Finite Group Databasebeta

← #38767791

Subgroups — #38767791

Order 25472.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#27164647C211yesmaximal, characteristic, complemented
2#27164171C211yesmaximal, characteristic
3#27164168C211yesmaximal, characteristic
4#27163936C211yesmaximal, characteristic
5#27163754C211yesmaximal, characteristic, complemented
6#27163743C211yesmaximal, characteristic, complemented
7#328×C199C211yesmaximal, characteristic, complemented
8C22×D796C2211yescharacteristic
9C2×#15691875—21no—
10C2×#15691873—21no—
11#20263954—21no—
12#20263953—21no—
13#20263945—21no—
14#20263945—21no—
15#20263934C2211yescharacteristic
16#20263932—21no—
17#20263911—21no—
18C2×Dic199⋊C4C2211yescharacteristic
19C2×(C2×D199)⋊C4—21no—
20C2×(C2×D199)⋊C4—21no—
21#20263877—21no—
22#20263875—21no—
23C22⋊(C199⋊C8)C2211yescharacteristic
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199—21no—
28C2×Dic398—21no—
29C2×Dic398D411yescharacteristic
30C2×D796—21no—
31C2×D796—21no—
32C2×D796—42no—
33C22×Dic199D411yescharacteristic
34C22×Dic199D411yescharacteristic
35#15691875—42no—
36#15691873—42no—
37C2×C199⋊C8—21no—
38C4⋊Dic199—21no—
39C4⋊Dic199D411yescharacteristic
40Dic199⋊C4—21no—
41Dic199⋊C4—21no—
42Dic199⋊C4—21no—
43Dic199⋊C4—21no—
44Dic199⋊C4—21no—
45Dic199⋊C4—21no—
46Dic199⋊C4—21no—
47C22⋊Dic199—21no—
48(C2×D199)⋊C4—42no—
49(C2×D199)⋊C4—42no—
50C4×Dic199—21no—
51C4×Dic199—21no—
52C2×Q8×C199D411yescharacteristic
53C22×C4×C199—21no—
54C22×C4×C199C2311yescharacteristic
55C2×C8×C199—21no—
56C4⋊C4×C199—21no—
57C4⋊C4×C199D411yescharacteristic
58C22⋊C4×C199—21no—
59C22×D199—22no—
60C22×D199—41no—
61C22×D199—41no—
62C22×D199—41no—
63C22×D199—41no—
64C22×D199—41no—
65C22×D199—41no—
66Dic398—41no—
67Dic398—81no—
68D796—22no—
69D796—22no—
70D796—41no—
71D796—41no—
72C2×Dic199—21no—
73C2×Dic199—21no—
74C2×Dic199—21no—
75C2×Dic199—21no—
76C2×Dic199—21no—
77C2×Dic199—41no—
78C2×Dic199SD1612yescomplemented
79C199⋊C8—41no—
80C23×C199C2×D411yescharacteristic
81Q8×C199—41no—
82C2×C4×C199—21no—
83C2×C4×C199—21no—
84C2×C4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199—41no—
87C2×C4×C199—41no—
88C2×C4×C199C2×D411yescharacteristic
89C2×C4×C199C2×D411yescharacteristic
90C8×C199—41no—
91C22×C199—21no—
92C22×C199—21no—
93C22×C199#12011yescharacteristic
94C22×C199C2×SD1611yescharacteristic
95C22×C199#13711yescharacteristic
96C2×D199—42no—
97C2×D199—42no—
98C2×D199—42no—
99C2×D199—81no—
100C2×D199—81no—
101C2×D199—81no—
102C2×D199—81no—
103C4×C199—21no—
104C4×C199—21no—
105C4×C199—41no—
106C4×C199—41no—
107C4×C199—81no—
108Dic199—22no—
109Dic199—41no—
110Dic199—41no—
111Dic199—41no—
112C2×C199—21no—
113C2×C199—21no—
114C2×C199#44811yescharacteristic
115C2×C199#45411yescharacteristic
