristic
10C2×Dic788C2211yescharacteristic
11C2×#15691346C2212yes—
12C2×#15691342C2211yescharacteristic
13#20263325C2212yes—
14#20263319—22no—
15#20263313—22no—
16#20263313C2212yes—
17#20263307—22no—
18#20263305C2212yes—
19#20263303—22no—
20#20263291C2212yes—
21D4×Dic197C2212yes—
22#20263285C2211yescharacteristic
23#20263260C2211yescharacteristic
24#20263259C2212yes—
25#20263255C2212yes—
26#20263255C2212yes—
27#20263251C2212yes—
28#20263249C2212yes—
29#20263247C2212yes—
30C2×D8×C197C2211yescharacteristic
31C22×C8×C197C2211yescharacteristic
32#121×C197C2212yes—
33#107×C197C2211yescharacteristic
34#102×C197C2212yes—
35C2×Dic394—22no—
36C2×Dic394C2312yes—
37C22×Dic197—22no—
38Dic788—41no—
39#15691346—42no—
40#15691342—41no—
41C2×#13096802C2312yes—
42C4⋊Dic197D411yescharacteristic
43C4⋊Dic197D411yescharacteristic
44C4⋊Dic197C2312yes—
45#15691337—22no—
46#15691337—22no—
47#15691337—22no—
48C22⋊Dic197—22no—
49C22⋊Dic197—22no—
50C22⋊Dic197—22no—
51C22⋊Dic197—22no—
52C22⋊Dic197D412yes—
53C4×Dic197—22no—
54C4×Dic197C2312yes—
55C2×D4×C197—22no—
56C2×D4×C197C2312yes—
57C22×C4×C197C2311yescharacteristic
58D8×C197—41no—
59C2×C8×C197—41no—
60C2×C8×C197C2311yescharacteristic
61C2×C8×C197C2311yescharacteristic
62C4⋊C4×C197C2312yes—
63C22⋊C4×C197—22no—
64Dic394—22no—
65Dic394—22no—
66Dic394—82no—
67C2×Dic197—22no—
68C2×Dic197—22no—
69C2×Dic197—82no—
70C2×Dic197C2×D412yes—
71C2×Dic197C2×D412yes—
72#13096802—22no—
73C23×C197—22no—
74C23×C197C2×D411yescharacteristic
75D4×C197—22no—
76D4×C197—22no—
77D4×C197—82no—
78C2×C4×C197—21no—
79C2×C4×C197—21no—
80C2×C4×C197—22no—
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197C2411yescharacteristic
83C8×C197—21no—
84C8×C197—21no—
85C22×C197—21no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197—42no—
89C22×C197—42no—
90C22×C197—42no—
91C22×C197C22×D411yescharacteristic
92C4×C197—21no—
93C4×C197—42no—
94C4×C197H(5,𝔽2)11yescharacteristic
95C4×C1972−1+411yescharacteristic
96Dic197—22no—
97Dic197—22no—
98Dic197—42no—
99Dic197—42no—
100C2×C197—41no—
101C2×C197—82no—
102C2×C197#53711yescharacteristic
103C2×C197#57611yescharacteristic
104C2×C197#57811yescharacteristic
105C197#326211yescharacteristic, complemented, Sylow
106#3262—1971nomaximal, Sylow
107#542—1972no—
108#477—1972no—
109#474—1972no—
110#468—1972no—
111#467—1971no—
112#466D19711yescharacteristic
113#464—1972no—
114#463—1972no—
115#424—1971no—
116C2×Q16—1971no—
117C2×SD16—1972no—
118C2×D8C2×D19711yescharacteristic
119C2×M16—1971no—
120C22×C8C2×D19711yescharacteristic
121C4⋊Q8—1972no—
122#126—3942no—
123#124—3942no—
124#124—1972no—
125#123—3942no—
126C22⋊Q8—3942no—
127C22⋊Q8—1972no—
128#121C2×D19712yes—
129C4×Q8—1972no—
130C4×D4—1972no—
131#117—1971no—
132#107—1971no—
133#107—1972no—
134#107C2×D19711yescharacteristic
135C4⋊C8—1972no—
136C4⋊C8—1972no—
137#103—1972no—
138#103—1972no—
139#102—1972no—
140#102C2×D19712yes—
141C2×Q8—3942no—
142C2×Q8—1972no—
143C2×D4—22no—
144C2×D4C22×D19712yes—
145C22×C4—3942no—
146C22×C4C22×D19711yescharacteristic
147Q16—7881no—
148SD16—7882no—
149D8—41no—
150M16—7881no—
151C2×C8—41no—
152C2×C8—1972no—
153C2×C8C22×D19711yescharacteristic
154C2×C8C22×D19711yescharacteristic
155C4⋊C4—3942no—
156C4⋊C4—3942no—
157C4⋊C4—3942no—
158C4⋊C4—1972no—
159C4⋊C4—1971no—
160C4⋊C4—1971no—
161C4⋊C4C22×D19712yes—
162C22⋊C4—22no—
163C22⋊C4—3942no—
164C22⋊C4—3942no—
165C22⋊C4—3942no—
166C22⋊C4—3942no—
167C22⋊C4—1972no—
168C42—3942no—
169C42—1972no—
170C23—22no—
171C23D4×D19711yescharacteristic
172Q8—15762no—
173Q8—3942no—
174Q8—3942no—
175D4—22no—
176D4—22no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—15762no—
182C2×C4—3942no—
183C2×C4—3942no—
184C2×C4—1972no—
185C2×C4—1972no—
186C2×C4D4×D19711yescharacteristic
187C2×C4C23×D19711yescharacteristic
188C8—21no—
189C8—21no—
190C8—3942no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22C2×D4×D19711yescharacteristic
198C4—21no—
199C4—42no—
200C4—7882no—
201C4—7882no—
202C4—3942no—
203C4—3942no—
204C4#2026337111yescharacteristic
205C4#2026337311yescharacteristic
206C2—41no—
207C2—82no—
208C2#2716165611yescharacteristic
209C2#2716188011yescharacteristic
210C2#2716189311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radical#466×C197C2
socleC22×C197C22×D4
hypercenter#466D197
hypocenterC197#3262
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #27161696C2semidirect
2. #27161373C2semidirect
3. #27161357C2semidirect
4. #27161323C2semidirect
5. #27161320C2semidirect
6. #27161300C2semidirect
7. #27161292C2semidirect
8. #27161105C2semidirect
105. C197#3262semidirect

Maximal subgroups

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

subgroupindex
#271616962
#271613732
#271613572
#271613232
#271613202
#271613002
#271612922
#271611052
#466×C1972
#3262197