Finite Group Databasebeta

← #38769078

Subgroups — #38769078

Order 25472.

302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 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#27164501C211yesmaximal, characteristic
2#27164060C211yesmaximal, characteristic, complemented
3#27164019C211yesmaximal, characteristic
4#27163888C211yesmaximal, characteristic, complemented
5#27163786C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7C2×#104×C199C211yesmaximal, characteristic
8C2×#15691870C2211yescharacteristic
9Q8×Dic199—21no—
10D4×Dic199—22no—
11D4×Dic199—21no—
12#20263914—21no—
13#20263914C2211yescharacteristic
14C2×C4⋊Dic199—21no—
15C2×C4⋊Dic199—22no—
16C2×C4⋊Dic199C2211yescharacteristic
17C2×C4⋊Dic199C2211yescharacteristic
18#20263889—21no—
19#20263887—21no—
20#20263879—21no—
21#20263876—21no—
22C2×#41×C199C2211yescharacteristic
23C2×M16×C199C2211yescharacteristic
24C2×C42×C199C2211yescharacteristic
25#104×C199C414yescomplemented
26C22×Dic199—22no—
27C22×Dic199—21no—
28C22×Dic199D412yes—
29#15691870—21no—
30#15691870—21no—
31C2×C199⋊C8—21no—
32C4⋊Dic199—22no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199—44no—
38C4⋊Dic199D411yescharacteristic
39C4⋊Dic199D411yescharacteristic
40C22⋊Dic199—22no—
41C22⋊Dic199—21no—
42C4×Dic199—22no—
43C4×Dic199—21no—
44#41×C199—21no—
45#41×C199—21no—
46#41×C199C2×C412yes—
47#41×C199Q812yes—
48C2×Q8×C199D411yescharacteristic
49C2×D4×C199—21no—
50C2×D4×C199D411yescharacteristic
51C22×C4×C199—21no—
52C22×C4×C199—21no—
53C22×C4×C199C2311yescharacteristic
54M16×C199—21no—
55M16×C199C2×C412yescomplemented
56C2×C8×C199—21no—
57C42×C199—21no—
58C42×C199C2×C412yescomplemented
59C2×Dic199—21no—
60C2×Dic199—22no—
61C2×Dic199—21no—
62C2×Dic199—42no—
63C2×Dic199—42no—
64C2×Dic199—41no—
65C2×Dic199—82no—
66C199⋊C8—41no—
67C23×C199—21no—
68C23×C199C2×D411yescharacteristic
69Q8×C199—21no—
70Q8×C199C4⋊C412yescomplemented
71D4×C199—22no—
72D4×C199—22no—
73D4×C199—21no—
74D4×C199C4⋊C412yescomplemented
75C2×C4×C199—22no—
76C2×C4×C199—22no—
77C2×C4×C199—22no—
78C2×C4×C199—21no—
79C2×C4×C199—22no—
80C2×C4×C199—22no—
81C2×C4×C199C22×C411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×Q811yescharacteristic
85C2×C4×C199#4111yescharacteristic
86C2×C4×C199#4111yescharacteristic
87C8×C199—22no—
88C22×C199—21no—
89C22×C199—21no—
90C22×C199—22no—
91C22×C199—41no—
92C22×C199—41no—
93C22×C199C4×D411yescharacteristic
94C22×C199#12011yescharacteristic
95C22×C199C22⋊Q811yescharacteristic
96C4×C199—24no—
97C4×C199—22no—
98C4×C199C2×C4⋊C411yescharacteristic
99C4×C199C4×D411yescharacteristic
100C4×C199C22⋊Q811yescharacteristic
101C4×C199#12311yescharacteristic
102Dic199—42no—
103Dic199—41no—
104Dic199—41no—
105Dic199—81no—
106C2×C199—21no—
107C2×C199—21no—
108C2×C199—42no—
109C2×C199#38511yescharacteristic
110C2×C199#45311yescharacteristic
111C2×C199#45611yescharacteristic
112C199#185911yescharacteristic, complemented, Sylow
113#1859—1991nomaximal, Sylow
114#520—1991no—
115#428—1991no—
116C2×#104D19911yescharacteristic
