Finite Group Databasebeta

← #31796804

Subgroups — #31796804

Order 17536.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×D137⋊M16C211yesmaximal, characteristic
2C2×#17984151C212yesmaximal, complemented
3#22578345C211yesmaximal, characteristic
4#22578276C212yesmaximal, complemented
5C22⋊Q8×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7D137⋊M16—24no—
8C2×#14555519—21no—
9C2×#14555518C2212yes—
10C2×C4⋊#10587739C2212yes—
11C2×C22⋊#10587739—21no—
12#17984151—24no—
13(C2×#10587739)⋊C4—21no—
14C2×Q8×D137C2211yescharacteristic
15C22×C4×D137C2211yescharacteristic
16#17984068C411yescharacteristic, complemented
17#17984067C411yescharacteristic, complemented
18#17984066—21no—
19#17984065—21no—
20#17984064C411yescharacteristic, complemented
21C4⋊C4×D137—21no—
22C4⋊C4×D137C2211yescharacteristic
23C22⋊C4×D137—21no—
24C22⋊Q8×C137C411yescharacteristic, complemented
25C22×#10587739—22no—
26#14555520—24no—
27#14555519—24no—
28#14555518—22no—
29#14555518D414yes—
30C2×#11764047—22no—
31C4⋊#10587739—22no—
32C4⋊#10587739—22no—
33C22⋊#10587739—42no—
34C23×D137D411yescharacteristic
35C2×Dic274—21no—
36C2×Dic274C2×C411yescharacteristic
37Q8×D137—22no—
38Q8×D137—22no—
39C22×Dic137D411yescharacteristic
40C2×C4×D137—22no—
41C2×C4×D137—22no—
42C2×C4×D137—21no—
43C2×C4×D137—21no—
44C2×C4×D137C2×C411yescharacteristic
45C2×C4×D137D411yescharacteristic
46C2×C4×D137D411yescharacteristic
47C2×C4×D137C2311yescharacteristic
48C4⋊Dic137—21no—
49C4⋊Dic137C2×C411yescharacteristic
50#14555489—21no—
51#14555489—21no—
52#14555489—21no—
53C22⋊Dic137—21no—
54#14555487—21no—
55#14555487—21no—
56C2×Q8×C137C2×C411yescharacteristic
57C22×C4×C137C2×C411yescharacteristic
58C4⋊C4×C137—21no—
59C4⋊C4×C137C2×C411yescharacteristic
60C22⋊C4×C137—21no—
61C2×#10587739—42no—
62C2×#10587739—42no—
63C2×#10587739—42no—
64#11764047—24no—
65C22×D137—22no—
66C22×D137—21no—
67C22×D137—21no—
68C22×D137—22no—
69C22×D137C22⋊C411yescharacteristic
70C22×D137C2×D411yescharacteristic
71Dic274—22no—
72Dic274—41no—
73Dic274—41no—
74C2×Dic137—21no—
75C2×Dic137—21no—
76C2×Dic137—21no—
77C2×Dic137—21no—
78C2×Dic137C22⋊C411yescharacteristic
79C2×Dic137C2×D411yescharacteristic
80C4×D137—22no—
81C4×D137—22no—
82C4×D137—21no—
83C4×D137—21no—
84C4×D137—42no—
85C4×D137—41no—
86C4×D137—41no—
87C4×D137C2×D412yes—
88C4×D137#4112yes—
89C23×C137C22⋊C411yescharacteristic
90Q8×C137—22no—
91C2×C4×C137—22no—
92C2×C4×C137—21no—
93C2×C4×C137—21no—
94C2×C4×C137C22⋊C411yescharacteristic
95C2×C4×C137C22×C411yescharacteristic
96C22×C137—21no—
97C22×C137—21no—
98C22×C137—21no—
99C22×C137C2×C22⋊C411yescharacteristic
100C2×D137—21no—
101C2×D137—21no—
102C2×D137—21no—
103C2×D137—42no—
104C2×D137—42no—
105C2×D137#12112yes—
106C2×D137AGL(1,ℤ/8)11yescharacteristic
107C2×D137#13712yes—
108C2×D137#13711yescharacteristic
109#10587739—82no—
110C4×C137—21no—
111C4×C137—41no—
112C4×C137—41no—
113C4×C137C4×D412yes—
114Dic137—21no—
115Dic137—41no—
116Dic137—41no—
117Dic137#12011yescharacteristic
118Dic137#12311yescharacteristic
