Finite Group Databasebeta

← #30670448

Subgroups — #30670448

Order 16768.

596 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16862219C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
3#22553239C211yesmaximal, characteristic, complemented
4#22553238C211yesmaximal, characteristic, complemented
5#22553237C211yesmaximal, characteristic, complemented
6#22553212C212yesmaximal, complemented
7#22553190C212yesmaximal, complemented
8#22553188C212yesmaximal, complemented
9#22553176C212yesmaximal, complemented
10#22553159C211yesmaximal, characteristic, complemented
11#397×C131C211yesmaximal, characteristic, complemented
12C22×D524C2211yescharacteristic
13C22×C22⋊D131C2211yescharacteristic
14C22×C22⋊D131C2211yescharacteristic
15C22×C4×D131C2211yescharacteristic
16C22×C4×D131C2212yes—
17#16862219—24no—
18C2×C4⋊Dic131C2211yescharacteristic
19C2×Dic131⋊C4C2211yescharacteristic
20C2×Dic131⋊C4C2212yes—
21C4⋊C4×D131—24no—
22C2×C22⋊Dic131C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×C4×Dic131C2212yes—
28C131⋊#95C2211yescharacteristic
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C22⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2212yes—
35#95×C131C2211yescharacteristic
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C23×D131C2311yescharacteristic
39C2×D524—22no—
40C2×D524—24no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131—24no—
43C2×C22⋊D131—22no—
44C2×C22⋊D131—24no—
45C22×Dic131C2312yes—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C2×C4×D131—24no—
50C2×C4×D131—24no—
51C2×C4×D131—42no—
52C2×C4×D131—42no—
53C2×C4×D131D414yes—
54C4⋊Dic131—22no—
55Dic131⋊C4—22no—
56Dic131⋊C4—24no—
57C22⋊Dic131—42no—
58(C2×D131)⋊C4—24no—
59(C2×D131)⋊C4—22no—
60(C2×D131)⋊C4—42no—
61(C2×D131)⋊C4—42no—
62C4×Dic131—24no—
63C24×C131C2311yescharacteristic
64C22×C4×C131C2312yes—
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C22×C4×C131C2311yescharacteristic
68C4⋊C4×C131—22no—
69C22⋊C4×C131—22no—
70C22⋊C4×C131—42no—
71C22×D131—22no—
72C22×D131—21no—
73C22×D131—21no—
74C22×D131—21no—
75C22×D131—22no—
76C22×D131—22no—
77C22×D131—21no—
78C22×D131—21no—
79C22×D131C2×D412yes—
80C22×D131#4114yes—
81D524—44no—
82C22⋊D131—44no—
83C22⋊D131—44no—
84C2×Dic131—21no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131C2×D412yes—
93C2×Dic131#4114yes—
94C4×D131—28no—
95C4×D131—48no—
96C23×C131—21no—
97C23×C131—22no—
98C23×C131—22no—
99C23×C131—21no—
100C23×C131—21no—
101C23×C131C2411yescharacteristic
102C2×C4×C131—22no—
103C2×C4×C131—21no—
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—22no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131C2×D412yes—
112C22×C131—41no—
113C22×C131—41no—
114C22×C131—41no—
115C22×C131—42no—
116C22×C131—42no—
117C22×C131C22×D411yescharacteristic
118C22×C131C2×#4112yes—
119C22×C131C2×#4112yes—
120C22×C131H(5,𝔽2)11yescharacteristic
121C22×C131H(5,𝔽2)11yescharacteristic
122C2×D131—22no—
123C2×D131—22no—
124C2×D131—24no—
125C2×D131—24no—
126C2×D131—41no—
127C2×D131—41no—
128C2×D131—41no—
129C2×D131—42no—
130C2×D131—42no—
131C2×D131#12414yes—
132C4×C131—22no—
133C4×C131—41no—
134C4×C131—42no—
135C4×C131—42no—
136Dic131—41no—
137Dic131—42no—
138Dic131—42no—
139Dic131#12314yes—
140C2×C131—81no—
141C2×C131C2×#12311yescharacteristic
142C2×C131C2×#12411yescharacteristic
143C2×C131#53511yescharacteristic
144C2×C131#55312yes—
145C2×C131#55312yes—
146D131—24no—
