Finite Group Databasebeta

← #31782401

Subgroups — #31782401

Order 17536.

228 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 79 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×#17984094C211yesmaximal, characteristic
2C2×#17984027C211yesmaximal, characteristic, complemented
3C2×C16×D137C211yesmaximal, characteristic, complemented
4C22⋊#14555485C211yesmaximal, characteristic, complemented
5#22577182C211yesmaximal, characteristic, complemented
6#22577182C211yesmaximal, characteristic, complemented
7C22⋊C16×C137C211yesmaximal, characteristic, complemented
8C2×#14555510C411yescharacteristic
9#17984094—22no—
10#17984094—22no—
11C2×#14555494C411yescharacteristic
12C2×#14555493C411yescharacteristic
13C2×#14555493C411yescharacteristic
14C2×C8×D137C2211yescharacteristic
15C2×C8×D137C2211yescharacteristic
16#17984027—22no—
17C2×#14555485C2211yescharacteristic
18C2×#14555485C2211yescharacteristic
19C16×D137—22no—
20C22×C8×C137C2211yescharacteristic
21C2×C16×C137C2211yescharacteristic
22C2×C16×C137C2211yescharacteristic
23#14555510—22no—
24#14555510—22no—
25C2×Dic274C811yescharacteristic
26C2×D548C811yescharacteristic
27C2×C22⋊D137C811yescharacteristic
28C2×C22⋊D137C811yescharacteristic
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137C2×C411yescharacteristic
31#14555494—22no—
32#14555493—22no—
33#14555493—22no—
34C2×#11764040C2×C411yescharacteristic
35C2×#11764040C2×C411yescharacteristic
36C8×D137—21no—
37C8×D137—21no—
38C8×D137D412yes—
39C8×D137D412yes—
40#14555485—21no—
41#14555485—21no—
42C22×C4×C137C2×C411yescharacteristic
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C2×C8×C137C2×C411yescharacteristic
46C2×C8×C137C2311yescharacteristic
47C16×C137—21no—
48C16×C137—21no—
49C22×D137C2×C811yescharacteristic
50C22×D137C2×C811yescharacteristic
51Dic274—22no—
52D548—22no—
53C22⋊D137—22no—
54C22⋊D137—22no—
55C2×Dic137C2×C811yescharacteristic
56C2×Dic137C2×C811yescharacteristic
57C4×D137—21no—
58C4×D137—21no—
59C4×D137C22⋊C412yes—
60C4×D137M1612yes—
61#11764040—21no—
62#11764040C22⋊C412yes—
63C23×C137C2×C811yescharacteristic
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137C2×C811yescharacteristic
67C2×C4×C137C22×C411yescharacteristic
68C8×C137—21no—
69C8×C137C2×D411yescharacteristic
70C8×C137C2×D411yescharacteristic
71C22×C137—21no—
72C22×C137—21no—
73C22×C137—21no—
74C22×C137C22×C811yescharacteristic
75C2×D137—21no—
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—21no—
80C2×D137C22⋊C812yes—
81C4×C137—21no—
82C4×C137C2×C22⋊C411yescharacteristic
83C4×C137C2×M1611yescharacteristic
84Dic137—21no—
85Dic137C22⋊C812yes—
86C2×C137—41no—
87C2×C137C2×C22⋊C811yescharacteristic
88C2×C137#50411yescharacteristic
89C2×C137#50411yescharacteristic
90D137—22no—
91D137—41no—
92C137#211311yescharacteristic, complemented, Sylow
93#2113—1371nomaximal, Sylow
94C2×#131—1371no—
95C2×M32—1371no—
96C22×C16—1371no—
97C22⋊C16—1371no—
98C22⋊C16—1371no—
99C22⋊C16—1371no—
100C22⋊C16D13711yescharacteristic, complemented
101C2×#41—1371no—
102#131—2742no—
103#131—2742no—
104C2×M16—1371no—
105C2×M16—1371no—
106C2×M16—1371no—
107C22×C8—1371no—
108C22×C8—1371no—
109C22×C8C2×D13711yescharacteristic
110M32—2742no—
111C2×C16—2742no—
112C2×C16—1371no—
113C2×C16—1371no—
114C2×C16C2×D13711yescharacteristic
115C2×C16C2×D13711yescharacteristic
116#41—2742no—
117#41—2742no—
118C2×Q8—1371no—
119C2×D4—1371no—
120C2×D4—1371no—
121C2×D4—1371no—
122C22×C4—1371no—
123C22×C4—1371no—
124C22×C4C4×D13711yescharacteristic
125M16—2742no—
126M16—2742no—
127M16—2742no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2741no—
131C2×C8—2741no—
132C2×C8—1371no—
133C2×C8—1371no—
134C2×C8—1372no—
135C2×C8—1372no—
136C2×C8C4×D13711yescharacteristic
137C2×C8C22×D13711yescharacteristic
138C16—21no—
139C16—21no—
140C16—2741no—
141C16—2741no—
142C23—1371no—
143C23—1371no—
144C23C8×D13711yescharacteristic
145Q8—2742no—
146D4—2742no—
147D4—2742no—
148D4—2742no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—2741no—
152C2×C4—2741no—
153C2×C4—1371no—
154C2×C4—1371no—
155C2×C4—1372no—
156C2×C4—1372no—
157C2×C4C8×D13711yescharacteristic
158C2×C4C2×C4×D13711yescharacteristic
159C8—21no—
160C8—2741no—
161C8—1372no—
162C8D4×D13711yescharacteristic
163C8D4×D13711yescharacteristic
164C22—21no—
165C22—21no—
166C22—21no—
167C22—2741no—
168C22—2741no—
169C22—2741no—
170C22—2741no—
171C22—2741no—
172C22—1372no—
173C22C2×C8×D13711yescharacteristic
174C4—21no—
175C4—2741no—
176C4—1372no—
177C4C22⋊C4×D13711yescharacteristic
178C4M16×D13711yescharacteristic
179C2—41no—
180C2—2742no—
181C2—5481no—
182C2C22⋊C8×D13711yescharacteristic
183C2#2257781911yescharacteristic
184C2#2257782111yescharacteristic
subgroupG/H
center, FrattiniC2×C8C22×D137
commutator, socleC22×C137C22×C8
FittingC22⋊C16×C137C2
prime radicalC2×#14555510C4
cosocleC2×C8×C137C23
hypercenterC22⋊C16D137
hypocenterC137#2113
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C2×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#17984027C2semidirect
3. C2×C16×D137C2semidirect
4. C22⋊#14555485C2semidirect
5. #22577182C2semidirect
6. #22577182C2semidirect
7. C22⋊C16×C137C2semidirect
92. C137#2113semidirect
100. C22⋊C16D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840942
C2×#179840272
C2×C16×D1372
C22⋊#145554852
#225771822
C22⋊C16×C1372
#2113137