Finite Group Databasebeta

← #34052738

Subgroups — #34052738

Order 19072.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#18011898C211yesmaximal, characteristic, complemented
2C2×#18011897C211yesmaximal, characteristic, complemented
3#23749540C212yesmaximal, complemented
4#23749539C212yesmaximal, complemented
5#23749538C211yesmaximal, characteristic, complemented
6#23749537C212yesmaximal, complemented
7#23749537C211yesmaximal, characteristic, complemented
8#23749533C212yesmaximal, complemented
9#23749533C211yesmaximal, characteristic, complemented
10#23749489C211yesmaximal, characteristic, complemented
11#392×C149C211yesmaximal, characteristic, complemented
12C22×Dic298C2211yescharacteristic
13C22×Dic298C2211yescharacteristic
14C22×C22⋊D149C2211yescharacteristic
15C22×C22⋊D149C2212yes—
16#18011898—24no—
17#18011897—24no—
18C2×#14558715C2212yes—
19C2×C22⋊Dic149C2212yes—
20C2×C22⋊Dic149C2211yescharacteristic
21C2×C22⋊Dic149C2212yes—
22C2×C22⋊Dic149C2212yes—
23C2×C22⋊Dic149C2212yes—
24C2×#14558713C2211yescharacteristic
25C2×#14558713C2211yescharacteristic
26C2×#14558713C2212yes—
27C2×C4×Dic149C2211yescharacteristic
28#18011818C2211yescharacteristic
29#18011818C2212yes—
30#18011818C2212yes—
31#18011818C2212yes—
32C22×D4×C149C2211yescharacteristic
33C22×D4×C149C2212yes—
34C2×C22⋊C4×C149C2211yescharacteristic
35C2×C22⋊C4×C149C2212yes—
36#95×C149C2211yescharacteristic
37C23×D149C2311yescharacteristic
38C2×Dic298—22no—
39C2×Dic298—22no—
40C2×Dic298—42no—
41C2×Dic298—42no—
42C2×C22⋊D149—22no—
43C2×C22⋊D149—42no—
44C2×C22⋊D149—42no—
45C2×C22⋊D149—42no—
46C2×C22⋊D149—42no—
47C22×Dic149C2312yes—
48C22×Dic149C2311yescharacteristic
49C22×Dic149C2312yes—
50C22×Dic149C2311yescharacteristic
51C22×Dic149C2311yescharacteristic
52#14558715—42no—
53C22⋊Dic149—22no—
54C22⋊Dic149—24no—
55C22⋊Dic149—24no—
56C22⋊Dic149—42no—
57C22⋊Dic149—42no—
58#14558713—22no—
59#14558713—22no—
60#14558713—42no—
61C4×Dic149D414yescomplemented
62C24×C149C2311yescharacteristic
63C24×C149C2312yes—
64C2×D4×C149—22no—
65C2×D4×C149—42no—
66C2×D4×C149—42no—
67C2×D4×C149—42no—
68C2×D4×C149—42no—
69C22×C4×C149C2311yescharacteristic
70C22×C4×C149C2312yes—
71C22×C4×C149C2311yescharacteristic
72C22⋊C4×C149—22no—
73C22⋊C4×C149—42no—
74C22×D149—21no—
75C22×D149—22no—
76C22×D149—21no—
77C22×D149—22no—
78C22×D149—21no—
79Dic298—44no—
80Dic298—44no—
81C22⋊D149—44no—
82C22⋊D149—48no—
83C2×Dic149—22no—
84C2×Dic149—21no—
85C2×Dic149—21no—
86C2×Dic149—21no—
87C2×Dic149—22no—
88C2×Dic149—22no—
89C2×Dic149—22no—
90C2×Dic149—22no—
91C2×Dic149—22no—
92C2×Dic149—22no—
93C2×Dic149—22no—
94C2×Dic149C2×D412yes—
95C2×Dic149C2×D412yes—
96C23×C149—21no—
97C23×C149—22no—
98C23×C149—22no—
99C23×C149—22no—
100C23×C149—22no—
101C23×C149—22no—
102C23×C149—22no—
103C23×C149—21no—
104C23×C149—22no—
105C23×C149—22no—
106C23×C149—21no—
107C23×C149—22no—
108C23×C149C2411yescharacteristic
109D4×C149—44no—
110D4×C149—48no—
111C2×C4×C149—22no—
112C2×C4×C149—21no—
113C2×C4×C149—22no—
114C2×C4×C149—22no—
115C2×C4×C149—22no—
116C2×C4×C149—22no—
117C2×C4×C149C2×D412yes—
118C22×C149—42no—
119C22×C149—42no—
120C22×C149—41no—
121C22×C149—41no—
122C22×C149—41no—
123C22×C149—42no—
124C22×C149—42no—
125C22×C149—42no—
126C22×C149—42no—
127C22×C149—42no—
128C22×C149—42no—
129C22×C149—42no—
130C22×C149C22×D411yescharacteristic
131C22×C149H(5,𝔽2)12yes—
132C22×C149H(5,𝔽2)12yes—
133C22×C149H(5,𝔽2)11yescharacteristic
134C22×C1492−1+411yescharacteristic
135C2×D149—41no—
136C2×D149—41no—
137C2×D149—41no—
138C2×D149—42no—
139C2×D149—42no—
140C4×C149—22no—
141C4×C149—41no—
