Finite Group Databasebeta

← #28373498

Subgroups — #28373498

Order 13952.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812732C212yesmaximal, complemented
2C2×#16812727C212yesmaximal, complemented
3#20295576C211yesmaximal, characteristic, complemented
4C2×#16812687C211yesmaximal, characteristic, complemented
5#20295420C212yesmaximal, complemented
6#20295415C212yesmaximal, complemented
7#20295410C214yesmaximal, complemented
8C4×M16×C109C211yesmaximal, characteristic, complemented
9C2×#13182029C2212yes—
10C22×Dic218C2211yescharacteristic
11C22×D436C2211yescharacteristic
12#16812732—28no—
13#16812727—28no—
14C2×Dic436C2214yescomplemented
15C2×#13182018C2214yes—
16C2×#13182018C2214yescomplemented
17C2×D872C2214yes—
18#16812699C2211yescharacteristic, complemented
19#16812696C2211yescharacteristic, complemented
20#16812687—22no—
21#16812687C2212yes—
22#16812687C2212yes—
23#16812687C2212yescomplemented
24#16812673—22no—
25C4×D436—22no—
26C2×#13182006—22no—
27C2×M16×C109C2212yes—
28C2×C42×C109C2211yescharacteristic
29#97×C109C2212yescomplemented
30C4×C8×C109C2212yescomplemented
31C23×D109—21no—
32#13182029—24no—
33#13182029—24no—
34C2×Dic218—24no—
35C2×Dic218—21no—
36C2×Dic218C2312yes—
37C2×Dic218C2312yes—
38C2×D436—24no—
39C2×D436—22no—
40C2×D436—21no—
41C2×D436C2312yes—
42C2×D436C2312yes—
43C2×C22⋊D109—22no—
44C22×Dic109—21no—
45C2×C4×D109—22no—
46Dic436—28no—
47#13182018—28no—
48#13182018—28no—
49D872—28no—
50C4⋊Dic109—22no—
51#13182006—24no—
52#13182006—24no—
53#13182006—24no—
54C22×C4×C109D412yes—
55C22×C4×C109C2311yescharacteristic
56M16×C109D418yescomplemented
57C2×C8×C109C2314yes—
58C42×C109D412yes—
59C42×C109C2311yescharacteristic
60C42×C109C2311yescharacteristic
61C23×C109C2×D411yescharacteristic
62C22×D109—22no—
63C22×D109—22no—
64C22×D109—44no—
65C22×D109—42no—
66C2×C4×C109—42no—
67C2×C4×C109C2×D412yes—
68C2×C4×C109C2×D412yes—
69C2×C4×C109C2×D411yescharacteristic
70C2×C4×C109C2×D414yes—
71C2×C4×C109C2411yescharacteristic
72C22⋊D109—44no—
73C2×Dic109—22no—
74C2×Dic109—22no—
75C2×Dic109—42no—
76D436—22no—
77D436—24no—
78D436—22no—
79D436—44no—
80D436—42no—
81C4×D109—44no—
82Dic218—22no—
83Dic218—24no—
84Dic218—22no—
85Dic218—42no—
86C8×C109C2×D418yes—
87C22×C109—22no—
88C22×C109#12712yes—
89C22×C109C22×D411yescharacteristic
90C2×D109—42no—
91C2×D109—42no—
92C2×D109—44no—
93C2×D109—44no—
94C2×D109—84no—
95C4×C109—22no—
96C4×C109—22no—
97C4×C109#12712yes—
98C4×C109C22×D412yes—
99Dic109—42no—
100Dic109—42no—
101C2×C109—22no—
102C2×C109C2×#12711yescharacteristic
103C2×C109#57512yes—
104D109—82no—
105D109—82no—
106#3153—1091nomaximal, Sylow
107C109#315311yescharacteristic, complemented, Sylow
108C2×#137—1092no—
109C2×AGL(1,ℤ/8)—1092no—
110#532—1091no—
111C2×#124—1091no—
112#497—1092no—
113#496—1092no—
