Finite Group Databasebeta

← #31796215

Subgroups — #31796215

Order 17536.

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

normal subgroups only

This group has 284 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
1C2×#17984145C211yesmaximal, characteristic, complemented
2#22578346C211yesmaximal, characteristic, complemented
3#22578257C212yesmaximal, complemented
4#22578254C212yesmaximal, complemented
5#22577910C211yesmaximal, characteristic
6C22×#11764047—21no—
7C2×C22⋊#10587739—21no—
8#17984145—21no—
9#17984145C2212yes—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12C22×C4×D137C2211yescharacteristic
13#17984069—21no—
14#17984064C411yescharacteristic, complemented
15#17984064C2211yescharacteristic
16#17984059C411yescharacteristic, complemented
17#17984059C2211yescharacteristic
18#17984056—21no—
19C4×C22⋊D137—21no—
20C4×C22⋊D137—21no—
21#17984047C411yescharacteristic, complemented
22C2×#41×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26C22⋊#10587739—42no—
27C23×D137D411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C22⋊D137—21no—
30C22×Dic137D411yescharacteristic
31C2×C4×D137—42no—
32C2×C4×D137C2×C411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137C2311yescharacteristic
36C4⋊Dic137C2×C411yescharacteristic
37C4⋊Dic137C2×C411yescharacteristic
38#14555489—21no—
39#14555489—21no—
40C22⋊Dic137—21no—
41C22⋊Dic137—21no—
42C22⋊Dic137—21no—
43#14555487—21no—
44#14555487—21no—
45C4×Dic137—21no—
46#41×C137—42no—
47C2×Q8×C137C2×C411yescharacteristic
48C2×D4×C137—21no—
49C2×D4×C137C2×C411yescharacteristic
50C22×C4×C137—21no—
51C22×C4×C137C2×C411yescharacteristic
52C2×#10587739—42no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55#11764047—42no—
56C22×D137—22no—
57C22×D137—22no—
58C22×D137—41no—
59C22×D137C22⋊C411yescharacteristic
60C22×D137C2×D411yescharacteristic
61C22⋊D137—42no—
62C22⋊D137—42no—
63C2×Dic137—22no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C4×D137—41no—
69C4×D137—41no—
70C4×D137—41no—
71C4×D137—41no—
72C23×C137—21no—
73C23×C137C22⋊C411yescharacteristic
74Q8×C137—41no—
75D4×C137—41no—
76D4×C137—42no—
77C2×C4×C137—22no—
78C2×C4×C137—21no—
79C2×C4×C137—42no—
80C2×C4×C137C22⋊C411yescharacteristic
81C2×C4×C137C22×C411yescharacteristic
82C22×C137—21no—
83C22×C137—21no—
84C22×C137—41no—
85C22×C137—41no—
86C22×C137—41no—
87C22×C137#9912yes—
88C22×C137C2×C22⋊C411yescharacteristic
89C2×D137—21no—
90C2×D137—21no—
91C2×D137—21no—
92C2×D137—21no—
93C2×D137—21no—
94C2×D137—44no—
95C2×D137#13512yes—
96#10587739—82no—
97C4×C137—21no—
98C4×C137—21no—
99C4×C137—41no—
100Dic137—21no—
101Dic137—21no—
102Dic137—41no—
103Dic137—41no—
104C2×C137—22no—
105C2×C137—81no—
106C2×C137C2×#9911yescharacteristic
107C2×C137#41611yescharacteristic
108C2×C137#42111yescharacteristic
109D137—22no—
110D137—41no—
111C137#151611yescharacteristic, complemented, Sylow
112#1516—1371nomaximal, Sylow
113#525—1371no—
114C2×C22⋊C8—1371no—
115#381—1371no—
116#328—1372no—
117#327—1372no—
118C2×#41#1058773911yescharacteristic, complemented
119C23×C4—1371no—
120C22×C8—2741no—
121#123—2741no—
122C22⋊Q8—1371no—
123C22⋊Q8—1371no—
124#121—1371no—
125#121—1371no—
126#120—2741no—
127C4×D4—2741no—
128C4×D4—2741no—
129#117—1371no—
130C2×C22⋊C4—2741no—
131C22⋊C8—2741no—
132C22⋊C8—1372no—
133#95—2741no—
134#95—1372no—
135C24—1371no—
136#41—42no—
137C2×Q8C2×#1058773911yescharacteristic
138C2×D4—21no—
139C2×D4—2741no—
140C2×D4—2741no—
141C2×D4C2×#1058773911yescharacteristic
142C22×C4—21no—
143C22×C4—2742no—
144C22×C4—5482no—
145C22×C4—1371no—
146C22×C4—1371no—
147C22×C4—1371no—
148C22×C4—1371no—
149C22×C4—1371no—
150C22×C4C2×#1058773911yescharacteristic
151C2×C8—2742no—
152C2×C8—5482no—
153C4⋊C4—2741no—
154C4⋊C4—2741no—
155C4⋊C4—1371no—
156C4⋊C4—1371no—
157C22⋊C4—2741no—
158C22⋊C4—2741no—
159C22⋊C4—2741no—
160C22⋊C4—2741no—
161C22⋊C4—2741no—
162C22⋊C4—5482no—
163C42—2741no—
164C23—21no—
165C23—2742no—
166C23—2742no—
167C23—5481no—
168C23—1371no—
169C23—1371no—
170C23C22⋊#1058773911yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—5482no—
175D4—5482no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—2742no—
180C2×C4—2741no—
181C2×C4—2741no—
182C2×C4—5481no—
183C2×C4—5481no—
184C2×C4—5481no—
185C2×C4—5481no—
186C2×C4—5482no—
187C2×C4—5482no—
188C2×C4—5482no—
189C2×C4—1371no—
190C2×C4—1371no—
191C2×C4C22⋊#1058773911yescharacteristic
192C2×C4C22×#1058773911yescharacteristic
193C8—5482no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—2741no—
200C22—2741no—
201C22—2741no—
202C22—2741no—
203C22—2741no—
204C22—5484no—
205C22—1372no—
206C22#1798414612yes—
207C22C2×C22⋊#1058773911yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—2741no—
212C4—2741no—
213C4—5481no—
214C4—5481no—
215C4—10962no—
216C2—22no—
217C2—81no—
218C2—2742no—
219C2—5481no—
220C2C2×#1798414611yescharacteristic
221C2#2257839311yescharacteristic
222C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×#41×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577910C2
cosocleC2×C4×D137C23
hypercenterC2×#41#10587739
hypocenterC137#1516
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#17984145C2semidirect
2. #22578346C2semidirect
3. #22578257C2semidirect
4. #22578254C2semidirect
14. #17984064C4semidirect
16. #17984059C4semidirect
21. #17984047C4semidirect
22. C2×#41×C137C4semidirect
111. C137#1516semidirect
118. C2×#41#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841452
#225783462
#225782572
#225782542
#225779102
#1516137