Finite Group Databasebeta

← #31779342

Subgroups — #31779342

Order 17536.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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#22577913C211yesmaximal, characteristic
2C22⋊C8×D137C211yesmaximal, characteristic, complemented
3#22577265C211yesmaximal, characteristic, complemented
4#22577141C211yesmaximal, characteristic, complemented
5#22577139C211yesmaximal, characteristic, complemented
6#22577137C211yesmaximal, characteristic, complemented
7#327×C137C211yesmaximal, characteristic, complemented
8C2×#14555509C411yescharacteristic, complemented
9C22×C4×D137C2211yescharacteristic
10C2×C8×D137—21no—
11#17984070—21no—
12#17984068C411yescharacteristic, complemented
13#17984067C2211yescharacteristic
14#17984059C411yescharacteristic, complemented
15#17984056—21no—
16D4×Dic137—21no—
17#17984048—21no—
18#17984046C411yescharacteristic, complemented
19C2×#14555487—21no—
20C22⋊#11764040C2211yescharacteristic
21#17984000—21no—
22#17983995—21no—
23#17983995C2211yescharacteristic
24#121×C137C2211yescharacteristic
25C22⋊C8×C137C2211yescharacteristic
26#95×C137C2211yescharacteristic
27C23×D137D411yescharacteristic
28#14555509—42no—
29C2×Dic274C2×C411yescharacteristic
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137—21no—
33C22×Dic137—21no—
34C22×Dic137D411yescharacteristic
35C2×C4×D137—42no—
36C2×C4×D137C2×C411yescharacteristic
37C2×C4×D137C2×C411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40C2×#11764040—21no—
41C8×D137—42no—
42C4⋊Dic137—21no—
43C4⋊Dic137C2×C411yescharacteristic
44#14555489—21no—
45C22⋊Dic137—21no—
46C22⋊Dic137—21no—
47#14555487—21no—
48#14555487—21no—
49#14555487—42no—
50C4×Dic137—21no—
51C2×D4×C137—21no—
52C2×D4×C137C2×C411yescharacteristic
53C22×C4×C137—21no—
54C22×C4×C137C2311yescharacteristic
55C2×C8×C137—21no—
56C4⋊C4×C137C2×C411yescharacteristic
57C22⋊C4×C137—21no—
58C22×D137—21no—
59C22×D137—21no—
60C22×D137—22no—
61C22×D137—41no—
62C22×D137C22⋊C411yescharacteristic
63C22×D137C22⋊C411yescharacteristic
64Dic274—41no—
65C22⋊D137—42no—
66C22⋊D137—42no—
67C2×Dic137—21no—
68C2×Dic137—21no—
69C2×Dic137—41no—
70C2×Dic137—42no—
71C2×Dic137—41no—
72C2×Dic137—41no—
73C2×Dic137—41no—
74C2×Dic137C22⋊C411yescharacteristic
75C2×Dic137C22⋊C411yescharacteristic
76C4×D137—22no—
77C4×D137—22no—
78C4×D137—41no—
79C4×D137—41no—
80#11764040—41no—
81C23×C137—21no—
82C23×C137C2×D411yescharacteristic
83D4×C137—41no—
84D4×C137—41no—
85D4×C137—41no—
86C2×C4×C137—21no—
87C2×C4×C137—41no—
88C2×C4×C137—41no—
89C2×C4×C137—41no—
90C2×C4×C137—41no—
91C2×C4×C137C22×C411yescharacteristic
92C2×C4×C137C2×D411yescharacteristic
93C8×C137—41no—
94C22×C137—21no—
95C22×C137—21no—
96C22×C137—41no—
97C22×C137—41no—
98C22×C137—41no—
99C22×C137C2×C22⋊C411yescharacteristic
100C22×C137#13511yescharacteristic
101C22×C137#13511yescharacteristic
102C2×D137—21no—
103C2×D137—21no—
104C2×D137—21no—
105C2×D137—21no—
106C2×D137—21no—
107C2×D137—42no—
108C2×D137—42no—
109C2×D137#9912yes—
110C4×C137—21no—
111C4×C137—21no—
112C4×C137—41no—
113C4×C137—81no—
114Dic137—21no—
115Dic137—21no—
116Dic137—41no—
117Dic137—41no—
118Dic137—81no—
119C2×C137—21no—
120C2×C137—21no—
121C2×C137—81no—
