Finite Group Databasebeta

← #28376184

Subgroups — #28376184

Order 13952.

350 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 113 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#20295840C211yesmaximal, characteristic, complemented
2#20295839C211yesmaximal, characteristic, complemented
3#20295469C211yesmaximal, characteristic, complemented
4#20295468C212yesmaximal, complemented
5#20295465C212yesmaximal, complemented
6#20295457C212yesmaximal, complemented
7#20295456C211yesmaximal, characteristic, complemented
8C2×#16812645C211yesmaximal, characteristic, complemented
9Q32×D109C212yesmaximal, complemented
10#20295444C211yesmaximal, characteristic, complemented
11#510×C109C211yesmaximal, characteristic, complemented
12#16812747—21no—
13#16812746—21no—
14#16812731—22no—
15#16812730—21no—
16#16812729—21no—
17#16812728—22no—
18#16812722C2212yes—
19#16812721C2211yescharacteristic
20#16812718C2212yes—
21C2×Dic436C2211yescharacteristic
22Q16×D109C2212yes—
23Q16×D109C2212yes—
24#16812705C2211yescharacteristic
25Dic872C2212yes—
26#16812645C2211yescharacteristic
27#16812645C2212yes—
28#16812645C2212yes—
29#16812644C2212yes—
30#16812643C2211yescharacteristic
31#16812642C2211yescharacteristic
32#16812640C2211yescharacteristic
33#16812638C2212yes—
34C2×#13182004C2211yescharacteristic
35C16×D109C2212yes—
36#135×C109C2211yescharacteristic
37C2×Q16×C109C2211yescharacteristic
38Q32×C109C2212yes—
39SD32×C109C2212yes—
40M32×C109C2211yescharacteristic
41#13182030—22no—
42#13182030—41no—
43#13182029—21no—
44#13182029—42no—
45#13182029—41no—
46#13182029D411yescharacteristic
47#13182028—22no—
48#13182028—42no—
49C2×Dic218—21no—
50C2×Dic218—41no—
51Q8×D109—22no—
52Q8×D109—22no—
53Q8×D109—41no—
54Dic436—21no—
55Dic436C2311yescharacteristic
56Dic436C2312yes—
57#13182019—22no—
58#13182019—22no—
59#13182018—21no—
60#13182017—22no—
61#13182016—22no—
62D872C2311yescharacteristic
63#13182013D411yescharacteristic
64#13182012D412yes—
65C8×D109C2312yes—
66#13182004C2311yescharacteristic
67#13182004C2311yescharacteristic
68#41×C109—21no—
69C2×Q8×C109—21no—
70Q16×C109—21no—
71Q16×C109C2312yes—
72Q16×C109C2311yescharacteristic
73SD16×C109—21no—
74D8×C109C2311yescharacteristic
75C2×C8×C109C2311yescharacteristic
76C16×C109C2312yes—
77Q8×C109—21no—
78Q8×C109—22no—
79Q8×C109—41no—
80D4×C109—21no—
81D4×C109—41no—
82C2×C4×C109—41no—
83C2×C4×C109—41no—
84C2×C4×C109C2×D411yescharacteristic
85C22⋊D109—41no—
86C22⋊D109—42no—
87C22⋊D109D812yes—
88C2×Dic109—41no—
89C2×Dic109—42no—
90D436—21no—
91D436—42no—
92D436D811yescharacteristic
93C4×D109—42no—
94C4×D109—42no—
95C4×D109—41no—
96C4×D109—42no—
97C4×D109C2×D412yes—
98Dic218—22no—
99Dic218—21no—
100Dic218—41no—
101Dic218—42no—
102Dic218—42no—
103Dic218D811yescharacteristic
104C8×C109C2×D411yescharacteristic
105C8×C109C2411yescharacteristic
106#11724784C2×D412yes—
107C22×C109—41no—
108C22×C109C2×D811yescharacteristic
109C2×D109—41no—
110C2×D109C2×D812yes—
111C4×C109—41no—
112C4×C109—42no—
113C4×C109C2×D811yescharacteristic
114C4×C109C22×D411yescharacteristic
115Dic109—41no—
116Dic109—42no—
