Finite Group Databasebeta

← #31796562

Subgroups — #31796562

Order 17536.

324 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 98 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×Q8×#10587739C211yesmaximal, characteristic
2C2×#17984151C211yesmaximal, characteristic
3C2×#17984142C211yesmaximal, characteristic
4C4⋊C4×#10587739C211yesmaximal, characteristic
5#22578277C211yesmaximal, characteristic
6#22578276C211yesmaximal, characteristic
7#103×D137C211yesmaximal, characteristic
8C2×#14555518C2211yescharacteristic
9Q8×#10587739—22no—
10Q8×#10587739C414yescomplemented
11C2×C4⋊#10587739—21no—
12C2×C4⋊#10587739C2211yescharacteristic
13C2×C4⋊#10587739C2211yescharacteristic
14C2×C4×#10587739—21no—
15C2×C4×#10587739—21no—
16C2×C4×#10587739C2211yescharacteristic
17#17984151C414yescomplemented
18#17984142—22no—
19(C2×#10587739)⋊C4—21no—
20C2×Q8×D137C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic
22C4⋊C4×D137C2211yescharacteristic
23#17984013C411yescharacteristic, complemented
24#17984012C411yescharacteristic, complemented
25#17984011C411yescharacteristic, complemented
26#103×C137C411yescharacteristic, complemented
27C22×#10587739—21no—
28C22×#10587739—21no—
29C22×#10587739D411yescharacteristic
30C22×#10587739D411yescharacteristic
31#14555518—21no—
32#14555518C2×C412yes—
33C2×#11764047—21no—
34C4⋊#10587739—24no—
35C4⋊#10587739—21no—
36C4⋊#10587739—21no—
37C4⋊#10587739C2×C412yes—
38C4⋊#10587739C2×C412yes—
39C4×#10587739—24no—
40C4×#10587739—42no—
41C4×#10587739C2×C412yes—
42C4×#10587739D412yes—
43C2×Dic274—21no—
44C2×Dic274C2×C411yescharacteristic
45Q8×D137—21no—
46Q8×D137—21no—
47Q8×D137C2×C412yes—
48Q8×D137C2×C412yes—
49C2×C4×D137—21no—
50C2×C4×D137—21no—
51C2×C4×D137C2311yescharacteristic
52C2×#11764040C2×C411yescharacteristic
53C8×D137—21no—
54C8×D137—21no—
55C4⋊Dic137C2×C411yescharacteristic
56#14555489—21no—
57C2×Q8×C137C2×C411yescharacteristic
58C2×C8×C137C2×C411yescharacteristic
59C4⋊C4×C137C2×C411yescharacteristic
60C2×#10587739—21no—
61C2×#10587739—21no—
62C2×#10587739—21no—
63C2×#10587739—21no—
64C2×#10587739—22no—
65C2×#10587739—22no—
66C2×#10587739—41no—
67C2×#10587739—41no—
68C2×#10587739—41no—
69C2×#10587739—41no—
70C2×#10587739C22⋊C412yes—
71C2×#10587739C22⋊C412yes—
72#11764047—22no—
73C22×D137C2×D411yescharacteristic
74Dic274—21no—
75Dic274—22no—
76Dic274—22no—
77Dic274C4212yescomplemented
78C2×Dic137—21no—
79C2×Dic137—21no—
80C2×Dic137#4111yescharacteristic
81C4×D137—22no—
82C4×D137—22no—
83C4×D137—41no—
84C4×D137—41no—
85C4×D137C22×C411yescharacteristic
86C4×D137C22×C411yescharacteristic
87C4×D137C2×D411yescharacteristic
88C4×D137#4111yescharacteristic
89#11764040—21no—
90Q8×C137—21no—
91Q8×C137C4212yescomplemented
92C2×C4×C137—21no—
93C2×C4×C137—21no—
94C2×C4×C137C22×C411yescharacteristic
95C8×C137—21no—
96C22×C137C4×D411yescharacteristic
97C2×D137C2×C22⋊C411yescharacteristic
98C2×D137C4×D411yescharacteristic
99C2×D137AGL(1,ℤ/8)11yescharacteristic
100C2×D137#13711yescharacteristic
101C2×D137#13711yescharacteristic
102C2×D137#13711yescharacteristic
103#10587739—22no—
104#10587739—22no—
105#10587739—42no—
106#10587739—42no—
107C4×C137—22no—
108C4×C137—41no—
109C4×C137C2×C4211yescharacteristic
110C4×C137C4×D411yescharacteristic
111Dic137—22no—
112Dic137—41no—
113Dic137#11711yescharacteristic
