Finite Group Databasebeta

← #34053989

Subgroups — #34053989

Order 19072.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18011958C211yesmaximal, characteristic
2C2×#18011941C211yesmaximal, characteristic, complemented
3C22×Dic596C211yesmaximal, characteristic, complemented
4#23750201C211yesmaximal, characteristic, complemented
5#23749788C214yesmaximal, complemented
6#23749782C214yesmaximal, complemented
7#23749633C211yesmaximal, characteristic, complemented
8#23749591C211yesmaximal, characteristic, complemented
9#416×C149C211yesmaximal, characteristic, complemented
10#18011958—24no—
11#18011958—24no—
12C2×#14558736—21no—
13C2×#14558736C2211yescharacteristic
14C2×#14558735—22no—
15C2×#14558735C2212yes—
16C22×Dic298—21no—
17C22×Dic298C2211yescharacteristic
18C22×Dic298C2211yescharacteristic
19C2×Q8×D149C2212yes—
20#18011941—24no—
21C2×Dic596—24no—
22C2×Dic596C2214yes—
23C2×#14558726C2212yes—
24C2×#14558724C2212yes—
25C2×#14558720C2211yescharacteristic
26#18011911—21no—
27#18011899—21no—
28#18011895—22no—
29#18011891C2214yes—
30#18011834C2214yes—
31#18011823C2214yes—
32C2×#41×C149C2211yescharacteristic
33C22×C8×C149C2211yescharacteristic
34#117×C149C2211yescharacteristic
35#103×C149C2212yes—
36#102×C149C2212yes—
37#14558736—28no—
38#14558736—22no—
39#14558736—22no—
40#14558735—216no—
41#14558735—24no—
42#14558735—24no—
43C2×Dic298—28no—
44C2×Dic298—22no—
45C2×Dic298—21no—
46C2×Dic298—22no—
47C2×Dic298—21no—
48C2×Dic298—22no—
49C2×Dic298—22no—
50C2×Dic298D414yes—
51C2×Dic298D411yescharacteristic
52C2×Dic298C2312yes—
53C2×Dic298C2312yes—
54Q8×D149—24no—
55Q8×D149—24no—
56C2×D596D411yescharacteristic
57C2×C22⋊D149—22no—
58C2×C22⋊D149D412yes—
59C22×Dic149—22no—
60C22×Dic149—21no—
61C22×Dic149—21no—
62C2×C4×D149—22no—
63C2×C4×D149C2312yes—
64Dic596—28no—
65#14558726—24no—
66#14558724—24no—
67#14558720—22no—
68C2×#11924746C2312yes—
69C4⋊Dic149—24no—
70#14558713—24no—
71#41×C149—22no—
72#41×C149D414yes—
73C2×Q8×C149C2311yescharacteristic
74C2×D4×C149—21no—
75C2×D4×C149C2311yescharacteristic
76C22×C4×C149—21no—
77C22×C4×C149C2311yescharacteristic
78C2×C8×C149—22no—
79C2×C8×C149C2312yes—
80C4⋊C4×C149C2312yes—
81C22⋊C4×C149—21no—
82C42×C149—21no—
83C22×D149C2×D412yes—
84Dic298—28no—
85Dic298—24no—
86Dic298—22no—
87Dic298—22no—
88Dic298—22no—
89Dic298—22no—
90Dic298—42no—
91Dic298C2×D414yes—
92D596—22no—
93C22⋊D149—28no—
94C22⋊D149—24no—
95C2×Dic149—28no—
96C2×Dic149—24no—
97C2×Dic149—22no—
98C2×Dic149—22no—
99C2×Dic149—42no—
100C2×Dic149C2×D412yes—
101C4×D149—28no—
102C4×D149—22no—
103C4×D149—22no—
104#11924746—22no—
105C23×C149—21no—
106C23×C149C2×D411yescharacteristic
107Q8×C149—21no—
108Q8×C149C2×D412yes—
109D4×C149—24no—
110D4×C149—21no—
111D4×C149C2×D412yes—
112C2×C4×C149—21no—
113C2×C4×C149—24no—
114C2×C4×C149—22no—
115C2×C4×C149C2×D411yescharacteristic
116C2×C4×C149C2×D412yes—
117C2×C4×C149C2×D412yes—
118C2×C4×C149C2411yescharacteristic
119C8×C149—22no—
120C22×C149—22no—
121C22×C149—22no—
122C22×C149—42no—
123C22×C149#12012yes—
124C22×C149C22×D411yescharacteristic
125C2×D149—22no—
126C2×D149—24no—
127C4×C149—22no—
128C4×C149—42no—
129C4×C149#12012yes—
130C4×C149C22×D411yescharacteristic
131C4×C149C22×D411yescharacteristic
132Dic149—24no—
133Dic149—22no—
134Dic149—42no—
135C2×C149—22no—
136C2×C149—42no—
137C2×C149C2×#12011yescharacteristic
