Finite Group Databasebeta

← #38767571

Subgroups — #38767571

Order 25472.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164569C211yesmaximal, characteristic
2#27164074C211yesmaximal, characteristic, complemented
3#27164040C211yesmaximal, characteristic, complemented
4#27163764C211yesmaximal, characteristic, complemented
5#27163761C211yesmaximal, characteristic, complemented
6#27163761C211yesmaximal, characteristic, complemented
7#332×C199C211yesmaximal, characteristic, complemented
8C2×#15691887—21no—
9C2×#15691869—21no—
10#20263952C2211yescharacteristic
11#20263944—21no—
12#20263930—21no—
13#20263930C412yes—
14#20263917—21no—
15#20263917C412yes—
16C42×D199C2211yescharacteristic
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19Dic199⋊C8—21no—
20(C2×D199)⋊C8—21no—
21#20263865—21no—
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25#15691887—42no—
26#15691887—42no—
27C2×D796—22no—
28C2×D796—21no—
29C2×D796—21no—
30C2×D796C2×C412yes—
31C2×C4×D199—22no—
32C2×C4×D199C2×C411yescharacteristic
33C2×C4×D199D411yescharacteristic
34C2×C4×D199D411yescharacteristic
35#15691869—42no—
36C2×C199⋊C8—21no—
37C2×C199⋊C8—21no—
38(C2×D199)⋊C4—22no—
39(C2×D199)⋊C4—22no—
40C4×Dic199C2×C411yescharacteristic
41C4×Dic199D411yescharacteristic
42C4×Dic199D411yescharacteristic
43C2×Q8×C199—21no—
44C2×C8×C199—21no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199C2×C412yes—
48C42×C199C2311yescharacteristic
49C22×D199—22no—
50C22×D199C22⋊C411yescharacteristic
51D796—42no—
52D796—44no—
53D796—41no—
54D796—41no—
55D796—41no—
56D796—41no—
57C2×Dic199C22⋊C411yescharacteristic
58C2×Dic199C22⋊C411yescharacteristic
59C2×Dic199C22⋊C411yescharacteristic
60C4×D199—21no—
61C4×D199—21no—
62C4×D199—21no—
63C4×D199—21no—
64C4×D199—41no—
65C4×D199—44no—
66C199⋊C8—41no—
67C199⋊C8—41no—
68Q8×C199—41no—
69Q8×C199—41no—
70C2×C4×C199—22no—
71C2×C4×C199C22×C411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C8×C199—41no—
75C8×C199—41no—
76C22×C199C2×C22⋊C411yescharacteristic
77C2×D199—21no—
78C2×D199—21no—
79C2×D199—21no—
80C2×D199—42no—
81C2×D199—42no—
82C2×D199—42no—
83C4×C199—21no—
84C4×C199—42no—
85C4×C199AGL(1,ℤ/8)11yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87C4×C199AGL(1,ℤ/8)11yescharacteristic
88C4×C199#13711yescharacteristic
89Dic199—21no—
90Dic199—21no—
91Dic199—21no—
92C2×C199#41311yescharacteristic
93C2×C199#41711yescharacteristic
94C2×C199#41811yescharacteristic
95D199—41no—
96D199—82no—
97C199#155711yescharacteristic, complemented, Sylow
98#1557—1991nomaximal, Sylow
99#532—1991no—
100#432—1991no—
101#423—1991no—
102#332—1991no—
103#332D19911yescharacteristic, complemented
104#331—1991no—
105#331—1991no—
106C2×#41—3981no—
107C2×M16—3981no—
108C4⋊Q8C2×D19911yescharacteristic
109#127—1991no—
110#124—3981no—
111#121—3981no—
112#121—1992no—
113C4×D4—3981no—
114C4×D4—1992no—
115C2×C42—1991no—
116C4⋊C8—3981no—
117C4⋊C8—1991no—
118C4⋊C8—1991no—
119C4⋊C8C2×D19911yescharacteristic
120C4⋊C8C2×D19911yescharacteristic
121C22⋊C8—3981no—
122#97—3981no—
123#41—7962no—
124#41—7962no—
125C2×Q8—21no—
126C2×D4—3982no—
127C2×D4—3981no—
128C2×D4—3981no—
129C2×D4—1992no—
130C22×C4—3982no—
131C22×C4—1991no—
132C22×C4—1991no—
133C22×C4—1991no—
134M16—7962no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—3981no—
138C2×C8—3981no—
139C4⋊C4—21no—
140C4⋊C4C4×D19912yes—
141C22⋊C4—3982no—
142C22⋊C4—3982no—
143C42—1991no—
144C42—1991no—
145C42—1991no—
146C42C22×D19911yescharacteristic
147C23—3982no—
148C23—1991no—
149Q8—41no—
150Q8—41no—
151D4—7962no—
152D4—7964no—
153D4—7961no—
154D4—7961no—
155D4—7961no—
156D4—7961no—
157C2×C4—22no—
158C2×C4—7961no—
159C2×C4—7964no—
160C2×C4—3981no—
161C2×C4—3981no—
162C2×C4—3981no—
163C2×C4—3981no—
164C2×C4—1991no—
165C2×C4—1991no—
166C2×C4—1991no—
167C2×C4C2×C4×D19911yescharacteristic
168C2×C4D4×D19911yescharacteristic
169C2×C4D4×D19911yescharacteristic
170C8—41no—
171C8—41no—
172C8—7961no—
173C8—7961no—
174C22—7962no—
175C22—7962no—
176C22—7962no—
177C22—3981no—
178C22—3981no—
179C22—3981no—
180C22C22⋊C4×D19911yescharacteristic
181C4—21no—
182C4—42no—
183C4—3981no—
184C4—3981no—
185C4—3981no—
186C4#2026397911yescharacteristic
187C4#2026398211yescharacteristic
188C4#2026398211yescharacteristic
189C4#2026398711yescharacteristic
190C2—7961no—
191C2—15922no—
192C2#2716400111yescharacteristic
193C2#2716401811yescharacteristic
194C2#2716402111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC2×C4×C199C22×C4
FrattiniC42C22×D199
Fitting#332×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164569C2
cosocleC42×C199C23
hypercenter#332D199
hypocenterC199#1557
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27164074C2semidirect
3. #27164040C2semidirect
4. #27163764C2semidirect
5. #27163761C2semidirect
6. #27163761C2semidirect
7. #332×C199C2semidirect
97. C199#1557semidirect
103. #332D199semidirect

Maximal subgroups

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

subgroupindex
#271645692
#271640742
#271640402
#271637642
#271637612
#332×C1992
#1557199