Finite Group Databasebeta

← #31793396

Subgroups — #31793396

Order 17536.

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

normal subgroups only

This group has 360 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#22578103C212yesmaximal, complemented
2#22577805C211yesmaximal, characteristic, complemented
3#22577725C212yesmaximal, complemented
4#22577715C212yesmaximal, complemented
5#22577703C212yesmaximal, complemented
6#22577653C212yesmaximal, complemented
7#22577482C211yesmaximal, characteristic, complemented
8#22577444C212yesmaximal, complemented
9C4×D8×C137C211yesmaximal, characteristic, complemented
10C2×#14555493C2212yes—
11#17984081—22no—
12#17984080C2212yes—
13#17984071—22no—
14#17984061—22no—
15#17984059C2212yes—
16#17984059C2212yes—
17C4×D548C2211yescharacteristic
18C4×C22⋊D137—22no—
19#17984046C2212yes—
20C4⋊C4×D137C2212yes—
21#17984022C2211yescharacteristic
22#17984022C2211yescharacteristic
23#17984021C2212yes—
24#17984020C2212yes—
25#17984019C2212yes—
26C4⋊#11764040C2211yescharacteristic
27#17984009C2212yes—
28#17984009C2212yes—
29#17984000C2212yes—
30C2×D8×C137C2211yescharacteristic
31C4×D4×C137C2212yes—
32#107×C137C2211yescharacteristic
33#102×C137C2212yes—
34C4×C8×C137C2211yescharacteristic
35C2×D548D411yescharacteristic
36C2×C22⋊D137—22no—
37C2×C22⋊D137—22no—
38C2×C4×D137—22no—
39C2×C4×D137C2312yes—
40#14555493—24no—
41C2×#11764040C2312yes—
42C4⋊Dic137D411yescharacteristic
43C4⋊Dic137C2312yes—
44C4⋊Dic137C2312yes—
45#14555489—22no—
46#14555489—22no—
47#14555489—22no—
48C22⋊Dic137—22no—
49C22⋊Dic137—22no—
50#14555487—22no—
51#14555487D412yes—
52C4×Dic137—22no—
53C2×D4×C137C2312yes—
54C22×C4×C137—22no—
55D8×C137—22no—
56C2×C8×C137C2311yescharacteristic
57C2×C8×C137C2311yescharacteristic
58C4⋊C4×C137C2312yes—
59C22⋊C4×C137—22no—
60C42×C137C2311yescharacteristic
61C22×D137C2×D412yes—
62D548—41no—
63C22⋊D137—44no—
64C22⋊D137—82no—
65C2×Dic137—22no—
66C2×Dic137—22no—
67C2×Dic137C2×D412yes—
68C4×D137—22no—
69C4×D137—22no—
70C4×D137—82no—
71#11764040—22no—
72C23×C137—22no—
73D4×C137—22no—
74D4×C137—22no—
75C2×C4×C137—22no—
76C2×C4×C137—82no—
77C2×C4×C137C2×D411yescharacteristic
78C2×C4×C137C2×D411yescharacteristic
79C2×C4×C137C2411yescharacteristic
80C8×C137—21no—
81C8×C137#4112yes—
82C22×C137—42no—
83C22×C137—42no—
84C22×C137—42no—
85C22×C137C22×D411yescharacteristic
86C2×D137—22no—
87C2×D137—22no—
88C2×D137—22no—
89C4×C137—21no—
90C4×C137—21no—
91C4×C137—42no—
92C4×C137C2×#4111yescharacteristic
93C4×C1372−1+411yescharacteristic
94Dic137—22no—
95Dic137—42no—
96Dic137—42no—
97C2×C137—82no—
98C2×C137#54711yescharacteristic
99C2×C137#57511yescharacteristic
100C2×C137#57611yescharacteristic
101D137—42no—
102C137#335511yescharacteristic, complemented, Sylow
103#3355—1371nomaximal, Sylow
104#553—1372no—
105#499—1371no—
106#482—1372no—
107#480—1372no—
108#478—1372no—
109#469—1372no—
110C4×D8D13711yescharacteristic, complemented
111#436—1371no—
112#428—1372no—
113C2×D8C2×D13711yescharacteristic
114C2×M16—1372no—
115#126—2742no—
116#125—1372no—
117#123—2742no—
118#121—2742no—
119#121—1372no—
120#121—1372no—
121C4×D4—2742no—
122C4×D4—1371no—
123C4×D4C2×D13712yes—
124#117—1372no—
125C2×C4⋊C4—1372no—
126#107—1371no—
127#107—1371no—
128#107—1372no—
129#107C2×D13711yescharacteristic
130#106—1372no—
131#106—1372no—
132C4⋊C8—1371no—
133#102—1372no—
134#102—1372no—
135#102C2×D13712yes—
136C22⋊C8—1372no—
137C4×C8C2×D13711yescharacteristic
138C2×D4—2742no—
139C2×D4—2742no—
140C2×D4—1371no—
141C2×D4C22×D13712yes—
142C22×C4—22no—
143C22×C4—2742no—
144C22×C4—1372no—
145D8—22no—
146M16—2744no—
147C2×C8—1372no—
148C2×C8C22×D13711yescharacteristic
149C2×C8C22×D13711yescharacteristic
150C4⋊C4—2742no—
151C4⋊C4—2742no—
152C4⋊C4—2742no—
153C4⋊C4—1372no—
154C4⋊C4—1372no—
155C4⋊C4—1371no—
156C4⋊C4C22×D13712yes—
157C22⋊C4—22no—
158C22⋊C4—2742no—
159C22⋊C4—2742no—
160C22⋊C4—2742no—
161C22⋊C4—1372no—
162C42—2742no—
163C42C22×D13711yescharacteristic
164C23—22no—
165C23—1372no—
166D4—22no—
167D4—22no—
168D4—5481no—
169D4—5484no—
170D4—10962no—
171C2×C4—22no—
172C2×C4—82no—
173C2×C4—2742no—
174C2×C4—2742no—
175C2×C4—2742no—
176C2×C4—2742no—
177C2×C4—10962no—
178C2×C4—1372no—
179C2×C4D4×D13711yescharacteristic
180C2×C4D4×D13711yescharacteristic
181C2×C4C23×D13711yescharacteristic
182C8—21no—
183C8—2742no—
184C8#1455551012yes—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—2742no—
189C22—2742no—
190C22—2742no—
191C22C2×D4×D13711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—2742no—
196C4—5482no—
197C4—5482no—
198C4C2×#1455551011yescharacteristic
199C4#1798413511yescharacteristic
200C2—82no—
201C2—5482no—
202C2#2257805811yescharacteristic
203C2#2257820011yescharacteristic
204C2#2257820411yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×D8×C137C2
socleC22×C137C22×D4
hypercenterC4×D8D137
hypocenterC137#3355
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22578103C2semidirect
2. #22577805C2semidirect
3. #22577725C2semidirect
4. #22577715C2semidirect
5. #22577703C2semidirect
6. #22577653C2semidirect
7. #22577482C2semidirect
8. #22577444C2semidirect
9. C4×D8×C137C2semidirect
102. C137#3355semidirect
110. C4×D8D137semidirect

Maximal subgroups

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

subgroupindex
#225781032
#225778052
#225777252
#225777152
#225777032
#225776532
#225774822
#225774442
C4×D8×C1372
#3355137