Finite Group Databasebeta

← #34047956

Subgroups — #34047956

Order 19072.

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

normal subgroups only

This group has 560 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×#18011871C211yesmaximal, characteristic
2C2×#18011869C211yesmaximal, characteristic, complemented
3#23749563C211yesmaximal, characteristic, complemented
4#23749557C211yesmaximal, characteristic, complemented
5#23749551C212yesmaximal, complemented
6#23749550C211yesmaximal, characteristic, complemented
7#23749541C212yesmaximal
8C4⋊C4×Dic149C212yesmaximal, complemented
9C22⋊C4×Dic149C212yesmaximal, complemented
10#23749472C211yesmaximal, characteristic, complemented
11#399×C149C211yesmaximal, characteristic, complemented
12C22×C22⋊D149C2211yescharacteristic
13C23×Dic149C2211yescharacteristic
14C22×C4×D149C2211yescharacteristic
15#18011871—24no—
16#18011869—24no—
17C2×C4⋊Dic149C2212yes—
18C2×C4⋊Dic149C2211yescharacteristic
19C2×#14558715C2211yescharacteristic
20C2×C22⋊Dic149C2212yes—
21C2×#14558713C2212yes—
22C2×#14558713C2212yes—
23C2×#14558713C2211yescharacteristic
24C2×C4×Dic149C2212yes—
25C2×C4×Dic149C2212yes—
26C2×C4×Dic149C2211yescharacteristic
27#18011818C2211yescharacteristic
28#18011818C2212yes—
29#18011818C2212yes—
30#18011818C2212yes—
31#18011818C2212yes—
32C2×C4⋊C4×C149C2212yes—
33C2×C4⋊C4×C149C2211yescharacteristic
34C2×C22⋊C4×C149C2212yes—
35C2×C22⋊C4×C149C2211yescharacteristic
36#95×C149C2211yescharacteristic
37C23×D149C2311yescharacteristic
38C2×C22⋊D149—22no—
39C2×C22⋊D149—22no—
40C2×C22⋊D149—22no—
41C22×Dic149—22no—
42C22×Dic149—24no—
43C22×Dic149C2311yescharacteristic
44C22×Dic149C2312yes—
45C22×Dic149C2312yes—
46C22×Dic149C2311yescharacteristic
47C22×Dic149C2311yescharacteristic
48C2×C4×D149—24no—
49C2×C4×D149D414yes—
50C4⋊Dic149—24no—
51C4⋊Dic149—22no—
52#14558715—22no—
53C22⋊Dic149—24no—
54#14558713—24no—
55#14558713—22no—
56#14558713—24no—
57C4×Dic149—24no—
58C4×Dic149—22no—
59C4×Dic149—24no—
60C24×C149C2311yescharacteristic
61C22×C4×C149C2311yescharacteristic
62C22×C4×C149C2312yes—
63C22×C4×C149C2311yescharacteristic
64C22×C4×C149C2312yes—
65C4⋊C4×C149—22no—
66C4⋊C4×C149—24no—
67C22⋊C4×C149—22no—
68C22⋊C4×C149—24no—
69C22×D149—22no—
70C22×D149—21no—
71C22×D149—21no—
72C22×D149—22no—
73C22×D149C2×D412yes—
74C22⋊D149—28no—
75C2×Dic149—21no—
76C2×Dic149—21no—
77C2×Dic149—21no—
78C2×Dic149—22no—
79C2×Dic149—22no—
80C2×Dic149—22no—
81C2×Dic149—22no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149—22no—
85C2×Dic149—44no—
86C2×Dic149C2×D412yes—
87C2×Dic149#4112yes—
88C2×Dic149#4114yes—
89C4×D149—28no—
90C23×C149—21no—
91C23×C149—22no—
92C23×C149—21no—
93C23×C149—22no—
94C23×C149#4112yes—
95C23×C149C2411yescharacteristic
96C2×C4×C149—22no—
97C2×C4×C149—22no—
98C2×C4×C149—22no—
99C2×C4×C149—21no—
100C2×C4×C149—22no—
101C2×C4×C149—22no—
102C2×C4×C149—22no—
103C2×C4×C149—22no—
104C2×C4×C149C2×D412yes—
105C2×C4×C149#4114yes—
106C22×C149—22no—
107C22×C149—24no—
108C22×C149—41no—
109C22×C149—42no—
110C22×C149—41no—
111C22×C149C22×D411yescharacteristic
112C22×C149C2×#4112yes—
113C22×C149C2×#4111yescharacteristic
114C22×C149C2×#4112yes—
115C22×C149C2×#4111yescharacteristic
116C2×D149—22no—
117C2×D149—22no—
118C2×D149—22no—
119C2×D149—42no—
120C2×D149—42no—
121C4×C149—22no—
122C4×C149—24no—
123C4×C149—41no—
124C4×C149—42no—
125Dic149—22no—
126Dic149—41no—
127Dic149—42no—
128Dic149—42no—
129Dic149#12314yes—
