Finite Group Databasebeta

← #354×D197 #38744371

Subgroups — #354×D197 #38744371

Order 25216.

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

normal subgroups only

This group has 284 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#99×D197C212yesmaximal
3#27160874C211yesmaximal, characteristic, complemented
4#27160873C212yesmaximal, complemented
5#354×C197C211yesmaximal, characteristic, complemented
6C2×Q8×D197C411yescharacteristic, complemented
7C2×D4×D197C2211yescharacteristic
8#20263315C411yescharacteristic, complemented
9#20263314C411yescharacteristic, complemented
10#20263313C2211yescharacteristic
11#20263312—22no—
12C22⋊C4×D197—22no—
13C22⋊C4×D197—22no—
14C42×D197C411yescharacteristic, complemented
15#20263241C2212yes—
16#20263240—22no—
17#124×C197C2211yescharacteristic
18#99×C197C2212yes—
19C23×D197D412yes—
20C2×Dic394—21no—
21C2×Dic394C2×C411yescharacteristic, complemented
22Q8×D197—41no—
23Q8×D197—41no—
24C2×D788C2×C411yescharacteristic
25C2×C22⋊D197D412yes—
26D4×D197—41no—
27D4×D197—41no—
28C2×C4×D197—21no—
29C2×C4×D197—21no—
30C2×C4×D197—42no—
31C2×C4×D197C2×C411yescharacteristic
32C22⋊Dic197—22no—
33C22⋊Dic197—22no—
34#15691335—22no—
35#15691335—22no—
36#15691335—42no—
37C4×Dic197—21no—
38C4×Dic197C2×C411yescharacteristic, complemented
39C2×Q8×C197C2×C411yescharacteristic, complemented
40C2×D4×C197C2311yescharacteristic
41C22⋊C4×C197—22no—
42C22⋊C4×C197—22no—
43C42×C197C2×C411yescharacteristic, complemented
44C22×D197—22no—
45C22×D197—22no—
46C22×D197—44no—
47C22×D197C22⋊C411yescharacteristic
48C22×D197C22⋊C412yes—
49Dic394—42no—
50Dic394—41no—
51D788—41no—
52C22⋊D197—42no—
53C2×Dic197—21no—
54C2×Dic197—21no—
55C2×Dic197—42no—
56C2×Dic197C22⋊C411yescharacteristic
57C4×D197—21no—
58C4×D197—21no—
59C4×D197—42no—
60C4×D197—41no—
61C4×D197—41no—
62C4×D197—84no—
63C23×C197C2×D412yes—
64Q8×C197—41no—
65D4×C197—41no—
66C2×C4×C197—21no—
67C2×C4×C197—21no—
68C2×C4×C197—42no—
69C2×C4×C197C22×C411yescharacteristic
70C22×C197—22no—
71C22×C197—42no—
72C22×C197C2×C22⋊C411yescharacteristic
73C2×D197—22no—
74C2×D197—22no—
75C2×D197—44no—
76C2×D197—42no—
77C2×D197—82no—
78C2×D197#9911yescharacteristic
79C2×D197#9911yescharacteristic
80C4×C197—21no—
81C4×C197—41no—
82C4×C197—41no—
83C4×C197—82no—
84Dic197—21no—
85Dic197—41no—
86Dic197—41no—
87Dic197—82no—
88C2×C197—21no—
89C2×C197—42no—
90C2×C197C2×#9911yescharacteristic
91D197—21no—
92D197—42no—
93D197#35412yesdirect product factor
94C197C2×#35411yescharacteristic, complemented, Sylow
95C2×#354—1971nomaximal, Sylow
96C2×#124—1971no—
97C2×#99—1972no—
98#354—1971no—
99#354—1972no—
100#354D19711yescharacteristic, direct product factor
101C22×Q8—1971no—
102C22×D4—1971no—
103#124—3942no—
104#124—1971no—
105#124—1971no—
106#124—1971no—
107#124C2×D19711yescharacteristic
108C2×C22⋊C4—3942no—
109C2×C22⋊C4—3942no—
110C2×C42—1971no—
111#99—3942no—
112#99—1972no—
113#99C2×D19712yes—
114C24—1972no—
115C2×Q8—7881no—
116C2×Q8—7881no—
117C2×Q8—3941no—
118C2×Q8—1971no—
119C2×Q8C4×D19711yescharacteristic, complemented
120C2×D4—7881no—
121C2×D4—7881no—
122C2×D4—1972no—
123C2×D4—1971no—
124C2×D4C22×D19711yescharacteristic
125C22×C4—7882no—
126C22×C4—3941no—
127C22×C4—3941no—
128C22×C4—1971no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—7882no—
132C22⋊C4—3942no—
133C22⋊C4—3942no—
134C22⋊C4—3942no—
135C22⋊C4—3942no—
136C42—3941no—
137C42—1971no—
138C42C4×D19711yescharacteristic, complemented
139C23—7884no—
140C23—3942no—
141C23—3942no—
142C23—1972no—
143C23—1971no—
144C23D4×D19712yes—
145Q8—41no—
146Q8—7881no—
147Q8—7882no—
148D4—41no—
149D4—7881no—
150D4—7882no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—7882no—
155C2×C4—7881no—
156C2×C4—7881no—
157C2×C4—7882no—
158C2×C4—15764no—
159C2×C4—3941no—
160C2×C4—3941no—
161C2×C4—3941no—
162C2×C4—3941no—
163C2×C4—1971no—
164C2×C4C2×C4×D19711yescharacteristic
165C22—22no—
166C22—42no—
167C22—7882no—
168C22—7884no—
169C22—15762no—
170C22—3942no—
171C22—3942no—
172C22—1971no—
173C22—1971no—
174C22C22⋊C4×D19711yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—7881no—
180C4—7881no—
181C4—15762no—
182C4—3941no—
183C2—21no—
184C2—42no—
185C2—7882no—
186C2—3941no—
187C2—1972no—
188C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#354×C197C2
socleC2×C197C2×#99
prime radicalC2×D4×D197C22
cosocleC2×D4×C197C23
hypercenter#354D197
hypocenterC197C2×#354
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #27160874C2semidirect
4. #27160873C2semidirect
5. #354×C197C2semidirect
6. C2×Q8×D197C4semidirect
8. #20263315C4semidirect
9. #20263314C4semidirect
14. C42×D197C4semidirect
21. C2×Dic394C2×C4semidirect
38. C4×Dic197C2×C4semidirect
39. C2×Q8×C197C2×C4semidirect
43. C42×C197C2×C4semidirect
93. D197#354direct (N × H)
94. C197C2×#354semidirect
100. #354D197direct (N × H)
119. C2×Q8C4×D197semidirect
138. C42C4×D197semidirect

Maximal subgroups

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

subgroupindex
#124×D1972
#99×D1972
#271608742
#271608732
#354×C1972
C2×#354197