114#495—1094no—
115C4×M16D10911yescharacteristic, complemented
116C2×#41—1092no—
117C22×Q8—1091no—
118C22×D4—1091no—
119#137—2188no—
120AGL(1,ℤ/8)—2188no—
121C2×Q16—1094no—
122C2×SD16—1094no—
123C2×SD16—1094no—
124C2×D8—1094no—
125C2×M16C2×D10912yes—
126C4⋊Q8—1091no—
127#127—1091no—
128#124—1092no—
129#124—1092no—
130#124—1092no—
131#124—2182no—
132#121—2182no—
133C4×D4—2182no—
134C2×C22⋊C4—2182no—
135C2×C42C2×D10911yescharacteristic
136#97C2×D10912yescomplemented
137C4×C8C2×D10912yescomplemented
138C24—2181no—
139#41—2184no—
140#41—2184no—
141C2×Q8—1092no—
142C2×Q8—1092no—
143C2×Q8—2184no—
144C2×Q8—2181no—
145C2×D4—1092no—
146C2×D4—1092no—
147C2×D4—2182no—
148C2×D4—2182no—
149C2×D4—2181no—
150C2×D4—2184no—
151C22×C4—2181no—
152C22×C4—2182no—
153C22×C4D43612yes—
154C22×C4C22×D10911yescharacteristic
155Q16—2188no—
156SD16—2188no—
157SD16—2188no—
158D8—2188no—
159M16D43618yescomplemented
160C2×C8C22×D10914yes—
161C4⋊C4—2182no—
162C22⋊C4—2184no—
163C22⋊C4—2184no—
164C22⋊C4—2184no—
165C42D43612yes—
166C42C22×D10911yescharacteristic
167C42C22×D10911yescharacteristic
168C23—4364no—
169C23—4362no—
170C23—2182no—
171C23—2182no—
172C23C2×D43611yescharacteristic
173Q8—4362no—
174Q8—2182no—
175Q8—2184no—
176Q8—2182no—
177D4—4364no—
178D4—4364no—
179D4—4362no—
180D4—2182no—
181D4—2184no—
182D4—2182no—
183C2×C4—42no—
184C2×C4—4362no—
185C2×C4—4364no—
186C2×C4—2182no—
187C2×C4—2182no—
188C2×C4C2×D43612yes—
189C2×C4C2×D43612yes—
190C2×C4C2×D43611yescharacteristic
191C2×C4C2×D43614yes—
192C2×C4C23×D10911yescharacteristic
193C8C2×D43618yes—
194C22—22no—
195C22—8724no—
196C22—4362no—
197C22—4362no—
198C22—4364no—
199C22—4364no—
200C22#1681269612yes—
201C22C22×D43611yescharacteristic
202C4—22no—
203C4—22no—
204C4—4362no—
205C4—4362no—
206C4#1681269612yes—
207C4C22×D43612yes—
208C2—22no—
209C2—8722no—
210C2—8722no—
211C2C2×#1681269611yescharacteristic
212C2#2029582612yes—
subgroupG/H
centerC22C22×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC4×M16×C109C2
socleC22×C109C22×D4
hypercenterC4×M16D109
hypocenterC109#3153
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C2×#16812727C2semidirect
3. #20295576C2semidirect
4. C2×#16812687C2semidirect
5. #20295420C2semidirect
6. #20295415C2semidirect
7. #20295410C2semidirect
8. C4×M16×C109C2semidirect
14. C2×Dic436C22semidirect
16. C2×#13182018C22semidirect
18. #16812699C22semidirect
19. #16812696C22semidirect
23. #16812687C22semidirect
29. #97×C109C22semidirect
30. C4×C8×C109C22semidirect
56. M16×C109D4semidirect
107. C109#3153semidirect
115. C4×M16D109semidirect
136. #97C2×D109semidirect
137. C4×C8C2×D109semidirect
159. M16D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×#168127272
#202955762
C2×#168126872
#202954202
#202954152
#202954102
C4×M16×C1092
#3153109