Finite Group Databasebeta

← #31796659

Subgroups — #31796659

Order 17536.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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#22578404C211yesmaximal, characteristic, complemented
2C2×#17984150C212yesmaximal
3#22578347C211yesmaximal, characteristic, complemented
4#22578276C212yesmaximal, complemented
5C4×D4×D137C211yesmaximal, characteristic
6C2×#14555518C2212yes—
7C2×C4⋊#10587739C2212yes—
8#17984155—21no—
9C4⋊#11764047—21no—
10#17984150—24no—
11(C2×#10587739)⋊C4—21no—
12(C2×#10587739)⋊C4—21no—
13C2×D4×D137C2211yescharacteristic
14C22×C4×D137—21no—
15C4×D548C411yescharacteristic, complemented
16D4×Dic137C411yescharacteristic, complemented
17#17984050C411yescharacteristic, complemented
18C4×C22⋊D137—21no—
19#17984048—21no—
20C4⋊C4×D137C2211yescharacteristic
21C22⋊C4×D137—21no—
22C42×D137C2211yescharacteristic
23C4×D4×C137C411yescharacteristic, complemented
24C22×#10587739—22no—
25#14555518—22no—
26#14555518—22no—
27C2×#11764047—22no—
28C4⋊#10587739—22no—
29C4⋊#10587739—22no—
30C23×D137—21no—
31C2×D548C2×C411yescharacteristic
32C2×C22⋊D137—21no—
33D4×D137—21no—
34D4×D137—21no—
35D4×D137D412yes—
36D4×D137Q812yes—
37C22×Dic137—21no—
38C2×C4×D137—21no—
39C2×C4×D137—21no—
40C2×C4×D137—21no—
41C2×C4×D137—44no—
42C2×C4×D137D411yescharacteristic
43C2×C4×D137D411yescharacteristic
44C2×C4×D137C2311yescharacteristic
45C4⋊Dic137C2×C411yescharacteristic
46#14555489—21no—
47C22⋊Dic137—21no—
48#14555487—21no—
49#14555487—21no—
50C4×Dic137C2×C411yescharacteristic
51C4×Dic137D411yescharacteristic
52C4×Dic137D411yescharacteristic
53C2×D4×C137C2×C411yescharacteristic
54C22×C4×C137—21no—
55C4⋊C4×C137C2×C411yescharacteristic
56C22⋊C4×C137—21no—
57C42×C137C2×C411yescharacteristic
58C2×#10587739—42no—
59C2×#10587739—42no—
60C2×#10587739—42no—
61#11764047—42no—
62C22×D137—21no—
63C22×D137—22no—
64C22×D137—22no—
65C22×D137—42no—
66C22×D137—42no—
67C22×D137C2×D411yescharacteristic
68D548—21no—
69D548C4⋊C412yes—
70C22⋊D137—22no—
71C22⋊D137—22no—
72C2×Dic137—21no—
73C2×Dic137—42no—
74C2×Dic137C22⋊C411yescharacteristic
75C2×Dic137C22⋊C411yescharacteristic
76C2×Dic137C2×D411yescharacteristic
77C4×D137—22no—
78C4×D137—22no—
79C4×D137—42no—
80C4×D137—42no—
81C4×D137—42no—
82C4×D137C2×D411yescharacteristic
83C4×D137C2×Q811yescharacteristic
84C4×D137#4111yescharacteristic
85C4×D137#4111yescharacteristic
86C23×C137—21no—
87D4×C137—21no—
88D4×C137C4⋊C412yes—
89C2×C4×C137—21no—
90C2×C4×C137—42no—
91C2×C4×C137C22⋊C411yescharacteristic
92C2×C4×C137C22⋊C411yescharacteristic
93C2×C4×C137C22×C411yescharacteristic
94C22×C137—22no—
95C22×C137—41no—
96C22×C137—41no—
97C22×C137C2×C22⋊C411yescharacteristic
98C2×D137—22no—
99C2×D137—41no—
100C2×D137—41no—
101C2×D137—44no—
102C2×D137—44no—
103C2×D137#12111yescharacteristic
104C2×D137C22⋊Q811yescharacteristic
105C2×D137#13512yes—
106C2×D137AGL(1,ℤ/8)12yes—
107#10587739—82no—
108C4×C137—21no—
109C4×C137—21no—
110C4×C137—41no—
111C4×C137C2×C4⋊C411yescharacteristic
112C4×C137#11711yescharacteristic
113Dic137—21no—
114Dic137—21no—
115Dic137—41no—
116Dic137#12111yescharacteristic
