Finite Group Databasebeta

← #34047976

Subgroups — #34047976

Order 19072.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#18011876C211yesmaximal, characteristic, complemented
2C2×#18011871C211yesmaximal, characteristic
3#23749566C212yesmaximal, complemented
4#23749551C211yesmaximal, characteristic, complemented
5#23749540C212yesmaximal
6#23749538C211yesmaximal, characteristic, complemented
7#23749537C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic149C212yesmaximal, complemented
9#23749472C212yesmaximal, complemented
10#23749470C211yesmaximal, characteristic, complemented
11#397×C149C211yesmaximal, characteristic, complemented
12C22×Dic298C2211yescharacteristic
13C22×C22⋊D149C2211yescharacteristic
14C23×Dic149C2211yescharacteristic
15C22×C4×D149C2211yescharacteristic
16#18011876—24no—
17#18011871—24no—
18C2×#14558715C2212yes—
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×C149C2211yescharacteristic
33C2×C22⋊C4×C149C2212yes—
34C2×C22⋊C4×C149C2211yescharacteristic
35#95×C149C2211yescharacteristic
36#95×C149C2212yes—
37C23×D149C2311yescharacteristic
38C2×Dic298—24no—
39C2×Dic298—22no—
40C2×C22⋊D149—22no—
41C2×C22⋊D149—22no—
42C2×C22⋊D149—22no—
43C22×Dic149—22no—
44C22×Dic149—24no—
45C22×Dic149C2311yescharacteristic
46C22×Dic149C2312yes—
47C22×Dic149C2311yescharacteristic
48C22×Dic149C2312yes—
49C22×Dic149C2311yescharacteristic
50C2×C4×D149—22no—
51C2×C4×D149—24no—
52#14558715—22no—
53#14558715—24no—
54C22⋊Dic149—24no—
55#14558713—24no—
56#14558713—24no—
57#14558713—22no—
58C4×Dic149—24no—
59C4×Dic149—24no—
60C4×Dic149D414yes—
61C24×C149C2311yescharacteristic
62C22×C4×C149C2311yescharacteristic
63C22×C4×C149C2312yes—
64C22×C4×C149C2312yes—
65C22×C4×C149C2311yescharacteristic
66C4⋊C4×C149—22no—
67C22⋊C4×C149—24no—
68C22⋊C4×C149—22no—
69C22×D149—21no—
70C22×D149—22no—
71C22×D149—21no—
72C22×D149—22no—
73C22×D149#4112yes—
74Dic298—28no—
75C22⋊D149—28no—
76C2×Dic149—22no—
77C2×Dic149—21no—
78C2×Dic149—21no—
79C2×Dic149—22no—
80C2×Dic149—21no—
81C2×Dic149—22no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149—22no—
85C2×Dic149—44no—
86C2×Dic149C2×D412yes—
87C2×Dic149C2×D412yes—
88C2×Dic149#4114yes—
89C2×Dic149#4114yes—
90C4×D149—44no—
91C23×C149—21no—
92C23×C149—22no—
93C23×C149—21no—
94C23×C149—22no—
95C23×C149#4112yes—
96C23×C149C2411yescharacteristic
97C2×C4×C149—22no—
98C2×C4×C149—21no—
99C2×C4×C149—22no—
100C2×C4×C149—22no—
101C2×C4×C149—22no—
102C2×C4×C149—22no—
103C2×C4×C149—22no—
104C2×C4×C149—22no—
105C2×C4×C149—22no—
106C2×C4×C149C2×D412yes—
107C22×C149—22no—
108C22×C149—24no—
109C22×C149—41no—
110C22×C149—41no—
111C22×C149—42no—
112C22×C149C22×D411yescharacteristic
113C22×C149C2×#4111yescharacteristic
114C22×C149C2×#4112yes—
115C22×C149C2×#4112yes—
116C22×C149C2×#4111yescharacteristic
117C2×D149—22no—
118C2×D149—24no—
119C2×D149—41no—
120C2×D149—41no—
121C2×D149—42no—
122C4×C149—22no—
123C4×C149—42no—
124C4×C149—41no—
125C4×C149—42no—
126Dic149—22no—
127Dic149—24no—
128Dic149—41no—
129Dic149—42no—
130Dic149#12414yes—
131C2×C149—42no—
132C2×C149#52511yescharacteristic
133C2×C149#52511yescharacteristic
134C2×C149C2×#12411yescharacteristic
135C2×C149#52912yes—
136C2×C149#52912yes—
137D149—42no—
