Finite Group Databasebeta

← #34052735

Subgroups — #34052735

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×#18011897C212yesmaximal, complemented
2#23749542C211yesmaximal, characteristic, complemented
3#23749541C212yesmaximal, complemented
4#23749538C212yesmaximal, complemented
5#23749537C212yesmaximal, complemented
6#23749534C212yesmaximal, complemented
7#23749533C212yesmaximal, complemented
8#23749490C211yesmaximal, characteristic, complemented
9#394×C149C211yesmaximal, characteristic, complemented
10C22×Dic298C2212yes—
11C22×D596C2211yescharacteristic
12C22×C22⋊D149C2212yes—
13C22×C22⋊D149C2212yes—
14#18011897—28no—
15C2×C4⋊Dic149C2211yescharacteristic
16C2×C22⋊Dic149C2212yes—
17C2×C22⋊Dic149C2212yes—
18C2×#14558713C2212yes—
19C2×#14558713C2212yes—
20C2×#14558713C2212yes—
21C2×#14558713C2212yes—
22C2×C4×Dic149C2211yescharacteristic
23#18011818C2212yes—
24#18011818C2212yes—
25#18011818C2211yescharacteristic
26#18011818C2212yes—
27C22×D4×C149C2211yescharacteristic
28C2×C4⋊C4×C149C2211yescharacteristic
29C2×C22⋊C4×C149C2212yes—
30C2×C22⋊C4×C149C2212yes—
31#95×C149C2211yescharacteristic
32C23×D149C2312yes—
33C2×Dic298—24no—
34C2×Dic298—42no—
35C2×Dic298—42no—
36C2×D596—42no—
37C2×D596—41no—
38C2×C22⋊D149—24no—
39C2×C22⋊D149—42no—
40C2×C22⋊D149—42no—
41C2×C22⋊D149—42no—
42C2×C22⋊D149—42no—
43C2×C22⋊D149—42no—
44C22×Dic149C2312yes—
45C22×Dic149C2312yes—
46C22×Dic149C2312yes—
47C4⋊Dic149—41no—
48C22⋊Dic149—24no—
49C22⋊Dic149—42no—
50#14558713—24no—
51#14558713—24no—
52#14558713—42no—
53#14558713—42no—
54C4×Dic149D414yescomplemented
55C24×C149C2312yes—
56C2×D4×C149—41no—
57C2×D4×C149—42no—
58C22×C4×C149C2311yescharacteristic
59C22×C4×C149C2312yes—
60C22×C4×C149C2311yescharacteristic
61C22×C4×C149C2311yescharacteristic
62C4⋊C4×C149—41no—
63C22⋊C4×C149—24no—
64C22⋊C4×C149—42no—
65C22×D149—22no—
66C22×D149—22no—
67C22×D149—22no—
68C22×D149—22no—
69C22×D149—22no—
70C22×D149—22no—
71C22×D149—22no—
72Dic298—48no—
73D596—44no—
74C22⋊D149—48no—
75C22⋊D149—48no—
76C2×Dic149—22no—
77C2×Dic149—22no—
78C2×Dic149—22no—
79C2×Dic149—22no—
80C2×Dic149—22no—
81C2×Dic149—22no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149C2×D414yes—
85C23×C149—22no—
86C23×C149—22no—
87C23×C149—22no—
88C23×C149—22no—
89C23×C149—22no—
90C23×C149—22no—
91C23×C149—22no—
92C23×C149C2411yescharacteristic
93D4×C149—44no—
94C2×C4×C149—21no—
95C2×C4×C149—21no—
96C2×C4×C149—22no—
97C2×C4×C149—22no—
98C2×C4×C149—21no—
99C2×C4×C149—22no—
100C2×C4×C149—22no—
101C2×C4×C149—21no—
102C2×C4×C149—22no—
103C2×C4×C149C2×D412yes—
104C22×C149—42no—
105C22×C149—42no—
106C22×C149—42no—
107C22×C149—42no—
108C22×C149—42no—
109C22×C149—42no—
110C22×C149—42no—
111C22×C149C22×D411yescharacteristic
112C22×C149H(5,𝔽2)12yes—
113C22×C149H(5,𝔽2)11yescharacteristic
114C22×C149H(5,𝔽2)12yes—
115C22×C1492−1+411yescharacteristic
116C2×D149—42no—
117C2×D149—42no—
118C2×D149—42no—
119C2×D149—42no—
120C2×D149—42no—
121C2×D149—42no—
122C2×D149—42no—
123C4×C149—22no—
124C4×C149—42no—
125C4×C149—41no—
126C4×C149—41no—
127Dic149—24no—
128Dic149—42no—
129Dic149—42no—
130C2×C149—82no—
131C2×C149#53512yes—
