Finite Group Databasebeta

← #31782441

Subgroups — #31782441

Order 17536.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#22577975C211yesmaximal, characteristic
2#22577384C211yesmaximal, characteristic
3#22577375C211yesmaximal, characteristic
4#22577194C211yesmaximal, characteristic, complemented
5#22577193C211yesmaximal, characteristic, complemented
6#22577192C211yesmaximal, characteristic, complemented
7#352×C137C211yesmaximal, characteristic, complemented
8C2×#14555509C2211yescharacteristic
9C22×Dic274C411yescharacteristic, complemented
10M16×D137—21no—
11#17984087—21no—
12#17984074—21no—
13#17984072C411yescharacteristic, complemented
14#17984071C411yescharacteristic, complemented
15#17984069C2211yescharacteristic
16#17984066—21no—
17#17984065—21no—
18#17984044—21no—
19#17984044C411yescharacteristic, complemented
20#17984006—21no—
21#17984005C2211yescharacteristic
22#17984003C2211yescharacteristic
23#17984002—21no—
24#123×C137C2211yescharacteristic
25#100×C137C2211yescharacteristic
26#99×C137C2211yescharacteristic
27#14555509—41no—
28#14555509—41no—
29C2×Dic274—21no—
30C2×Dic274—21no—
31C2×Dic274—41no—
32C2×Dic274—41no—
33C2×Dic274D411yescharacteristic
34C2×C22⋊D137C2×C411yescharacteristic
35C2×C22⋊D137D411yescharacteristic
36C22×Dic137C2×C411yescharacteristic
37C22×Dic137C2×C411yescharacteristic
38C22×Dic137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40#14555494—21no—
41#14555493—41no—
42C8×D137—41no—
43C4⋊Dic137—21no—
44#14555489—21no—
45#14555489—21no—
46#14555489—21no—
47#14555489—41no—
48C22⋊Dic137—21no—
49C22⋊Dic137—21no—
50C22⋊Dic137C2×C411yescharacteristic
51#14555487—21no—
52#14555487—21no—
53C4×Dic137—21no—
54C4×Dic137—41no—
55C2×D4×C137C2311yescharacteristic
56C22×C4×C137C2×C411yescharacteristic
57M16×C137—21no—
58C4⋊C4×C137—21no—
59C22⋊C4×C137—21no—
60C22⋊C4×C137—21no—
61C22⋊C4×C137C2×C411yescharacteristic
62C22×D137C22⋊C411yescharacteristic
63Dic274—41no—
64Dic274—42no—
65Dic274—41no—
66Dic274—41no—
67C22⋊D137—41no—
68C22⋊D137—41no—
69C2×Dic137—21no—
70C2×Dic137—21no—
71C2×Dic137—21no—
72C2×Dic137—21no—
73C2×Dic137—21no—
74C2×Dic137—21no—
75C2×Dic137—41no—
76C2×Dic137—41no—
77C2×Dic137C22⋊C411yescharacteristic
78C2×Dic137C22⋊C411yescharacteristic
79C2×Dic137C22⋊C411yescharacteristic
80C4×D137—21no—
81C4×D137—21no—
82#11764040—41no—
83C23×C137C22×C411yescharacteristic
84C23×C137C2×D411yescharacteristic
85D4×C137—41no—
86C2×C4×C137—21no—
87C2×C4×C137—21no—
88C2×C4×C137—41no—
89C2×C4×C137—41no—
90C2×C4×C137C2×D411yescharacteristic
91C8×C137—41no—
92C22×C137—21no—
93C22×C137—21no—
94C22×C137—41no—
95C22×C137—41no—
96C22×C137C2×C22⋊C411yescharacteristic
97C2×D137—21no—
98C2×D137—41no—
99C4×C137—21no—
100C4×C137—41no—
101C4×C137—41no—
102C4×C137—81no—
103Dic137—21no—
104Dic137—21no—
105Dic137—41no—
106Dic137—41no—
107Dic137—81no—
108Dic137#9911yescharacteristic
109Dic137#9911yescharacteristic
