Finite Group Databasebeta

← #31792244

Subgroups — #31792244

Order 17536.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#22578020C212yesmaximal, complemented
2#22577697C212yesmaximal, complemented
3#22577681C212yesmaximal, complemented
4#22577647C212yesmaximal, complemented
5#22577644C211yesmaximal, characteristic, complemented
6#22577624C212yesmaximal, complemented
7#22577616C212yesmaximal, complemented
8#22577429C211yesmaximal, characteristic, complemented
9#466×C137C211yesmaximal, characteristic
10C2×Dic548C2211yescharacteristic
11C2×#14555498C2212yes—
12C2×#14555494C2211yescharacteristic
13#17984087C2212yes—
14#17984081—22no—
15#17984075—22no—
16#17984075C2212yes—
17#17984069—22no—
18#17984067C2212yes—
19#17984065—22no—
20#17984053C2212yes—
21D4×Dic137C2212yes—
22#17984047C2211yescharacteristic
23#17984022C2211yescharacteristic
24#17984021C2212yes—
25#17984017C2212yes—
26#17984017C2212yes—
27#17984013C2212yes—
28#17984011C2212yes—
29#17984009C2212yes—
30C2×D8×C137C2211yescharacteristic
31C22×C8×C137C2211yescharacteristic
32#121×C137C2212yes—
33#107×C137C2211yescharacteristic
34#102×C137C2212yes—
35C2×Dic274—22no—
36C2×Dic274C2312yes—
37C22×Dic137—22no—
38Dic548—41no—
39#14555498—42no—
40#14555494—41no—
41C2×#11764040C2312yes—
42C4⋊Dic137D411yescharacteristic
43C4⋊Dic137D411yescharacteristic
44C4⋊Dic137C2312yes—
45#14555489—22no—
46#14555489—22no—
47#14555489—22no—
48C22⋊Dic137—22no—
49C22⋊Dic137—22no—
50C22⋊Dic137—22no—
51C22⋊Dic137—22no—
52C22⋊Dic137D412yes—
53C4×Dic137—22no—
54C4×Dic137C2312yes—
55C2×D4×C137—22no—
56C2×D4×C137C2312yes—
57C22×C4×C137C2311yescharacteristic
58D8×C137—41no—
59C2×C8×C137—41no—
60C2×C8×C137C2311yescharacteristic
61C2×C8×C137C2311yescharacteristic
62C4⋊C4×C137C2312yes—
63C22⋊C4×C137—22no—
64Dic274—22no—
65Dic274—22no—
66Dic274—82no—
67C2×Dic137—22no—
68C2×Dic137—22no—
69C2×Dic137—82no—
70C2×Dic137C2×D412yes—
71C2×Dic137C2×D412yes—
72#11764040—22no—
73C23×C137—22no—
74C23×C137C2×D411yescharacteristic
75D4×C137—22no—
76D4×C137—22no—
77D4×C137—82no—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—22no—
81C2×C4×C137C2×D411yescharacteristic
82C2×C4×C137C2411yescharacteristic
83C8×C137—21no—
84C8×C137—21no—
85C22×C137—21no—
86C22×C137—21no—
87C22×C137—21no—
88C22×C137—42no—
89C22×C137—42no—
90C22×C137—42no—
91C22×C137C22×D411yescharacteristic
92C4×C137—21no—
93C4×C137—42no—
94C4×C137H(5,𝔽2)11yescharacteristic
95C4×C1372−1+411yescharacteristic
96Dic137—22no—
97Dic137—22no—
98Dic137—42no—
99Dic137—42no—
100C2×C137—41no—
101C2×C137—82no—
102C2×C137#53711yescharacteristic
103C2×C137#57611yescharacteristic
104C2×C137#57811yescharacteristic
105C137#326211yescharacteristic, complemented, Sylow
106#3262—1371nomaximal, Sylow
107#542—1372no—
108#477—1372no—
109#474—1372no—
110#468—1372no—
111#467—1371no—
112#466D13711yescharacteristic
113#464—1372no—
114#463—1372no—
115#424—1371no—
116C2×Q16—1371no—
117C2×SD16—1372no—
118C2×D8C2×D13711yescharacteristic
119C2×M16—1371no—
120C22×C8C2×D13711yescharacteristic
121C4⋊Q8—1372no—
122#126—2742no—
123#124—2742no—
124#124—1372no—
125#123—2742no—
126C22⋊Q8—2742no—
127C22⋊Q8—1372no—
128#121C2×D13712yes—
129C4×Q8—1372no—
130C4×D4—1372no—
131#117—1371no—
132#107—1371no—
133#107—1372no—
134#107C2×D13711yescharacteristic
135C4⋊C8—1372no—
136C4⋊C8—1372no—
137#103—1372no—
138#103—1372no—
139#102—1372no—
140#102C2×D13712yes—
141C2×Q8—2742no—
142C2×Q8—1372no—
143C2×D4—22no—
144C2×D4C22×D13712yes—
145C22×C4—2742no—
146C22×C4C22×D13711yescharacteristic
147Q16—5481no—
148SD16—5482no—
149D8—41no—
150M16—5481no—
151C2×C8—41no—
152C2×C8—1372no—
153C2×C8C22×D13711yescharacteristic
154C2×C8C22×D13711yescharacteristic
155C4⋊C4—2742no—
156C4⋊C4—2742no—
157C4⋊C4—2742no—
158C4⋊C4—1372no—
159C4⋊C4—1371no—
160C4⋊C4—1371no—
161C4⋊C4C22×D13712yes—
162C22⋊C4—22no—
163C22⋊C4—2742no—
164C22⋊C4—2742no—
165C22⋊C4—2742no—
166C22⋊C4—2742no—
167C22⋊C4—1372no—
168C42—2742no—
169C42—1372no—
170C23—22no—
171C23D4×D13711yescharacteristic
172Q8—2742no—
173Q8—2742no—
174Q8—10962no—
175D4—22no—
176D4—22no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—2742no—
182C2×C4—2742no—
183C2×C4—10962no—
184C2×C4—1372no—
185C2×C4—1372no—
186C2×C4D4×D13711yescharacteristic
187C2×C4C23×D13711yescharacteristic
188C8—21no—
189C8—21no—
190C8—2742no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22C2×D4×D13711yescharacteristic
198C4—21no—
199C4—42no—
200C4—2742no—
201C4—2742no—
202C4—5482no—
203C4—5482no—
204C4#1798413311yescharacteristic
205C4#1798413511yescharacteristic
206C2—41no—
207C2—82no—
208C2#2257798011yescharacteristic
209C2#2257820411yescharacteristic
210C2#2257821711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radical#466×C137C2
socleC22×C137C22×D4
hypercenter#466D137
hypocenterC137#3262
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22578020C2semidirect
2. #22577697C2semidirect
3. #22577681C2semidirect
4. #22577647C2semidirect
5. #22577644C2semidirect
6. #22577624C2semidirect
7. #22577616C2semidirect
8. #22577429C2semidirect
105. C137#3262semidirect

Maximal subgroups

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

subgroupindex
#225780202
#225776972
#225776812
#225776472
#225776442
#225776242
#225776162
#225774292
#466×C1372
#3262137