Finite Group Databasebeta

← #34077112

Subgroups — #34077112

Order 19328.

578 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 133 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×#18012713C211yesmaximal, characteristic, complemented
2#23752151C211yesmaximal, characteristic, complemented
3#23752148C211yesmaximal, characteristic, complemented
4#23752134C212yesmaximal, complemented
5#23752129C211yesmaximal, characteristic, complemented
6#23752128C212yesmaximal, complemented
7#23752127C212yesmaximal, complemented
8#23752121C211yesmaximal, characteristic, complemented
9#23752120C212yesmaximal, complemented
10#23752072C211yesmaximal, characteristic, complemented
11C2×#123×C151C211yesmaximal, characteristic, complemented
12C22×D604C2211yescharacteristic
13C22×C22⋊D151C2212yes—
14C22×C22⋊D151C2212yes—
15#18012713—24no—
16C2×C4⋊Dic151C2211yescharacteristic
17C2×C4⋊Dic151C2212yes—
18C2×Dic151⋊C4C2212yes—
19C2×C22⋊Dic151C2212yes—
20C2×C22⋊Dic151C2212yes—
21C2×(C2×D151)⋊C4C2212yes—
22C2×(C2×D151)⋊C4C2212yes—
23C2×(C2×D151)⋊C4C2212yes—
24C2×(C2×D151)⋊C4C2212yes—
25C151⋊#95C2211yescharacteristic
26C151⋊#95C2211yescharacteristic
27C151⋊#95C2212yes—
28C151⋊#95C2212yes—
29C22×D4×C151C2211yescharacteristic
30C23×C4×C151C2211yescharacteristic
31#123×C151—24no—
32C2×C4⋊C4×C151C2211yescharacteristic
33C2×C4⋊C4×C151C2211yescharacteristic
34C2×C22⋊C4×C151C2211yescharacteristic
35C2×C22⋊C4×C151C2211yescharacteristic
36C2×C22⋊C4×C151C2211yescharacteristic
37C23×D151C2312yes—
38C2×D604—22no—
39C2×D604—42no—
40C2×C22⋊D151—24no—
41C2×C22⋊D151—42no—
42C2×C22⋊D151—42no—
43C2×C22⋊D151—42no—
44C2×C22⋊D151—42no—
45C2×C22⋊D151—42no—
46C22×Dic151C2312yes—
47C22×Dic151C2312yes—
48C22×Dic151C2312yes—
49C4⋊Dic151—22no—
50C4⋊Dic151—42no—
51Dic151⋊C4—42no—
52C22⋊Dic151—42no—
53C22⋊Dic151—42no—
54(C2×D151)⋊C4—24no—
55(C2×D151)⋊C4—42no—
56(C2×D151)⋊C4—42no—
57(C2×D151)⋊C4—42no—
58C24×C151C2311yescharacteristic
59C24×C151C2311yescharacteristic
60C2×D4×C151—22no—
61C2×D4×C151—42no—
62C22×C4×C151—42no—
63C22×C4×C151D414yescomplemented
64C22×C4×C151C2311yescharacteristic
65C22×C4×C151C2311yescharacteristic
66C22×C4×C151C2311yescharacteristic
67C22×C4×C151C2311yescharacteristic
68C22×C4×C151C2311yescharacteristic
69C4⋊C4×C151—22no—
70C4⋊C4×C151—22no—
71C22⋊C4×C151—22no—
72C22⋊C4×C151—22no—
73C22⋊C4×C151—22no—
74C22×D151—22no—
75C22×D151—22no—
76C22×D151—22no—
77C22×D151—22no—
78C22×D151—22no—
79C22×D151—22no—
80C22×D151—22no—
81D604—44no—
82C22⋊D151—48no—
83C22⋊D151—48no—
84C2×Dic151—22no—
85C2×Dic151—22no—
86C2×Dic151—22no—
87C2×Dic151—22no—
88C2×Dic151—22no—
89C2×Dic151—22no—
90C2×Dic151—22no—
91C2×Dic151—22no—
92C2×Dic151—22no—
93C23×C151—21no—
94C23×C151—22no—
95C23×C151—22no—
96C23×C151—21no—
97C23×C151—21no—
98C23×C151—22no—
99C23×C151—21no—
100C23×C151—22no—
101C23×C151—21no—
102C23×C151C2×D412yes—
103C23×C151C2411yescharacteristic
104D4×C151—44no—
105C2×C4×C151—22no—
106C2×C4×C151—22no—
107C2×C4×C151—21no—
108C2×C4×C151—21no—
109C2×C4×C151—21no—
110C2×C4×C151—21no—
111C2×C4×C151—22no—
112C2×C4×C151—22no—
113C2×C4×C151—21no—
114C2×C4×C151—44no—
115C2×C4×C151C2×D412yes—
116C2×C4×C151C2×D412yes—
117C22×C151—22no—
118C22×C151—22no—
119C22×C151—22no—
120C22×C151—42no—
121C22×C151—42no—
122C22×C151—41no—
123C22×C151—41no—
124C22×C151—41no—
125C22×C151—42no—
126C22×C151—42no—
127C22×C151C22×D411yescharacteristic
128C22×C151H(5,𝔽2)12yes—
129C22×C151H(5,𝔽2)12yes—
130C22×C151H(5,𝔽2)11yescharacteristic
131C22×C1512−1+411yescharacteristic
132C2×D151—42no—
133C2×D151—42no—
134C2×D151—42no—
135C2×D151—42no—
136C2×D151—42no—
137C2×D151—42no—
138C2×D151—42no—
139C4×C151—22no—
140C4×C151—22no—
141C4×C151—41no—
142C4×C151—41no—
143C4×C151—41no—
144Dic151—42no—
145Dic151—42no—
