Finite Group Databasebeta

← #34047980

Subgroups — #34047980

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×#18011873C211yesmaximal, characteristic
3#23749563C211yesmaximal, characteristic, complemented
4#23749554C212yesmaximal, complemented
5#23749543C211yesmaximal, characteristic
6#23749542C211yesmaximal, characteristic
7#23749538C212yesmaximal, complemented
8#23749472C212yesmaximal, complemented
9#23749472C212yesmaximal, complemented
10#23749470C211yesmaximal, characteristic, complemented
11#400×C149C211yesmaximal, characteristic, complemented
12C22×Dic298C2211yescharacteristic
13C22×D596C2211yescharacteristic
14C22×C4×D149C2212yes—
15#18011876—24no—
16#18011873—24no—
17C2×#14558715C2211yescharacteristic
18C2×#14558715C2211yescharacteristic
19C2×#14558713C2212yes—
20C2×#14558713C2212yes—
21C2×#14558713C2212yes—
22C2×#14558713C2212yes—
23C2×#14558713C2212yes—
24C2×C4×Dic149C2212yes—
25C2×C4×Dic149C2211yescharacteristic
26C2×C4×Dic149C2211yescharacteristic
27C2×C4×Dic149C2211yescharacteristic
28#18011818C2212yes—
29#18011818C2211yescharacteristic
30#18011818C2212yes—
31#18011818C2212yes—
32C2×C4⋊C4×C149C2211yescharacteristic
33C2×C4⋊C4×C149C2211yescharacteristic
34#95×C149C2211yescharacteristic
35#95×C149C2212yes—
36#95×C149C2212yes—
37C23×D149C2312yes—
38C2×Dic298—22no—
39C2×Dic298—22no—
40C2×Dic298—22no—
41C2×D596—24no—
42C2×D596—22no—
43C22×Dic149C2312yes—
44C22×Dic149C2311yescharacteristic
45C22×Dic149C2311yescharacteristic
46C22×Dic149C2311yescharacteristic
47C22×Dic149C2311yescharacteristic
48C2×C4×D149—24no—
49C2×C4×D149—24no—
50C2×C4×D149—24no—
51#14558715—22no—
52#14558715—22no—
53#14558713—24no—
54#14558713—24no—
55#14558713—24no—
56#14558713—24no—
57#14558713—24no—
58C4×Dic149—22no—
59C4×Dic149—22no—
60C4×Dic149—24no—
61C4×Dic149D414yes—
62C22×C4×C149C2312yes—
63C22×C4×C149C2312yes—
64C22×C4×C149C2311yescharacteristic
65C22×C4×C149C2311yescharacteristic
66C22×C4×C149C2311yescharacteristic
67C4⋊C4×C149—22no—
68C4⋊C4×C149—22no—
69C22×D149—22no—
70C22×D149—22no—
71C22×D149—22no—
72C22×D149—22no—
73C22×D149—22no—
74C22×D149—22no—
75C22×D149#4114yes—
76Dic298—28no—
77D596—28no—
78C2×Dic149—21no—
79C2×Dic149—21no—
80C2×Dic149—21no—
81C2×Dic149—21no—
82C2×Dic149—21no—
83C2×Dic149—22no—
84C2×Dic149—22no—
85C2×Dic149—22no—
86C2×Dic149—22no—
87C2×Dic149C2×D412yes—
88C2×Dic149C2×D412yes—
89C2×Dic149#4112yes—
90C2×Dic149#4114yes—
91C4×D149—48no—
92C23×C149C2411yescharacteristic
93C2×C4×C149—21no—
94C2×C4×C149—22no—
95C2×C4×C149—21no—
96C2×C4×C149—22no—
97C2×C4×C149—21no—
98C2×C4×C149—22no—
99C2×C4×C149—22no—
100C2×C4×C149—22no—
101C2×C4×C149—22no—
102C2×C4×C149—22no—
103C2×C4×C149—22no—
104C2×C4×C149C2×D412yes—
105C2×C4×C149#4112yes—
106C22×C149C22×D411yescharacteristic
107C22×C149C2×#4112yes—
108C22×C149C2×#4111yescharacteristic
109C22×C149C2×#4112yes—
110C22×C149C2×#4111yescharacteristic
111C2×D149—24no—
112C2×D149—24no—
113C2×D149—24no—
114C2×D149—42no—
115C2×D149—42no—
116C2×D149—42no—
117C2×D149—42no—
118C4×C149—22no—
119C4×C149—22no—
120C4×C149—41no—
121C4×C149—42no—
122C4×C149—42no—
123Dic149—22no—
124Dic149—22no—
125Dic149—41no—
126Dic149—42no—
