Finite Group Databasebeta

← #30670923

Subgroups — #30670923

Order 16768.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16862218C212yesmaximal, complemented
2C2×C4×C22⋊D131C211yesmaximal, characteristic, complemented
3#22553243C211yesmaximal, characteristic, complemented
4#22553238C211yesmaximal, characteristic, complemented
5#22553226C211yesmaximal, characteristic, complemented
6#22553220C211yesmaximal, characteristic, complemented
7#22553200C212yesmaximal, complemented
8#22553192C212yesmaximal, complemented
9#22553188C212yesmaximal, complemented
10C2×C131⋊#95C211yesmaximal, characteristic, complemented
11C2×#121×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2212yes—
13C22×C22⋊D131C2211yescharacteristic
14C23×Dic131C2212yes—
15C22×C4×D131C2211yescharacteristic
16#16862218—28no—
17C4×C22⋊D131—24no—
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2211yescharacteristic
20C2×C22⋊Dic131C2211yescharacteristic
21C2×C22⋊Dic131C2212yes—
22C2×C22⋊Dic131C2212yes—
23C2×C22⋊Dic131C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2211yescharacteristic
26C2×C4×Dic131C2211yescharacteristic
27C131⋊#95C2212yes—
28C131⋊#95C2211yescharacteristic
29C131⋊#95C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C22×D4×C131C2212yes—
33C22×D4×C131C2211yescharacteristic
34C23×C4×C131C2211yescharacteristic
35#121×C131—24no—
36C2×C4⋊C4×C131C2211yescharacteristic
37C2×C22⋊C4×C131C2212yes—
38C23×D131C2311yescharacteristic
39C2×C22⋊D131—24no—
40C2×C22⋊D131—24no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131—22no—
43C2×C22⋊D131—42no—
44C2×C22⋊D131D414yescomplemented
45C22×Dic131—24no—
46C22×Dic131—24no—
47C22×Dic131—42no—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2312yes—
50C22×Dic131C2311yescharacteristic
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2312yes—
53C2×C4×D131—22no—
54C2×C4×D131—42no—
55Dic131⋊C4—24no—
56Dic131⋊C4—22no—
57C22⋊Dic131—24no—
58C22⋊Dic131—24no—
59C22⋊Dic131—22no—
60C22⋊Dic131—42no—
61(C2×D131)⋊C4—24no—
62(C2×D131)⋊C4—22no—
63C4×Dic131—22no—
64C24×C131C2312yes—
65C24×C131C2311yescharacteristic
66C2×D4×C131—24no—
67C2×D4×C131—24no—
68C2×D4×C131—24no—
69C2×D4×C131—22no—
70C2×D4×C131—42no—
71C22×C4×C131—42no—
72C22×C4×C131D414yescomplemented
73C22×C4×C131C2312yes—
74C22×C4×C131C2311yescharacteristic
75C22×C4×C131C2311yescharacteristic
76C4⋊C4×C131—22no—
77C22⋊C4×C131—24no—
78C22×D131—21no—
79C22×D131—22no—
80C22×D131—21no—
81C22×D131—22no—
82C22×D131C2×D412yes—
83C22⋊D131—28no—
84C22⋊D131—48no—
85C2×Dic131—21no—
86C2×Dic131—22no—
87C2×Dic131—21no—
88C2×Dic131—22no—
89C2×Dic131—21no—
90C2×Dic131—22no—
91C2×Dic131—21no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—48no—
97C2×Dic131C2×D412yes—
98C2×Dic131#4114yes—
99C4×D131—44no—
100C23×C131—22no—
101C23×C131—22no—
102C23×C131—22no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—22no—
108C23×C131—22no—
109C23×C131C2×D412yes—
110C23×C131C2×D412yes—
111C23×C131#4112yes—
112C23×C131C2411yescharacteristic
113D4×C131—44no—
114D4×C131—48no—
115C2×C4×C131—22no—
116C2×C4×C131—22no—
117C2×C4×C131—22no—
118C2×C4×C131—22no—
119C2×C4×C131—22no—
120C2×C4×C131—44no—
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2×D412yes—
123C22×C131—22no—
124C22×C131—22no—
125C22×C131—24no—
126C22×C131—24no—
127C22×C131—42no—
128C22×C131—42no—
129C22×C131—42no—
130C22×C131—42no—
131C22×C131—42no—
132C22×C131—42no—
133C22×C131—42no—
134C22×C131#12114yes—
135C22×C131C22×D411yescharacteristic
136C22×C131C22×D411yescharacteristic
137C22×C131C2×#4112yes—
138C22×C131C2×#4111yescharacteristic
139C22×C131H(5,𝔽2)12yes—
140C2×D131—22no—
141C2×D131—24no—
142C2×D131—41no—
143C2×D131—41no—
144C2×D131—42no—
145C4×C131—22no—
146C4×C131—22no—
147C4×C131—42no—
148Dic131—22no—
149Dic131—24no—
150Dic131—41no—
151Dic131—41no—
152Dic131—42no—
