Finite Group Databasebeta

← #31779941

Subgroups — #31779941

Order 17536.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#22578056C211yesmaximal, characteristic
2#22577755C211yesmaximal, characteristic, complemented
3#22577694C211yesmaximal, characteristic, complemented
4#22577694C211yesmaximal, characteristic, complemented
5#22577130C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#332×C137C211yesmaximal, characteristic, complemented
8C2×#14555511—21no—
9#17984088C2211yescharacteristic
10#17984073—22no—
11#17984064—21no—
12#17984063—21no—
13C4×D548C2211yescharacteristic
14C4×D548C2211yescharacteristic
15C4⋊C4×D137—21no—
16#17984020—21no—
17#17984020—21no—
18#17984011—22no—
19#17984010—21no—
20#17984010—21no—
21C4×#11764040C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555511—42no—
26#14555511—42no—
27C2×Dic274—21no—
28C2×D548—22no—
29C2×D548D411yescharacteristic
30C2×D548D411yescharacteristic
31C2×C4×D137—22no—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×#11764040—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137D411yescharacteristic
38C4⋊Dic137D411yescharacteristic
39#14555489—22no—
40#14555487—22no—
41#14555487—21no—
42#14555487—21no—
43C2×Q8×C137—21no—
44C2×C8×C137—21no—
45C2×C8×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137D412yes—
48C42×C137C2311yescharacteristic
49C22×D137—21no—
50C22×D137—21no—
51Dic274—42no—
52D548—21no—
53D548—21no—
54D548—21no—
55D548—44no—
56D548SD1612yes—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137—41no—
60C4×D137—44no—
61C4×D137—41no—
62C4×D137—41no—
63C4×D137—41no—
64C4×D137—41no—
65#11764040—41no—
66Q8×C137—41no—
67Q8×C137—41no—
68C2×C4×C137—22no—
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—41no—
73C8×C137—41no—
74C22×C137#12011yescharacteristic
75C2×D137—22no—
76C2×D137—41no—
77C2×D137—41no—
78C2×D137—41no—
79C2×D137—41no—
80C2×D137—41no—
81C4×C137—21no—
82C4×C137—42no—
83C4×C137C2×SD1611yescharacteristic
84C4×C137#13511yescharacteristic
85C4×C137AGL(1,ℤ/8)11yescharacteristic
86C4×C137#13711yescharacteristic
87Dic137—41no—
88Dic137—41no—
89Dic137—81no—
90C2×C137#45011yescharacteristic
91C2×C137#45211yescharacteristic
92C2×C137#45411yescharacteristic
93D137—42no—
94D137—81no—
95C137#163511yescharacteristic, complemented, Sylow
96#1635—1371nomaximal, Sylow
97#547—1371no—
98#487—1371no—
99#476—1371no—
100#476—1371no—
101#332D13711yescharacteristic, complemented
102D4⋊C8—1371no—
103D4⋊C8—1371no—
104C2×#41—2741no—
105C4⋊Q8—1371no—
106C4⋊Q8C2×D13711yescharacteristic
107#123—2742no—
108C22⋊Q8—2741no—
109#121—2741no—
110C4×D4—1371no—
111C4×D4—1371no—
112C2×C4⋊C4—2741no—
113#106—2741no—
114#106—2741no—
115C4⋊C8C2×D13711yescharacteristic
116C4⋊C8C2×D13711yescharacteristic
117#103—2742no—
118#102—2741no—
119#102—2741no—
120C4×C8—1371no—
121#41—5482no—
122#41—5482no—
123C2×Q8—21no—
124C2×Q8—2741no—
125C2×D4—2742no—
126C2×D4—1371no—
127C2×D4—1371no—
128C22×C4—2742no—
129C22×C4—2741no—
130C22×C4—2741no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2741no—
134C4⋊C4—21no—
135C4⋊C4—2742no—
136C4⋊C4—2741no—
137C4⋊C4—2741no—
138C4⋊C4—1371no—
139C4⋊C4—1371no—
140C4⋊C4D54812yes—
141C22⋊C4—2742no—
142C22⋊C4—2741no—
143C22⋊C4—2741no—
144C42C22×D13711yescharacteristic
145C23—2741no—
146C23—2741no—
147Q8—41no—
148Q8—41no—
149Q8—5482no—
150D4—2741no—
151D4—2741no—
152D4—2741no—
153D4—5484no—
154D4—1372no—
155C2×C4—22no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—5484no—
159C2×C4—5481no—
160C2×C4—5481no—
161C2×C4—5481no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4D4×D13711yescharacteristic
165C2×C4D4×D13711yescharacteristic
166C2×C4C2×D54811yescharacteristic
167C8—41no—
168C8—41no—
169C8—5481no—
170C22—2742no—
171C22—5481no—
172C22—5481no—
173C22—5481no—
174C22—5481no—
175C22—5481no—
176C22#1798405811yescharacteristic
177C4—21no—
178C4—42no—
179C4—5481no—
180C4—5481no—
181C4—10961no—
182C4SD16×D13711yescharacteristic
183C4#1798410911yescharacteristic
184C4#1798411211yescharacteristic
185C4#1798412011yescharacteristic
186C2—5482no—
187C2—10961no—
188C2#2257755511yescharacteristic
189C2#2257756811yescharacteristic
190C2#2257757811yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#332×C137C2
socleC22×C137#120
prime radical#22578056C2
hypercenter#332D137
hypocenterC137#1635
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22577755C2semidirect
3. #22577694C2semidirect
4. #22577694C2semidirect
5. #22577130C2semidirect
6. #22577130C2semidirect
7. #332×C137C2semidirect
95. C137#1635semidirect
101. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225780562
#225777552
#225776942
#225771302
#332×C1372
#1635137