Finite Group Databasebeta

← #31796144

Subgroups — #31796144

Order 17536.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578419C211yesmaximal, characteristic, complemented
2#22578406C211yesmaximal, characteristic, complemented
3#22578253C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5#22577934C211yesmaximal, characteristic
6C2×#14555519—21no—
7#17984155—21no—
8C4⋊#11764047—21no—
9C4⋊#11764047C2212yes—
10C22⋊#11764047—21no—
11C4×#11764047C2212yes—
12#17984081—21no—
13#17984079C411yescharacteristic
14#17984075C411yescharacteristic
15#17984069—21no—
16#17984067C411yescharacteristic
17#17984053C2211yescharacteristic
18D4×Dic137—21no—
19D4×Dic137C2211yescharacteristic
20#17984044—21no—
21C2×C4×Dic137C2211yescharacteristic
22#121×C137C411yescharacteristic
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×Dic274C2×C411yescharacteristic
27C22×Dic137—21no—
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137C2×C411yescharacteristic
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35C22⋊Dic137—21no—
36C22⋊Dic137—21no—
37C22⋊Dic137—21no—
38C22⋊Dic137—21no—
39C4×Dic137—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C2×D4×C137—21no—
45C2×D4×C137C2×C411yescharacteristic
46C22×C4×C137C2×C411yescharacteristic
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49#11764047—42no—
50#11764047—42no—
51Dic274—22no—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C2×Dic137—21no—
56C2×Dic137—41no—
57C2×Dic137—41no—
58C2×Dic137—41no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C2×Dic137C2×D411yescharacteristic
63C23×C137—21no—
64C23×C137C22⋊C411yescharacteristic
65D4×C137—22no—
66D4×C137—42no—
67C2×C4×C137—22no—
68C2×C4×C137—21no—
69C2×C4×C137C22⋊C411yescharacteristic
70C2×C4×C137C22×C411yescharacteristic
71C22×C137—21no—
72C22×C137—21no—
73C22×C137—21no—
74C22×C137—41no—
75C22×C137—41no—
76C22×C137—41no—
77C22×C137C2×C22⋊C411yescharacteristic
78C4×C137—21no—
79C4×C137—41no—
80C4×C137#13112yes—
81Dic137—21no—
82Dic137—21no—
83Dic137—21no—
84Dic137—41no—
85Dic137—41no—
86Dic137AGL(1,ℤ/8)11yescharacteristic
87Dic137#13711yescharacteristic
88C2×C137—41no—
89C2×C137—81no—
90C2×C137#40811yescharacteristic
91C2×C137#41711yescharacteristic
92C2×C137#42111yescharacteristic
93C137#149511yescharacteristic, complemented, Sylow
94#1495—1371nomaximal, Sylow
95#529—1371no—
96#431—1371no—
97#423—1371no—
98Q8⋊C8—1372no—
99D4⋊C8—1372no—
100C2×M16—2741no—
101#126—2741no—
102#125—1371no—
103#124—1371no—
104#123—2741no—
105C22⋊Q8—1371no—
106#121#1058773911yescharacteristic
107C4×Q8—1371no—
108C4×D4—2741no—
109C4×D4—1371no—
110#117—2741no—
111C2×C42—1371no—
112C4⋊C8—2741no—
113C4⋊C8—2741no—
114C4⋊C8—1372no—
115C22⋊C8—2741no—
116C4×C8—1372no—
117C2×Q8—1371no—
118C2×D4—21no—
119C2×D4C2×#1058773911yescharacteristic
120C22×C4—2741no—
121C22×C4—1371no—
122C22×C4—1371no—
123C22×C4C2×#1058773911yescharacteristic
124M16—5482no—
125C2×C8—2742no—
126C2×C8—2742no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C4⋊C4—1371no—
132C4⋊C4C2×#1058773911yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—2741no—
135C22⋊C4—2741no—
136C22⋊C4—2741no—
137C22⋊C4—2741no—
138C42—2741no—
139C42—1371no—
140C42—1371no—
141C42—1371no—
142C42—1371no—
143C23—21no—
144C23C22⋊#1058773911yescharacteristic
145Q8—2742no—
146D4—22no—
147D4—42no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—2741no—
151C2×C4—2741no—
152C2×C4—2741no—
153C2×C4—2741no—
154C2×C4—5481no—
155C2×C4—5481no—
156C2×C4—5481no—
157C2×C4—1371no—
158C2×C4—1371no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4C22⋊#1058773911yescharacteristic
162C2×C4C22×#1058773911yescharacteristic
163C8—5482no—
164C8—5482no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊#1058773911yescharacteristic
172C4—21no—
173C4—41no—
174C4—2741no—
175C4—2741no—
176C4—2741no—
177C4—5481no—
178C4—5481no—
179C4—1371no—
180C4—1371no—
181C4#1798417512yes—
182C2—41no—
183C2—81no—
184C2#2257837611yescharacteristic
185C2#2257839411yescharacteristic
186C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting, prime radical#121×C137C4
socleC22×C137C2×C22⋊C4
cosocleC4×Dic137C23
hypercenter#121#10587739
hypocenterC137#1495
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22578419C2semidirect
2. #22578406C2semidirect
3. #22578253C2semidirect
4. D4⋊#11764047C2semidirect
93. C137#1495semidirect

Maximal subgroups

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

subgroupindex
#225784192
#225784062
#225782532
D4⋊#117640472
#225779342
#1495137