122C2×C137C2×#9911yescharacteristic
123C2×C137#41611yescharacteristic
124C2×C137#42111yescharacteristic
125D137—22no—
126D137—41no—
127C137#151611yescharacteristic, complemented, Sylow
128#1516—1371nomaximal, Sylow
129#525—1371no—
130C2×C22⋊C8—1371no—
131#381—1371no—
132#328—1371no—
133#328—1371no—
134#327—1371no—
135#327D13711yescharacteristic, complemented
136C2×#41—1371no—
137C23×C4—1371no—
138C22×C8—2741no—
139#123—2741no—
140C22⋊Q8—1371no—
141C22⋊Q8—1371no—
142#121—1371no—
143#121C2×D13711yescharacteristic
144#120—2741no—
145C4×D4—2741no—
146C4×D4—2741no—
147#117—1371no—
148C2×C22⋊C4—2741no—
149C22⋊C8—2741no—
150C22⋊C8—1371no—
151C22⋊C8C2×D13711yescharacteristic
152#95—2741no—
153#95—1371no—
154#95C2×D13711yescharacteristic
155C24—1371no—
156#41—5482no—
157C2×Q8—1371no—
158C2×D4—21no—
159C2×D4—2741no—
160C2×D4—2741no—
161C2×D4C4×D13711yescharacteristic
162C22×C4—21no—
163C22×C4—2741no—
164C22×C4—2741no—
165C22×C4—5482no—
166C22×C4—1371no—
167C22×C4—1371no—
168C22×C4—1371no—
169C22×C4—1371no—
170C22×C4—1371no—
171C22×C4C22×D13711yescharacteristic
172C2×C8—21no—
173C2×C8—2741no—
174C2×C8—5482no—
175C4⋊C4—2741no—
176C4⋊C4—2741no—
177C4⋊C4—1371no—
178C4⋊C4C4×D13711yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—2741no—
181C22⋊C4—2741no—
182C22⋊C4—2741no—
183C22⋊C4—2741no—
184C22⋊C4—5482no—
185C42—2741no—
186C23—21no—
187C23—2742no—
188C23—2741no—
189C23—2741no—
190C23—5481no—
191C23—1371no—
192C23—1371no—
193C23D4×D13711yescharacteristic
194Q8—5481no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—5482no—
199D4—5482no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—2742no—
206C2×C4—2742no—
207C2×C4—2741no—
208C2×C4—2741no—
209C2×C4—5481no—
210C2×C4—5481no—
211C2×C4—5481no—
212C2×C4—5482no—
213C2×C4—5481no—
214C2×C4—5481no—
215C2×C4—5481no—
216C2×C4—1371no—
217C2×C4—1371no—
218C2×C4C2×C4×D13711yescharacteristic
219C2×C4D4×D13711yescharacteristic
220C8—41no—
221C8—5481no—
222C22—21no—
223C22—21no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—2741no—
228C22—2741no—
229C22—2741no—
230C22—2741no—
231C22—2741no—
232C22—5482no—
233C22—5482no—
234C22—1372no—
235C22C22⋊C4×D13711yescharacteristic
236C22#1798410711yescharacteristic
237C22#1798410911yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—81no—
242C4—2741no—
243C4—2741no—
244C4—5481no—
245C4—5481no—
246C4—10961no—
247C2—21no—
248C2—21no—
249C2—81no—
250C2—2742no—
251C2—5481no—
252C2#99×D13711yescharacteristic
253C2#2257739611yescharacteristic
254C2#2257741811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#327×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577913C2
cosocleC22×C4×C137C23
hypercenter#327D137
hypocenterC137#1516
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D137C2semidirect
3. #22577265C2semidirect
4. #22577141C2semidirect
5. #22577139C2semidirect
6. #22577137C2semidirect
7. #327×C137C2semidirect
8. C2×#14555509C4semidirect
12. #17984068C4semidirect
14. #17984059C4semidirect
18. #17984046C4semidirect
127. C137#1516semidirect
135. #327D137semidirect

Maximal subgroups

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

subgroupindex
#225779132
C22⋊C8×D1372
#225772652
#225771412
#225771392
#225771372
#327×C1372
#1516137