Finite Group Databasebeta

← #31793365

Subgroups — #31793365

Order 17536.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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#22578094C212yesmaximal
2#22577805C211yesmaximal, characteristic, complemented
3#22577743C212yesmaximal, complemented
4#22577737C212yesmaximal, complemented
5#22577710C212yesmaximal, complemented
6#22577658C212yesmaximal, complemented
7#22577482C211yesmaximal, characteristic
8#22577444C212yesmaximal, complemented
9C4×Q16×C137C211yesmaximal, characteristic, complemented
10C2×#14555493C2212yes—
11#17984082—22no—
12#17984080C2212yes—
13#17984078—22no—
14#17984073—22no—
15#17984064C2212yes—
16#17984064C2212yes—
17C4×D548C2211yescharacteristic
18#17984046C2212yes—
19#17984045—22no—
20C4⋊C4×D137C2212yes—
21#17984022C2211yescharacteristic
22#17984022C2211yescharacteristic
23#17984021C2212yes—
24#17984020C2212yes—
25#17984019C2212yes—
26C4⋊#11764040C2211yescharacteristic
27#17984012C2212yes—
28#17984012C2212yes—
29#17984000C2212yes—
30C2×Q16×C137C2211yescharacteristic
31C4×Q8×C137C2212yes—
32#107×C137C2211yescharacteristic
33#103×C137C2212yes—
34C4×C8×C137C2211yescharacteristic
35C2×D548D411yescharacteristic
36C2×C4×D137—22no—
37C2×C4×D137C2312yes—
38#14555493—24no—
39C2×#11764040C2312yes—
40C4⋊Dic137D411yescharacteristic
41C4⋊Dic137C2312yes—
42C4⋊Dic137C2312yes—
43#14555489—22no—
44#14555489—22no—
45#14555489—22no—
46#14555489—22no—
47#14555489—22no—
48#14555487—22no—
49#14555487—22no—
50#14555487—22no—
51#14555487D412yes—
52C4×Dic137—22no—
53C2×Q8×C137C2312yes—
54Q16×C137—22no—
55C2×C8×C137C2311yescharacteristic
56C2×C8×C137C2311yescharacteristic
57C4⋊C4×C137—22no—
58C4⋊C4×C137C2312yes—
59C42×C137—22no—
60C42×C137C2311yescharacteristic
61C22×D137C2×D412yes—
62D548—41no—
63C2×Dic137—22no—
64C2×Dic137—22no—
65C2×Dic137C2×D412yes—
66C4×D137—22no—
67C4×D137—22no—
68C4×D137—44no—
69#11764040—22no—
70Q8×C137—22no—
71Q8×C137—22no—
72C2×C4×C137—22no—
73C2×C4×C137—22no—
74C2×C4×C137C2×D411yescharacteristic
75C2×C4×C137C2×D411yescharacteristic
76C2×C4×C137C2411yescharacteristic
77C8×C137—21no—
78C8×C137#4112yes—
79C22×C137C22×D411yescharacteristic
80C2×D137—22no—
81C2×D137—22no—
82C2×D137—22no—
83C4×C137—21no—
84C4×C137—21no—
85C4×C137—42no—
86C4×C137—42no—
87C4×C137C2×#4111yescharacteristic
88C4×C1372−1+411yescharacteristic
89Dic137—22no—
90Dic137—42no—
91Dic137—42no—
92C2×C137#54711yescharacteristic
93C2×C137#57511yescharacteristic
94C2×C137#57811yescharacteristic
95D137—42no—
96C137#335211yescharacteristic, complemented, Sylow
97#3352—1371nomaximal, Sylow
98#552—1372no—
99#499—1371no—
100#485—1372no—
101#484—1372no—
102#479—1372no—
103#470—1372no—
104C4×Q16D13711yescharacteristic, complemented
105#436—1371no—
106#428—1372no—
107C2×Q16C2×D13711yescharacteristic
108C2×M16—1372no—
109#126—2742no—
110#125—2742no—
111#125—1372no—
112#123—2742no—
113C22⋊Q8—1372no—
114C22⋊Q8—1372no—
115C4×Q8C2×D13712yes—
116C4×D4—1371no—
117#117—2742no—
118#117—1372no—
119C2×C4⋊C4—1372no—
120#107—1371no—
121#107—1371no—
122#107—1372no—
123#107C2×D13711yescharacteristic
124#106—1372no—
125#106—1372no—
126C4⋊C8—1371no—
127#103—1372no—
128#103—1372no—
129#103C2×D13712yes—
130C22⋊C8—1372no—
131C4×C8C2×D13711yescharacteristic
132C2×Q8C22×D13712yes—
133C2×D4—1371no—
134C22×C4—2742no—
135C22×C4—1372no—
136Q16—22no—
137M16—2744no—
138C2×C8—1372no—
139C2×C8C22×D13711yescharacteristic
140C2×C8C22×D13711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—2742no—
143C4⋊C4—2742no—
144C4⋊C4—2742no—
145C4⋊C4—2742no—
146C4⋊C4—2742no—
147C4⋊C4—1372no—
148C4⋊C4—1372no—
149C4⋊C4—1371no—
150C4⋊C4C22×D13712yes—
151C22⋊C4—2742no—
152C22⋊C4—2742no—
153C22⋊C4—2742no—
154C22⋊C4—1372no—
155C42—22no—
156C42—2742no—
157C42C22×D13711yescharacteristic
158C23—1372no—
159Q8—22no—
160Q8—22no—
161D4—5481no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—2742no—
165C2×C4—2742no—
166C2×C4—2742no—
167C2×C4—2742no—
168C2×C4—5484no—
169C2×C4—1372no—
170C2×C4D4×D13711yescharacteristic
171C2×C4D4×D13711yescharacteristic
172C2×C4C23×D13711yescharacteristic
173C8—21no—
174C8—2742no—
175C8#1455551012yes—
176C22—2742no—
177C22—2742no—
178C22—2742no—
179C22C2×D4×D13711yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—2742no—
185C4—5482no—
186C4—5482no—
187C4C2×#1455551011yescharacteristic
188C4#1798413511yescharacteristic
189C2—5482no—
190C2#2257805811yescharacteristic
191C2#2257820011yescharacteristic
192C2#2257821611yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×Q16×C137C2
socleC22×C137C22×D4
prime radicalC2×D548D4
hypercenterC4×Q16D137
hypocenterC137#3352
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #22577805C2semidirect
3. #22577743C2semidirect
4. #22577737C2semidirect
5. #22577710C2semidirect
6. #22577658C2semidirect
8. #22577444C2semidirect
9. C4×Q16×C137C2semidirect
96. C137#3352semidirect
104. C4×Q16D137semidirect

Maximal subgroups

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

subgroupindex
#225780942
#225778052
#225777432
#225777372
#225777102
#225776582
#225774822
#225774442
C4×Q16×C1372
#3352137