Finite Group Databasebeta

← #38769066

Subgroups — #38769066

Order 25472.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#27164495C211yesmaximal, characteristic
2#27164061C211yesmaximal, characteristic
3#27164060C211yesmaximal, characteristic
4C2×#20263873C211yesmaximal, characteristic
5#27163924C211yesmaximal, characteristic
6#27163778C211yesmaximal, characteristic
7#339×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×#15691870C2211yescharacteristic
10C4×Dic398—21no—
11C4×Dic398—21no—
12#20263920—21no—
13#20263920—21no—
14#20263914C2211yescharacteristic
15#20263911—22no—
16C2×C4⋊Dic199C2211yescharacteristic
17#20263889—21no—
18#20263888—21no—
19#20263887—21no—
20#20263886—21no—
21#20263873C414yescomplemented
22C199⋊#95—22no—
23C2×M16×C199C2211yescharacteristic
24#117×C199C2211yescharacteristic
25C2×C42×C199C2211yescharacteristic
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398C2×C412yes—
31C2×Dic398Q812yes—
32C22×Dic199—21no—
33C22×Dic199D412yes—
34#15691870—21no—
35#15691870C2×C412yes—
36C2×C199⋊C8—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199—21no—
39C4⋊Dic199D411yescharacteristic
40C4⋊Dic199D411yescharacteristic
41Dic199⋊C4—22no—
42Dic199⋊C4—22no—
43C22⋊Dic199—21no—
44C4×Dic199—22no—
45C4×Dic199—22no—
46C4×Dic199—21no—
47C22×C4×C199—21no—
48C22×C4×C199C2311yescharacteristic
49M16×C199—21no—
50M16×C199C2×C412yes—
51C2×C8×C199—21no—
52C4⋊C4×C199D411yescharacteristic
53C4⋊C4×C199D411yescharacteristic
54C22⋊C4×C199—21no—
55C42×C199—21no—
56C42×C199—21no—
57C42×C199—21no—
58Dic398—22no—
59Dic398—22no—
60Dic398—22no—
61Dic398—22no—
62C2×Dic199—22no—
63C2×Dic199—22no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—42no—
67C2×Dic199—41no—
68C2×Dic199C4⋊C414yes—
69C199⋊C8—22no—
70C23×C199C2×D411yescharacteristic
71C2×C4×C199—22no—
72C2×C4×C199—21no—
73C2×C4×C199—21no—
74C2×C4×C199—41no—
75C2×C4×C199—41no—
76C2×C4×C199C22×C411yescharacteristic
77C2×C4×C199C2×D411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C2×C4×C199C2×Q811yescharacteristic
80C2×C4×C199#4111yescharacteristic
81C2×C4×C199#4111yescharacteristic
82C8×C199—22no—
83C22×C199—21no—
84C22×C199—21no—
85C22×C199C2×C4⋊C411yescharacteristic
86C22×C199#12011yescharacteristic
87C22×C199#12311yescharacteristic
88C4×C199—42no—
89C4×C199—41no—
90C4×C199—41no—
91C4×C199C4×D411yescharacteristic
92C4×C199C4×D411yescharacteristic
93C4×C199C22⋊Q811yescharacteristic
94C4×C199C22⋊Q811yescharacteristic
95Dic199—24no—
96Dic199—41no—
97Dic199—41no—
98Dic199—81no—
99C2×C199—21no—
100C2×C199—21no—
101C2×C199#38511yescharacteristic
102C2×C199#45411yescharacteristic
103C2×C199#45611yescharacteristic
104C199#185711yescharacteristic, complemented, Sylow
105#1857—1991nomaximal, Sylow
106#519—1991no—
107#428—1991no—
108#428—1991no—
109C2×#101—1991no—
110#391—1991no—
111#339—1991no—
112#339D19911yescharacteristic
113C22×Q8—1991no—
114C2×M16—1991no—
115C2×M16C2×D19911yescharacteristic
116C4×Q8—3981no—
117C4×Q8—3981no—
118C4×Q8—3981no—
119C4×Q8—3981no—
120#117—3982no—
121#117—1991no—
122#117C2×D19911yescharacteristic
123C2×C4⋊C4—1991no—
124C2×C42C2×D19911yescharacteristic
125#107—3981no—
126#107—3981no—
127#106—3981no—
128#106—3981no—
129#101—1994no—
130#95—3982no—
131C2×Q8—3981no—
132C2×Q8—3981no—
133C2×Q8—3981no—
134C2×Q8—3981no—
135C2×Q8—1992no—
136C2×Q8—1992no—
137C22×C4—21no—
138C22×C4—3981no—
139C22×C4—1992no—
140C22×C4C22×D19911yescharacteristic
141M16—21no—
142M16—3981no—
143M16—1992no—
144M16C4×D19912yes—
145C2×C8—21no—
146C2×C8—3981no—
147C4⋊C4—3982no—
148C4⋊C4—3982no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C4⋊C4—1991no—
152C4⋊C4—1991no—
153C4⋊C4D79611yescharacteristic
154C4⋊C4D79611yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3981no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—3982no—
161C42—3982no—
162C42—3981no—
163C23D4×D19911yescharacteristic
164Q8—3982no—
165Q8—3982no—
166Q8—3982no—
167Q8—3982no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—7962no—
174C2×C4—7961no—
175C2×C4—3981no—
176C2×C4—3981no—
177C2×C4—3982no—
178C2×C4—3982no—
179C2×C4—1994no—
180C2×C4C2×C4×D19911yescharacteristic
181C2×C4D4×D19911yescharacteristic
182C2×C4C2×D79611yescharacteristic
183C2×C4Q8×D19911yescharacteristic
184C2×C4#1569188511yescharacteristic
185C2×C4#1569188611yescharacteristic
186C8—22no—
187C8—3982no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D19911yescharacteristic
191C22#2026392511yescharacteristic
192C22#2026393711yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—7961no—
197C4—7961no—
198C4—15921no—
199C4—3984no—
200C4(C22⋊D199)⋊C411yescharacteristic
201C4C4×D79611yescharacteristic
202C4#2026393311yescharacteristic
203C4#2026393511yescharacteristic
204C2—21no—
205C2—21no—
206C2#2716389211yescharacteristic
207C2#2716419211yescharacteristic
208C2#2716420211yescharacteristic
subgroupG/H
centerC22#20263925
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#339×C199C2
socleC22×C199#120
prime radicalC23×C199C2×D4
cosocleC22×C4×C199C23
hypercenter#339D199
hypocenterC199#1857
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
21. #20263873C4semidirect
104. C199#1857semidirect

Maximal subgroups

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

subgroupindex
#271644952
#271640612
#271640602
C2×#202638732
#271639242
#271637782
#339×C1992
#1857199