132C2×C149#53711yescharacteristic
133C2×C149H(3,𝔽4)11yescharacteristic
134C2×C149H(3,𝔽4)11yescharacteristic
135C2×C149#56212yes—
136D149—82no—
137C149#281911yescharacteristic, complemented, Sylow
138#2819—1491nomaximal, Sylow
139C2×#124—1492no—
140#394—1491no—
141#394—1492no—
142#394D14911yescharacteristic, complemented
143#393—1492no—
144#393—1492no—
145#392—1492no—
146#392—1492no—
147#382—1491no—
148C22×Q8—1492no—
149C22×D4—1492no—
150C22×D4—1492no—
151C22×D4—1491no—
152C22×D4C2×D14911yescharacteristic
153#124—2988no—
154C2×C4⋊C4—1491no—
155C2×C4⋊C4C2×D14911yescharacteristic
156C2×C22⋊C4—1492no—
157C2×C22⋊C4—1492no—
158C2×C22⋊C4—1492no—
159C2×C22⋊C4—1492no—
160C2×C22⋊C4—1492no—
161C2×C22⋊C4—1492no—
162C2×C22⋊C4C2×D14912yes—
163C2×C22⋊C4C2×D14912yes—
164C2×C42—1491no—
165#95—1492no—
166#95—1492no—
167#95—1492no—
168#95—1491no—
169#95C2×D14911yescharacteristic
170C24—1492no—
171C24C22×D14912yes—
172C2×Q8—2984no—
173C2×Q8—5962no—
174C2×Q8—5962no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—2984no—
178C2×D4—5962no—
179C2×D4—5961no—
180C2×D4—5962no—
181C2×D4—5962no—
182C2×D4—5962no—
183C2×D4—5962no—
184C2×D4—5962no—
185C22×C4—1492no—
186C22×C4—1492no—
187C22×C4—1492no—
188C22×C4C22×D14911yescharacteristic
189C22×C4C22×D14911yescharacteristic
190C22×C4C22×D14912yes—
191C22×C4C22×D14911yescharacteristic
192C4⋊C4—41no—
193C4⋊C4—5961no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—2984no—
197C22⋊C4—2984no—
198C22⋊C4—2984no—
199C22⋊C4—5962no—
200C22⋊C4—5962no—
201C22⋊C4—5962no—
202C42—1494no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—2982no—
211C23—2982no—
212C23—2982no—
213C23—2982no—
214C23—2982no—
215C23—2982no—
216C23—2982no—
217C23C23×D14911yescharacteristic
218Q8—5968no—
219D4—44no—
220D4—5964no—
221D4—5968no—
222D4—5968no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—2982no—
235C2×C4—2982no—
236C2×C4—2982no—
237C2×C4—2982no—
238C2×C4—2982no—
239C2×C4—2982no—
240C2×C4—1494no—
241C2×C4D4×D14912yes—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—5962no—
250C22—5962no—
251C22—5962no—
252C22—5962no—
253C22—5962no—
254C22—5962no—
255C22—5962no—
256C22C2×D4×D14911yescharacteristic
257C22#1801195612yes—
258C22#1801195611yescharacteristic
259C22#1801195712yes—
260C22#1801195811yescharacteristic
261C4—22no—
262C4—42no—
263C4—41no—
264C4—41no—
265C4—2984no—
266C4—5962no—
267C4—5962no—
268C2—82no—
269C2—11922no—
270C2#2375019112yes—
271C2#2375020211yescharacteristic
272C2#2375036711yescharacteristic
273C2#2375036811yescharacteristic
274C2#2375037212yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#394×C149C2
hypercenter#394D149
hypocenterC149#2819
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#18011897C2semidirect
2. #23749542C2semidirect
3. #23749541C2semidirect
4. #23749538C2semidirect
5. #23749537C2semidirect
6. #23749534C2semidirect
7. #23749533C2semidirect
8. #23749490C2semidirect
9. #394×C149C2semidirect
54. C4×Dic149D4semidirect
137. C149#2819semidirect
142. #394D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118972
#237495422
#237495412
#237495382
#237495372
#237495342
#237495332
#237494902
#394×C1492
#2819149