Finite Group Databasebeta

← #31780102

Subgroups — #31780102

Order 17536.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#22578157C211yesmaximal, characteristic
2#22577813C211yesmaximal, characteristic, complemented
3#22577812C211yesmaximal, characteristic, complemented
4#22577797C211yesmaximal, characteristic
5#22577144C211yesmaximal, characteristic, complemented
6#22577142C211yesmaximal, characteristic
7#330×C137C211yesmaximal, characteristic, complemented
8C2×Dic548—21no—
9C2×#14555499—21no—
10#17984088C2211yescharacteristic
11#17984087—21no—
12#17984083—21no—
13#17984083—21no—
14#17984080C2211yescharacteristic
15#17984076C2211yescharacteristic
16#17984073—21no—
17#17984068—21no—
18#17984066—21no—
19#17984022—21no—
20#17984021—21no—
21#17984020—21no—
22#17984019—21no—
23#17983999C2211yescharacteristic
24#125×C137C2211yescharacteristic
25#97×C137C2211yescharacteristic
26#97×C137C2211yescharacteristic
27C2×Dic274—21no—
28C2×Dic274—21no—
29C2×Dic274D411yescharacteristic
30C2×D548D411yescharacteristic
31C2×C4×D137—21no—
32Dic548—42no—
33#14555499—42no—
34C2×#11764040—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137D411yescharacteristic
39C4⋊Dic137D411yescharacteristic
40#14555489—21no—
41#14555489—21no—
42#14555489—21no—
43#14555487—21no—
44#14555487—21no—
45#14555487—21no—
46#14555487—21no—
47#14555487—21no—
48C4×Dic137—21no—
49C2×C8×C137—21no—
50C2×C8×C137—21no—
51C4⋊C4×C137—21no—
52C4⋊C4×C137—21no—
53C4⋊C4×C137D411yescharacteristic
54C4⋊C4×C137D411yescharacteristic
55C42×C137C2311yescharacteristic
56C22×D137—21no—
57Dic274—41no—
58Dic274—41no—
59Dic274—41no—
60Dic274—81no—
61D548—41no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—41no—
66C4×D137—81no—
67#11764040—22no—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—22no—
74C8×C137—22no—
75C22×C137#12011yescharacteristic
76C2×D137—41no—
77C2×D137—41no—
78C2×D137—41no—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—41no—
83C4×C137—41no—
84Dic137—41no—
85Dic137—41no—
86Dic137—41no—
87Dic137—81no—
88C2×C137#45311yescharacteristic
89C2×C137#45611yescharacteristic
90C2×C137#45611yescharacteristic
91D137—81no—
92C137#166011yescharacteristic, complemented, Sylow
93#1660—1371nomaximal, Sylow
94#563—1371no—
95#501—1371no—
96#501—1371no—
97#497—1371no—
98#330—1371no—
99#330D13711yescharacteristic, complemented
100#329—1371no—
101C2×Q16—2741no—
102C2×SD16—2741no—
103C4⋊Q8—2741no—
104C4⋊Q8—1371no—
105#126—2741no—
106#126—2741no—
107#125—1371no—
108#125C2×D13711yescharacteristic
109#124—1371no—
110#123—2741no—
111C22⋊Q8—2741no—
112C22⋊Q8—2741no—
113#107—2741no—
114#107—2741no—
115#106—2741no—
116#106—2741no—
117#97—1371no—
118#97C2×D13711yescharacteristic
119#97C2×D13711yescharacteristic
120C2×Q8—2741no—
121C2×Q8—2741no—
122C2×Q8—1371no—
123C2×D4—1371no—
124C22×C4—2741no—
125Q16—5482no—
126SD16—5482no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—2741no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—2741no—
133C4⋊C4—2741no—
134C4⋊C4—2741no—
135C4⋊C4—2741no—
136C4⋊C4—2741no—
137C4⋊C4—2741no—
138C4⋊C4—1371no—
139C4⋊C4—1371no—
140C4⋊C4D54811yescharacteristic
141C4⋊C4D54811yescharacteristic
142C22⋊C4—2741no—
143C22⋊C4—2741no—
144C22⋊C4—2741no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C42—2741no—
148C42C22×D13711yescharacteristic
149C23—2741no—
150Q8—5481no—
151Q8—5481no—
152Q8—5481no—
153Q8—10961no—
154D4—5481no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—5481no—
161C2×C4—10961no—
162C2×C4D4×D13711yescharacteristic
163C2×C4D4×D13711yescharacteristic
164C2×C4C2×D54811yescharacteristic
165C8—22no—
166C8—22no—
167C8—2742no—
168C22—5481no—
169C22—5481no—
170C22—5481no—
171C22#1798405811yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—5481no—
178C4—5481no—
179C4—5481no—
180C4—10961no—
181C2—10961no—
182C2#2257757211yescharacteristic
183C2#2257758711yescharacteristic
184C2#2257758811yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#330×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#330D137
hypocenterC137#1660
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #22577813C2semidirect
3. #22577812C2semidirect
5. #22577144C2semidirect
7. #330×C137C2semidirect
92. C137#1660semidirect
99. #330D137semidirect

Maximal subgroups

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

subgroupindex
#225781572
#225778132
#225778122
#225777972
#225771442
#225771422
#330×C1372
#1660137