Finite Group Databasebeta

← #34045754

Subgroups — #34045754

Order 19072.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#23750425C211yesmaximal, characteristic
2C2×#18011944C211yesmaximal, characteristic, complemented
3#23750201C211yesmaximal, characteristic, complemented
4#23749641C211yesmaximal, characteristic, complemented
5#23749635C211yesmaximal, characteristic, complemented
6#23749598C211yesmaximal, characteristic, complemented
7#344×C149C211yesmaximal, characteristic, complemented
8C2×#14558736C2211yescharacteristic
9C2×#14558736C2211yescharacteristic
10C22×Dic298C2211yescharacteristic
11#18011944—22no—
12#18011944—22no—
13#18011944—21no—
14#18011944—21no—
15#18011939—21no—
16#18011939—21no—
17#18011933—22no—
18C2×Dic596—21no—
19C2×#14558725—21no—
20#18011911—21no—
21#18011899—21no—
22#18011895—22no—
23#18011891—22no—
24#18011870C2211yescharacteristic
25#18011837—22no—
26#18011836—22no—
27#18011825—22no—
28C2×M16×C149C2211yescharacteristic
29C2×M16×C149C2211yescharacteristic
30#117×C149C2211yescharacteristic
31#14558736—21no—
32#14558736—21no—
33#14558736—21no—
34#14558736—21no—
35#14558736—21no—
36#14558736—21no—
37#14558736—21no—
38#14558736—21no—
39C2×Dic298—21no—
40C2×Dic298—21no—
41C2×Dic298—21no—
42C2×Dic298—21no—
43C2×Dic298—21no—
44C2×Dic298D411yescharacteristic
45C2×Dic298D411yescharacteristic
46C2×Dic298D412yes—
47C2×D596D411yescharacteristic
48C2×D596D411yescharacteristic
49C2×C22⋊D149—21no—
50C2×C22⋊D149—21no—
51C22×Dic149—21no—
52C2×C4×D149—21no—
53C2×C4×D149—21no—
54Dic596—24no—
55Dic596—22no—
56#14558725—24no—
57#14558725—22no—
58#14558725—22no—
59D1192—22no—
60C4⋊Dic149—24no—
61C4⋊Dic149—21no—
62C22⋊Dic149—21no—
63#14558713—24no—
64C4×Dic149—21no—
65C22×C4×C149C2311yescharacteristic
66M16×C149—21no—
67M16×C149—21no—
68M16×C149—21no—
69M16×C149D412yes—
70C2×C8×C149—21no—
71C2×C8×C149D412yes—
72C4⋊C4×C149—21no—
73C22⋊C4×C149D412yes—
74C42×C149—21no—
75C22×D149—21no—
76C22×D149—21no—
77Dic298—22no—
78Dic298—21no—
79Dic298—21no—
80Dic298—21no—
81Dic298—22no—
82Dic298—21no—
83Dic298—41no—
84Dic298—41no—
85D596—21no—
86D596—21no—
87D596—21no—
88D596—21no—
89C22⋊D149—41no—
90C22⋊D149—41no—
91C22⋊D149—41no—
92C22⋊D149—41no—
93C2×Dic149—21no—
94C2×Dic149—22no—
95C2×Dic149—21no—
96C2×Dic149—41no—
97C2×Dic149—41no—
98C2×Dic149—41no—
99C4×D149—41no—
100C4×D149—41no—
101C4×D149—41no—
102C4×D149—41no—
103C23×C149C2×D411yescharacteristic
104C2×C4×C149—22no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2×D411yescharacteristic
108C2×C4×C149C2×D411yescharacteristic
109C2×C4×C149C2×D411yescharacteristic
110C2×C4×C149#4111yescharacteristic
111C8×C149—22no—
112C8×C149—22no—
113C22×C149—41no—
114C22×C149#12011yescharacteristic
115C22×C149#12111yescharacteristic
116C22×C149#12711yescharacteristic
117C2×D149—42no—
118C2×D149—41no—
119C2×D149—41no—
120C2×D149—81no—
121C4×C149—42no—
122C4×C149#12011yescharacteristic
123C4×C149#12011yescharacteristic
124C4×C149#12111yescharacteristic
125C4×C149#12111yescharacteristic
126Dic149—41no—
127Dic149—41no—
128Dic149—42no—
129Dic149—81no—
130C2×C149—21no—
131C2×C149—21no—
132C2×C149—21no—
133C2×C149#39211yescharacteristic
134D149—81no—
135D149—81no—
136C149#200911yescharacteristic, complemented, Sylow