127Dic149#12414yes—
128C2×C149#52511yescharacteristic
129C2×C149#52511yescharacteristic
130C2×C149C2×#12411yescharacteristic
131C2×C149#52912yes—
132C2×C149#52912yes—
133D149—44no—
134C149#239811yescharacteristic, complemented, Sylow
135#2398—1491nomaximal, Sylow
136C2×C4×Q8—1491no—
137C2×C4×D4—1491no—
138#400—1491no—
139#400D14911yescharacteristic, complemented
140#397—1492no—
141#394—1491no—
142#394—1491no—
143#393—1492no—
144C4×C22⋊C4—1492no—
145C4×C22⋊C4—1492no—
146#376—1491no—
147C22×Q8—1491no—
148C22×D4—1491no—
149C23×C4—1492no—
150C4×Q8—2984no—
151C4×D4—2984no—
152C2×C4⋊C4—1491no—
153C2×C4⋊C4—1491no—
154C2×C4⋊C4C2×D14911yescharacteristic
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×C42—1491no—
162C2×C42—1491no—
163C2×C42—1492no—
164C2×C42—1491no—
165#95—1492no—
166#95—1491no—
167#95—1492no—
168#95—1492no—
169#95C2×D14911yescharacteristic
170#95C2×D14912yes—
171#95C2×D14912yes—
172C24—1492no—
173C2×Q8—2982no—
174C2×Q8—2982no—
175C2×Q8—2982no—
176C2×D4—2984no—
177C2×D4—2982no—
178C22×C4—2984no—
179C22×C4—2984no—
180C22×C4—2984no—
181C22×C4—1492no—
182C22×C4—1491no—
183C22×C4—1491no—
184C22×C4—1491no—
185C22×C4—1491no—
186C22×C4C22×D14912yes—
187C22×C4C22×D14912yes—
188C22×C4C22×D14911yescharacteristic
189C22×C4C22×D14911yescharacteristic
190C22×C4C22×D14911yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—2982no—
194C4⋊C4—2982no—
195C22⋊C4—2984no—
196C22⋊C4—2984no—
197C22⋊C4—2984no—
198C22⋊C4—2984no—
199C22⋊C4—2984no—
200C42—2982no—
201C42—2982no—
202C42—2984no—
203C42—1494no—
204C23—2982no—
205C23—2982no—
206C23—2982no—
207C23—2982no—
208C23—2982no—
209C23—2982no—
210C23—1494no—
211C23C23×D14911yescharacteristic
212Q8—2988no—
213D4—2988no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—2981no—
226C2×C4—2981no—
227C2×C4—2981no—
228C2×C4—2981no—
229C2×C4—2981no—
230C2×C4—2982no—
231C2×C4—2982no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—5968no—
235C2×C4—1492no—
236C2×C4—1492no—
237C2×C4—1492no—
238C2×C4—1494no—
239C2×C4D4×D14912yes—
240C2×C4#1455873712yes—
241C22—2984no—
242C22—2984no—
243C22—2984no—
244C22—5962no—
245C22—5962no—
246C22—5962no—
247C22—5962no—
248C22C2×D4×D14911yescharacteristic
249C22#41×D14911yescharacteristic
250C22#41×D14912yes—
251C22#41×D14912yes—
252C22C2×#1455873711yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—2982no—
259C4—2982no—
260C4—5962no—
261C4—5961no—
262C4—1494no—
263C2—5964no—
264C2#2375013311yescharacteristic
265C2#2375013811yescharacteristic
266C2#124×D14911yescharacteristic
267C2#2375015812yes—
268C2#2375016012yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#400×C149C2
prime radicalC22×D596C22
hypercenter#400D149
hypocenterC149#2398
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#18011876C2semidirect
3. #23749563C2semidirect
4. #23749554C2semidirect
7. #23749538C2semidirect
8. #23749472C2semidirect
9. #23749472C2semidirect
10. #23749470C2semidirect
11. #400×C149C2semidirect
134. C149#2398semidirect
139. #400D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118762
C2×#180118732
#237495632
#237495542
#237495432
#237495422
#237495382
#237494722
#237494702
#400×C1492
#2398149