Finite Group Databasebeta

← #38767938

Subgroups — #38767938

Order 25472.

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

normal subgroups only

This group has 256 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#27164673C211yesmaximal, characteristic, complemented
2#27164364C211yesmaximal, characteristic, complemented
3#27164210C211yesmaximal, characteristic, complemented
4#27163763C211yesmaximal, characteristic, complemented
5D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#459×C199C211yesmaximal, characteristic, complemented
8C2×#15691886—21no—
9C2×#15691871—21no—
10#20263955C2211yescharacteristic
11#20263937—22no—
12#20263934—21no—
13#20263926—21no—
14C4×D796C2211yescharacteristic
15C4⋊C4×D199—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263876—21no—
18#20263876—22no—
19C4×C199⋊C8C2211yescharacteristic
20C2×D8×C199—21no—
21#127×C199C2211yescharacteristic
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25#15691886—42no—
26#15691886—42no—
27C2×Dic398—21no—
28C2×D796D411yescharacteristic
29C2×C22⋊D199—22no—
30C2×C4×D199—22no—
31C2×C4×D199—21no—
32#15691871—42no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199D412yes—
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—22no—
39C22⋊Dic199—22no—
40(C2×D199)⋊C4—21no—
41C2×D4×C199—21no—
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199D411yescharacteristic
45C22×C4×C199—21no—
46D8×C199—42no—
47C2×C8×C199—21no—
48C4⋊C4×C199D411yescharacteristic
49C22⋊C4×C199—21no—
50C42×C199C2311yescharacteristic
51C22×D199—21no—
52Dic398—41no—
53Dic398—41no—
54D796—21no—
55D796D812yescomplemented
56C22⋊D199—44no—
57C2×Dic199—21no—
58C2×Dic199—22no—
59C4×D199—44no—
60C4×D199—41no—
61C4×D199—41no—
62C199⋊C8—41no—
63C199⋊C8—41no—
64C23×C199—21no—
65C23×C199—41no—
66D4×C199—21no—
67D4×C199—21no—
68D4×C199—42no—
69D4×C199—41no—
70D4×C199—41no—
71D4×C199—41no—
72D4×C199—41no—
73C2×C4×C199—21no—
74C2×C4×C199—41no—
75C2×C4×C199—41no—
76C2×C4×C199C2×D411yescharacteristic
77C2×C4×C199C2×D411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C8×C199—41no—
80C22×C199—41no—
81C22×C199—41no—
82C22×C199—41no—
83C22×C199—81no—
84C22×C199—81no—
85C22×C199—81no—
86C22×C199#12011yescharacteristic
87C2×D199—22no—
88C2×D199—41no—
89C2×D199—41no—
90C4×C199—21no—
91C4×C199—41no—
92C4×C199C2×D811yescharacteristic
93C4×C199#13511yescharacteristic
94C4×C199AGL(1,ℤ/8)11yescharacteristic
95C4×C199AGL(1,ℤ/8)11yescharacteristic
96Dic199—42no—
97Dic199—41no—
98C2×C199—81no—
99C2×C199—161no—
100C2×C199#44711yescharacteristic
101C2×C199#44911yescharacteristic
102C2×C199#45411yescharacteristic
103D199—42no—
104C199#162511yescharacteristic, complemented, Sylow
105#1625—1991nomaximal, Sylow
106#547—1991no—
107#486—1991no—
108#459—1991no—
109#459D19911yescharacteristic, complemented
110#332—1991no—
111D4⋊C8—1991no—
112D4⋊C8—1991no—
113C2×#41—3981no—
114C2×D8—21no—
115C2×D8—3981no—
116C4⋊Q8—1991no—
117#127C2×D19911yescharacteristic
118#123—3982no—
119C22⋊Q8—3981no—
120#121—3981no—
121C4×D4—1991no—
122C4×D4C2×D19911yescharacteristic
123C2×C4⋊C4—3981no—
124C4⋊C8—1991no—
125C4⋊C8C2×D19911yescharacteristic
126#102—21no—
127#102—3981no—
128#102—3982no—
129C4×C8—1991no—
130#41—7962no—
131#41—7962no—
132C2×Q8—3981no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3982no—
137C2×D4—1991no—
138C2×D4C22⋊D19911yescharacteristic
139C22×C4—21no—
140C22×C4—3982no—
141C22×C4—3981no—
142D8—42no—
143D8—7962no—
144C2×C8—21no—
145C2×C8—3981no—
146C2×C8—3981no—
147C4⋊C4—3982no—
148C4⋊C4—3981no—
149C4⋊C4—1991no—
150C4⋊C4—1992no—
151C4⋊C4C22⋊D19911yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3982no—
154C22⋊C4—3981no—
155C42C22×D19911yescharacteristic
156C23—21no—
157C23—41no—
158C23—3981no—
159Q8—7961no—
160Q8—7961no—
161D4—21no—
162D4—21no—
163D4—42no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—7964no—
169D4—3981no—
170D4—1992no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—7964no—
175C2×C4—7961no—
176C2×C4—7961no—
177C2×C4—3981no—
178C2×C4—3982no—
179C2×C4D4×D19911yescharacteristic
180C2×C4D4×D19911yescharacteristic
181C2×C4C2×C22⋊D19911yescharacteristic
182C8—41no—
183C8—7961no—
184C8—7961no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—7961no—
192C22—7961no—
193C22—3982no—
194C22#2026392311yescharacteristic
195C4—21no—
196C4—41no—
197C4—7961no—
198C4—7962no—
199C4D8×D19911yescharacteristic
200C4#2026397511yescharacteristic
201C4#2026397911yescharacteristic
202C4#2026398111yescharacteristic
203C2—81no—
204C2—161no—
205C2—7962no—
206C2#2716415011yescharacteristic
207C2#2716416111yescharacteristic
208C2#2716418911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#459×C199C2
socleC22×C199#120
hypercenter#459D199
hypocenterC199#1625
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #27164673C2semidirect
2. #27164364C2semidirect
3. #27164210C2semidirect
4. #27163763C2semidirect
5. D4⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #459×C199C2semidirect
55. D796D8semidirect
104. C199#1625semidirect
109. #459D199semidirect

Maximal subgroups

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

subgroupindex
#271646732
#271643642
#271642102
#271637632
D4⋊(C199⋊C8)2
#271637452
#459×C1992
#1625199