147D131—81no—
148C131#250411yescharacteristic, complemented, Sylow
149#2504—1311nomaximal, Sylow
150C2×#121—1311no—
151C22×C4⋊C4—1311no—
152#397—1311no—
153#397—1311no—
154#397—1311no—
155#397D13111yescharacteristic, complemented
156#390—1312no—
157#386—1312no—
158#386—1312no—
159#382—1312no—
160C4×C22⋊C4—1311no—
161C22×D4—1311no—
162C22×D4—1311no—
163C22×D4—1311no—
164C23×C4—1311no—
165C23×C4—1312no—
166#121—2624no—
167C2×C4⋊C4—2624no—
168C2×C4⋊C4—1312no—
169C2×C4⋊C4—1311no—
170C2×C4⋊C4—1311no—
171C2×C4⋊C4C2×D13111yescharacteristic
172C2×C22⋊C4—1311no—
173C2×C22⋊C4—1312no—
174C2×C22⋊C4—1312no—
175C2×C22⋊C4—1312no—
176C2×C22⋊C4—1312no—
177C2×C22⋊C4C2×D13111yescharacteristic
178C2×C22⋊C4C2×D13112yes—
179C2×C42—1312no—
180#95—1311no—
181#95—1312no—
182#95—1312no—
183#95—1312no—
184#95C2×D13111yescharacteristic
185#95C2×D13112yes—
186C24—1311no—
187C24—1311no—
188C24C22×D13111yescharacteristic
189C2×D4—2622no—
190C2×D4—2624no—
191C2×D4—2622no—
192C2×D4—2624no—
193C2×D4—2622no—
194C2×D4—2624no—
195C22×C4—2624no—
196C22×C4—2624no—
197C22×C4—5242no—
198C22×C4—5242no—
199C22×C4—1311no—
200C22×C4—1312no—
201C22×C4—1312no—
202C22×C4—1314no—
203C22×C4—1311no—
204C22×C4C22×D13111yescharacteristic
205C22×C4C22×D13112yes—
206C22×C4C22×D13112yes—
207C22×C4C22×D13111yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—2624no—
210C4⋊C4—2622no—
211C4⋊C4—2622no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—2624no—
215C22⋊C4—2622no—
216C22⋊C4—5242no—
217C22⋊C4—5242no—
218C22⋊C4—5242no—
219C42—2624no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—2622no—
226C23—2621no—
227C23—2621no—
228C23—2621no—
229C23—2622no—
230C23—2622no—
231C23—2621no—
232C23—2621no—
233C23—1312no—
234C23—1314no—
235C23C23×D13111yescharacteristic
236D4—5244no—
237D4—5244no—
238D4—5244no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—2628no—
249C2×C4—2621no—
250C2×C4—2622no—
251C2×C4—2622no—
252C2×C4—2622no—
253C2×C4—2622no—
254C2×C4—2622no—
255C2×C4—2622no—
256C2×C4—2622no—
257C2×C4—5248no—
258C2×C4—1312no—
259C2×C4—1314no—
260C2×C4D4×D13112yes—
261C22—41no—
262C22—42no—
263C22—42no—
264C22—41no—
265C22—41no—
266C22—2622no—
267C22—2624no—
268C22—2624no—
269C22—2622no—
270C22—5241no—
271C22—5242no—
272C22—5242no—
273C22—5241no—
274C22—5241no—
275C22—1314no—
276C22C2×D4×D13111yescharacteristic
277C22#41×D13112yes—
278C22#41×D13112yes—
279C22#1686229111yescharacteristic
280C22#1686229211yescharacteristic
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—5242no—
286C4—5242no—
287C4—5241no—
288C4—1314no—
289C2—2624no—
290C2—81no—
291C2—10481no—
292C2#123×D13111yescharacteristic
293C2#124×D13111yescharacteristic
294C2#2255387811yescharacteristic
295C2#2255400712yes—
296C2#2255401312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#397×C131C2
prime radicalC2×#16862219C2
hypercenter#397D131
hypocenterC131#2504
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D131C2semidirect
3. #22553239C2semidirect
4. #22553238C2semidirect
5. #22553237C2semidirect
6. #22553212C2semidirect
7. #22553190C2semidirect
8. #22553188C2semidirect
9. #22553176C2semidirect
10. #22553159C2semidirect
11. #397×C131C2semidirect
148. C131#2504semidirect
155. #397D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622192
C2×C4⋊C4×D1312
#225532392
#225532382
#225532372
#225532122
#225531902
#225531882
#225531762
#225531592
#397×C1312
#2504131