117Dic137C22⋊Q811yescharacteristic
118C2×C137—42no—
119C2×C137#38011yescharacteristic
120C2×C137#41611yescharacteristic
121C2×C137#41811yescharacteristic
122D137—42no—
123D137#46312yes—
124D137#47812yes—
125C137#180411yescharacteristic, complemented, Sylow
126#1804—1371nomaximal, Sylow
127C2×C4×D4—1371no—
128C2×C4⋊C8—1371no—
129C2×#102—1372no—
130#382—1371no—
131#340—1372no—
132C22×D4—1371no—
133C23×C4—2741no—
134C22×C8—1372no—
135C4×D4—2741no—
136C4×D4—2741no—
137C4×D4—1371no—
138C4×D4—1371no—
139C4×D4—1371no—
140C4×D4#1058773911yescharacteristic, complemented
141C2×C4⋊C4—1372no—
142C2×C4⋊C4—1371no—
143C2×C22⋊C4—2741no—
144C2×C42—1371no—
145C4⋊C8—2741no—
146C4⋊C8—2741no—
147#102—2744no—
148#95—2741no—
149#95—2741no—
150C24—2741no—
151C2×D4—2741no—
152C2×D4—2741no—
153C2×D4—2741no—
154C2×D4—1372no—
155C2×D4—1372no—
156C2×D4—1371no—
157C2×D4C2×#1058773911yescharacteristic
158C22×C4—21no—
159C22×C4—2741no—
160C22×C4—2741no—
161C22×C4—2741no—
162C22×C4—2741no—
163C22×C4—2742no—
164C22×C4—5484no—
165C22×C4—1371no—
166C22×C4—1371no—
167C22×C4—1371no—
168C2×C8—2742no—
169C2×C8—2742no—
170C2×C8—2742no—
171C4⋊C4—2741no—
172C4⋊C4—2742no—
173C4⋊C4—2742no—
174C4⋊C4—1371no—
175C4⋊C4C2×#1058773911yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—2741no—
178C22⋊C4—2741no—
179C22⋊C4—2741no—
180C42—1371no—
181C42—1371no—
182C42—1371no—
183C42C2×#1058773911yescharacteristic
184C23—21no—
185C23—2742no—
186C23—2742no—
187C23—2741no—
188C23—5482no—
189C23—5482no—
190C23—1371no—
191D4—21no—
192D4—2742no—
193D4—2742no—
194D4—2741no—
195D4—1372no—
196D4C4⋊#1058773912yes—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—2742no—
200C2×C4—2742no—
201C2×C4—2741no—
202C2×C4—5482no—
203C2×C4—5482no—
204C2×C4—5482no—
205C2×C4—5482no—
206C2×C4—5482no—
207C2×C4—5482no—
208C2×C4—5482no—
209C2×C4—1371no—
210C2×C4—1371no—
211C2×C4—1371no—
212C2×C4—1371no—
213C2×C4—1371no—
214C2×C4—1371no—
215C2×C4—1371no—
216C2×C4C22⋊#1058773911yescharacteristic
217C2×C4C22⋊#1058773911yescharacteristic
218C2×C4C22×#1058773911yescharacteristic
219C8—5482no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—2742no—
224C22—5484no—
225C22—5484no—
226C22—5481no—
227C22—5481no—
228C22—1371no—
229C22—1371no—
230C22—1372no—
231C22—1372no—
232C22C2×C22⋊#1058773911yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—2741no—
237C4—2741no—
238C4—5481no—
239C4—10962no—
240C4—1371no—
241C4—1371no—
242C4C2×C4⋊#1058773911yescharacteristic
243C4#1798416711yescharacteristic
244C2—42no—
245C2—5482no—
246C2—1372no—
247C2—1372no—
248C2#2257834511yescharacteristic
249C2#2257839311yescharacteristic
250C2#2257839511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C22
cosocleC2×C4×D137C23
hypercenterC4×D4#10587739
hypocenterC137#1804
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578404C2semidirect
3. #22578347C2semidirect
4. #22578276C2semidirect
15. C4×D548C4semidirect
16. D4×Dic137C4semidirect
17. #17984050C4semidirect
23. C4×D4×C137C4semidirect
125. C137#1804semidirect
140. C4×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784042
C2×#179841502
#225783472
#225782762
C4×D4×D1372
#1804137