Finite Group Databasebeta

← #38767260

Subgroups — #38767260

Order 25472.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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#27164550C211yesmaximal, characteristic
2C4⋊C8×D199C211yesmaximal, characteristic, complemented
3C4×#15691869C211yesmaximal, characteristic, complemented
4#27163750C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic, complemented
8C2×#15691869—21no—
9C2×C8×D199—21no—
10#20263948—21no—
11#20263946C411yescharacteristic
12#20263942C411yescharacteristic
13#20263937—21no—
14#20263935C411yescharacteristic
15#20263926C411yescharacteristic
16C4×Dic398C2211yescharacteristic
17C4×C22⋊D199—21no—
18#20263914—21no—
19C42×D199C2211yescharacteristic
20C4⋊(C199⋊C8)C2211yescharacteristic
21Dic199⋊C8—21no—
22#20263865—21no—
23C4×C199⋊C8C2211yescharacteristic
24C4×D4×C199C2211yescharacteristic
25C4⋊C8×C199C2211yescharacteristic
26C4×C8×C199C2211yescharacteristic
27C2×Dic398C2×C411yescharacteristic
28C2×C22⋊D199—21no—
29C2×C4×D199C2×C411yescharacteristic
30C2×C4×D199D411yescharacteristic
31C2×C4×D199D411yescharacteristic
32#15691869—24no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C8×D199—42no—
36C4⋊Dic199—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199C2×C411yescharacteristic
39Dic199⋊C4—21no—
40Dic199⋊C4—21no—
41C22⋊Dic199—21no—
42C22⋊Dic199—21no—
43C22⋊Dic199—21no—
44(C2×D199)⋊C4—21no—
45C4×Dic199—21no—
46C4×Dic199C2×C411yescharacteristic
47C4×Dic199D411yescharacteristic
48C4×Dic199D411yescharacteristic
49C2×D4×C199C2×C411yescharacteristic
50C22×C4×C199—21no—
51C2×C8×C199—21no—
52C2×C8×C199—21no—
53C4⋊C4×C199C2×C411yescharacteristic
54C22⋊C4×C199—21no—
55C42×C199C2311yescharacteristic
56C22×D199C22⋊C411yescharacteristic
57Dic398—21no—
58Dic398—21no—
59C22⋊D199—42no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199C22⋊C411yescharacteristic
63C2×Dic199C22⋊C411yescharacteristic
64C2×Dic199C22⋊C411yescharacteristic
65C4×D199—22no—
66C4×D199—21no—
67C4×D199—21no—
68C4×D199—21no—
69C4×D199—21no—
70C199⋊C8—22no—
71C199⋊C8—41no—
72C23×C199—21no—
73D4×C199—21no—
74D4×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199—41no—
77C2×C4×C199—41no—
78C2×C4×C199C22×C411yescharacteristic
79C2×C4×C199C2×D411yescharacteristic
80C2×C4×C199C2×D411yescharacteristic
81C8×C199—22no—
82C8×C199—41no—
83C22×C199—41no—
84C22×C199—41no—
85C22×C199—41no—
86C22×C199C2×C22⋊C411yescharacteristic
87C2×D199—21no—
88C2×D199—21no—
89C2×D199#10412yes—
90C4×C199—21no—
91C4×C199—41no—
92C4×C199#13111yescharacteristic
93C4×C199#13111yescharacteristic
94C4×C199AGL(1,ℤ/8)11yescharacteristic
95C4×C199#13711yescharacteristic
96Dic199—21no—
97Dic199—21no—
98Dic199—21no—
99Dic199—41no—
100Dic199—41no—
101C2×C199—81no—
102C2×C199#40811yescharacteristic
103C2×C199#41711yescharacteristic
104C2×C199C2×#10411yescharacteristic
105D199—22no—
106C199#148311yescharacteristic, complemented, Sylow
107#1483—1991nomaximal, Sylow
108#529—1991no—
109C2×C4⋊C8—1991no—
110C4×M16—1991no—
111Q8⋊C8—1991no—
112Q8⋊C8—1991no—
113D4⋊C8—1991no—
114D4⋊C8D19911yescharacteristic, complemented
115C2×M16—3981no—
116C22×C8—3981no—
117#126—3981no—
118#125—1991no—
119#124—1991no—
120#123—3981no—
121C22⋊Q8—1991no—
122#121—1991no—
123C4×Q8—1991no—
124C4×D4—3981no—
125C4×D4C2×D19911yescharacteristic
126#117—3981no—
127C2×C42—1991no—
128C4⋊C8—3981no—
129C4⋊C8—1991no—
130C4⋊C8C2×D19911yescharacteristic
131#97—3981no—
132C4×C8—1991no—
133C4×C8C2×D19911yescharacteristic
134C2×Q8—1991no—
135C2×D4—3981no—
136C2×D4C4×D19911yescharacteristic
137C22×C4—21no—
138C22×C4—1991no—
139C22×C4—1991no—
140C22×C4—1991no—
141M16—3984no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—7962no—
145C2×C8—3981no—
146C2×C8—3981no—
147C4⋊C4—3981no—
148C4⋊C4—3981no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C4⋊C4—1991no—
152C4⋊C4C4×D19911yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—3981no—
155C22⋊C4—3981no—
156C22⋊C4—3981no—
157C22⋊C4—3981no—
158C42—3981no—
159C42—1991no—
160C42—1991no—
161C42—1991no—
162C42C22×D19911yescharacteristic
163C23—21no—
164C23—1991no—
165Q8—3981no—
166Q8—3981no—
167D4—21no—
168D4—21no—
169D4—7962no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—3982no—
174C2×C4—3981no—
175C2×C4—3981no—
176C2×C4—3981no—
177C2×C4—3981no—
178C2×C4—3981no—
179C2×C4—3981no—
180C2×C4—1991no—
181C2×C4—1991no—
182C2×C4—1991no—
183C2×C4C2×C4×D19911yescharacteristic
184C2×C4D4×D19911yescharacteristic
185C2×C4D4×D19911yescharacteristic
186C8—22no—
187C8—41no—
188C8—7961no—
189C8—3982no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—3981no—
194C22—3981no—
195C22—1992no—
196C22C22⋊C4×D19911yescharacteristic
197C4—21no—
198C4—41no—
199C4—7961no—
200C4—7961no—
201C4—3981no—
202C4—3981no—
203C4—3981no—
204C4#2026396111yescharacteristic
205C4#2026396311yescharacteristic
206C4#2026397911yescharacteristic
207C4#2026398411yescharacteristic
208C2—81no—
209C2—3982no—
210C2#2716398311yescharacteristic
211C2#2716401611yescharacteristic
212C2#104×D19911yescharacteristic
subgroupG/H
centerC2×C4D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263926C4
cosocleC42×C199C23
hypercenterD4⋊C8D199
hypocenterC199#1483
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×D199C2semidirect
3. C4×#15691869C2semidirect
4. #27163750C2semidirect
5. #27163748C2semidirect
6. D4⋊(C199⋊C8)C2semidirect
7. D4⋊C8×C199C2semidirect
106. C199#1483semidirect
114. D4⋊C8D199semidirect

Maximal subgroups

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

subgroupindex
#271645502
C4⋊C8×D1992
C4×#156918692
#271637502
#271637482
D4⋊(C199⋊C8)2
D4⋊C8×C1992
#1483199