Finite Group Databasebeta

← #38769708

Subgroups — #38769708

Order 25472.

312 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×#15691876C211yesmaximal, characteristic
2C2×#20263888C211yesmaximal, characteristic
3C2×#20263880C211yesmaximal, characteristic
4C2×C8×Dic199C211yesmaximal, characteristic
5#27163925C211yesmaximal, characteristic
6#27163923C211yesmaximal, characteristic
7C2×#103×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×#15691876—22no—
10C2×#15691876—22no—
11C2×#15691876C414yescomplemented
12C22×C199⋊C8C2211yescharacteristic
13C2×C4⋊Dic199C2211yescharacteristic
14C2×C4×Dic199C2211yescharacteristic
15#20263888—22no—
16#20263880—22no—
17C8×Dic199—22no—
18C199⋊#95—22no—
19C199⋊#95—22no—
20C22×Q8×C199C2211yescharacteristic
21C22×C8×C199C2211yescharacteristic
22C2×C4⋊C4×C199C2211yescharacteristic
23#103×C199—22no—
24C2×Dic398—21no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398C2×C412yes—
30C22×Dic199—21no—
31C22×Dic199—21no—
32C22×Dic199D412yes—
33#15691876—28no—
34C2×C199⋊C8C2×C412yes—
35C2×C199⋊C8D412yes—
36C2×C199⋊C8D412yes—
37C4⋊Dic199—21no—
38C4⋊Dic199—21no—
39C4×Dic199—21no—
40C4×Dic199—21no—
41C2×Q8×C199—21no—
42C2×Q8×C199—21no—
43C2×Q8×C199—21no—
44C2×Q8×C199—21no—
45C2×Q8×C199—21no—
46C2×Q8×C199C2×C412yes—
47C22×C4×C199—21no—
48C22×C4×C199—21no—
49C22×C4×C199C2311yescharacteristic
50C2×C8×C199—21no—
51C2×C8×C199—21no—
52C2×C8×C199D412yes—
53C4⋊C4×C199—21no—
54C4⋊C4×C199—21no—
55Dic398—22no—
56Dic398—22no—
57Dic398—41no—
58Dic398—41no—
59C2×Dic199—22no—
60C2×Dic199—22no—
61C2×Dic199—22no—
62C2×Dic199—41no—
63C2×Dic199—41no—
64C2×Dic199—41no—
65C2×Dic199—42no—
66C2×Dic199Q1614yescomplemented
67C199⋊C8C22⋊C414yes—
68C23×C199C2×D411yescharacteristic
69Q8×C199—22no—
70Q8×C199—22no—
71Q8×C199—41no—
72Q8×C199—41no—
73C2×C4×C199—22no—
74C2×C4×C199—41no—
75C2×C4×C199—41no—
76C2×C4×C199—41no—
77C2×C4×C199—42no—
78C2×C4×C199C22×C411yescharacteristic
79C2×C4×C199C2×D411yescharacteristic
80C2×C4×C199C2×D411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C2×C4×C199#4111yescharacteristic
83C2×C4×C199#4111yescharacteristic
84C8×C199—22no—
85C22×C199C4×D411yescharacteristic
86C22×C199#12111yescharacteristic
87C22×C199#12711yescharacteristic
88C22×C199C2×Q1612yes—
89C22×C199#13512yes—
90C4×C199—42no—
91C4×C199—81no—
92C4×C199C2×C22⋊C411yescharacteristic
93C4×C199C4×D411yescharacteristic
94C4×C199#12111yescharacteristic
95C4×C199#12411yescharacteristic
96Dic199—24no—
97Dic199—42no—
98Dic199—81no—
99C2×C199#39011yescharacteristic
100C2×C199C4×Q1612yes—
101C2×C199#46712yes—
102C2×C199#49411yescharacteristic
103C2×C199#49511yescharacteristic
104C199#196911yescharacteristic, complemented, Sylow
105#1969—1991nomaximal, Sylow
106C22×Q16—1991no—
107C2×#107—1991no—
108C2×#103—1991no—
109C2×#103D19911yescharacteristic
110C2×C4×C8—1991no—
111#391—1991no—
112#391—1991no—
113C22×Q8—1991no—
114C22×Q8C2×D19911yescharacteristic
115C2×Q16—3982no—
116C2×Q16—3982no—
117C2×Q16—1994no—
118C22×C8—1991no—
119C22×C8C2×D19911yescharacteristic
120C2×C4⋊C4—1991no—
121C2×C4⋊C4C2×D19911yescharacteristic
122C2×C42—1991no—
123#107—3982no—
124#103—22no—
125#103—3982no—
126C4×C8—3982no—
127#95—3982no—
128#95—3982no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—3981no—
135C2×Q8—3981no—
136C2×Q8—3981no—
137C2×Q8—3981no—
138C2×Q8—3981no—
139C2×Q8—1992no—
140C2×Q8C4×D19912yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—3981no—
144C22×C4—3981no—
145C22×C4—1992no—
146C22×C4C22×D19911yescharacteristic
147Q16—3988no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—1992no—
151C2×C8—1992no—
152C2×C8—1992no—
153C2×C8C22⋊D19912yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—3981no—
157C4⋊C4—3981no—
158C42—3981no—
159C42—3981no—
160C23D4×D19911yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—7961no—
166Q8—7961no—
167Q8—3982no—
168Q8—3982no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—7961no—
175C2×C4—7961no—
176C2×C4—7961no—
177C2×C4—7962no—
178C2×C4—3982no—
179C2×C4—3982no—
180C2×C4—3982no—
181C2×C4—1994no—
182C2×C4C2×C4×D19911yescharacteristic
183C2×C4D4×D19911yescharacteristic
184C2×C4D4×D19911yescharacteristic
185C2×C4C2×C22⋊D19911yescharacteristic
186C2×C4#1569188511yescharacteristic
187C2×C4#1569188611yescharacteristic
188C8—22no—
189C8—1994no—
190C22(C22⋊D199)⋊C411yescharacteristic
191C22#2026392811yescharacteristic
192C22#2026395111yescharacteristic
193C22Q16×D19912yes—
194C22#2026397612yes—
195C4—42no—
196C4—81no—
197C4—7962no—
198C4—15921no—
199C4—3984no—
200C4C22⋊C4×D19911yescharacteristic
201C4C4×C22⋊D19911yescharacteristic
202C4#2026392711yescharacteristic
203C4#2026394111yescharacteristic
204C2#2716391811yescharacteristic
205C2#2716411012yes—
206C2#2716425812yes—
207C2#2716439611yescharacteristic
208C2#2716439911yescharacteristic
subgroupG/H
centerC23D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#103×C199C2
socle, prime radicalC23×C199C2×D4
cosocleC22×C4×C199C23
hypercenterC2×#103D199
hypocenterC199#1969
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
11. C2×#15691876C4semidirect
66. C2×Dic199Q16semidirect
104. C199#1969semidirect

Maximal subgroups

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

subgroupindex
C22×#156918762
C2×#202638882
C2×#202638802
C2×C8×Dic1992
#271639252
#271639232
C2×#103×C1992
#1969199