Finite Group Databasebeta

← #31796207

Subgroups — #31796207

Order 17536.

286 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578375C211yesmaximal, characteristic, complemented
2#22578345C211yesmaximal, characteristic, complemented
3#22578257C212yesmaximal, complemented
4#22578255C212yesmaximal, complemented
5#22577916C211yesmaximal, characteristic
6C2×#14555519—21no—
7C2×C4⋊#10587739—21no—
8C2×C22⋊#10587739—21no—
9#17984145—21no—
10#17984145C2212yes—
11(C2×#10587739)⋊C4C2212yes—
12C2×#14555511C411yescharacteristic, complemented
13C22×C4×D137C2211yescharacteristic
14#17984073—21no—
15#17984064C2211yescharacteristic
16#17984063C2211yescharacteristic
17#17984062C411yescharacteristic, complemented
18#17984058—21no—
19#17984050—21no—
20#17984048—21no—
21#17984046C411yescharacteristic, complemented
22C22⋊Q8×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24#14555519—42no—
25C2×#11764047—22no—
26C4⋊#10587739—42no—
27C22⋊#10587739—42no—
28C23×D137D411yescharacteristic
29#14555511—42no—
30C2×D548—21no—
31C2×D548—21no—
32C2×D548C2×C411yescharacteristic
33C2×C22⋊D137—21no—
34C22×Dic137D411yescharacteristic
35C2×C4×D137—21no—
36C2×C4×D137—42no—
37C2×C4×D137C2×C411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40C2×C4×D137C2311yescharacteristic
41C4⋊Dic137C2×C411yescharacteristic
42#14555489—21no—
43#14555487—21no—
44#14555487—21no—
45#14555487—21no—
46#14555487—21no—
47C4×Dic137—21no—
48C2×Q8×C137C2×C411yescharacteristic
49C22×C4×C137C2×C411yescharacteristic
50C4⋊C4×C137—21no—
51C4⋊C4×C137C2×C411yescharacteristic
52C22⋊C4×C137—21no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55C2×#10587739—42no—
56#11764047—42no—
57C22×D137—21no—
58C22×D137—21no—
59C22×D137—21no—
60C22×D137—41no—
61C22×D137—41no—
62C22×D137C22⋊C411yescharacteristic
63C22×D137C2×D411yescharacteristic
64D548—41no—
65D548—42no—
66D548—41no—
67D548—41no—
68C22⋊D137—42no—
69C2×Dic137—22no—
70C2×Dic137—21no—
71C2×Dic137C22⋊C411yescharacteristic
72C2×Dic137C2×D411yescharacteristic
73C4×D137—22no—
74C4×D137—22no—
75C4×D137—41no—
76C4×D137—41no—
77C4×D137—42no—
78C23×C137C22⋊C411yescharacteristic
79Q8×C137—41no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—41no—
83C2×C4×C137C22⋊C411yescharacteristic
84C2×C4×C137C22×C411yescharacteristic
85C22×C137—21no—
86C22×C137—21no—
87C22×C137—21no—
88C22×C137C2×C22⋊C411yescharacteristic
89C2×D137—21no—
90C2×D137—21no—
91C2×D137—21no—
92C2×D137—21no—
93C2×D137—21no—
94C2×D137—42no—
95C2×D137—42no—
96C2×D137—41no—
97C2×D137—41no—
98C2×D137—41no—
99C2×D137AGL(1,ℤ/8)11yescharacteristic
100C2×D137#13711yescharacteristic
101#10587739—82no—
102C4×C137—21no—
103C4×C137—21no—
104C4×C137—41no—
105C4×C137—41no—
106Dic137—21no—
107Dic137—41no—
108Dic137#9912yes—
109C2×C137—41no—
110C2×C137C2×#9911yescharacteristic
111C2×C137#41711yescharacteristic
112C2×C137#42111yescharacteristic
113D137—21no—
114D137—21no—
115D137—41no—
116D137—81no—
117C137#151311yescharacteristic, complemented, Sylow
118#1513—1371nomaximal, Sylow
119#525—1371no—
120#407—1371no—
121#380—1371no—
122#328—1372no—
123#327—1372no—
124C2×#41—1371no—
125C23×C4—1371no—
126C2×M16—2741no—
127#123—2741no—
128C22⋊Q8—1371no—
129C22⋊Q8#1058773911yescharacteristic, complemented
130#121—1371no—
131#121—1371no—
132#120—2741no—
133C4×D4—2741no—
134C4×D4—2741no—
135#117—1371no—
136C2×C4⋊C4—2741no—
137C2×C22⋊C4—2741no—
138C22⋊C8—2741no—
139C22⋊C8—1372no—
140#95—1372no—
141C24—1371no—
142#41—5482no—
143C2×Q8C2×#1058773911yescharacteristic
144C2×D4—2741no—
145C2×D4—2741no—
146C2×D4—2741no—
147C2×D4—1371no—
148C22×C4—2741no—
149C22×C4—2742no—
150C22×C4—5482no—
151C22×C4—1371no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4—1371no—
155C22×C4—1371no—
156C22×C4C2×#1058773911yescharacteristic
157M16—5482no—
158C2×C8—2742no—
159C4⋊C4—21no—
160C4⋊C4—2741no—
161C4⋊C4—5482no—
162C4⋊C4—1371no—
163C4⋊C4C2×#1058773911yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2741no—
166C22⋊C4—2741no—
167C22⋊C4—2741no—
168C22⋊C4—2741no—
169C22⋊C4—5482no—
170C42—2741no—
171C23—2741no—
172C23—2741no—
173C23—2741no—
174C23—5481no—
175C23—5481no—
176C23—1371no—
177C23—1371no—
178C23C22⋊#1058773911yescharacteristic
179Q8—41no—
180D4—5481no—
181D4—5482no—
182D4—5481no—
183D4—5481no—
184D4—5482no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—2742no—
189C2×C4—2742no—
190C2×C4—2742no—
191C2×C4—2741no—
192C2×C4—5481no—
193C2×C4—5481no—
194C2×C4—5482no—
195C2×C4—5482no—
196C2×C4—5482no—
197C2×C4—5482no—
198C2×C4—1371no—
199C2×C4—1371no—
200C2×C4C22⋊#1058773911yescharacteristic
201C2×C4C22×#1058773911yescharacteristic
202C8—5482no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—2741no—
207C22—2741no—
208C22—2741no—
209C22—2741no—
210C22—2741no—
211C22—5482no—
212C22—5482no—
213C22—5481no—
214C22—5481no—
215C22—5481no—
216C22—1371no—
217C22—1371no—
218C22C2×C22⋊#1058773911yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—2741no—
224C4—5481no—
225C4—10962no—
226C4—1372no—
227C2—41no—
228C2—2741no—
229C2—2741no—
230C2—5481no—
231C2—10961no—
232C2D137⋊#9911yescharacteristic
233C2#2257839411yescharacteristic
234C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577916C2
cosocleC2×C4×D137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1513
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22578375C2semidirect
2. #22578345C2semidirect
3. #22578257C2semidirect
4. #22578255C2semidirect
12. C2×#14555511C4semidirect
17. #17984062C4semidirect
21. #17984046C4semidirect
22. C22⋊Q8×C137C4semidirect
117. C137#1513semidirect
129. C22⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783752
#225783452
#225782572
#225782552
#225779162
#1513137