Finite Group Databasebeta

← #354×D199 #38770509

Subgroups — #354×D199 #38770509

Order 25472.

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×D199C211yesmaximal, characteristic
2#99×D199C212yesmaximal
3#27163813C211yesmaximal, characteristic, complemented
4#27163812C212yesmaximal, complemented
5#354×C199C211yesmaximal, characteristic, complemented
6C2×Q8×D199C411yescharacteristic, complemented
7C2×D4×D199C2211yescharacteristic
8#20263944C411yescharacteristic, complemented
9#20263943C411yescharacteristic, complemented
10#20263942C2211yescharacteristic
11#20263941—22no—
12C22⋊C4×D199—22no—
13C22⋊C4×D199—22no—
14C42×D199C411yescharacteristic, complemented
15#20263870C2212yes—
16#20263869—22no—
17#124×C199C2211yescharacteristic
18#99×C199C2212yes—
19C23×D199D412yes—
20C2×Dic398—21no—
21C2×Dic398C2×C411yescharacteristic, complemented
22Q8×D199—41no—
23Q8×D199—41no—
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199D412yes—
26D4×D199—41no—
27D4×D199—41no—
28C2×C4×D199—21no—
29C2×C4×D199—21no—
30C2×C4×D199—42no—
31C2×C4×D199C2×C411yescharacteristic
32C22⋊Dic199—22no—
33C22⋊Dic199—22no—
34(C2×D199)⋊C4—22no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—42no—
37C4×Dic199—21no—
38C4×Dic199C2×C411yescharacteristic, complemented
39C2×Q8×C199C2×C411yescharacteristic, complemented
40C2×D4×C199C2311yescharacteristic
41C22⋊C4×C199—22no—
42C22⋊C4×C199—22no—
43C42×C199C2×C411yescharacteristic, complemented
44C22×D199—22no—
45C22×D199—22no—
46C22×D199—44no—
47C22×D199C22⋊C411yescharacteristic
48C22×D199C22⋊C412yes—
49Dic398—42no—
50Dic398—41no—
51D796—41no—
52C22⋊D199—42no—
53C2×Dic199—21no—
54C2×Dic199—21no—
55C2×Dic199—42no—
56C2×Dic199C22⋊C411yescharacteristic
57C4×D199—21no—
58C4×D199—21no—
59C4×D199—42no—
60C4×D199—41no—
61C4×D199—41no—
62C4×D199—84no—
63C23×C199C2×D412yes—
64Q8×C199—41no—
65D4×C199—41no—
66C2×C4×C199—21no—
67C2×C4×C199—21no—
68C2×C4×C199—42no—
69C2×C4×C199C22×C411yescharacteristic
70C22×C199—22no—
71C22×C199—42no—
72C22×C199C2×C22⋊C411yescharacteristic
73C2×D199—22no—
74C2×D199—22no—
75C2×D199—44no—
76C2×D199—42no—
77C2×D199—82no—
78C2×D199#9911yescharacteristic
79C2×D199#9911yescharacteristic
80C4×C199—21no—
81C4×C199—41no—
82C4×C199—41no—
83C4×C199—82no—
84Dic199—21no—
85Dic199—41no—
86Dic199—41no—
87Dic199—82no—
88C2×C199—21no—
89C2×C199—42no—
90C2×C199C2×#9911yescharacteristic
91D199—21no—
92D199—42no—
93D199#35412yesdirect product factor
94C199C2×#35411yescharacteristic, complemented, Sylow
95C2×#354—1991nomaximal, Sylow
96C2×#124—1991no—
97C2×#99—1992no—
98#354—1991no—
99#354—1992no—
100#354D19911yescharacteristic, direct product factor
101C22×Q8—1991no—
102C22×D4—1991no—
103#124—3982no—
104#124—1991no—
105#124—1991no—
106#124—1991no—
107#124C2×D19911yescharacteristic
108C2×C22⋊C4—3982no—
109C2×C22⋊C4—3982no—
110C2×C42—1991no—
111#99—3982no—
112#99—1992no—
113#99C2×D19912yes—
114C24—1992no—
115C2×Q8—7961no—
116C2×Q8—7961no—
117C2×Q8—3981no—
118C2×Q8—1991no—
119C2×Q8C4×D19911yescharacteristic, complemented
120C2×D4—7961no—
121C2×D4—7961no—
122C2×D4—1992no—
123C2×D4—1991no—
124C2×D4C22×D19911yescharacteristic
125C22×C4—7962no—
126C22×C4—3981no—
127C22×C4—3981no—
128C22×C4—1991no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—7962no—
132C22⋊C4—3982no—
133C22⋊C4—3982no—
134C22⋊C4—3982no—
135C22⋊C4—3982no—
136C42—3981no—
137C42—1991no—
138C42C4×D19911yescharacteristic, complemented
139C23—7964no—
140C23—3982no—
141C23—3982no—
142C23—1992no—
143C23—1991no—
144C23D4×D19912yes—
145Q8—41no—
146Q8—7961no—
147Q8—7962no—
148D4—41no—
149D4—7961no—
150D4—7962no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—7962no—
155C2×C4—7961no—
156C2×C4—7961no—
157C2×C4—7962no—
158C2×C4—15924no—
159C2×C4—3981no—
160C2×C4—3981no—
161C2×C4—3981no—
162C2×C4—3981no—
163C2×C4—1991no—
164C2×C4C2×C4×D19911yescharacteristic
165C22—22no—
166C22—42no—
167C22—7962no—
168C22—7964no—
169C22—15922no—
170C22—3982no—
171C22—3982no—
172C22—1991no—
173C22—1991no—
174C22C22⋊C4×D19911yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—7961no—
180C4—7961no—
181C4—15922no—
182C4—3981no—
183C2—21no—
184C2—42no—
185C2—7962no—
186C2—3981no—
187C2—1992no—
188C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#354×C199C2
socleC2×C199C2×#99
prime radicalC2×D4×D199C22
cosocleC2×D4×C199C23
hypercenter#354D199
hypocenterC199C2×#354
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #27163813C2semidirect
4. #27163812C2semidirect
5. #354×C199C2semidirect
6. C2×Q8×D199C4semidirect
8. #20263944C4semidirect
9. #20263943C4semidirect
14. C42×D199C4semidirect
21. C2×Dic398C2×C4semidirect
38. C4×Dic199C2×C4semidirect
39. C2×Q8×C199C2×C4semidirect
43. C42×C199C2×C4semidirect
93. D199#354direct (N × H)
94. C199C2×#354semidirect
100. #354D199direct (N × H)
119. C2×Q8C4×D199semidirect
138. C42C4×D199semidirect

Maximal subgroups

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

subgroupindex
#124×D1992
#99×D1992
#271638132
#271638122
#354×C1992
C2×#354199