117Dic109C2×D812yes—
118C2×C109—21no—
119C2×C109—81no—
120C2×C109C22×D811yescharacteristic
121D109—22no—
122D109—81no—
123#3417—1091nomaximal, Sylow
124C109#341711yescharacteristic, complemented, Sylow
125#578—1091no—
126#578—1091no—
127#510—1092no—
128#510—1092no—
129#510—1091no—
130#510D10911yescharacteristic, complemented
131#508—1091no—
132#508—1092no—
133C2×Q32—1091no—
134C2×Q32—1092no—
135#504—1091no—
1362−1+4—2181no—
1372−1+4—2181no—
138#137—2182no—
139#137—2182no—
140#137—2181no—
141#137—2181no—
142#135—1092no—
143#135—1092no—
144#135—1091no—
145#135C2×D10911yescharacteristic
146C2×Q16—1092no—
147C2×Q16—1092no—
148C2×Q16—1091no—
149C2×Q16C2×D10911yescharacteristic
150#131—1091no—
151Q32—1091no—
152Q32—1092no—
153Q32—1092no—
154Q32—1092no—
155Q32C2×D10912yes—
156SD32—1091no—
157SD32—1091no—
158SD32—1092no—
159SD32C2×D10912yes—
160D16—1091no—
161M32—1092no—
162M32C2×D10911yescharacteristic
163C2×C16—1091no—
164C2×C16—1092no—
165#41—21no—
166#41—1091no—
167#41—4362no—
168#41—4361no—
169#41—4362no—
170#41—4361no—
171#41—2182no—
172#41—2182no—
173#41—2181no—
174C2×Q8—21no—
175C2×Q8—4361no—
176C2×Q8—4361no—
177C2×Q8—2182no—
178C2×Q8—2182no—
179C2×Q8—2181no—
180Q16—21no—
181Q16—1091no—
182Q16—1092no—
183Q16—2182no—
184Q16—2182no—
185Q16—2181no—
186Q16C22×D10911yescharacteristic
187Q16C22×D10912yes—
188SD16—21no—
189SD16—2182no—
190SD16—2182no—
191SD16—2181no—
192D8—1091no—
193D8C22×D10911yescharacteristic
194M16—1091no—
195M16—1092no—
196C2×C8—1092no—
197C2×C8C22×D10911yescharacteristic
198C16—1091no—
199C16—1091no—
200C16C22×D10912yes—
201Q8—21no—
202Q8—22no—
203Q8—41no—
204Q8—1091no—
205Q8—4362no—
206Q8—4362no—
207Q8—4361no—
208Q8—2182no—
209Q8—2181no—
210D4—21no—
211D4—41no—
212D4—1092no—
213D4—1091no—
214D4—4362no—
215D4—4362no—
216D4—4361no—
217D4—2181no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—1092no—
221C2×C4—4362no—
222C2×C4—4362no—
223C2×C4—4362no—
224C2×C4—4362no—
225C2×C4—4361no—
226C2×C4—4361no—
227C2×C4D4×D10911yescharacteristic
228C8—1092no—
229C8D4×D10911yescharacteristic
230C8C23×D10911yescharacteristic
231C22—41no—
232C22—1092no—
233C22—4361no—
234C22D8×D10911yescharacteristic
235C4—41no—
236C4—42no—
237C4—1092no—
238C4—4361no—
239C4—4362no—
240C4D8×D10911yescharacteristic
241C4C2×D4×D10911yescharacteristic
242C2—21no—
243C2—81no—
244C2—8721no—
245C2—2182no—
246C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#510×C109C2
socleC2×C109C22×D8
hypercenter#510D109
hypocenterC109#3417
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #20295840C2semidirect
2. #20295839C2semidirect
3. #20295469C2semidirect
4. #20295468C2semidirect
5. #20295465C2semidirect
6. #20295457C2semidirect
7. #20295456C2semidirect
8. C2×#16812645C2semidirect
9. Q32×D109C2semidirect
10. #20295444C2semidirect
11. #510×C109C2semidirect
124. C109#3417semidirect
130. #510D109semidirect

Maximal subgroups

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

subgroupindex
#202958402
#202958392
#202954692
#202954682
#202954652
#202954572
#202954562
C2×#168126452
Q32×D1092
#202954442
#510×C1092
#3417109