Finite Group Databasebeta

← #38767921

Subgroups — #38767921

Order 25472.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164728C211yesmaximal, characteristic, complemented
2#27164390C211yesmaximal, characteristic, complemented
3#27164210C211yesmaximal, characteristic
4#27163766C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#460×C199C211yesmaximal, characteristic, complemented
8C2×#15691874—22no—
9C2×#15691871—21no—
10#20263955C2211yescharacteristic
11#20263947—21no—
12#20263933—22no—
13#20263931—21no—
14C4×D796—21no—
15C4×D796C2211yescharacteristic
16C4⋊C4×D199—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263876—21no—
19C4×C199⋊C8C2211yescharacteristic
20C2×SD16×C199—21no—
21#127×C199C2211yescharacteristic
22C4×Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25C2×Dic398D412yes—
26C2×D796D411yescharacteristic
27C2×C4×D199—22no—
28C2×C4×D199—21no—
29#15691874—44no—
30#15691871—42no—
31C2×C199⋊C8—21no—
32C2×C199⋊C8—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199D411yescharacteristic
36Dic199⋊C4—22no—
37Dic199⋊C4—22no—
38(C2×D199)⋊C4—22no—
39(C2×D199)⋊C4—21no—
40C2×Q8×C199D411yescharacteristic
41C2×D4×C199—21no—
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44SD16×C199—42no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C42×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51Dic398—42no—
52D796—21no—
53D796SD1612yescomplemented
54C2×Dic199—21no—
55C2×Dic199—22no—
56C4×D199—41no—
57C4×D199—41no—
58C4×D199—82no—
59C199⋊C8—22no—
60C199⋊C8—41no—
61C23×C199—41no—
62Q8×C199—21no—
63Q8×C199—21no—
64D4×C199—42no—
65D4×C199—41no—
66D4×C199—41no—
67D4×C199—41no—
68D4×C199—41no—
69C2×C4×C199—21no—
70C2×C4×C199—21no—
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C8×C199—41no—
75C22×C199—81no—
76C22×C199—81no—
77C22×C199—81no—
78C22×C199#12011yescharacteristic
79C2×D199—22no—
80C2×D199—41no—
81C2×D199—41no—
82C4×C199—21no—
83C4×C199—41no—
84C4×C199—41no—
85C4×C199C2×SD1611yescharacteristic
86C4×C199#13511yescharacteristic
87C4×C199AGL(1,ℤ/8)11yescharacteristic
88C4×C199#13711yescharacteristic
89Dic199—41no—
90Dic199—42no—
91C2×C199—161no—
92C2×C199#44811yescharacteristic
93C2×C199#44911yescharacteristic
94C2×C199#45311yescharacteristic
95D199—42no—
96C199#162211yescharacteristic, complemented, Sylow
97#1622—1991nomaximal, Sylow
98#554—1991no—
99#492—1991no—
100#460D19911yescharacteristic, complemented
101#459—1991no—
102#333—1991no—
103Q8⋊C8—1991no—
104D4⋊C8—1991no—
105C2×SD16—21no—
106C2×SD16—3982no—
107C2×D8—3981no—
108C4⋊Q8—1991no—
109#127C2×D19911yescharacteristic
110#125—3981no—
111C22⋊Q8—3982no—
112C22⋊Q8—3981no—
113C4×Q8C2×D19911yescharacteristic
114C4×D4—3981no—
115C4×D4—1991no—
116C2×C4⋊C4—3981no—
117C4⋊C8—1991no—
118C4⋊C8C2×D19911yescharacteristic
119#102—21no—
120#102—3981no—
121C4×C8—1991no—
122C2×Q8—1992no—
123C2×Q8C22⋊D19911yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—1991no—
128C22×C4—3982no—
129C22×C4—3981no—
130SD16—42no—
131SD16—7964no—
132D8—7962no—
133C2×C8—21no—
134C2×C8—3981no—
135C2×C8—3981no—
136C4⋊C4—21no—
137C4⋊C4—3982no—
138C4⋊C4—3982no—
139C4⋊C4—3981no—
140C4⋊C4—3981no—
141C4⋊C4—1991no—
142C4⋊C4C22⋊D19911yescharacteristic
143C22⋊C4—3982no—
144C22⋊C4—3981no—
145C42—21no—
146C42C22×D19911yescharacteristic
147C23—41no—
148C23—3981no—
149Q8—21no—
150Q8—21no—
151Q8—7962no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—3981no—
158D4—1992no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—7961no—
162C2×C4—7961no—
163C2×C4—15922no—
164C2×C4—3981no—
165C2×C4—3982no—
166C2×C4D4×D19911yescharacteristic
167C2×C4D4×D19911yescharacteristic
168C2×C4C2×C22⋊D19911yescharacteristic
169C8—41no—
170C8—7961no—
171C8—3982no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—7961no—
176C22—7961no—
177C22—3982no—
178C22#2026392311yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—7961no—
183C4—7962no—
184C4SD16×D19911yescharacteristic
185C4#2026397511yescharacteristic
186C4#2026398111yescharacteristic
187C4#2026398411yescharacteristic
188C2—161no—
189C2—7962no—
190C2#2716415511yescharacteristic
191C2#2716416111yescharacteristic
192C2#2716418411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#460×C199C2
socleC22×C199#120
prime radical#27164210C2
hypercenter#460D199
hypocenterC199#1622
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #27164728C2semidirect
2. #27164390C2semidirect
4. #27163766C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #460×C199C2semidirect
53. D796SD16semidirect
96. C199#1622semidirect
100. #460D199semidirect

Maximal subgroups

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

subgroupindex
#271647282
#271643902
#271642102
#271637662
Q8⋊(C199⋊C8)2
#271637452
#460×C1992
#1622199