114Dic137C4×D411yescharacteristic
115C2×C137C4×C22⋊C411yescharacteristic
116C2×C137#44011yescharacteristic
117C2×C137#44111yescharacteristic
118D137#41711yescharacteristic
119D137#41911yescharacteristic
120D137#44011yescharacteristic
121D137#44111yescharacteristic
122C137#176311yescharacteristic, complemented, Sylow
123#1763—1371nomaximal, Sylow
124C2×C4×Q8—1371no—
125C2×#103—1371no—
126C2×#103—1371no—
127C2×#97—1371no—
128C4×C4⋊C4—1371no—
129#340—1371no—
130#340—1371no—
131C22×Q8—1371no—
132C22×C8—1371no—
133C22×C8—1371no—
134C4×Q8—2742no—
135C4×Q8—1374no—
136C2×C4⋊C4—2741no—
137C2×C4⋊C4—1371no—
138C2×C4⋊C4—1371no—
139C2×C4⋊C4—1371no—
140C2×C42—2741no—
141C2×C42—2741no—
142C2×C42—1371no—
143#103—1374no—
144#103—1371no—
145#103—1371no—
146#103—1371no—
147#103#1058773911yescharacteristic, complemented
148#97—2742no—
149#95—2741no—
150C2×Q8—2741no—
151C2×Q8—2741no—
152C2×Q8—2741no—
153C2×Q8—1372no—
154C2×Q8—1372no—
155C2×Q8—1371no—
156C2×Q8C2×#1058773911yescharacteristic
157C22×C4—2741no—
158C22×C4—2741no—
159C22×C4—2741no—
160C22×C4—2741no—
161C22×C4—1371no—
162C22×C4—1371no—
163C22×C4—1371no—
164C2×C8—2741no—
165C2×C8—2741no—
166C2×C8—2741no—
167C2×C8—2741no—
168C2×C8—1372no—
169C2×C8—1371no—
170C2×C8C2×#1058773911yescharacteristic
171C4⋊C4—2741no—
172C4⋊C4—2744no—
173C4⋊C4—2741no—
174C4⋊C4—2741no—
175C4⋊C4—1372no—
176C4⋊C4—1372no—
177C4⋊C4—1371no—
178C4⋊C4C2×#1058773911yescharacteristic
179C42—2744no—
180C42—5482no—
181C42—1372no—
182C42—1372no—
183C23—1371no—
184Q8—21no—
185Q8—2742no—
186Q8—2742no—
187Q8—2741no—
188Q8—1372no—
189Q8C4×#1058773912yescomplemented
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—2742no—
193C2×C4—2742no—
194C2×C4—2741no—
195C2×C4—2741no—
196C2×C4—2741no—
197C2×C4—2741no—
198C2×C4—2741no—
199C2×C4—2742no—
200C2×C4—2742no—
201C2×C4—2741no—
202C2×C4—5481no—
203C2×C4—5481no—
204C2×C4—5481no—
205C2×C4—5481no—
206C2×C4—5481no—
207C2×C4—5481no—
208C2×C4—1372no—
209C2×C4—1372no—
210C2×C4—1371no—
211C2×C4—1371no—
212C2×C4—1371no—
213C2×C4—1371no—
214C2×C4—1371no—
215C2×C4C22×#1058773911yescharacteristic
216C8—21no—
217C8—2742no—
218C8—2741no—
219C22—1371no—
220C22—1371no—
221C22—1371no—
222C22—1371no—
223C22—1371no—
224C22—1371no—
225C22D4×#1058773911yescharacteristic
226C4—22no—
227C4—41no—
228C4—2742no—
229C4—2742no—
230C4—2742no—
231C4—5482no—
232C4—5482no—
233C4—5481no—
234C4—1371no—
235C4—1371no—
236C4C2×C4×#1058773911yescharacteristic
237C4D4×#1058773911yescharacteristic
238C2—1371no—
239C2—1371no—
240C2—1371no—
241C2—1371no—
242C2C22⋊C4×#1058773911yescharacteristic
243C2#2257844211yescharacteristic
244C2#2257844311yescharacteristic
subgroupG/H
centerC22D4×#10587739
commutatorC4×C137C2×C42
FrattiniC2×C4C22×#10587739
Fitting#103×C137C4
socleC22×C137C4×D4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenter#103#10587739
hypocenterC137#1763
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
10. Q8×#10587739C4semidirect
17. #17984151C4semidirect
23. #17984013C4semidirect
24. #17984012C4semidirect
25. #17984011C4semidirect
26. #103×C137C4semidirect
77. Dic274C42semidirect
91. Q8×C137C42semidirect
122. C137#1763semidirect
147. #103#10587739semidirect
189. Q8C4×#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#105877392
C2×#179841512
C2×#179841422
C4⋊C4×#105877392
#225782772
#225782762
#103×D1372
#1763137