Finite Group Databasebeta

← #31797041

Subgroups — #31797041

Order 17536.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#22578390C211yesmaximal, characteristic, complemented
2#22578383C211yesmaximal, characteristic, complemented
3#22578306C212yesmaximal, complemented
4#22578303C212yesmaximal, complemented
5#22577909C211yesmaximal, characteristic
6C2×#14555519—21no—
7#17984167—21no—
8#17984149—21no—
9#17984148C2212yes—
10#17984146—21no—
11#17984146C2212yes—
12C2×#14555509C2211yescharacteristic
13C23×Dic137C411yescharacteristic, complemented
14#17984072C2211yescharacteristic
15#17984069C411yescharacteristic, complemented
16#17984065—21no—
17#17984060—21no—
18#17984057C2211yescharacteristic
19D4×Dic137—21no—
20#17984048—21no—
21#17984044C411yescharacteristic, complemented
22#120×C137C411yescharacteristic, complemented
23#14555519—22no—
24#14555519—41no—
25C2×#11764047—41no—
26C4⋊#10587739—41no—
27C22⋊#10587739—22no—
28C4×#10587739—41no—
29#14555509—41no—
30#14555509—41no—
31C2×Dic274D411yescharacteristic
32C2×C22⋊D137—21no—
33C2×C22⋊D137D411yescharacteristic
34C2×C22⋊D137C2311yescharacteristic
35C22×Dic137—21no—
36C22×Dic137—21no—
37C22×Dic137—41no—
38C22×Dic137—41no—
39C22×Dic137C2×C411yescharacteristic
40C22×Dic137C2×C411yescharacteristic
41C22×Dic137D411yescharacteristic
42C2×C4×D137D411yescharacteristic
43C4⋊Dic137—21no—
44#14555489—21no—
45#14555489—21no—
46C22⋊Dic137—21no—
47C22⋊Dic137—21no—
48C22⋊Dic137C2×C411yescharacteristic
49#14555487—21no—
50C4×Dic137—21no—
51C24×C137C2×C411yescharacteristic
52C2×D4×C137—21no—
53C2×D4×C137C2×C411yescharacteristic
54C22⋊C4×C137—21no—
55C22⋊C4×C137C2×C411yescharacteristic
56C2×#10587739—42no—
57#11764047—42no—
58C22×D137C2×D411yescharacteristic
59Dic274—41no—
60C22⋊D137—41no—
61C22⋊D137—41no—
62C22⋊D137—42no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137—21no—
69C2×Dic137—42no—
70C2×Dic137—41no—
71C2×Dic137—41no—
72C2×Dic137—41no—
73C2×Dic137C22⋊C411yescharacteristic
74C2×Dic137C22⋊C411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C4×D137—21no—
77C4×D137—21no—
78C23×C137—21no—
79C23×C137—42no—
80C23×C137—41no—
81C23×C137C22⋊C411yescharacteristic
82C23×C137C22×C411yescharacteristic
83D4×C137—41no—
84D4×C137—42no—
85C2×C4×C137—21no—
86C2×C4×C137C22⋊C411yescharacteristic
87C22×C137—22no—
88C22×C137—21no—
89C22×C137—21no—
90C22×C137—41no—
91C22×C137—41no—
92C22×C137—41no—
93C22×C137—42no—
94C22×C137—42no—
95C22×C137—41no—
96C22×C137C2×C22⋊C411yescharacteristic
97C2×D137—21no—
98C2×D137—41no—
99#10587739—82no—
100C4×C137—21no—
101C4×C137—41no—
102Dic137—21no—
103Dic137—21no—
104Dic137—41no—
105Dic137—41no—
106Dic137#9911yescharacteristic
107Dic137#9911yescharacteristic
108C2×C137—21no—
109C2×C137—41no—
110C2×C137—42no—
111C2×C137—41no—
112C2×C137C2×#9911yescharacteristic
113D137—41no—
114C137#212011yescharacteristic, complemented, Sylow
115#2120—1371nomaximal, Sylow
116#525—1371no—
117#413—1371no—
118#410—1371no—
119#352—1372no—
120AΣL(1,𝔽16)—1372no—
121C2×#41—1371no—
122C23×C4—1371no—
123C2×M16—2741no—
124#123—1371no—
125#123—1371no—
126C22⋊Q8—2741no—
127#121—2741no—
128#120—1371no—
129#120#1058773911yescharacteristic, complemented
130C4×D4—2741no—
131C4×D4—2741no—
132#117—2741no—
133#117—1371no—
134#101—2741no—
135#100—1372no—
136#99—2741no—
137#99—1372no—
138C24C2×#1058773911yescharacteristic
139#41—5481no—
140#41—5481no—
141C2×Q8—1371no—
142C2×D4—21no—
143C2×D4—2741no—
144C2×D4—1371no—
145C2×D4—1371no—
146C2×D4C2×#1058773911yescharacteristic
147C22×C4—2741no—
148C22×C4—2741no—
149C22×C4—5481no—
150C22×C4—5481no—
151C22×C4—1371no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4—1371no—
155M16—2742no—
156M16—5481no—
157C2×C8—5481no—
158C4⋊C4—2741no—
159C4⋊C4—2741no—
160C4⋊C4—2741no—
161C4⋊C4—5481no—
162C22⋊C4—21no—
163C22⋊C4—2741no—
164C22⋊C4—2741no—
165C22⋊C4—2741no—
166C22⋊C4—2742no—
167C22⋊C4—1371no—
168C22⋊C4C2×#1058773911yescharacteristic
169C42—2741no—
170C42—5481no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—1371no—
175C23C22⋊#1058773911yescharacteristic
176C23C22×#1058773911yescharacteristic
177Q8—5481no—
178D4—41no—
179D4—42no—
180D4—5481no—
181D4—5481no—
182D4—5482no—
183C2×C4—21no—
184C2×C4—2741no—
185C2×C4—2741no—
186C2×C4—2741no—
187C2×C4—2741no—
188C2×C4—2741no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—2741no—
192C2×C4—5482no—
193C2×C4—5481no—
194C2×C4—5481no—
195C2×C4—5481no—
196C2×C4—5482no—
197C2×C4—1371no—
198C2×C4—1371no—
199C2×C4—1371no—
200C2×C4C22⋊#1058773911yescharacteristic
201C8—5482no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—2741no—
212C22—5481no—
213C22C2×C22⋊#1058773911yescharacteristic
214C4—21no—
215C4—41no—
216C4—2741no—
217C4—2741no—
218C4—5481no—
219C4—5481no—
220C4—10962no—
221C4—1371no—
222C4—1371no—
223C2—21no—
224C2—42no—
225C2—41no—
226C2—41no—
227C2—5481no—
228C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting#120×C137C4
socleC2×C137C2×#99
prime radical#22577909C2
cosocleC2×C22⋊D137C23
hypercenter#120#10587739
hypocenterC137#2120
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #22578390C2semidirect
2. #22578383C2semidirect
3. #22578306C2semidirect
4. #22578303C2semidirect
13. C23×Dic137C4semidirect
15. #17984069C4semidirect
21. #17984044C4semidirect
22. #120×C137C4semidirect
114. C137#2120semidirect
129. #120#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783902
#225783832
#225783062
#225783032
#225779092
#2120137