Finite Group Databasebeta

← #31780114

Subgroups — #31780114

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#22578147C211yesmaximal, characteristic
2#22577772C211yesmaximal, characteristic, complemented
3#22577763C211yesmaximal, characteristic, complemented
4#22577762C211yesmaximal, characteristic, complemented
5#22577144C211yesmaximal, characteristic, complemented
6#22577142C211yesmaximal, characteristic, complemented
7#329×C137C211yesmaximal, characteristic, complemented
8#17984088C2211yescharacteristic
9#17984080C2211yescharacteristic
10#17984076C2211yescharacteristic
11#17984072—21no—
12#17984070—21no—
13#17984065—21no—
14#17984064—21no—
15#17984061—21no—
16#17984060—21no—
17#17984013—21no—
18#17984013—21no—
19#17984013—21no—
20#17984012—21no—
21#17984010—21no—
22#17984009—21no—
23#17983999C2211yescharacteristic
24#124×C137C2211yescharacteristic
25#97×C137C2211yescharacteristic
26#97×C137C2211yescharacteristic
27C2×Dic274—21no—
28C2×Dic274D411yescharacteristic
29C2×D548D411yescharacteristic
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137—21no—
33C2×C4×D137—21no—
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—
43C22⋊Dic137—21no—
44#14555487—21no—
45#14555487—21no—
46#14555487—21no—
47C2×Q8×C137D411yescharacteristic
48C2×D4×C137D411yescharacteristic
49C2×C8×C137—21no—
50C2×C8×C137—21no—
51C22⋊C4×C137—21no—
52C22⋊C4×C137—21no—
53C42×C137C2311yescharacteristic
54C22×D137—21no—
55Dic274—41no—
56Dic274—42no—
57D548—41no—
58C22⋊D137—81no—
59C22⋊D137—81no—
60C2×Dic137—21no—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—41no—
64C2×Dic137—81no—
65C4×D137—81no—
66#11764040—41no—
67C23×C137—21no—
68Q8×C137—41no—
69D4×C137—41no—
70C2×C4×C137—21no—
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C2×C4×C137C2×D411yescharacteristic
74C8×C137—41no—
75C8×C137—41no—
76C22×C137—41no—
77C22×C137—41no—
78C22×C137—41no—
79C22×C137#12011yescharacteristic
80C2×D137—41no—
81C2×D137—41no—
82C2×D137—41no—
83C4×C137—21no—
84C4×C137—21no—
85C4×C137—21no—
86C4×C137—41no—
87Dic137—41no—
88Dic137—41no—
89Dic137—41no—
90Dic137—81no—
91C2×C137—81no—
92C2×C137#45411yescharacteristic
93C2×C137#45511yescharacteristic
94C2×C137#45511yescharacteristic
95D137—81no—
96C137#166211yescharacteristic, complemented, Sylow
97#1662—1371nomaximal, Sylow
98#562—1371no—
99#491—1371no—
100#489—1371no—
101#489—1371no—
102#330—1371no—
103#329—1371no—
104#329D13711yescharacteristic, complemented
105C4⋊Q8—1371no—
106#125—1371no—
107#124—1371no—
108#124C2×D13711yescharacteristic
109#123—2741no—
110#123—2741no—
111C22⋊Q8—2741no—
112C22⋊Q8—2741no—
113#121—2741no—
114#121—2741no—
115#103—2741no—
116#103—2741no—
117#103—2741no—
118#103—2741no—
119#102—2741no—
120#102—2741no—
121#97—1371no—
122#97C2×D13711yescharacteristic
123#97C2×D13711yescharacteristic
124C2×Q8—2741no—
125C2×Q8—1371no—
126C2×Q8D54811yescharacteristic
127C2×D4—2741no—
128C2×D4—2741no—
129C2×D4—1371no—
130C2×D4D54811yescharacteristic
131C22×C4—2741no—
132C22×C4—2741no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2741no—
136C4⋊C4—2741no—
137C4⋊C4—2741no—
138C4⋊C4—2741no—
139C4⋊C4—2741no—
140C4⋊C4—2741no—
141C4⋊C4—2741no—
142C4⋊C4—1371no—
143C4⋊C4—1371no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C22⋊C4—2741no—
149C22⋊C4—2741no—
150C42C22×D13711yescharacteristic
151C23—21no—
152C23—2741no—
153Q8—41no—
154Q8—5481no—
155Q8—5482no—
156D4—41no—
157D4—5481no—
158D4—10961no—
159D4—10961no—
160C2×C4—21no—
161C2×C4—2741no—
162C2×C4—2741no—
163C2×C4—2741no—
164C2×C4—5481no—
165C2×C4—10961no—
166C2×C4—10961no—
167C2×C4D4×D13711yescharacteristic
168C2×C4D4×D13711yescharacteristic
169C2×C4C2×D54811yescharacteristic
170C8—41no—
171C8—41no—
172C8—5481no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—5481no—
177C22—5481no—
178C22—5481no—
179C22#1798405811yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—5481no—
185C4—5481no—
186C4—5481no—
187C4—10961no—
188C2—81no—
189C2—10961no—
190C2#2257757711yescharacteristic
191C2#2257758211yescharacteristic
192C2#2257758311yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#329×C137C2
socleC22×C137#120
prime radical#22578147C2
hypercenter#329D137
hypocenterC137#1662
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22577772C2semidirect
3. #22577763C2semidirect
4. #22577762C2semidirect
5. #22577144C2semidirect
6. #22577142C2semidirect
7. #329×C137C2semidirect
96. C137#1662semidirect
104. #329D137semidirect

Maximal subgroups

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

subgroupindex
#225781472
#225777722
#225777632
#225777622
#225771442
#225771422
#329×C1372
#1662137