138C2×C149#57812yes—
139D149—42no—
140C149#301111yescharacteristic, complemented, Sylow
141#3011—1491nomaximal, Sylow
142C2×2−1+4—1491no—
143C2×#137—1491no—
144C22×Q16—1491no—
145#536—1491no—
146#452—1494no—
147#451—1494no—
148#419—1491no—
149#416D14911yescharacteristic, complemented
150#408—1491no—
1512−1+4—2984no—
1522−1+4—2984no—
153C2×#41—2981no—
154C2×#41—2982no—
155C2×#41—1492no—
156C2×#41—1491no—
157C2×#41C2×D14911yescharacteristic
158C22×Q8—2981no—
159C22×Q8—1492no—
160C22×Q8—1491no—
161C22×Q8—1491no—
162#137—2984no—
163C2×Q16—2984no—
164C2×Q16—1492no—
165C2×Q16—1494no—
166C2×SD16—1492no—
167C2×M16—1491no—
168C22×C8C2×D14911yescharacteristic
169C4⋊Q8—2981no—
170#124—2981no—
171#123—2982no—
172C22⋊Q8—1494no—
173#117C2×D14911yescharacteristic
174#103—1494no—
175#103C2×D14912yes—
176#102C2×D14912yes—
177C22⋊C8—1494no—
178#41—22no—
179#41—2988no—
180#41—29816no—
181#41—2984no—
182#41—2984no—
183#41—2982no—
184#41—2982no—
185#41D59614yes—
186C2×Q8—2988no—
187C2×Q8—2982no—
188C2×Q8—2984no—
189C2×Q8—2984no—
190C2×Q8—2981no—
191C2×Q8—2981no—
192C2×Q8—2982no—
193C2×Q8—2982no—
194C2×Q8—2982no—
195C2×Q8—1494no—
196C2×Q8—1492no—
197C2×Q8—1492no—
198C2×Q8—1491no—
199C2×Q8C22×D14911yescharacteristic
200C2×D4—21no—
201C2×D4—2982no—
202C2×D4—1492no—
203C2×D4—1491no—
204C2×D4C22×D14911yescharacteristic
205C22×C4—21no—
206C22×C4—2982no—
207C22×C4—2981no—
208C22×C4—2982no—
209C22×C4—2981no—
210C22×C4—1492no—
211C22×C4C22×D14911yescharacteristic
212Q16—2988no—
213Q16—2984no—
214SD16—2984no—
215M16—2982no—
216C2×C8—22no—
217C2×C8—1492no—
218C2×C8C22×D14912yes—
219C4⋊C4—2984no—
220C4⋊C4C22×D14912yes—
221C22⋊C4—21no—
222C22⋊C4—2984no—
223C42—21no—
224C23—21no—
225C23—1492no—
226C23D4×D14911yescharacteristic
227Q8—21no—
228Q8—2988no—
229Q8—2982no—
230Q8—2984no—
231Q8—2982no—
232Q8—2982no—
233Q8—2982no—
234Q8—5962no—
235Q8—1494no—
236Q8C2×D59612yes—
237D4—24no—
238D4—21no—
239D4—2988no—
240D4—2984no—
241D4—2982no—
242D4C2×D59612yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—2988no—
247C2×C4—2988no—
248C2×C4—2982no—
249C2×C4—2984no—
250C2×C4—2982no—
251C2×C4—2982no—
252C2×C4—2982no—
253C2×C4—5962no—
254C2×C4—1492no—
255C2×C4D4×D14912yes—
256C2×C4D4×D14911yescharacteristic
257C2×C4C2×D59612yes—
258C2×C4C23×D14911yescharacteristic
259C8—22no—
260C8—2982no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—2984no—
265C22—2982no—
266C22#1801188112yes—
267C22C2×D4×D14911yescharacteristic
268C4—22no—
269C4—42no—
270C4—2984no—
271C4—2982no—
272C4—5962no—
273C4#1801188112yes—
274C4C2×D4×D14911yescharacteristic
275C4C22×D59611yescharacteristic
276C2—22no—
277C2—42no—
278C2—5962no—
279C2C2×#1801188111yescharacteristic
280C2#2375043912yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#416×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011958C2
hypercenter#416D149
hypocenterC149#3011
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#18011941C2semidirect
3. C22×Dic596C2semidirect
4. #23750201C2semidirect
5. #23749788C2semidirect
6. #23749782C2semidirect
7. #23749633C2semidirect
8. #23749591C2semidirect
9. #416×C149C2semidirect
140. C149#3011semidirect
149. #416D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
C2×#180119412
C22×Dic5962
#237502012
#237497882
#237497822
#237496332
#237495912
#416×C1492
#3011149