Finite Group Databasebeta

← #30670867

Subgroups — #30670867

Order 16768.

604 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16862224C212yesmaximal, complemented
2C2×#16862211C211yesmaximal, characteristic, complemented
3C2×#16862208C211yesmaximal, characteristic
4#22553243C211yesmaximal, characteristic, complemented
5#22553230C211yesmaximal, characteristic
6#22553229C211yesmaximal, characteristic
7#22553202C212yesmaximal, complemented
8#22553201C212yesmaximal, complemented
9#22553176C211yesmaximal, characteristic, complemented
10#22553171C212yesmaximal, complemented
11#398×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×D524C2211yescharacteristic
14C22×C4×D131C2212yes—
15#16862224—28no—
16#16862211—24no—
17#16862208—24no—
18C2×Dic131⋊C4C2211yescharacteristic
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2212yes—
21C2×Dic131⋊C4C2211yescharacteristic
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×C4×Dic131C2211yescharacteristic
28C2×C4×Dic131C2211yescharacteristic
29C2×C4×Dic131C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C131⋊#95C2211yescharacteristic
33C2×C4⋊C4×C131C2212yes—
34C2×C4⋊C4×C131C2211yescharacteristic
35C2×C4⋊C4×C131C2211yescharacteristic
36#95×C131C2212yes—
37#95×C131C2211yescharacteristic
38C23×D131C2312yes—
39C2×Dic262—22no—
40C2×Dic262—22no—
41C2×Dic262D414yescomplemented
42C2×D524—24no—
43C2×D524—22no—
44C22×Dic131C2312yes—
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2311yescharacteristic
49C2×C4×D131—24no—
50C2×C4×D131—24no—
51C2×C4×D131—24no—
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54Dic131⋊C4—24no—
55Dic131⋊C4—22no—
56(C2×D131)⋊C4—24no—
57(C2×D131)⋊C4—24no—
58(C2×D131)⋊C4—24no—
59(C2×D131)⋊C4—24no—
60(C2×D131)⋊C4—24no—
61C4×Dic131—22no—
62C4×Dic131—22no—
63C4×Dic131D414yes—
64C22×C4×C131C2312yes—
65C22×C4×C131C2312yes—
66C22×C4×C131C2311yescharacteristic
67C22×C4×C131C2311yescharacteristic
68C22×C4×C131C2311yescharacteristic
69C4⋊C4×C131—22no—
70C4⋊C4×C131—22no—
71C4⋊C4×C131—24no—
72C22×D131—22no—
73C22×D131—22no—
74C22×D131—22no—
75C22×D131—22no—
76C22×D131—22no—
77C22×D131—22no—
78C22×D131#4114yes—
79Dic262—28no—
80D524—28no—
81C2×Dic131—21no—
82C2×Dic131—21no—
83C2×Dic131—22no—
84C2×Dic131—21no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131C2×D412yes—
90C2×Dic131C2×D412yes—
91C2×Dic131C2×D412yes—
92C2×Dic131C2×D412yes—
93C2×Dic131#4112yes—
94C4×D131—48no—
95C23×C131C2411yescharacteristic
96C2×C4×C131—21no—
97C2×C4×C131—22no—
98C2×C4×C131—21no—
99C2×C4×C131—21no—
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131C2×D412yes—
108C2×C4×C131C2×D412yes—
109C22×C131C22×D411yescharacteristic
110C22×C131C22×D411yescharacteristic
111C22×C131C2×#4111yescharacteristic
112C22×C131C2×#4112yes—
113C22×C1312−1+412yes—
114C2×D131—24no—
115C2×D131—24no—
116C2×D131—24no—
117C2×D131—42no—
118C2×D131—42no—
119C2×D131—42no—
120C2×D131—42no—
121C4×C131—22no—
122C4×C131—22no—
123C4×C131—42no—
124C4×C131—41no—
125C4×C131—42no—
126Dic131—22no—
127Dic131—22no—
128Dic131—41no—
129Dic131—42no—
130Dic131#12114yes—
131C2×C131C2×#12111yescharacteristic
132C2×C131#52511yescharacteristic
133C2×C131#53611yescharacteristic
134C2×C131#54812yes—
