Finite Group Databasebeta

← #16829146

Subgroups — #16829146

Order 3712.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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×#90785C211yesmaximal, characteristic, complemented
2C2×#11745269C211yesmaximal, characteristic
3#14271739C211yesmaximal, characteristic
4#14271737C211yesmaximal, characteristic
5#14271420C214yesmaximal, complemented
6#14271403C214yesmaximal, complemented
7#14271199C211yesmaximal, characteristic, complemented
8#14271173C211yesmaximal, characteristic, complemented
9C29×#419C211yesmaximal, characteristic, complemented
10C22×Q8×C29C2211yescharacteristic
11C2×#90805C2211yescharacteristic
12C22×Dic58C2211yescharacteristic
13C2×M16×C29C2211yescharacteristic
14C29×#117C2211yescharacteristic
15#11745309—21no—
16#11745308C2214yes—
17#11745306—21no—
18C2×#90785—24no—
19C2×#90785C2212yes—
20C2×#90785C2212yes—
21#11745291—22no—
22#11745288C2211yescharacteristic
23C22×#6597C2211yescharacteristic
24#11745269—24no—
25C2×#90772C2212yes—
26C2×C8×D29C2212yes—
27#11745259C2214yes—
28#11745248—22no—
29#11745237—21no—
30#11745231—21no—
31C29×#103C2214yes—
32#11745164C2214yes—
33#11745156C2212yes—
34#11745155C2212yes—
35C2×Q8×C29—21no—
36C2×Q8×C29—22no—
37C2×Q8×C29—22no—
38C2×Q8×C29C2312yes—
39C22×C4×C29—21no—
40C22×C4×C29C2311yescharacteristic
41C2×C22⋊D29D412yes—
42C22×Dic29—21no—
43#90805—22no—
44#90805—22no—
45C2×D116D411yescharacteristic
46C2×C4×D29C2312yes—
47C2×Dic58—22no—
48C2×Dic58—22no—
49C2×Dic58—21no—
50C2×Dic58D411yescharacteristic
51C2×Dic58C2312yes—
52M16×C29—22no—
53C2×C8×C29C2312yes—
54C4⋊C4×C29C2312yes—
55C22⋊C4×C29—21no—
56C42×C29—21no—
57C22⋊Dic29—21no—
58#90785—24no—
59#90785—24no—
60#90781—24no—
61#90781—24no—
62C4⋊Dic29—24no—
63C4⋊Dic29C2312yes—
64#90779—24no—
65C4×Dic29—21no—
66C2×#6597D414yes—
67C2×#6597C2311yescharacteristic
68C2×#6597C2311yescharacteristic
69#90772—24no—
70C8×D29—24no—
71C23×C29C2×D411yescharacteristic
72C22×D29C2×D412yes—
73Q8×C29—22no—
74Q8×C29—22no—
75Q8×C29—42no—
76C2×C4×C29—22no—
77C2×C4×C29—22no—
78C2×C4×C29—42no—
79C2×C4×C29C2×D411yescharacteristic
80C2×C4×C29C2×D412yes—
81C2×C4×C29#4112yes—
82C2×C4×C29C2411yescharacteristic
83C22⋊D29—24no—
84C2×Dic29—22no—
85C2×Dic29—22no—
86C2×Dic29—42no—
87C2×Dic29C2×D412yes—
88D116—22no—
89C4×D29—22no—
90C4×D29—22no—
91Dic58—22no—
92Dic58—22no—
93Dic58—22no—
94Dic58—42no—
95C8×C29—22no—
96#6597C2×D412yes—
97#6597C2×D412yes—
98#3059—291nomaximal, Sylow
99C22×C29—22no—
100C22×C29#12112yes—
101C22×C29C22×D411yescharacteristic
102C2×D29—22no—
103C2×D29—24no—
104C4×C29—42no—
105C4×C29—42no—
106C4×C29#12112yes—
107C4×C29C22×D411yescharacteristic
108C4×C29C2×#4111yescharacteristic
109Dic29—22no—
110Dic29—42no—
111Dic29—42no—
112C22×Q16—291no—
113C2×#131—291no—
114#536—291no—
115#536—291no—
116#470—294no—
117#467—294no—
118#427—291no—
119#419—291no—
120#419D2911yescharacteristic, complemented
121C2×C29—22no—
122C2×C29C2×#12111yescharacteristic
123C2×C29#57812yes—
124D29—42no—
125C2×#41—291no—
126C22×Q8—291no—
127C22×Q8C2×D2911yescharacteristic
128C2×Q16—292no—
129C2×Q16—292no—
130C2×Q16—584no—
131#131—584no—
132C2×M16—292no—
133C2×M16C2×D2911yescharacteristic
134C22×C8—291no—
135C22×C8—292no—
136C4⋊Q8—581no—
137C4⋊Q8—581no—
138#124—581no—
139#124—581no—
140#123—582no—
141#123—582no—
142C22⋊Q8—294no—
143C22⋊Q8—294no—
144#117—291no—
145#117C2×D2911yescharacteristic
146#107—292no—
147#106—292no—
148#103—294no—
149#103C2×D2914yes—
150C29#305911yescharacteristic, complemented, Sylow
151#41—582no—
152#41—582no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—291no—
157C2×Q8—292no—
158C2×Q8—581no—
159C2×Q8—582no—
160C2×Q8—582no—
161C2×Q8C22×D2912yes—
162C2×D4—292no—
163C2×D4—291no—
164C22×C4—21no—
165C22×C4—292no—
166C22×C4—581no—
167C22×C4C22×D2911yescharacteristic
168Q16—584no—
169Q16—584no—
170M16—22no—
171M16—584no—
172C2×C8—294no—
173C2×C8—291no—
174C2×C8—291no—
175C2×C8—584no—
176C2×C8C22×D2912yes—
177C4⋊C4—292no—
178C4⋊C4—584no—
179C4⋊C4—584no—
180C4⋊C4C22×D2912yes—
181C22⋊C4—21no—
182C22⋊C4—584no—
183C22⋊C4—584no—
184C22⋊C4—581no—
185C42—21no—
186C42—581no—
187C23—292no—
188C23D4×D2911yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—582no—
193Q8—582no—
194Q8—582no—
195Q8—1162no—
196D4—584no—
197D4—582no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—292no—
202C2×C4—582no—
203C2×C4—582no—
204C2×C4—582no—
205C2×C4—582no—
206C2×C4—1162no—
207C2×C4#9080512yes—
208C2×C4D4×D2912yes—
209C2×C4D4×D2911yescharacteristic
210C2×C4C23×D2911yescharacteristic
211C8—22no—
212C8—292no—
213C8—292no—
214C22—22no—
215C22—582no—
216C22—584no—
217C22#1174524612yes—
218C22C2×D4×D2911yescharacteristic
219C4—42no—
220C4—42no—
221C4—582no—
222C4—1162no—
223C4—1162no—
224C4#1174524612yes—
225C4C2×#9080511yescharacteristic
226C4C2×D4×D2911yescharacteristic
227C2—22no—
228C2—1162no—
229C2C2×#1174524611yescharacteristic
230C2#1427197912yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#419C2
socleC22×C29C22×D4
prime radicalC2×#90805C22
hypercenter#419D29
hypocenterC29#3059
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#90785C2semidirect
5. #14271420C2semidirect
6. #14271403C2semidirect
7. #14271199C2semidirect
8. #14271173C2semidirect
9. C29×#419C2semidirect
120. #419D29semidirect
150. C29#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#907852
C2×#117452692
#142717392
#142717372
#142714202
#142714032
#142711992
#142711732
C29×#4192
#305929