Finite Group Databasebeta

← #355×D197 #38744388

Subgroups — #355×D197 #38744388

Order 25216.

260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 57 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#124×D197C211yesmaximal, characteristic
2#101×D197C212yesmaximal, complemented
3#27160876C211yesmaximal, characteristic, complemented
4#27160875C212yesmaximal, complemented
5#355×C197C211yesmaximal, characteristic, complemented
6C2×Q8×D197C2211yescharacteristic
7C2×D4×D197C411yescharacteristic
8M16×D197—22no—
9#20263315C2211yescharacteristic
10#20263314C411yescharacteristic
11#20263313C411yescharacteristic
12#20263312—22no—
13C22⋊C4×D197—22no—
14C42×D197C411yescharacteristic
15#20263245C2212yescomplemented
16#20263244—22no—
17#124×C197C2211yescharacteristic
18#101×C197C2212yescomplemented
19C23×D197—21no—
20C2×Dic394C2×C411yescharacteristic
21C2×Dic394D412yes—
22Q8×D197—41no—
23Q8×D197—41no—
24C2×D788C2×C411yescharacteristic
25C2×C22⋊D197—21no—
26D4×D197—41no—
27D4×D197—41no—
28C2×C4×D197—21no—
29C2×C4×D197C2×C411yescharacteristic
30C2×C4×D197D412yes—
31#15691342—22no—
32#15691341—42no—
33C8×D197—42no—
34C22⋊Dic197—22no—
35#15691335—22no—
36#15691335—22no—
37C4×Dic197—21no—
38C4×Dic197C2×C411yescharacteristic
39C2×Q8×C197C2311yescharacteristic
40C2×D4×C197C2×C411yescharacteristic
41M16×C197—22no—
42C22⋊C4×C197—22no—
43C42×C197C2×C411yescharacteristic
44C22×D197—21no—
45C22×D197—44no—
46C22×D197—41no—
47C22×D197—41no—
48C22×D197C22⋊C411yescharacteristic
49Dic394—41no—
50Dic394—42no—
51D788—41no—
52C22⋊D197—42no—
53C2×Dic197—21no—
54C2×Dic197C22⋊C411yescharacteristic
55C2×Dic197C22⋊C412yes—
56C4×D197—22no—
57C4×D197—22no—
58C4×D197—21no—
59C4×D197—21no—
60C4×D197—42no—
61#13096802—42no—
62C23×C197—21no—
63Q8×C197—41no—
64D4×C197—41no—
65C2×C4×C197—21no—
66C2×C4×C197C22×C411yescharacteristic
67C2×C4×C197C2×D412yes—
68C8×C197—42no—
69C22×C197—42no—
70C22×C197—41no—
71C22×C197C2×C22⋊C411yescharacteristic
72C2×D197—22no—
73C2×D197—42no—
74C2×D197—41no—
75C2×D197—82no—
76C2×D197—82no—
77C2×D197#9911yescharacteristic
78C2×D197#9911yescharacteristic
79C4×C197—22no—
80C4×C197—21no—
81C4×C197—41no—
82Dic197—21no—
83Dic197—22no—
84Dic197—41no—
85C2×C197—21no—
86C2×C197—81no—
87C2×C197C2×#9911yescharacteristic
88D197—21no—
89D197—81no—
90D197#35512yesdirect product factor
91C197C2×#35511yescharacteristic, complemented, Sylow
92C2×#355—1971nomaximal, Sylow
93C2×#124—1971no—
94C2×#101—1972no—
95#355—1971no—
96#355—1972no—
97#355D19711yescharacteristic, direct product factor
98C22×Q8—1971no—
99C22×D4—1971no—
100C2×M16—3942no—
101#124—3942no—
102#124—1971no—
103#124—1971no—
104#124—1971no—
105#124C2×D19711yescharacteristic
106C2×C22⋊C4—3942no—
107C2×C42—1971no—
108#101—3942no—
109#101—1972no—
110#101C2×D19712yescomplemented
111C24—3941no—
112C2×Q8—7881no—
113C2×Q8—7881no—
114C2×Q8—1971no—
115C2×Q8—1972no—
116C2×Q8C22×D19711yescharacteristic
117C2×D4—7881no—
118C2×D4—7881no—
119C2×D4—3941no—
120C2×D4—1971no—
121C2×D4C4×D19711yescharacteristic
122C22×C4—3941no—
123C22×C4—1971no—
124C22×C4—1972no—
125M16—22no—
126M16—7882no—
127M16—3942no—
128C2×C8—7882no—
129C22⋊C4—22no—
130C22⋊C4—3942no—
131C22⋊C4—3942no—
132C22⋊C4—3942no—
133C42—3941no—
134C42—1971no—
135C42C4×D19711yescharacteristic
136C23—21no—
137C23—7884no—
138C23—7881no—
139C23—7881no—
140C23—3941no—
141C23—1971no—
142Q8—41no—
143Q8—7882no—
144Q8—7881no—
145D4—41no—
146D4—7881no—
147D4—7882no—
148C2×C4—21no—
149C2×C4—7882no—
150C2×C4—3942no—
151C2×C4—3942no—
152C2×C4—3941no—
153C2×C4—3941no—
154C2×C4—3941no—
155C2×C4—1971no—
156C2×C4—1972no—
157C2×C4C2×C4×D19711yescharacteristic
158C2×C4D4×D19712yes—
159C8—42no—
160C8—7882no—
161C22—42no—
162C22—41no—
163C22—7882no—
164C22—7881no—
165C22—15762no—
166C22—15762no—
167C22—3942no—
168C22—1971no—
169C22—1971no—
170C22C22⋊C4×D19711yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—7881no—
175C4—3941no—
176C4—3942no—
177C2—21no—
178C2—81no—
179C2—15761no—
180C2—3941no—
181C2—1972no—
182C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×Q8C22×D197
Fitting#355×C197C2
socleC2×C197C2×#99
prime radicalC2×D4×D197C4
cosocleC2×Q8×C197C23
hypercenter#355D197
hypocenterC197C2×#355
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D197C2semidirect
3. #27160876C2semidirect
4. #27160875C2semidirect
5. #355×C197C2semidirect
15. #20263245C22semidirect
18. #101×C197C22semidirect
90. D197#355direct (N × H)
91. C197C2×#355semidirect
97. #355D197direct (N × H)
110. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
#124×D1972
#101×D1972
#271608762
#271608752
#355×C1972
C2×#355197