135C2×C131#55212yes—
136D131—44no—
137C131#253611yescharacteristic, complemented, Sylow
138#2536—1311nomaximal, Sylow
139C2×C22⋊Q8—1312no—
140C2×C4×Q8—1311no—
141C2×C4×D4—1311no—
142#398—1311no—
143#398D13111yescharacteristic, complemented
144#394—1311no—
145#394—1311no—
146#388—1312no—
147#388—1312no—
148#382—1311no—
149#380—1312no—
150C22×Q8—1311no—
151C22×D4—1311no—
152C23×C4—1312no—
153C22⋊Q8—2628no—
154C4×Q8—2624no—
155C4×D4—2624no—
156C2×C4⋊C4—1311no—
157C2×C4⋊C4—1311no—
158C2×C4⋊C4—1312no—
159C2×C4⋊C4—1312no—
160C2×C4⋊C4C2×D13111yescharacteristic
161C2×C4⋊C4C2×D13111yescharacteristic
162C2×C4⋊C4C2×D13112yes—
163C2×C22⋊C4—1312no—
164C2×C22⋊C4—1312no—
165C2×C22⋊C4—1312no—
166C2×C22⋊C4—1312no—
167C2×C22⋊C4—1312no—
168C2×C42—1311no—
169C2×C42—1311no—
170C2×C42—1311no—
171#95—1312no—
172#95—1312no—
173#95—1311no—
174#95C2×D13112yes—
175#95C2×D13111yescharacteristic
176C24—1312no—
177C2×Q8—2622no—
178C2×Q8—2622no—
179C2×Q8—1314no—
180C2×D4—2624no—
181C2×D4—2622no—
182C22×C4—2624no—
183C22×C4—2624no—
184C22×C4—2624no—
185C22×C4—1312no—
186C22×C4—1311no—
187C22×C4—1311no—
188C22×C4—1311no—
189C22×C4—1311no—
190C22×C4C22×D13112yes—
191C22×C4C22×D13112yes—
192C22×C4C22×D13111yescharacteristic
193C22×C4C22×D13111yescharacteristic
194C22×C4C22×D13111yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—2622no—
199C4⋊C4—2624no—
200C4⋊C4—2624no—
201C4⋊C4—2622no—
202C22⋊C4—2624no—
203C22⋊C4—2624no—
204C22⋊C4—2624no—
205C22⋊C4—2624no—
206C22⋊C4—2624no—
207C42—2622no—
208C42—2622no—
209C42—1314no—
210C23—2622no—
211C23—2622no—
212C23—2622no—
213C23—2622no—
214C23—2622no—
215C23—2622no—
216C23—1314no—
217C23C23×D13111yescharacteristic
218Q8—2628no—
219D4—2628no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—2621no—
232C2×C4—2621no—
233C2×C4—2622no—
234C2×C4—2621no—
235C2×C4—2622no—
236C2×C4—2622no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—5248no—
240C2×C4—1312no—
241C2×C4—1312no—
242C2×C4—1312no—
243C2×C4—1312no—
244C2×C4—1312no—
245C2×C4D4×D13112yes—
246C2×C4D4×D13112yes—
247C22—2624no—
248C22—2624no—
249C22—2624no—
250C22—5242no—
251C22—5242no—
252C22—5242no—
253C22—5242no—
254C22C2×D4×D13111yescharacteristic
255C22C2×D4×D13111yescharacteristic
256C22#41×D13112yes—
257C22#41×D13111yescharacteristic
258C22#1686229412yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—2622no—
265C4—2622no—
266C4—5242no—
267C4—5241no—
268C4—1314no—
269C2—5244no—
270C2#121×D13111yescharacteristic
271C2#2255382011yescharacteristic
272C2#2255388711yescharacteristic
273C2#2255397212yes—
274C2#2255399412yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#398×C131C2
prime radicalC22×D524C22
hypercenter#398D131
hypocenterC131#2536
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16862224C2semidirect
2. C2×#16862211C2semidirect
4. #22553243C2semidirect
7. #22553202C2semidirect
8. #22553201C2semidirect
9. #22553176C2semidirect
10. #22553171C2semidirect
11. #398×C131C2semidirect
41. C2×Dic262D4semidirect
137. C131#2536semidirect
143. #398D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622242
C2×#168622112
C2×#168622082
#225532432
#225532302
#225532292
#225532022
#225532012
#225531762
#225531712
#398×C1312
#2536131