130C2×C149—42no—
131C2×C149C2×#12311yescharacteristic
132C2×C149#52511yescharacteristic
133C2×C149#52912yes—
134C2×C149#52912yes—
135C2×C149#53211yescharacteristic
136D149—42no—
137C149#239511yescharacteristic, complemented, Sylow
138#2395—1491nomaximal, Sylow
139C2×C4×D4—1491no—
140C2×#117—1491no—
141#400—1491no—
142#399D14911yescharacteristic, complemented
143#398—1491no—
144#397—1492no—
145#397—1491no—
146#394—1492no—
147C4×C4⋊C4—1492no—
148C4×C22⋊C4—1492no—
149C4×C22⋊C4—1491no—
150C22×D4—1491no—
151C23×C4—1491no—
152C23×C4—1491no—
153C4×D4—2984no—
154#117—2984no—
155C2×C4⋊C4—1491no—
156C2×C4⋊C4—1491no—
157C2×C4⋊C4—1492no—
158C2×C4⋊C4C2×D14911yescharacteristic
159C2×C4⋊C4C2×D14912yes—
160C2×C22⋊C4—1492no—
161C2×C22⋊C4—1492no—
162C2×C22⋊C4—1491no—
163C2×C22⋊C4—1492no—
164C2×C22⋊C4C2×D14911yescharacteristic
165C2×C22⋊C4C2×D14912yes—
166C2×C42—1492no—
167C2×C42—1491no—
168C2×C42—1492no—
169#95—1492no—
170#95—1491no—
171#95—1492no—
172#95—1492no—
173#95—1492no—
174#95C2×D14911yescharacteristic
175C24—1491no—
176C24C22×D14911yescharacteristic
177C2×D4—2982no—
178C2×D4—2982no—
179C2×D4—2982no—
180C22×C4—2984no—
181C22×C4—2982no—
182C22×C4—2984no—
183C22×C4—1492no—
184C22×C4—1491no—
185C22×C4—1492no—
186C22×C4—1491no—
187C22×C4—1494no—
188C22×C4—1491no—
189C22×C4C22×D14911yescharacteristic
190C22×C4C22×D14912yes—
191C22×C4C22×D14911yescharacteristic
192C22×C4C22×D14912yes—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—2984no—
196C4⋊C4—2982no—
197C4⋊C4—2982no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—2984no—
201C22⋊C4—2982no—
202C22⋊C4—2984no—
203C22⋊C4—2984no—
204C42—2984no—
205C42—2982no—
206C42—2984no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—2982no—
212C23—2981no—
213C23—2982no—
214C23—2981no—
215C23—1492no—
216C23#1455873712yes—
217C23C23×D14911yescharacteristic
218D4—2988no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—2988no—
228C2×C4—2981no—
229C2×C4—2981no—
230C2×C4—2982no—
231C2×C4—2981no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—2982no—
235C2×C4—2982no—
236C2×C4—2982no—
237C2×C4—2982no—
238C2×C4—5964no—
239C2×C4—1492no—
240C2×C4—1492no—
241C2×C4—1494no—
242C2×C4D4×D14912yes—
243C2×C4#1455873514yes—
244C22—24no—
245C22—22no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—2982no—
250C22—2982no—
251C22—2982no—
252C22—5962no—
253C22—5962no—
254C22C2×D4×D14911yescharacteristic
255C22#41×D14911yescharacteristic
256C22#41×D14912yes—
257C22C2×#1455873512yes—
258C22C2×#1455873711yescharacteristic
259C4—24no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—2982no—
264C4—5962no—
265C4—5961no—
266C4—5962no—
267C4—1494no—
268C2—42no—
269C2—5962no—
270C2#123×D14911yescharacteristic
271C2#2375013811yescharacteristic
272C2#2375015212yes—
273C2#2375016112yes—
274C2#2375017311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#399×C149C2
prime radicalC22×C22⋊D149C22
hypercenter#399D149
hypocenterC149#2395
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#18011869C2semidirect
3. #23749563C2semidirect
4. #23749557C2semidirect
5. #23749551C2semidirect
6. #23749550C2semidirect
8. C4⋊C4×Dic149C2semidirect
9. C22⋊C4×Dic149C2semidirect
10. #23749472C2semidirect
11. #399×C149C2semidirect
137. C149#2395semidirect
142. #399D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118712
C2×#180118692
#237495632
#237495572
#237495512
#237495502
#237495412
C4⋊C4×Dic1492
C22⋊C4×Dic1492
#237494722
#399×C1492
#2395149