Finite Group Databasebeta

← #38744396

Subgroups — #38744396

Order 25216.

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

normal subgroups only

This group has 250 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#27161630C211yesmaximal, characteristic
2#27161062C212yesmaximal, complemented
3#27160878C211yesmaximal, characteristic, complemented
4#27160875C212yesmaximal, complemented
5#356×C197C211yesmaximal, characteristic, complemented
6C2×#15691359C411yescharacteristic
7C2×#15691359C2211yescharacteristic
8M16×D197—22no—
9#20263323C411yescharacteristic
10#20263315C411yescharacteristic
11#20263315C2211yescharacteristic
12#20263301—22no—
13#20263288—22no—
14C42×D197C411yescharacteristic
15#20263245C2212yes—
16#20263242—22no—
17C4⋊Q8×C197C2211yescharacteristic
18#101×C197C2212yes—
19#15691359—41no—
20#15691359—41no—
21#15691359—41no—
22#15691359—41no—
23C2×D788—22no—
24C2×D788—21no—
25C2×D788C2×C411yescharacteristic
26C2×D788C2×C411yescharacteristic
27C2×D788D412yes—
28C2×C4×D197—21no—
29C2×C4×D197—21no—
30C2×C4×D197C2×C411yescharacteristic
31C2×C4×D197D412yes—
32#15691342—22no—
33#15691341—42no—
34C8×D197—42no—
35#15691335—22no—
36#15691335—22no—
37C4×Dic197—21no—
38C4×Dic197C2×C411yescharacteristic
39C2×Q8×C197C2×C411yescharacteristic
40C2×Q8×C197C2311yescharacteristic
41M16×C197—22no—
42C4⋊C4×C197—22no—
43C42×C197C2×C411yescharacteristic
44C22×D197—21no—
45C22×D197C22⋊C412yes—
46C22×D197C22⋊C411yescharacteristic
47D788—41no—
48D788—42no—
49D788—41no—
50D788—44no—
51D788—42no—
52C2×Dic197—21no—
53C2×Dic197C22⋊C411yescharacteristic
54C4×D197—22no—
55C4×D197—22no—
56C4×D197—21no—
57C4×D197—21no—
58C4×D197—42no—
59C4×D197—41no—
60C4×D197—41no—
61#13096802—42no—
62Q8×C197—41no—
63Q8×C197—41no—
64C2×C4×C197—21no—
65C2×C4×C197—21no—
66C2×C4×C197C22×C411yescharacteristic
67C2×C4×C197C2×D412yes—
68C8×C197—42no—
69C22×C197C2×C22⋊C411yescharacteristic
70C2×D197—21no—
71C2×D197—22no—
72C2×D197—41no—
73C2×D197—42no—
74C2×D197—42no—
75C2×D197—41no—
76C4×C197—22no—
77C4×C197—22no—
78C4×C197—21no—
79C4×C197—41no—
80Dic197—41no—
81Dic197#9911yescharacteristic
82Dic197#9911yescharacteristic
83C2×C197—21no—
84C2×C197C2×#9911yescharacteristic
85D197—41no—
86D197—42no—
87D197—81no—
88C197#213211yescharacteristic, complemented, Sylow
89#2132—1971nomaximal, Sylow
90#532—1971no—
91#413—1972no—
92#356—1971no—
93#356D19711yescharacteristic, complemented
94#355—1972no—
95C2×#41—1971no—
96C2×#41—1971no—
97C2×M16—3942no—
98C4⋊Q8C2×D19711yescharacteristic
99#127—1971no—
100#124—1971no—
101#124—1971no—
102#121—3942no—
103C4×D4—3942no—
104C2×C42—1971no—
105#101—1972no—
106#101C2×D19712yes—
107#100—3942no—
108#41—7881no—
109#41—7881no—
110#41—7881no—
111#41—7881no—
112C2×Q8C4×D19711yescharacteristic
113C2×Q8C22×D19711yescharacteristic
114C2×D4—3942no—
115C2×D4—3941no—
116C2×D4—1971no—
117C2×D4—1971no—
118C2×D4—1972no—
119C22×C4—3941no—
120C22×C4—3941no—
121C22×C4—1971no—
122C22×C4—1972no—
123M16—22no—
124M16—7882no—
125M16—3942no—
126C2×C8—7882no—
127C4⋊C4—22no—
128C22⋊C4—3942no—
129C22⋊C4—3942no—
130C42—3941no—
131C42—1971no—
132C42C4×D19711yescharacteristic
133C23—3941no—
134C23—1972no—
135C23—1971no—
136Q8—41no—
137Q8—41no—
138D4—7881no—
139D4—7884no—
140D4—7882no—
141D4—7881no—
142D4—7882no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—7882no—
146C2×C4—7881no—
147C2×C4—7881no—
148C2×C4—3942no—
149C2×C4—3942no—
150C2×C4—3941no—
151C2×C4—3941no—
152C2×C4—3941no—
153C2×C4—1971no—
154C2×C4C2×C4×D19711yescharacteristic
155C2×C4D4×D19712yes—
156C8—42no—
157C8—7882no—
158C22—7881no—
159C22—7882no—
160C22—7882no—
161C22—7881no—
162C22—3941no—
163C22—3942no—
164C22C22⋊C4×D19711yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—7881no—
170C4—1971no—
171C4—1971no—
172C2—21no—
173C2—7881no—
174C2—7882no—
175C2—15761no—
176C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×Q8C22×D197
Fitting#356×C197C2
socleC2×C197C2×#99
prime radical#27161630C2
cosocleC2×Q8×C197C23
hypercenter#356D197
hypocenterC197#2132
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #27161062C2semidirect
3. #27160878C2semidirect
4. #27160875C2semidirect
5. #356×C197C2semidirect
88. C197#2132semidirect
93. #356D197semidirect

Maximal subgroups

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

subgroupindex
#271616302
#271610622
#271608782
#271608752
#356×C1972
#2132197