Finite Group Databasebeta

← #30670922

Subgroups — #30670922

Order 16768.

724 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16862221C212yesmaximal, complemented
2C2×#16862221C211yesmaximal, characteristic, complemented
3C2×#16862208C211yesmaximal, characteristic, complemented
4#22553238C211yesmaximal, characteristic, complemented
5#22553229C211yesmaximal, characteristic, complemented
6#22553222C211yesmaximal, characteristic, complemented
7#22553202C212yesmaximal, complemented
8#22553190C212yesmaximal, complemented
9#22553190C212yesmaximal, complemented
10#95×D131C211yesmaximal, characteristic, complemented
11#398×C131C211yesmaximal, characteristic, complemented
12C22×D524C2212yes—
13C22×D524C2212yes—
14C22×D524C2211yescharacteristic
15C22×D524C2211yescharacteristic
16C22×C4×D131C2212yes—
17C22×C4×D131C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic
19#16862221—28no—
20#16862221—24no—
21#16862208—24no—
22C2×(C2×D131)⋊C4C2211yescharacteristic
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2212yes—
29C2×C4×Dic131C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2211yescharacteristic
32C131⋊#95C2212yes—
33C2×C4⋊C4×C131C2212yes—
34C2×C4⋊C4×C131C2211yescharacteristic
35C2×C4⋊C4×C131C2211yescharacteristic
36#95×C131C2211yescharacteristic
37#95×C131C2212yes—
38C23×D131C2312yes—
39C23×D131C2311yescharacteristic
40C23×D131C2311yescharacteristic
41C2×D524—22no—
42C2×D524—22no—
43C2×D524—24no—
44C2×D524—24no—
45C2×D524—24no—
46C2×D524—24no—
47C2×D524—22no—
48C2×D524—24no—
49C2×D524—42no—
50C2×D524—42no—
51C2×D524D414yescomplemented
52C22×Dic131C2312yes—
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2311yescharacteristic
55C2×C4×D131—24no—
56C2×C4×D131—22no—
57C2×C4×D131—24no—
58C2×C4×D131—42no—
59C2×C4×D131—42no—
60C2×C4×D131—42no—
61C2×C4×D131D414yescomplemented
62(C2×D131)⋊C4—22no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—24no—
65(C2×D131)⋊C4—22no—
66(C2×D131)⋊C4—24no—
67(C2×D131)⋊C4—24no—
68(C2×D131)⋊C4—42no—
69C4×Dic131—22no—
70C22×C4×C131C2311yescharacteristic
71C22×C4×C131C2312yes—
72C22×C4×C131C2311yescharacteristic
73C22×C4×C131C2311yescharacteristic
74C22×C4×C131C2312yes—
75C4⋊C4×C131—22no—
76C4⋊C4×C131—24no—
77C4⋊C4×C131—22no—
78C22×D131—22no—
79C22×D131—22no—
80C22×D131—21no—
81C22×D131—21no—
82C22×D131—22no—
83C22×D131—22no—
84C22×D131—22no—
85C22×D131—22no—
86C22×D131—22no—
87C22×D131—22no—
88C22×D131—22no—
89C22×D131—22no—
90C22×D131—22no—
91C22×D131C2×D412yes—
92C22×D131C2×D412yes—
93C22×D131C2×D412yes—
94C22×D131#4112yes—
95D524—28no—
96D524—48no—
97D524—44no—
98D524—48no—
99C2×Dic131—22no—
100C2×Dic131—21no—
101C2×Dic131—22no—
102C2×Dic131—22no—
103C2×Dic131—22no—
104C2×Dic131—22no—
105C2×Dic131C2×D412yes—
106C4×D131—44no—
107C4×D131—44no—
108C4×D131—48no—
109C23×C131C2411yescharacteristic
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131—21no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—21no—
116C2×C4×C131—22no—
117C2×C4×C131—22no—
118C2×C4×C131—22no—
119C2×C4×C131—22no—
120C2×C4×C131C2×D412yes—
121C2×C4×C131C2×D412yes—
122C2×C4×C131#4114yes—
123C22×C131C22×D411yescharacteristic
124C22×C131C22×D411yescharacteristic
125C22×C131C2×#4111yescharacteristic
126C22×C131C2×#4112yes—
127C22×C131H(5,𝔽2)12yes—
128C2×D131—22no—
129C2×D131—22no—
130C2×D131—24no—
131C2×D131—24no—
132C2×D131—22no—
133C2×D131—24no—
134C2×D131—41no—
135C2×D131—41no—
136C2×D131—42no—
137C2×D131—42no—
138C2×D131—42no—
139C2×D131—42no—
140C2×D131—42no—
141C2×D131—42no—
142C2×D131—42no—
143C2×D131—42no—
144C2×D131#12114yes—
145C4×C131—22no—
146C4×C131—22no—
147C4×C131—24no—
148C4×C131—42no—
149C4×C131—41no—
150Dic131—22no—
151Dic131—41no—
152Dic131—42no—
153C2×C131C2×#12111yescharacteristic