153C2×C131—24no—
154C2×C131—82no—
155C2×C131C2×#12111yescharacteristic
156C2×C131#53211yescharacteristic
157C2×C131#53411yescharacteristic
158C2×C131#54612yes—
159C2×C131#55312yes—
160D131—42no—
161C131#254311yescharacteristic, complemented, Sylow
162#2543—1311nomaximal, Sylow
163C2×#121—1312no—
164C2×#121D13111yescharacteristic, complemented
165C2×C4×D4—1311no—
166#398—1311no—
167#397—1311no—
168#394—1311no—
169#392—1311no—
170#388—1312no—
171#386—1312no—
172#386—1312no—
173C2×#95—1311no—
174C22×D4—1312no—
175C22×D4—1311no—
176C22×D4C2×D13112yes—
177C22×D4C2×D13111yescharacteristic
178C23×C4—1312no—
179C23×C4—1311no—
180C23×C4C2×D13111yescharacteristic
181#121—24no—
182#121—2628no—
183C4×D4—2624no—
184C2×C4⋊C4—1311no—
185C2×C4⋊C4—1312no—
186C2×C4⋊C4C2×D13111yescharacteristic
187C2×C22⋊C4—1311no—
188C2×C22⋊C4—1312no—
189C2×C22⋊C4—1312no—
190C2×C22⋊C4—1311no—
191C2×C22⋊C4—1312no—
192C2×C22⋊C4—1312no—
193C2×C22⋊C4C2×D13112yes—
194C2×C42—1311no—
195#95—1312no—
196#95—1311no—
197#95—1311no—
198#95—1312no—
199#95—1312no—
200C24—1311no—
201C24C22×D13112yes—
202C24C22×D13111yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—2622no—
209C2×D4—2622no—
210C2×D4—2624no—
211C2×D4—2624no—
212C2×D4—5242no—
213C2×D4—1314no—
214C22×C4—42no—
215C22×C4—2624no—
216C22×C4—2624no—
217C22×C4—2622no—
218C22×C4—5242no—
219C22×C4—5242no—
220C22×C4—1312no—
221C22×C4—1311no—
222C22×C4—1311no—
223C22×C4—1312no—
224C22×C4—1311no—
225C22×C4C22⋊D13114yescomplemented
226C22×C4C22×D13112yes—
227C22×C4C22×D13111yescharacteristic
228C22×C4C22×D13111yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—2624no—
231C4⋊C4—2622no—
232C22⋊C4—24no—
233C22⋊C4—2624no—
234C22⋊C4—2622no—
235C22⋊C4—2624no—
236C22⋊C4—2624no—
237C22⋊C4—2622no—
238C22⋊C4—5242no—
239C42—2622no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—2621no—
250C23—2622no—
251C23—2621no—
252C23—2622no—
253C23—1312no—
254C23D4×D13112yes—
255C23C2×C22⋊D13112yes—
256C23#1452351212yes—
257C23C23×D13111yescharacteristic
258D4—44no—
259D4—48no—
260D4—2628no—
261D4—5248no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—2622no—
269C2×C4—2621no—
270C2×C4—2621no—
271C2×C4—2622no—
272C2×C4—2622no—
273C2×C4—2621no—
274C2×C4—2621no—
275C2×C4—2622no—
276C2×C4—2622no—
277C2×C4—2622no—
278C2×C4—2622no—
279C2×C4—5244no—
280C2×C4—5248no—
281C2×C4—1312no—
282C2×C4—1314no—
283C2×C4C2×C22⋊D13112yes—
284C2×C4C2×C22⋊D13112yes—
285C22—22no—
286C22—22no—
287C22—24no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—2624no—
297C22—2622no—
298C22—5241no—
299C22—5241no—
300C22—5242no—
301C22#1686221714yes—
302C22C2×D4×D13111yescharacteristic
303C22C22×C22⋊D13111yescharacteristic
304C22#41×D13112yes—
305C22C2×#1452351211yescharacteristic
306C22#1686229112yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—2624no—
311C4—2622no—
312C4—5241no—
313C4—5241no—
314C4—5242no—
315C2—24no—
316C2—82no—
317C2—5242no—
318C2C2×#1686221711yescharacteristic
319C2#2255385811yescharacteristic
320C2#2255387111yescharacteristic
321C2#2255395112yes—
322C2#2255400512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#121×C131C2
hypercenterC2×#121D131
hypocenterC131#2543
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16862218C2semidirect
2. C2×C4×C22⋊D131C2semidirect
3. #22553243C2semidirect
4. #22553238C2semidirect
5. #22553226C2semidirect
6. #22553220C2semidirect
7. #22553200C2semidirect
8. #22553192C2semidirect
9. #22553188C2semidirect
10. C2×C131⋊#95C2semidirect
11. C2×#121×C131C2semidirect
44. C2×C22⋊D131D4semidirect
72. C22×C4×C131D4semidirect
161. C131#2543semidirect
164. C2×#121D131semidirect
225. C22×C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622182
C2×C4×C22⋊D1312
#225532432
#225532382
#225532262
#225532202
#225532002
#225531922
#225531882
C2×C131⋊#952
C2×#121×C1312
#2543131