117#417—1991no—
118#384—1991no—
119#343—1991no—
120#339—1991no—
121C2×#41C2×D19911yescharacteristic
122C2×M16—1991no—
123C2×M16C2×D19911yescharacteristic
124C4×Q8—3981no—
125C4×D4—3981no—
126C4×D4—3982no—
127#117—3981no—
128#117—1991no—
129C2×C4⋊C4—3982no—
130C2×C4⋊C4—3981no—
131C2×C4⋊C4—1991no—
132C2×C4⋊C4—1991no—
133C2×C42C2×D19911yescharacteristic
134#107—3981no—
135#106—3981no—
136#104Dic19914yescomplemented
137#103—3981no—
138#102—3981no—
139#41—21no—
140#41—21no—
141#41C2×Dic19912yes—
142#41Dic39812yes—
143C2×Q8D79611yescharacteristic
144C2×D4—21no—
145C2×D4D79611yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3981no—
149C22×C4—3982no—
150C22×C4—1992no—
151C22×C4C22×D19911yescharacteristic
152M16—21no—
153M16—3981no—
154M16—3981no—
155M16C2×Dic19912yescomplemented
156C2×C8—21no—
157C2×C8—3981no—
158C4⋊C4—7964no—
159C4⋊C4—3982no—
160C4⋊C4—3981no—
161C4⋊C4—3981no—
162C4⋊C4—3981no—
163C4⋊C4—3981no—
164C4⋊C4—1991no—
165C4⋊C4—1991no—
166C22⋊C4—3982no—
167C22⋊C4—3981no—
168C42—21no—
169C42—3982no—
170C42—3981no—
171C42C2×Dic19912yescomplemented
172C23—21no—
173C23D4×D19911yescharacteristic
174Q8—21no—
175Q8C4⋊Dic19912yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4C4⋊Dic19912yescomplemented
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—7962no—
187C2×C4—7962no—
188C2×C4—7961no—
189C2×C4—15922no—
190C2×C4—3981no—
191C2×C4—3982no—
192C2×C4—3981no—
193C2×C4C22×Dic19911yescharacteristic
194C2×C4D4×D19911yescharacteristic
195C2×C4C2×D79611yescharacteristic
196C2×C4C2×Dic39811yescharacteristic
197C2×C4#1569188511yescharacteristic
198C2×C4#1569188511yescharacteristic
199C8—22no—
200C8—7961no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22D4×Dic19911yescharacteristic
207C22#2026392511yescharacteristic
208C22#2026393211yescharacteristic
209C4—24no—
210C4—22no—
211C4—7962no—
212C4—7961no—
213C4—7961no—
214C4—15921no—
215C4C2×C4⋊Dic19911yescharacteristic
216C4D4×Dic19911yescharacteristic
217C4#2026393211yescharacteristic
218C4#2026393911yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#2716389311yescharacteristic
223C2#2716418711yescharacteristic
224C2#2716420211yescharacteristic
subgroupG/H
centerC22#20263925
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#104×C199C2
socleC22×C199#120
prime radicalC2×#41×C199C22
cosocleC22×C4×C199C23
hypercenterC2×#104D199
hypocenterC199#1859
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #27164060C2semidirect
4. #27163888C2semidirect
5. #27163786C2semidirect
6. #27163778C2semidirect
25. #104×C199C4semidirect
55. M16×C199C2×C4semidirect
58. C42×C199C2×C4semidirect
70. Q8×C199C4⋊C4semidirect
74. D4×C199C4⋊C4semidirect
112. C199#1859semidirect
136. #104Dic199semidirect
155. M16C2×Dic199semidirect
171. C42C2×Dic199semidirect
175. Q8C4⋊Dic199semidirect
179. D4C4⋊Dic199semidirect

Maximal subgroups

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

subgroupindex
#271645012
#271640602
#271640192
#271638882
#271637862
#271637782
C2×#104×C1992
#1859199