Finite Group Databasebeta

← #38769951

Subgroups — #38769951

Order 25472.

370 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 93 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×Dic796C211yesmaximal, characteristic
2C22×#15691876C211yesmaximal, characteristic
3C2×#20263953C211yesmaximal, characteristic
4C2×#20263878C211yesmaximal, characteristic
5#27163923C211yesmaximal, characteristic
6#27163772C211yesmaximal, characteristic
7C2×#103×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C22×Dic398C2211yescharacteristic
10C2×Dic796—22no—
11C2×Dic796—22no—
12C2×Dic796—22no—
13C2×#15691876—22no—
14C2×#15691876—22no—
15C2×#15691876—22no—
16C22×C199⋊C8C2211yescharacteristic
17#20263953—22no—
18#20263953—22no—
19C2×Dic199⋊C4—22no—
20C2×C4×Dic199C2211yescharacteristic
21#20263878—22no—
22C199⋊#95—22no—
23C22×Q8×C199C2211yescharacteristic
24C22×C8×C199C2211yescharacteristic
25C2×C4⋊C4×C199C2211yescharacteristic
26#103×C199—22no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398—21no—
31C2×Dic398—21no—
32C2×Dic398—21no—
33C2×Dic398—21no—
34C2×Dic398—21no—
35C2×Dic398—21no—
36C2×Dic398—21no—
37C2×Dic398D412yes—
38C2×Dic398D412yes—
39C22×Dic199—21no—
40C22×Dic199—21no—
41C22×Dic199D412yes—
42Dic796—28no—
43#15691876—28no—
44C2×C199⋊C8—21no—
45C2×C199⋊C8—21no—
46C2×C199⋊C8D412yes—
47Dic199⋊C4—28no—
48C4×Dic199—21no—
49C4×Dic199—21no—
50C2×Q8×C199—21no—
51C2×Q8×C199—21no—
52C2×Q8×C199—21no—
53C2×Q8×C199—21no—
54C2×Q8×C199—21no—
55C2×Q8×C199D412yes—
56C22×C4×C199—21no—
57C22×C4×C199—21no—
58C22×C4×C199C2311yescharacteristic
59C2×C8×C199—21no—
60C2×C8×C199—21no—
61C2×C8×C199D412yes—
62C4⋊C4×C199—21no—
63C4⋊C4×C199—21no—
64Dic398—22no—
65Dic398—22no—
66Dic398—22no—
67Dic398—22no—
68Dic398—41no—
69Dic398—41no—
70Dic398—41no—
71Dic398—41no—
72C2×Dic199—22no—
73C2×Dic199—22no—
74C2×Dic199—22no—
75C2×Dic199—22no—
76C2×Dic199—41no—
77C2×Dic199—41no—
78C2×Dic199—41no—
79C2×Dic199—41no—
80C2×Dic199Q1614yescomplemented
81C199⋊C8—22no—
82C23×C199C2×D411yescharacteristic
83Q8×C199—22no—
84Q8×C199—22no—
85Q8×C199—41no—
86Q8×C199—41no—
87C2×C4×C199—22no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199—42no—
92C2×C4×C199C2×D411yescharacteristic
93C2×C4×C199C2×D411yescharacteristic
94C2×C4×C199C2×D411yescharacteristic
95C2×C4×C199C2×D411yescharacteristic
96C2×C4×C199C2×D411yescharacteristic
97C2×C4×C199#4111yescharacteristic
98C8×C199—22no—
99C22×C199#12011yescharacteristic
100C22×C199#12111yescharacteristic
101C22×C199#12711yescharacteristic
102C22×C199C2×Q1612yes—
103C22×C199#13712yes—
104C4×C199—42no—
105C4×C199—81no—
106C4×C199#12011yescharacteristic
107C4×C199#12011yescharacteristic
108C4×C199#12111yescharacteristic
109C4×C199#12111yescharacteristic
110Dic199—24no—
111Dic199—42no—
112Dic199—42no—
113C2×C199#39211yescharacteristic
114C2×C199#45112yes—
115C2×C199#46212yes—
116C2×C199#49411yescharacteristic