116C2×C199B(4,𝔽2)11yescharacteristic
117D199—82no—
118C199#159911yescharacteristic, complemented, Sylow
119#1599—1991nomaximal, Sylow
120#544—1991no—
121#450—1991no—
122#450—1991no—
123#394—1991no—
124#328—1991no—
125#328D19911yescharacteristic, complemented
126#324—1991no—
127C22×D4—1991no—
128C2×SD16—3981no—
129C2×SD16—3981no—
130C4⋊Q8—3981no—
131C4⋊Q8—3981no—
132#125—3981no—
133#125—3981no—
134C22⋊Q8—3981no—
135C22⋊Q8—1991no—
136C22⋊Q8C2×D19911yescharacteristic
137#117—3981no—
138C2×C4⋊C4—1991no—
139C2×C22⋊C4—3981no—
140C2×C22⋊C4—3981no—
141#102—3981no—
142#102—3981no—
143C22⋊C8—1991no—
144C22⋊C8C2×D19911yescharacteristic
145#95C2×D19911yescharacteristic
146C24—3981no—
147C2×Q8—3981no—
148C2×Q8—1991no—
149C2×Q8D79611yescharacteristic
150C2×D4—7962no—
151C2×D4—3981no—
152C2×D4—3981no—
153C22×C4—21no—
154C22×C4—1991no—
155C22×C4—1991no—
156C22×C4C22×D19911yescharacteristic
157SD16—7962no—
158SD16—7962no—
159C2×C8—21no—
160C2×C8—3981no—
161C4⋊C4—21no—
162C4⋊C4—3981no—
163C4⋊C4—3981no—
164C4⋊C4—3981no—
165C4⋊C4—3981no—
166C4⋊C4—3981no—
167C4⋊C4—3981no—
168C4⋊C4—3981no—
169C4⋊C4—3981no—
170C4⋊C4—1991no—
171C4⋊C4D79611yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—7962no—
174C22⋊C4—7962no—
175C22⋊C4—3981no—
176C42—3981no—
177C42—3981no—
178C23—7961no—
179C23—7961no—
180C23—7961no—
181C23—7961no—
182C23—7961no—
183C23—7961no—
184C23—3982no—
185C23D4×D19911yescharacteristic
186Q8—41no—
187Q8—7961no—
188Q8—15921no—
189D4—7961no—
190D4—7961no—
191D4—3982no—
192D4—3982no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—7961no—
200C2×C4—3981no—
201C2×C4—3981no—
202C2×C4—3981no—
203C2×C4—3981no—
204C2×C4—3981no—
205C2×C4—1992no—
206C2×C4D4×D19911yescharacteristic
207C2×C4C2×D79611yescharacteristic
208C8—41no—
209C8—7961no—
210C22—21no—
211C22—21no—
212C22—7962no—
213C22—7962no—
214C22—7962no—
215C22—15921no—
216C22—15921no—
217C22—15921no—
218C22—15921no—
219C22#2026392511yescharacteristic
220C22SD16×D19911yescharacteristic
221C22#2026398711yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—7961no—
228C4—7961no—
229C4—7961no—
230C4—3982no—
231C2—21no—
232C2—21no—
233C2—15922no—
234C2#2716415811yescharacteristic
235C2#2716419211yescharacteristic
236C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199#120
prime radicalC22×D796C22
hypercenter#328D199
hypocenterC199#1599
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27164647C2semidirect
5. #27163754C2semidirect
6. #27163743C2semidirect
7. #328×C199C2semidirect
78. C2×Dic199SD16semidirect
118. C199#1599semidirect
125. #328D199semidirect

Maximal subgroups

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

subgroupindex
#271646472
#271641712
#271641682
#271639362
#271637542
#271637432
#328×C1992
#1599199