119C2×C137—41no—
120C2×C137#38611yescharacteristic
121C2×C137#41711yescharacteristic
122C2×C137#41911yescharacteristic
123D137—41no—
124D137#46812yes—
125D137#47012yes—
126C137#189411yescharacteristic, complemented, Sylow
127#1894—1371nomaximal, Sylow
128C22×M16—1371no—
129C2×C22⋊Q8—1371no—
130C2×#103—1372no—
131#380—1371no—
132#340—1372no—
133C22×Q8—1371no—
134C23×C4—1371no—
135C2×M16—2741no—
136C2×M16—2741no—
137C2×M16—2744no—
138C22×C8—1372no—
139C22⋊Q8—2741no—
140C22⋊Q8—2741no—
141C22⋊Q8—1371no—
142C22⋊Q8—1371no—
143C22⋊Q8—1371no—
144C22⋊Q8#1058773911yescharacteristic, complemented
145C2×C4⋊C4—2741no—
146C2×C4⋊C4—1372no—
147C2×C4⋊C4—1371no—
148C2×C22⋊C4—2741no—
149C2×C22⋊C4—2741no—
150#103—2744no—
151#95—2741no—
152C24—1371no—
153C2×Q8—2741no—
154C2×Q8—2742no—
155C2×Q8—2742no—
156C2×Q8—1371no—
157C2×Q8C2×#1058773911yescharacteristic
158C22×C4—2742no—
159C22×C4—2742no—
160C22×C4—2741no—
161C22×C4—2741no—
162C22×C4—2742no—
163C22×C4—1371no—
164C22×C4—1371no—
165C22×C4—1371no—
166C22×C4—1371no—
167C22×C4—1371no—
168C22×C4C2×#1058773911yescharacteristic
169M16—2744no—
170M16—2744no—
171C2×C8—2742no—
172C2×C8—2742no—
173C2×C8—1374no—
174C4⋊C4—21no—
175C4⋊C4—2741no—
176C4⋊C4—2741no—
177C4⋊C4—2741no—
178C4⋊C4—2741no—
179C4⋊C4—2742no—
180C4⋊C4—2742no—
181C4⋊C4—1371no—
182C4⋊C4C2×#1058773911yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—2741no—
185C22⋊C4—2741no—
186C22⋊C4—2741no—
187C22⋊C4—5482no—
188C23—2741no—
189C23—2742no—
190C23—2741no—
191C23—2742no—
192C23—1371no—
193C23—1371no—
194C23C22⋊#1058773911yescharacteristic
195Q8—22no—
196Q8—2742no—
197Q8—5481no—
198Q8—5481no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—2742no—
203C2×C4—2741no—
204C2×C4—2741no—
205C2×C4—2741no—
206C2×C4—2741no—
207C2×C4—2742no—
208C2×C4—2741no—
209C2×C4—2741no—
210C2×C4—5482no—
211C2×C4—5481no—
212C2×C4—5481no—
213C2×C4—5482no—
214C2×C4—5482no—
215C2×C4—5482no—
216C2×C4—1371no—
217C2×C4—1371no—
218C2×C4—1372no—
219C2×C4—1372no—
220C2×C4C22⋊#1058773911yescharacteristic
221C2×C4C22×#1058773911yescharacteristic
222C8—2744no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—2741no—
227C22—2741no—
228C22—2741no—
229C22—5482no—
230C22—5482no—
231C22—1372no—
232C22—1371no—
233C22—1371no—
234C22—1372no—
235C22C2×C22⋊#1058773911yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—2741no—
240C4—5481no—
241C4—5481no—
242C4—10962no—
243C4—1371no—
244C4—1371no—
245C4D4×#1058773912yes—
246C2—41no—
247C2—5481no—
248C2—1372no—
249C2—1372no—
250C2#2257835111yescharacteristic
251C2#2257839411yescharacteristic
252C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1894
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#17984151C2semidirect
4. #22578276C2semidirect
16. #17984068C4semidirect
17. #17984067C4semidirect
20. #17984064C4semidirect
24. C22⋊Q8×C137C4semidirect
126. C137#1894semidirect
144. C22⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×D137⋊M162
C2×#179841512
#225783452
#225782762
C22⋊Q8×D1372
#1894137