137#2009—1491nomaximal, Sylow
138#575—1491no—
139C2×#137—1491no—
140#536—1491no—
141#421—1491no—
142#419—1491no—
143#410—1491no—
144#344D14911yescharacteristic, complemented
145C2×#41—1491no—
146C2×#41—1491no—
147C22×Q8—1491no—
148#137—2982no—
149#137—2981no—
150#137—2981no—
151#137—2982no—
152AGL(1,ℤ/8)—2981no—
153AGL(1,ℤ/8)—2981no—
154#135—2982no—
155C2×Q16—2981no—
156C2×SD16—2981no—
157C2×M16C2×D14911yescharacteristic
158C2×M16C2×D14911yescharacteristic
159C4⋊Q8—2981no—
160#124—2981no—
161#123—2982no—
162C22⋊Q8—2982no—
163#117—1491no—
164#117C2×D14911yescharacteristic
165#104—2982no—
166#103—2982no—
167#99—2982no—
168#41—2981no—
169#41—2981no—
170#41—2981no—
171#41—2981no—
172#41—2981no—
173#41—2981no—
174#41—2981no—
175#41—2981no—
176C2×Q8—2981no—
177C2×Q8—2981no—
178C2×Q8—2981no—
179C2×Q8—2981no—
180C2×Q8—2981no—
181C2×Q8—1492no—
182C2×Q8—1491no—
183C2×Q8—1491no—
184C2×D4—2981no—
185C2×D4—2981no—
186C2×D4—1491no—
187C2×D4—1491no—
188C22×C4—2981no—
189C22×C4—2981no—
190C22×C4—2981no—
191C22×C4C22×D14911yescharacteristic
192Q16—2984no—
193Q16—2982no—
194SD16—2984no—
195SD16—2982no—
196SD16—2982no—
197D8—2982no—
198M16—21no—
199M16—21no—
200M16—21no—
201M16D59612yes—
202C2×C8—21no—
203C2×C8D59612yes—
204C4⋊C4—21no—
205C4⋊C4—2984no—
206C4⋊C4—2981no—
207C22⋊C4—2984no—
208C22⋊C4—2981no—
209C22⋊C4D59612yes—
210C42—21no—
211C42—2981no—
212C23—2981no—
213C23—2981no—
214C23C2×D59611yescharacteristic
215Q8—2982no—
216Q8—2981no—
217Q8—2981no—
218Q8—2981no—
219Q8—2982no—
220Q8—2981no—
221Q8—5961no—
222Q8—5961no—
223D4—2981no—
224D4—2981no—
225D4—2981no—
226D4—2981no—
227D4—5961no—
228D4—5961no—
229D4—5961no—
230D4—5961no—
231C2×C4—22no—
232C2×C4—2982no—
233C2×C4—2981no—
234C2×C4—2981no—
235C2×C4—5961no—
236C2×C4—5961no—
237C2×C4—5961no—
238C2×C4—5961no—
239C2×C4—5961no—
240C2×C4—5961no—
241C2×C4—5961no—
242C2×C4D4×D14911yescharacteristic
243C2×C4D4×D14911yescharacteristic
244C2×C4D4×D14911yescharacteristic
245C2×C4C2×D59611yescharacteristic
246C2×C4C2×D59611yescharacteristic
247C2×C4#1455873711yescharacteristic
248C8—22no—
249C8—22no—
250C22—41no—
251C22—5962no—
252C22—5961no—
253C22—5961no—
254C22—11921no—
255C22#1801188111yescharacteristic
256C22#1801188611yescharacteristic
257C22#1801190811yescharacteristic
258C4—42no—
259C4—5961no—
260C4—5961no—
261C4—5962no—
262C4—11921no—
263C4#1801188111yescharacteristic
264C4#1801188111yescharacteristic
265C4#1801188611yescharacteristic
266C4#1801188611yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—11921no—
271C2—11921no—
272C2#2374953511yescharacteristic
subgroupG/H
centerC2#23749535
commutator, cosocleC22×C4×C149C23
FrattiniC22×C4C22×D149
Fitting#344×C149C2
socleC2×C149#392
prime radical#23750425C2
hypercenter#344D149
hypocenterC149#2009
LCSG ▷ C22×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#18011944C2semidirect
3. #23750201C2semidirect
4. #23749641C2semidirect
5. #23749635C2semidirect
6. #23749598C2semidirect
7. #344×C149C2semidirect
136. C149#2009semidirect
144. #344D149semidirect

Maximal subgroups

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

subgroupindex
#237504252
C2×#180119442
#237502012
#237496412
#237496352
#237495982
#344×C1492
#2009149