146Dic151—42no—
147C2×C151—42no—
148C2×C151—81no—
149C2×C151#53411yescharacteristic
150C2×C151#53511yescharacteristic
151C2×C151#53711yescharacteristic
152C2×C151#56012yes—
153C2×C151#56212yes—
154D151—82no—
155C151#281311yescharacteristic, complemented, Sylow
156#2813—1511nomaximal, Sylow
157C2×#123D15111yescharacteristic, complemented
158C2×#121—1511no—
159#400—1511no—
160#399—1511no—
161#396—1512no—
162#394—1511no—
163#394—1512no—
164#394—1512no—
165#392—1512no—
166#392—1511no—
167#380—1511no—
168C22×D4—1512no—
169C22×D4—1512no—
170C22×D4—1511no—
171C22×D4C2×D15111yescharacteristic
172C23×C4C2×D15111yescharacteristic
173#123—24no—
174#121—3024no—
175C2×C4⋊C4—1511no—
176C2×C4⋊C4—1512no—
177C2×C4⋊C4—1512no—
178C2×C4⋊C4C2×D15111yescharacteristic
179C2×C4⋊C4C2×D15111yescharacteristic
180C2×C22⋊C4—1512no—
181C2×C22⋊C4—1512no—
182C2×C22⋊C4—1512no—
183C2×C22⋊C4—1512no—
184C2×C22⋊C4—1512no—
185C2×C22⋊C4—1512no—
186C2×C22⋊C4C2×D15111yescharacteristic
187C2×C22⋊C4C2×D15111yescharacteristic
188C2×C22⋊C4C2×D15111yescharacteristic
189#95—1511no—
190#95—1511no—
191#95—1512no—
192#95—1512no—
193C24—1512no—
194C24C22×D15111yescharacteristic
195C24C22×D15111yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—3024no—
199C2×D4—3022no—
200C2×D4—6042no—
201C2×D4—6042no—
202C2×D4—6042no—
203C2×D4—6042no—
204C2×D4—6042no—
205C2×D4—6042no—
206C22×C4—42no—
207C22×C4—1512no—
208C22×C4—1512no—
209C22×C4—1512no—
210C22×C4C22⋊D15114yescomplemented
211C22×C4C22×D15111yescharacteristic
212C22×C4C22×D15111yescharacteristic
213C22×C4C22×D15111yescharacteristic
214C22×C4C22×D15111yescharacteristic
215C22×C4C22×D15111yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—3022no—
219C4⋊C4—6042no—
220C4⋊C4—6042no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—3024no—
225C22⋊C4—6042no—
226C22⋊C4—6042no—
227C22⋊C4—6042no—
228C22⋊C4—6042no—
229C22⋊C4—6042no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—3022no—
240C23—3022no—
241C23—3022no—
242C23—3022no—
243C23—3022no—
244C23—3022no—
245C23—3022no—
246C23C2×C22⋊D15112yes—
247C23C23×D15111yescharacteristic
248D4—44no—
249D4—6044no—
250D4—6048no—
251D4—6048no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—3022no—
263C2×C4—3022no—
264C2×C4—3022no—
265C2×C4—3022no—
266C2×C4—3022no—
267C2×C4—3022no—
268C2×C4—3022no—
269C2×C4—3022no—
270C2×C4—3022no—
271C2×C4C2×C22⋊D15112yes—
272C2×C4C2×C22⋊D15112yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—6042no—
284C22—6042no—
285C22—6042no—
286C22—6042no—
287C22—6042no—
288C22—6042no—
289C22—6042no—
290C22C22×C22⋊D15111yescharacteristic
291C22#1801278412yes—
292C22#1801278512yes—
293C22#1801278511yescharacteristic
294C22#1801278611yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—6042no—
301C4—6042no—
302C4—6042no—
303C2—42no—
304C2—81no—
305C2—12082no—
306C2#2375277111yescharacteristic
307C2#2375277611yescharacteristic
308C2#2375279011yescharacteristic
309C2#2375295212yes—
310C2#2375295912yes—
subgroupG/H
center, FrattiniC23C23×D151
commutator, socle, cosocleC23×C151C24
FittingC2×#123×C151C2
hypercenterC2×#123D151
hypocenterC151#2813
LCSG ▷ C23×C151 ▷ C151
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#18012713C2semidirect
2. #23752151C2semidirect
3. #23752148C2semidirect
4. #23752134C2semidirect
5. #23752129C2semidirect
6. #23752128C2semidirect
7. #23752127C2semidirect
8. #23752121C2semidirect
9. #23752120C2semidirect
10. #23752072C2semidirect
11. C2×#123×C151C2semidirect
63. C22×C4×C151D4semidirect
155. C151#2813semidirect
157. C2×#123D151semidirect
210. C22×C4C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127132
#237521512
#237521482
#237521342
#237521292
#237521282
#237521272
#237521212
#237521202
#237520722
C2×#123×C1512
#2813151