154C2×C131#53211yescharacteristic
155C2×C131#53411yescharacteristic
156C2×C131#54612yes—
157C2×C131#55312yes—
158D131—24no—
159D131—42no—
160D131—82no—
161C131#254311yescharacteristic, complemented, Sylow
162#2543—1311nomaximal, Sylow
163C2×#121—1311no—
164C2×#121—1312no—
165C2×C4×D4—1311no—
166#398D13111yescharacteristic, complemented
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—1312no—
176C22×D4—1311no—
177C22×D4—1311no—
178C23×C4—1312no—
179C23×C4—1311no—
180C23×C4—1311no—
181#121—2628no—
182#121—2624no—
183C4×D4—2624no—
184C2×C4⋊C4C2×D13111yescharacteristic
185C2×C4⋊C4C2×D13112yes—
186C2×C4⋊C4C2×D13111yescharacteristic
187C2×C22⋊C4—1311no—
188C2×C22⋊C4—1311no—
189C2×C22⋊C4—1312no—
190C2×C22⋊C4—1312no—
191C2×C22⋊C4—1312no—
192C2×C22⋊C4—1312no—
193C2×C22⋊C4—1312no—
194C2×C42—1311no—
195#95—1312no—
196#95—1311no—
197#95—1312no—
198#95C2×D13111yescharacteristic
199#95C2×D13112yes—
200C24—1312no—
201C24—1311no—
202C24—1311no—
203C2×D4—2622no—
204C2×D4—2622no—
205C2×D4—2624no—
206C2×D4—2624no—
207C2×D4—2624no—
208C2×D4—2624no—
209C2×D4—2622no—
210C2×D4—2624no—
211C2×D4—5242no—
212C2×D4—5242no—
213C2×D4—1314no—
214C22×C4—2624no—
215C22×C4—2622no—
216C22×C4—2624no—
217C22×C4—5242no—
218C22×C4—5242no—
219C22×C4—5242no—
220C22×C4—1312no—
221C22×C4—1311no—
222C22×C4—1311no—
223C22×C4—1314no—
224C22×C4C22×D13112yes—
225C22×C4C22×D13111yescharacteristic
226C22×C4C22×D13111yescharacteristic
227C22×C4C22×D13111yescharacteristic
228C22×C4C22×D13112yes—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C22⋊C4—2622no—
233C22⋊C4—2624no—
234C22⋊C4—2624no—
235C22⋊C4—2622no—
236C22⋊C4—2624no—
237C22⋊C4—2624no—
238C22⋊C4—5242no—
239C42—2622no—
240C23—2622no—
241C23—2622no—
242C23—2621no—
243C23—2621no—
244C23—2622no—
245C23—2622no—
246C23—2622no—
247C23—2622no—
248C23—2622no—
249C23—2622no—
250C23—2622no—
251C23—2622no—
252C23—2622no—
253C23—1312no—
254C23—1312no—
255C23—1312no—
256C23—1312no—
257C23C23×D13111yescharacteristic
258D4—2628no—
259D4—5248no—
260D4—5244no—
261D4—5248no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—2622no—
273C2×C4—2621no—
274C2×C4—2622no—
275C2×C4—2622no—
276C2×C4—2622no—
277C2×C4—2622no—
278C2×C4—5244no—
279C2×C4—5248no—
280C2×C4—5244no—
281C2×C4—1312no—
282C2×C4D4×D13112yes—
283C2×C4D4×D13112yes—
284C2×C4#1452351414yes—
285C22—2622no—
286C22—2622no—
287C22—2624no—
288C22—2624no—
289C22—2624no—
290C22—2622no—
291C22—5241no—
292C22—5241no—
293C22—5242no—
294C22—5242no—
295C22—5242no—
296C22—5242no—
297C22—5242no—
298C22—5242no—
299C22—5242no—
300C22—5242no—
301C22—1314no—
302C22C2×D4×D13111yescharacteristic
303C22C2×D4×D13111yescharacteristic
304C22#41×D13111yescharacteristic
305C22C2×#1452351412yes—
306C22#1686229212yes—
307C4—22no—
308C4—24no—
309C4—22no—
310C4—41no—
311C4—42no—
312C4—2622no—
313C4—5241no—
314C4—5242no—
315C2—2624no—
316C2—5242no—
317C2—10482no—
318C2#121×D13111yescharacteristic
319C2#2255386311yescharacteristic
320C2#2255387311yescharacteristic
321C2#2255395312yes—
322C2#2255401512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#398×C131C2
hypercenter#398D131
hypocenterC131#2543
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16862221C2semidirect
2. C2×#16862221C2semidirect
3. C2×#16862208C2semidirect
4. #22553238C2semidirect
5. #22553229C2semidirect
6. #22553222C2semidirect
7. #22553202C2semidirect
8. #22553190C2semidirect
9. #22553190C2semidirect
10. #95×D131C2semidirect
11. #398×C131C2semidirect
51. C2×D524D4semidirect
61. C2×C4×D131D4semidirect
161. C131#2543semidirect
166. #398D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622212
C2×#168622082
#225532382
#225532292
#225532222
#225532022
#225531902
#95×D1312
#398×C1312
#2543131