142C4×C149—42no—
143Dic149—22no—
144Dic149—22no—
145Dic149—41no—
146Dic149—42no—
147Dic149—42no—
148C2×C149—82no—
149C2×C149—81no—
150C2×C149#53511yescharacteristic
151C2×C149#53511yescharacteristic
152C2×C149#53711yescharacteristic
153C2×C149H(3,𝔽4)12yes—
154C2×C149#56212yes—
155D149—81no—
156C149#281911yescharacteristic, complemented, Sylow
157#2819—1491nomaximal, Sylow
158C2×#124—1491no—
159C2×#124—1491no—
160#394—1492no—
161#394—1492no—
162#393—1492no—
163#393—1491no—
164#393—1491no—
165#392—1492no—
166#392—1491no—
167#392D14911yescharacteristic, complemented
168#382—1491no—
169C22×Q8—1491no—
170C22×Q8—1491no—
171C22×D4—1491no—
172C22×D4—1492no—
173C22×D4C2×D14911yescharacteristic
174C22×D4C2×D14912yes—
175#124—2984no—
176#124—2984no—
177C2×C4⋊C4—1492no—
178C2×C22⋊C4—1492no—
179C2×C22⋊C4—1492no—
180C2×C22⋊C4—1491no—
181C2×C22⋊C4—1491no—
182C2×C22⋊C4—1491no—
183C2×C22⋊C4—1492no—
184C2×C22⋊C4—1492no—
185C2×C22⋊C4—1492no—
186C2×C22⋊C4C2×D14911yescharacteristic
187C2×C22⋊C4C2×D14912yes—
188C2×C42—1491no—
189#95—1491no—
190#95—1492no—
191#95—1492no—
192#95—1492no—
193#95C2×D14911yescharacteristic
194C24—1491no—
195C24C22×D14911yescharacteristic
196C24C22×D14912yes—
197C2×Q8—2982no—
198C2×Q8—2982no—
199C2×Q8—5962no—
200C2×Q8—5962no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—2982no—
207C2×D4—5962no—
208C2×D4—5962no—
209C2×D4—5962no—
210C2×D4—5962no—
211C22×C4—1492no—
212C22×C4—1492no—
213C22×C4—1491no—
214C22×C4—1491no—
215C22×C4—1491no—
216C22×C4C22×D14912yes—
217C22×C4C22×D14911yescharacteristic
218C22×C4C22×D14911yescharacteristic
219C4⋊C4—5962no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—2982no—
223C22⋊C4—2982no—
224C22⋊C4—2984no—
225C22⋊C4—2984no—
226C22⋊C4—2982no—
227C22⋊C4—5962no—
228C22⋊C4—5962no—
229C22⋊C4—5962no—
230C42—1494no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—22no—
243C23—2981no—
244C23—2982no—
245C23—2982no—
246C23—2981no—
247C23—2981no—
248C23C23×D14911yescharacteristic
249Q8—5964no—
250Q8—5964no—
251D4—44no—
252D4—48no—
253D4—5964no—
254D4—5968no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—2981no—
262C2×C4—2982no—
263C2×C4—2981no—
264C2×C4—2981no—
265C2×C4—2982no—
266C2×C4—2982no—
267C2×C4—2982no—
268C2×C4—2982no—
269C2×C4—2982no—
270C2×C4—2982no—
271C2×C4—2982no—
272C2×C4—1492no—
273C2×C4—1492no—
274C2×C4D4×D14912yes—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—5961no—
288C22—5961no—
289C22—5961no—
290C22—5962no—
291C22—5962no—
292C22C2×D4×D14911yescharacteristic
293C22#1801195612yes—
294C22#1801195612yes—
295C22#1801195611yescharacteristic
296C22#1801195811yescharacteristic
297C4—22no—
298C4—41no—
299C4—42no—
300C4—2982no—
301C4—2982no—
302C4—5961no—
303C4—5962no—
304C4—5962no—
305C2—81no—
306C2—82no—
307C2—11921no—
308C2#2375018811yescharacteristic
309C2#2375019011yescharacteristic
310C2#2375020411yescharacteristic
311C2#2375036812yes—
312C2#2375037012yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#392×C149C2
hypercenter#392D149
hypocenterC149#2819
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#18011898C2semidirect
2. C2×#18011897C2semidirect
3. #23749540C2semidirect
4. #23749539C2semidirect
5. #23749538C2semidirect
6. #23749537C2semidirect
7. #23749537C2semidirect
8. #23749533C2semidirect
9. #23749533C2semidirect
10. #23749489C2semidirect
11. #392×C149C2semidirect
61. C4×Dic149D4semidirect
156. C149#2819semidirect
167. #392D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118982
C2×#180118972
#237495402
#237495392
#237495382
#237495372
#237495332
#237494892
#392×C1492
#2819149