Finite Group Databasebeta

← #28373496

Subgroups — #28373496

Order 13952.

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

normal subgroups only

This group has 572 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×#16812730C212yesmaximal, complemented
2C2×#16812725C212yesmaximal, complemented
3#20295576C211yesmaximal, characteristic, complemented
4#20295421C212yesmaximal, complemented
5#20295416C212yesmaximal, complemented
6#20295408C214yesmaximal, complemented
7C4×#13182013C211yesmaximal, characteristic, complemented
8C2×#124×C109C211yesmaximal, characteristic, complemented
9C2×#13182029C2212yes—
10#16812730—28no—
11#16812725—28no—
12C2×#13182019C2214yescomplemented
13C2×#13182017C2214yes—
14C2×#13182016C2214yescomplemented
15C2×#13182014C2214yes—
16C2×#13182013C2212yes—
17#16812699C2211yescharacteristic, complemented
18#16812696C2211yescharacteristic, complemented
19#16812687C2212yescomplemented
20#16812673—22no—
21C4×D436—22no—
22#16812610C2212yescomplemented
23C4×#11724784C2212yescomplemented
24C22×Q8×C109C2211yescharacteristic
25C22×D4×C109C2211yescharacteristic
26#124×C109—22no—
27#124×C109C2212yes—
28#124×C109C2212yes—
29C2×C22⋊C4×C109—22no—
30C2×C42×C109C2211yescharacteristic
31#13182029—24no—
32#13182029—24no—
33C2×Dic218C2312yes—
34C2×D436—22no—
35C2×D436C2312yes—
36C2×C22⋊D109—22no—
37C2×C4×D109—22no—
38#13182019—28no—
39#13182017—28no—
40#13182016—28no—
41#13182014—28no—
42#13182013D418yescomplemented
43C2×#11724784C2314yes—
44C4⋊Dic109—22no—
45#13182006—24no—
46C24×C109—21no—
47C2×Q8×C109—21no—
48C2×Q8×C109—24no—
49C2×Q8×C109C2312yes—
50C2×D4×C109—24no—
51C2×D4×C109—21no—
52C2×D4×C109C2312yes—
53C22×C4×C109—21no—
54C22×C4×C109D412yes—
55C22×C4×C109C2311yescharacteristic
56C22⋊C4×C109—24no—
57C22⋊C4×C109—24no—
58C42×C109D412yes—
59C42×C109C2311yescharacteristic
60C42×C109C2311yescharacteristic
61C23×C109—22no—
62C23×C109—44no—
63C23×C109—42no—
64C23×C109C2×D411yescharacteristic
65C22×D109—22no—
66Q8×C109—24no—
67Q8×C109—42no—
68D4×C109—24no—
69D4×C109—42no—
70C2×C4×C109—22no—
71C2×C4×C109—42no—
72C2×C4×C109—42no—
73C2×C4×C109C2×D414yes—
74C2×C4×C109C2×D412yes—
75C2×C4×C109C2×D412yes—
76C2×C4×C109C2×D411yescharacteristic
77C2×C4×C109C2411yescharacteristic
78C22⋊D109—44no—
79C2×Dic109—22no—
80D436—22no—
81D436—22no—
82D436—44no—
83C4×D109—44no—
84Dic218—22no—
85Dic218—22no—
86#11724784C2×D418yes—
87C22×C109—22no—
88C22×C109—42no—
89C22×C109—44no—
90C22×C109—84no—
91C22×C109#12712yes—
92C22×C109C22×D411yescharacteristic
93C2×D109—42no—
94C2×D109—44no—
95C4×C109—22no—
96C4×C109—22no—
97C4×C109—42no—
98C4×C109#12712yes—
99C4×C109C22×D412yes—
100Dic109—42no—
101C2×C109—22no—
102C2×C109—82no—
103C2×C109C2×#12711yescharacteristic
104C2×C109#57512yes—
105D109—82no—
106#3153—1091nomaximal, Sylow
107C109#315311yescharacteristic, complemented, Sylow
108C2×#137—1092no—
109C2×AGL(1,ℤ/8)—1092no—
110#532—1091no—
111C2×#124D10911yescharacteristic, complemented
112#497—1092no—
113#496—1092no—
114#495—1094no—
115C4×M16—1091no—
116C2×#41—1092no—
117C22×Q8C2×D10911yescharacteristic
118C22×D4C2×D10911yescharacteristic
119#137—2188no—
120AGL(1,ℤ/8)—2188no—
121C2×Q16—1094no—
122C2×SD16—1094no—
123C2×SD16—1094no—
124C2×D8—1094no—
125C2×M16—1092no—
126C4⋊Q8—1091no—
127#127—1091no—
128#124—22no—
129#124—1092no—
130#124C2×D10912yes—
131#124C2×D10912yes—
132#121—2182no—
133C4×D4—2182no—
134C2×C22⋊C4—22no—
135C2×C42C2×D10911yescharacteristic
136#97—1092no—
137C4×C8—1092no—
138C24—21no—
139#41—2184no—
140#41—2184no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—1092no—
144C2×Q8C22×D10912yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—1092no—
148C2×D4—2182no—
149C2×D4—2182no—
150C2×D4C22×D10912yes—
151C22×C4—21no—
152C22×C4—2182no—
153C22×C4C22⋊D10912yes—
154C22×C4C22×D10911yescharacteristic
155Q16—2188no—
156SD16—2188no—
157SD16—2188no—
158D8—2188no—
159M16—1098no—
160C2×C8—1094no—
161C4⋊C4—2182no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—2184no—
165C42C22⋊D10912yes—
166C42C22×D10911yescharacteristic
167C42C22×D10911yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—2182no—
172C23C2×C22⋊D10911yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—2182no—
176Q8—2182no—
177D4—24no—
178D4—42no—
179D4—4364no—
180D4—4364no—
181D4—2182no—
182D4—2182no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—4364no—
187C2×C4—2182no—
188C2×C4D4×D10914yes—
189C2×C4C2×C22⋊D10912yes—
190C2×C4C2×C22⋊D10912yes—
191C2×C4C2×C22⋊D10911yescharacteristic
192C2×C4C23×D10911yescharacteristic
193C8—1098no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—4362no—
199C22—4364no—
200C22#1681269512yes—
201C22C22×C22⋊D10911yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—4362no—
206C4#1681269512yes—
207C4C2×D4×D10912yes—
208C2—22no—
209C2—82no—
210C2—8722no—
211C2C2×#1681269511yescharacteristic
212C2#2029582412yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#124×C109C2
socleC22×C109C22×D4
hypercenterC2×#124D109
hypocenterC109#3153
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16812730C2semidirect
2. C2×#16812725C2semidirect
3. #20295576C2semidirect
4. #20295421C2semidirect
5. #20295416C2semidirect
6. #20295408C2semidirect
7. C4×#13182013C2semidirect
8. C2×#124×C109C2semidirect
12. C2×#13182019C22semidirect
14. C2×#13182016C22semidirect
17. #16812699C22semidirect
18. #16812696C22semidirect
19. #16812687C22semidirect
22. #16812610C22semidirect
23. C4×#11724784C22semidirect
42. #13182013D4semidirect
107. C109#3153semidirect
111. C2×#124D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
C2×#168127252
#202955762
#202954212
#202954162
#202954082
C4×#131820132
C2×#124×C1092
#3153109