110C2×C137—21no—
111C2×C137—41no—
112C2×C137—41no—
113C2×C137C2×#9911yescharacteristic
114D137—41no—
115C137#212311yescharacteristic, complemented, Sylow
116#2123—1371nomaximal, Sylow
117#536—1371no—
118#413—1371no—
119#410—1371no—
120#352—1371no—
121#352—1371no—
122#352—1371no—
123#352D13711yescharacteristic, complemented
124C2×#41—1371no—
125C22×Q8—1371no—
126C2×M16—2741no—
127C4⋊Q8—2741no—
128#124—2741no—
129#123—1371no—
130#123—1371no—
131#123—1371no—
132#123C2×D13711yescharacteristic
133C22⋊Q8—2741no—
134C22⋊Q8—2741no—
135#117—2741no—
136#117—1371no—
137#101—2741no—
138#100—1371no—
139#100C2×D13711yescharacteristic
140#99—2741no—
141#99—1371no—
142#99C2×D13711yescharacteristic
143#41—5481no—
144#41—5481no—
145C2×Q8—2741no—
146C2×Q8—2741no—
147C2×Q8—5481no—
148C2×Q8—5481no—
149C2×Q8—1371no—
150C2×D4—1371no—
151C2×D4—1371no—
152C2×D4C22×D13711yescharacteristic
153C22×C4—1371no—
154C22×C4—1371no—
155C22×C4—1371no—
156C22×C4—1371no—
157C22×C4C4×D13711yescharacteristic
158M16—21no—
159M16—2741no—
160M16—5481no—
161C2×C8—5481no—
162C4⋊C4—21no—
163C4⋊C4—2741no—
164C4⋊C4—2741no—
165C4⋊C4—2741no—
166C4⋊C4—2741no—
167C4⋊C4—5481no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—2741no—
171C22⋊C4—2741no—
172C22⋊C4—2741no—
173C22⋊C4—2741no—
174C22⋊C4—1371no—
175C22⋊C4C4×D13711yescharacteristic
176C42—2741no—
177C42—5481no—
178C23—1371no—
179C23C2×C4×D13711yescharacteristic
180C23D4×D13711yescharacteristic
181Q8—5481no—
182Q8—5482no—
183Q8—5481no—
184Q8—5481no—
185D4—41no—
186D4—5481no—
187D4—5481no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—2741no—
193C2×C4—2741no—
194C2×C4—2741no—
195C2×C4—2741no—
196C2×C4—2741no—
197C2×C4—2741no—
198C2×C4—2741no—
199C2×C4—2741no—
200C2×C4—5481no—
201C2×C4—5481no—
202C2×C4—1371no—
203C2×C4—1371no—
204C2×C4—1371no—
205C2×C4D4×D13711yescharacteristic
206C8—41no—
207C8—5481no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—2741no—
213C22—5481no—
214C22C22⋊C4×D13711yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—2741no—
220C4—2741no—
221C4—5481no—
222C4—5481no—
223C4—10961no—
224C4—1371no—
225C4—1371no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—5481no—
230C2#99×D13711yescharacteristic
subgroupG/H
centerC2#99×D137
commutatorC23×C137C22×C4
FrattiniC2×D4C22×D137
Fitting#352×C137C2
socleC2×C137C2×#99
prime radicalC2×#14555509C22
cosocleC2×D4×C137C23
hypercenter#352D137
hypocenterC137#2123
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #22577194C2semidirect
5. #22577193C2semidirect
6. #22577192C2semidirect
7. #352×C137C2semidirect
9. C22×Dic274C4semidirect
13. #17984072C4semidirect
14. #17984071C4semidirect
19. #17984044C4semidirect
115. C137#2123semidirect
123. #352D137semidirect

Maximal subgroups

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

subgroupindex
#225779752
#225773842
#225773752
#225771942
#225771932
#225771922
#352×C1372
#2123137