138C149#239811yescharacteristic, complemented, Sylow
139#2398—1491nomaximal, Sylow
140C2×C4×Q8—1491no—
141C2×C4×D4—1491no—
142#400—1492no—
143#397—1491no—
144#397D14911yescharacteristic, complemented
145#394—1492no—
146#393—1491no—
147#393—1491no—
148C4×C22⋊C4—1492no—
149C4×C22⋊C4—1492no—
150#376—1491no—
151C22×Q8—1491no—
152C22×D4—1491no—
153C23×C4—1491no—
154C23×C4—1491no—
155C4×Q8—2984no—
156C4×D4—2984no—
157C2×C4⋊C4—1491no—
158C2×C4⋊C4—1492no—
159C2×C4⋊C4C2×D14911yescharacteristic
160C2×C22⋊C4—1492no—
161C2×C22⋊C4—1491no—
162C2×C22⋊C4—1492no—
163C2×C22⋊C4—1492no—
164C2×C22⋊C4C2×D14911yescharacteristic
165C2×C22⋊C4C2×D14912yes—
166C2×C42—1492no—
167C2×C42—1492no—
168C2×C42—1491no—
169#95—1491no—
170#95—1492no—
171#95—1492no—
172#95—1492no—
173#95—1492no—
174#95C2×D14911yescharacteristic
175#95C2×D14912yes—
176C24—1491no—
177C24C22×D14911yescharacteristic
178C2×Q8—2984no—
179C2×Q8—2982no—
180C2×D4—2982no—
181C2×D4—2982no—
182C2×D4—2982no—
183C22×C4—2982no—
184C22×C4—2984no—
185C22×C4—2982no—
186C22×C4—2984no—
187C22×C4—1492no—
188C22×C4—1491no—
189C22×C4—1492no—
190C22×C4—1491no—
191C22×C4—1491no—
192C22×C4C22×D14912yes—
193C22×C4C22×D14911yescharacteristic
194C22×C4C22×D14912yes—
195C22×C4C22×D14911yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—2982no—
198C4⋊C4—2984no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—2984no—
202C22⋊C4—2984no—
203C22⋊C4—2984no—
204C22⋊C4—2982no—
205C42—2984no—
206C42—2984no—
207C42—1494no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—2981no—
213C23—2982no—
214C23—2982no—
215C23—2981no—
216C23—1492no—
217C23#1455873512yes—
218C23C23×D14911yescharacteristic
219Q8—2988no—
220D4—2988no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—2982no—
231C2×C4—2981no—
232C2×C4—2982no—
233C2×C4—2981no—
234C2×C4—2981no—
235C2×C4—2982no—
236C2×C4—2982no—
237C2×C4—2982no—
238C2×C4—2982no—
239C2×C4—5964no—
240C2×C4—5964no—
241C2×C4—1492no—
242C2×C4—1494no—
243C2×C4—1492no—
244C2×C4—1494no—
245C2×C4D4×D14912yes—
246C22—24no—
247C22—22no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—2984no—
252C22—2982no—
253C22—5962no—
254C22—5961no—
255C22—5961no—
256C22C2×D4×D14911yescharacteristic
257C22#41×D14912yes—
258C22#41×D14912yes—
259C22#41×D14911yescharacteristic
260C22C2×#1455873511yescharacteristic
261C4—22no—
262C4—42no—
263C4—42no—
264C4—41no—
265C4—2984no—
266C4—2982no—
267C4—5962no—
268C4—5961no—
269C4—1494no—
270C2—42no—
271C2—5962no—
272C2#2375013311yescharacteristic
273C2#2375013511yescharacteristic
274C2#124×D14911yescharacteristic
275C2#2375015212yes—
276C2#2375015812yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#397×C149C2
prime radicalC22×C22⋊D149C22
hypercenter#397D149
hypocenterC149#2398
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#18011876C2semidirect
3. #23749566C2semidirect
4. #23749551C2semidirect
6. #23749538C2semidirect
7. #23749537C2semidirect
8. C22⋊C4×Dic149C2semidirect
9. #23749472C2semidirect
10. #23749470C2semidirect
11. #397×C149C2semidirect
138. C149#2398semidirect
144. #397D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118762
C2×#180118712
#237495662
#237495512
#237495402
#237495382
#237495372
C22⋊C4×Dic1492
#237494722
#237494702
#397×C1492
#2398149