117C2×C199#49711yescharacteristic
118C199#201611yescharacteristic, complemented, Sylow
119#2016—1991nomaximal, Sylow
120C22×Q16—1991no—
121C22×Q16—1991no—
122C2×C4⋊Q8—1991no—
123C2×#103—1991no—
124C2×#103D19911yescharacteristic
125#391—1991no—
126#336—1991no—
127C22×Q8—1991no—
128C22×Q8—1991no—
129C22×Q8C2×D19911yescharacteristic
130C2×Q16—3982no—
131C2×Q16—3982no—
132C2×Q16—3982no—
133C2×Q16—3982no—
134C2×Q16—3982no—
135C2×Q16—3982no—
136C22×C8—1991no—
137C22×C8C2×D19911yescharacteristic
138C4⋊Q8—3982no—
139C4⋊Q8—3982no—
140C2×C4⋊C4—3982no—
141C2×C4⋊C4C2×D19911yescharacteristic
142C2×C42—1991no—
143#103—22no—
144#103—3982no—
145#95—3982no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—3981no—
152C2×Q8—3981no—
153C2×Q8—3981no—
154C2×Q8—3981no—
155C2×Q8—3981no—
156C2×Q8—3981no—
157C2×Q8—3981no—
158C2×Q8—3981no—
159C2×Q8—3981no—
160C2×Q8—3981no—
161C2×Q8—1992no—
162C2×Q8—1992no—
163C2×Q8D79612yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—3981no—
167C22×C4—3981no—
168C22×C4—1992no—
169C22×C4C22×D19911yescharacteristic
170Q16—3988no—
171Q16—3988no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—3981no—
175C2×C8—3981no—
176C2×C8—1992no—
177C2×C8C22⋊D19912yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—3988no—
181C42—3981no—
182C42—3981no—
183C23D4×D19911yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8—7961no—
189Q8—7961no—
190Q8—7961no—
191Q8—7961no—
192Q8—3982no—
193Q8—3982no—
194Q8—3982no—
195Q8—3982no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—7961no—
202C2×C4—7961no—
203C2×C4—7961no—
204C2×C4—7961no—
205C2×C4—3982no—
206C2×C4—3982no—
207C2×C4—3982no—
208C2×C4—3982no—
209C2×C4—1994no—
210C2×C4D4×D19911yescharacteristic
211C2×C4D4×D19911yescharacteristic
212C2×C4D4×D19911yescharacteristic
213C2×C4C2×C22⋊D19911yescharacteristic
214C2×C4C2×D79611yescharacteristic
215C2×C4#1569188611yescharacteristic
216C8—22no—
217C8—3982no—
218C22#2026392511yescharacteristic
219C22#2026392811yescharacteristic
220C22#2026395111yescharacteristic
221C22Q16×D19912yes—
222C22#2026398412yes—
223C4—42no—
224C4—81no—
225C4—7962no—
226C4—7962no—
227C4—3984no—
228C4#2026392311yescharacteristic
229C4#2026392511yescharacteristic
230C4#2026392811yescharacteristic
231C4#2026392911yescharacteristic
232C2#2716392711yescharacteristic
233C2#2716417512yes—
234C2#2716422712yes—
235C2#2716439611yescharacteristic
236C2#2716440911yescharacteristic
subgroupG/H
centerC23D4×D199
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×#103×C199C2
socle, prime radicalC23×C199C2×D4
hypercenterC2×#103D199
hypocenterC199#2016
LCSG ▷ C22×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
80. C2×Dic199Q16semidirect
118. C199#2016semidirect

Maximal subgroups

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

subgroupindex
C22×Dic7962
C22×#156918762
C2×#202639532
C2×#202638782
#271639232
#271637722
C2×#103×C1992
#2016199