Finite Group Databasebeta

← #38770797

Subgroups — #38770797

Order 25472.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#27164815C211yesmaximal, characteristic, complemented
2#27164456C211yesmaximal, characteristic, complemented
3#27164453C211yesmaximal, characteristic, complemented
4#27164272C211yesmaximal, characteristic, complemented
5#27163827C211yesmaximal, characteristic, complemented
6#27163799C211yesmaximal, characteristic, complemented
7#360×C199C211yesmaximal, characteristic, complemented
8#41×D199—21no—
9#20263987—21no—
10#20263984—21no—
11#20263982—21no—
12#20263979—21no—
13#20263977C2211yescharacteristic
14#20263975—21no—
15C2×D1592C2211yescharacteristic, complemented
16C2×#15691869C2211yescharacteristic
17#20263930—21no—
18#20263900—21no—
19#20263898—21no—
20D3184—21no—
21#20263896—21no—
22#20263895C2211yescharacteristic, complemented
23#20263875—21no—
24#135×C199C2211yescharacteristic
25M32×C199C2211yescharacteristic, complemented
26#106×C199C2211yescharacteristic, complemented
27#15691887—21no—
28#15691886—21no—
29#15691886—41no—
30#15691885—21no—
31#15691885—41no—
32Q8×D199—21no—
33C2×D796—21no—
34C2×D796—41no—
35D4×D199—21no—
36D4×D199—41no—
37C2×C4×D199—41no—
38C2×C4×D199D411yescharacteristic
39Dic796D411yescharacteristic, complemented
40#15691876—21no—
41#15691875—21no—
42#15691874—21no—
43#15691873—21no—
44D1592—21no—
45D1592—21no—
46D1592D411yescharacteristic, complemented
47#15691871—21no—
48#15691869—21no—
49#15691869—21no—
50C2×C199⋊C8D411yescharacteristic
51(C2×D199)⋊C4—41no—
52C199⋊C16—21no—
53#41×C199—21no—
54Q16×C199D411yescharacteristic, complemented
55SD16×C199—21no—
56D8×C199D411yescharacteristic, complemented
57C2×C8×C199C2311yescharacteristic
58C4⋊C4×C199—21no—
59C16×C199—21no—
60C22×D199—41no—
61C22×D199—41no—
62C22×D199D811yescharacteristic
63Dic398—21no—
64Dic398—41no—
65D796—21no—
66D796—41no—
67D796—41no—
68D796—41no—
69D796—82no—
70C22⋊D199—41no—
71C22⋊D199—41no—
72C22⋊D199—41no—
73C2×Dic199—41no—
74C2×Dic199D811yescharacteristic
75C4×D199—21no—
76C4×D199—21no—
77C4×D199—41no—
78C4×D199—41no—
79C4×D199—41no—
80C199⋊C8—21no—
81Q8×C199—21no—
82D4×C199—21no—
83D4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199C2×D411yescharacteristic
87C8×C199C2×D411yescharacteristic
88C8×C199C2×D411yescharacteristic
89C22×C199—41no—
90C22×C199C2×D811yescharacteristic
91C2×D199—21no—
92C2×D199—41no—
93C2×D199—41no—
94C2×D199—82no—
95C2×D199—81no—
96C2×D199—81no—
97C2×D199—81no—
98C4×C199—41no—
99C4×C199—81no—
100C4×C199#12011yescharacteristic
101C4×C199AGL(1,ℤ/8)11yescharacteristic
102Dic199—21no—
103Dic199—41no—
104C2×C199—21no—
105C2×C199—81no—
106C2×C199#44711yescharacteristic
107D199—41no—
108D199—81no—
109D199—161no—
110C199#219011yescharacteristic, complemented, Sylow
111#2190—1991nomaximal, Sylow
112#575—1991no—
113#509—1991no—
114#509—1991no—
115#469—1991no—
116#360—1991no—
117#360D19911yescharacteristic, complemented
118#349—1991no—
119C2×#41—3981no—
120#137—3981no—
121#137—3981no—
122AGL(1,ℤ/8)—3981no—
123AGL(1,ℤ/8)—3981no—
124#135—3981no—
125#135—1991no—
126#135C2×D19911yescharacteristic
127C2×D8—1991no—
128C2×M16—1991no—
129#121—3981no—
130SD32—3981no—
131SD32—3981no—
132D16—3981no—
133D16—3981no—
134M32—1991no—
135M32C2×D19911yescharacteristic, complemented
136#106C2×D19911yescharacteristic, complemented
137#102—3981no—
138#41—21no—
139#41—7961no—
140#41—7961no—
141#41—3981no—
142#41—3981no—
143#41—3981no—
144C2×Q8—3981no—
145C2×D4—7961no—
146C2×D4—7961no—
147C2×D4—3981no—
148C2×D4—3981no—
149C22×C4—7961no—
150C22×C4—1991no—
151Q16—3981no—
152Q16—1991no—
153Q16D79611yescharacteristic, complemented
154SD16—21no—
155SD16—3981no—
156SD16—3981no—
157SD16—3981no—
158D8—3981no—
159D8—3981no—
160D8—3981no—
161D8—1991no—
162D8D79611yescharacteristic, complemented
163M16—3981no—
164M16—3981no—
165C2×C8—1991no—
166C2×C8C22×D19911yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—7961no—
169C16—21no—
170C16—3981no—
171C23—7961no—
172C23—7961no—
173C23—1991no—
174Q8—21no—
175Q8—7961no—
176Q8—3981no—
177D4—21no—
178D4—41no—
179D4—7961no—
180D4—7961no—
181D4—7961no—
182D4—7961no—
183D4—7961no—
184D4—7961no—
185D4—15922no—
186D4—3981no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—7961no—
190C2×C4—7961no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—3981no—
194C2×C4—3981no—
195C2×C4—1991no—
196C2×C4D4×D19911yescharacteristic
197C8—3981no—
198C8D4×D19911yescharacteristic
199C8C2×D79611yescharacteristic
200C22—41no—
201C22—7961no—
202C22—7961no—
203C22—15921no—
204C22—15921no—
205C22—15922no—
206C22—15921no—
207C22—3981no—
208C22D8×D19911yescharacteristic
209C4—41no—
210C4—81no—
211C4—7961no—
212C4—3981no—
213C4#2026392511yescharacteristic
214C4#2026398211yescharacteristic
215C2—21no—
216C2—81no—
217C2—7961no—
218C2—15921no—
219C2—31841no—
220C2#2716415311yescharacteristic
subgroupG/H
centerC2#27164153
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting#360×C199C2
socleC2×C199#447
hypercenter#360D199
hypocenterC199#2190
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #27164815C2semidirect
2. #27164456C2semidirect
3. #27164453C2semidirect
4. #27164272C2semidirect
5. #27163827C2semidirect
6. #27163799C2semidirect
7. #360×C199C2semidirect
15. C2×D1592C22semidirect
22. #20263895C22semidirect
25. M32×C199C22semidirect
26. #106×C199C22semidirect
39. Dic796D4semidirect
46. D1592D4semidirect
54. Q16×C199D4semidirect
56. D8×C199D4semidirect
110. C199#2190semidirect
117. #360D199semidirect
135. M32C2×D199semidirect
136. #106C2×D199semidirect
153. Q16D796semidirect
162. D8D796semidirect

Maximal subgroups

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

subgroupindex
#271648152
#271644562
#271644532
#271642722
#271638272
#271637992
#360×C1992
#2190199