Finite Group Databasebeta

← #28373513

Subgroups — #28373513

Order 13952.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16812725C212yesmaximal, complemented
2C2×#16812721C211yesmaximal, characteristic, complemented
3#20295593C212yesmaximal, complemented
4#20295414C214yesmaximal, complemented
5#20295409C212yesmaximal, complemented
6#20295401C212yesmaximal, complemented
7#20294981C211yesmaximal, characteristic, complemented
8C22×D8×C109C211yesmaximal, characteristic, complemented
9C2×#13182029C2212yes—
10#16812725—28no—
11#16812721—24no—
12C2×Dic436C2211yescharacteristic, complemented
13C2×#13182018C2212yescomplemented
14C2×#13182017C2214yescomplemented
15C2×D872C2211yescharacteristic, complemented
16C2×#13182014C2214yescomplemented
17C2×#13182013C2211yescharacteristic
18#16812695C2214yescomplemented
19#16812686C2214yescomplemented
20#16812670—24no—
21#16812668—24no—
22#16812658C2211yescharacteristic
23#16812609C2212yescomplemented
24C8×Dic109C2212yescomplemented
25C22×D4×C109C2212yes—
26C2×D8×C109—24no—
27C2×D8×C109C2212yes—
28C2×D8×C109C2212yes—
29C22×C8×C109C2211yescharacteristic
30#13182029—24no—
31#13182029—24no—
32C2×Dic218C2312yes—
33C2×D436C2312yes—
34C2×C22⋊D109—22no—
35C2×C22⋊D109—28no—
36C2×C4×D109—22no—
37Dic436—22no—
38#13182018—24no—
39#13182017—28no—
40D872—22no—
41#13182014—28no—
42#13182013D414yescomplemented
43C2×#11724784C2312yes—
44C4⋊Dic109D412yes—
45C22⋊Dic109—28no—
46C22⋊Dic109D412yes—
47C4×Dic109C2312yes—
48C24×C109—22no—
49C2×D4×C109—24no—
50C2×D4×C109—24no—
51C2×D4×C109—22no—
52C2×D4×C109C2314yes—
53C22×C4×C109C2311yescharacteristic
54D8×C109—24no—
55D8×C109—24no—
56C2×C8×C109D414yescomplemented
57C2×C8×C109C2311yescharacteristic
58C2×C8×C109C2311yescharacteristic
59C23×C109—24no—
60C23×C109—44no—
61C23×C109—84no—
62C23×C109C2×D411yescharacteristic
63C22×D109—22no—
64D4×C109—24no—
65D4×C109—24no—
66D4×C109—44no—
67C2×C4×C109C2×D411yescharacteristic
68C2×C4×C109C2×D412yes—
69C2×C4×C109C2×D412yes—
70C2×C4×C109C2411yescharacteristic
71C22⋊D109—44no—
72C22⋊D109—88no—
73C2×Dic109—22no—
74C2×Dic109C2×D414yes—
75D436—22no—
76D436—22no—
77C4×D109—44no—
78Dic218—22no—
79Dic218—22no—
80C8×C109C2×D412yes—
81C8×C109C2×D412yes—
82#11724784C2×D414yes—
83C22×C109—22no—
84C22×C109—44no—
85C22×C109—48no—
86C22×C109—88no—
87C22×C109#12712yes—
88C22×C109C22×D411yescharacteristic
89C2×D109—42no—
90C2×D109—44no—
91C4×C109#12712yes—
92C4×C109C22×D411yescharacteristic
93C4×C109C22×D411yescharacteristic
94Dic109—24no—
95Dic109—42no—
96C2×C109—22no—
97C2×C109—84no—
98C2×C109C2×#12711yescharacteristic
99C2×C109#57612yes—
100D109—82no—
101#3155—1091nomaximal, Sylow
102C109#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1092no—
104C2×#135—1091no—
105C22×D8D10911yescharacteristic, complemented
106#535—1092no—
107#496—1094no—
108#495—1092no—
109#493—1092no—
110#405—1091no—
111C2×#41—1092no—
112C22×D4C2×D10912yes—
113AGL(1,ℤ/8)—2188no—
114#135—2184no—
115C2×Q16—1091no—
116C2×SD16—1094no—
117C2×SD16—1092no—
118C2×D8—24no—
119C2×D8—1094no—
120C2×D8—1091no—
121C2×D8C2×D10912yes—
122C2×D8C2×D10912yes—
123C2×M16—1091no—
124C22×C8C2×D10911yescharacteristic
125#127—1094no—
126#124—1094no—
127#121—2184no—
128#120—2184no—
129#117—1091no—
130#97—1092no—
131C4×C8—1092no—
132C24—22no—
133#41—2184no—
134#41—2184no—
135C2×Q8—1092no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—1092no—
140C2×D4—2188no—
141C2×D4—2182no—
142C2×D4C22×D10914yes—
143C22×C4—2182no—
144C22×C4C22×D10911yescharacteristic
145Q16—2182no—
146SD16—2188no—
147SD16—2184no—
148D8—24no—
149D8—24no—
150D8—2188no—
151D8—2182no—
152M16—1094no—
153C2×C8—1092no—
154C2×C8C22⋊D10914yescomplemented
155C2×C8C22×D10911yescharacteristic
156C2×C8C22×D10911yescharacteristic
157C4⋊C4—1092no—
158C22⋊C4—1092no—
159C22⋊C4—2188no—
160C42—1092no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—2182no—
165C23D4×D10911yescharacteristic
166Q8—2182no—
167Q8—2182no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—8728no—
172D4—4364no—
173D4—2182no—
174D4—2182no—
175C2×C4—1094no—
176C2×C4—4364no—
177C2×C4—2182no—
178C2×C4D4×D10911yescharacteristic
179C2×C4D4×D10912yes—
180C2×C4C2×C22⋊D10912yes—
181C2×C4C23×D10911yescharacteristic
182C8—1094no—
183C8C2×C22⋊D10912yes—
184C8C2×C22⋊D10912yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—4362no—
190C22—4364no—
191C22#1681269512yes—
192C22C2×D4×D10911yescharacteristic
193C4—4362no—
194C4—2184no—
195C4#1681269512yes—
196C4C2×D4×D10911yescharacteristic
197C4C22×C22⋊D10911yescharacteristic
198C2—22no—
199C2—84no—
200C2—8722no—
201C2C2×#1681269511yescharacteristic
202C2#2029582712yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC22×D8×C109C2
socleC22×C109C22×D4
hypercenterC22×D8D109
hypocenterC109#3155
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16812725C2semidirect
2. C2×#16812721C2semidirect
3. #20295593C2semidirect
4. #20295414C2semidirect
5. #20295409C2semidirect
6. #20295401C2semidirect
7. #20294981C2semidirect
8. C22×D8×C109C2semidirect
12. C2×Dic436C22semidirect
13. C2×#13182018C22semidirect
14. C2×#13182017C22semidirect
15. C2×D872C22semidirect
16. C2×#13182014C22semidirect
18. #16812695C22semidirect
19. #16812686C22semidirect
23. #16812609C22semidirect
24. C8×Dic109C22semidirect
42. #13182013D4semidirect
56. C2×C8×C109D4semidirect
102. C109#3155semidirect
105. C22×D8D109semidirect
154. C2×C8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127252
C2×#168127212
#202955932
#202954142
#202954092
#202954012
#202949812
C22×D8×C1092
#3155109