Finite Group Databasebeta

← #31791177

Subgroups — #31791177

Order 17536.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#22577956C211yesmaximal, characteristic, complemented
2C2×#17984077C211yesmaximal, characteristic
3#22577764C214yesmaximal, complemented
4#22577759C214yesmaximal, complemented
5#22577412C211yesmaximal, characteristic, complemented
6#22577409C211yesmaximal, characteristic
7#22577407C211yesmaximal, characteristic
8M16×Dic137C211yesmaximal, characteristic, complemented
9#419×C137C211yesmaximal, characteristic, complemented
10C22×D548C2211yescharacteristic
11C2×#14555494C2211yescharacteristic
12#17984087C2212yescomplemented
13#17984079C2212yescomplemented
14#17984077—22no—
15#17984077C2212yes—
16#17984077C2212yes—
17#17984065—22no—
18C4×Dic274—22no—
19#17984047C2211yescharacteristic
20C2×#14555487—22no—
21C2×C4×Dic137C2211yescharacteristic
22#17984012C2214yescomplemented
23#17984010C2214yes—
24#17984008C2214yes—
25#17983998C2212yescomplemented
26C8×Dic137C2212yescomplemented
27C22×Q8×C137C2211yescharacteristic
28C2×M16×C137C2211yescharacteristic
29#117×C137C2211yescharacteristic
30#103×C137C2214yescomplemented
31C23×D137—21no—
32C2×Dic274—22no—
33C2×D548—21no—
34C2×D548—22no—
35C2×D548—22no—
36C2×D548C2312yes—
37C22×Dic137D412yes—
38#14555494—22no—
39C2×#11764040C2312yes—
40C4⋊Dic137—22no—
41C4⋊Dic137C2312yes—
42#14555489—22no—
43#14555489—22no—
44C22⋊Dic137—21no—
45#14555487—24no—
46#14555487—24no—
47C4×Dic137—21no—
48C4×Dic137D412yes—
49C4×Dic137C2311yescharacteristic
50C4×Dic137C2311yescharacteristic
51C2×Q8×C137—21no—
52C2×Q8×C137—22no—
53C2×Q8×C137—22no—
54C2×Q8×C137C2312yes—
55C22×C4×C137—21no—
56C22×C4×C137C2311yescharacteristic
57M16×C137—22no—
58C2×C8×C137C2312yes—
59C4⋊C4×C137C2312yes—
60C22⋊C4×C137—21no—
61C42×C137—21no—
62C22×D137—22no—
63C22×D137—44no—
64C22×D137—42no—
65Dic274—44no—
66D548—22no—
67D548—22no—
68D548—42no—
69C2×Dic137—22no—
70C2×Dic137—42no—
71C2×Dic137C2×D412yes—
72C2×Dic137C2×D412yes—
73#11764040—22no—
74C23×C137C2×D411yescharacteristic
75Q8×C137—22no—
76Q8×C137—22no—
77Q8×C137—42no—
78C2×C4×C137—22no—
79C2×C4×C137—22no—
80C2×C4×C137—42no—
81C2×C4×C137C2×D411yescharacteristic
82C2×C4×C137#4112yes—
83C2×C4×C137#4112yes—
84C2×C4×C137C2411yescharacteristic
85C8×C137—22no—
86C22×C137—22no—
87C22×C137#12412yes—
88C22×C137C22×D411yescharacteristic
89C2×D137—42no—
90C2×D137—44no—
91C2×D137—84no—
92C4×C137—42no—
93C4×C137—42no—
94C4×C137#12412yes—
95C4×C137C2×#4111yescharacteristic
96C4×C137C2×#4111yescharacteristic
97Dic137—22no—
98Dic137—22no—
99Dic137—42no—
100C2×C137—22no—
101C2×C137C2×#12411yescharacteristic
102C2×C137#57512yes—
103D137—82no—
104C137#314011yescharacteristic, complemented, Sylow
105#3140—1371nomaximal, Sylow
106#533—1371no—
107C2×#124—1371no—
108#489—1374no—
109#488—1374no—
110#419—1371no—
111#419D13711yescharacteristic, complemented
112#418—1371no—
113#418—1371no—
114C4×M16—1371no—
115C22×Q8C2×D13711yescharacteristic
116C22×D4—1371no—
117C2×M16—1371no—
118C2×M16C2×D13711yescharacteristic
119C4⋊Q8—1372no—
120#125—1372no—
121#124—2742no—
122#124—1372no—
123#124—1372no—
124C22⋊Q8—2742no—
125C4×Q8—2742no—
126#117—1371no—
127#117C2×D13711yescharacteristic
128C2×C22⋊C4—2742no—
129C2×C42—1371no—
130#103—1374no—
131#103C2×D13714yescomplemented
132#102—1374no—
133#102—1374no—
134#97—1372no—
135C4×C8—1372no—
136C24—2741no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—2742no—
141C2×Q8C22×D13712yes—
142C2×D4—2741no—
143C2×D4—2742no—
144C2×D4—2742no—
145C2×D4—1372no—
146C22×C4—21no—
147C22×C4—1372no—
148C22×C4C22×D13711yescharacteristic
149M16—22no—
150M16—2742no—
151C2×C8—1372no—
152C2×C8C22×D13712yes—
153C4⋊C4—2742no—
154C4⋊C4—2742no—
155C4⋊C4—2742no—
156C4⋊C4—1372no—
157C4⋊C4C22×D13712yes—
158C22⋊C4—21no—
159C22⋊C4—2744no—
160C22⋊C4—2744no—
161C22⋊C4—2741no—
162C42—21no—
163C42—2741no—
164C42—1372no—
165C42—1371no—
166C42—1371no—
167C23—2742no—
168C23—5484no—
169C23—5482no—
170C23D4×D13711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—5484no—
175D4—2742no—
176D4—2742no—
177D4—5482no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—2742no—
182C2×C4—5482no—
183C2×C4—1372no—
184C2×C4—1372no—
185C2×C4D4×D13711yescharacteristic
186C2×C4#1455550912yes—
187C2×C4#1455551012yes—
188C2×C4C23×D13711yescharacteristic
189C8—22no—
190C8—2742no—
191C22—22no—
192C22—5484no—
193C22—5482no—
194C22—10964no—
195C22#1798407412yes—
196C22C2×D4×D13711yescharacteristic
197C4—42no—
198C4—42no—
199C4—2742no—
200C4—2742no—
201C4—5482no—
202C4#1798407412yes—
203C4C2×#1455550911yescharacteristic
204C4C2×#1455551011yescharacteristic
205C2—22no—
206C2—10962no—
207C2C2×#1798407411yescharacteristic
208C2#2257820212yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#419×C137C2
socleC22×C137C22×D4
prime radicalC22×D548C22
hypercenter#419D137
hypocenterC137#3140
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #22577956C2semidirect
3. #22577764C2semidirect
4. #22577759C2semidirect
5. #22577412C2semidirect
8. M16×Dic137C2semidirect
9. #419×C137C2semidirect
12. #17984087C22semidirect
13. #17984079C22semidirect
22. #17984012C22semidirect
25. #17983998C22semidirect
26. C8×Dic137C22semidirect
30. #103×C137C22semidirect
104. C137#3140semidirect
111. #419D137semidirect
131. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779562
C2×#179840772
#225777642
#225777592
#225774122
#225774092
#225774072
M16×Dic1372
#419×C1372
#3140137