Finite Group Databasebeta

← #34056408

Subgroups — #34056408

Order 19072.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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×#18011932C212yesmaximal, complemented
2#23750279C211yesmaximal, characteristic, complemented
3#23750278C211yesmaximal, characteristic, complemented
4#23750004C211yesmaximal, characteristic, complemented
5#23749865C212yesmaximal, complemented
6#23749831C211yesmaximal, characteristic, complemented
7#23749790C212yesmaximal, complemented
8#23749784C212yesmaximal, complemented
9C4×#14558721C211yesmaximal, characteristic, complemented
10#23749690C211yesmaximal, characteristic, complemented
11#461×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2212yes—
13C2×#14558735C2212yes—
14#18011932—28no—
15C2×#14558726C2212yes—
16C2×#14558726C2212yes—
17C2×#14558725C2212yes—
18C2×#14558721C2211yescharacteristic
19C2×C8×D149C2212yes—
20#18011911C2211yescharacteristic
21#18011896—22no—
22#18011895—22no—
23#18011891C2212yes—
24#18011887C2212yes—
25#18011886—21no—
26#18011885—21no—
27C4×D596C2211yescharacteristic
28C4×D596C2211yescharacteristic
29C2×C4⋊Dic149—21no—
30C4⋊C4×D149—21no—
31#18011844C2211yescharacteristic
32#18011836C2212yes—
33#18011832C2211yescharacteristic
34#18011831C2211yescharacteristic
35#18011823C2212yes—
36C4×#11924746C2211yescharacteristic
37C2×SD16×C149C2212yes—
38C4⋊Q8×C149C2211yescharacteristic
39C4×D4×C149C2211yescharacteristic
40C4⋊C8×C149C2211yescharacteristic
41#103×C149C2212yes—
42#14558737—24no—
43#14558737—42no—
44#14558735—24no—
45#14558735—42no—
46C2×Dic298C2312yes—
47C2×D596—22no—
48C2×D596D411yescharacteristic
49C2×D596C2311yescharacteristic
50C2×C22⋊D149—22no—
51C22×Dic149—22no—
52C2×C4×D149—22no—
53C2×C4×D149—21no—
54C2×C4×D149C2312yes—
55#14558726—24no—
56#14558726—42no—
57#14558725—24no—
58#14558721D414yescomplemented
59C2×#11924746C2311yescharacteristic
60C2×#11924746C2311yescharacteristic
61C8×D149—24no—
62C4⋊Dic149—22no—
63C4⋊Dic149—21no—
64C4⋊Dic149—21no—
65C4⋊Dic149D411yescharacteristic
66C4⋊Dic149C2311yescharacteristic
67#14558715—22no—
68#14558713—22no—
69#14558713—22no—
70#14558713—21no—
71#14558713D412yes—
72C2×Q8×C149C2312yes—
73C2×D4×C149C2311yescharacteristic
74C22×C4×C149—21no—
75SD16×C149—24no—
76C2×C8×C149C2312yes—
77C4⋊C4×C149—21no—
78C4⋊C4×C149—21no—
79C4⋊C4×C149C2311yescharacteristic
80C22⋊C4×C149—21no—
81C42×C149C2311yescharacteristic
82C22×D149—21no—
83C22×D149C2×D412yes—
84Dic298—22no—
85Dic298—22no—
86D596—22no—
87D596—21no—
88D596—44no—
89D596C2×D412yes—
90C22⋊D149—44no—
91C2×Dic149—21no—
92C2×Dic149—22no—
93C2×Dic149—44no—
94C2×Dic149C2×D412yes—
95C4×D149—22no—
96C4×D149—22no—
97C4×D149—44no—
98C4×D149—42no—
99#11924746—21no—
100#11924746C2×D412yes—
101C23×C149—21no—
102Q8×C149—22no—
103Q8×C149—22no—
104D4×C149—21no—
105D4×C149C2×D412yes—
106C2×C4×C149—22no—
107C2×C4×C149—21no—
108C2×C4×C149—42no—
109C2×C4×C149C2×D411yescharacteristic
110C2×C4×C149C2×D411yescharacteristic
111C2×C4×C149C2411yescharacteristic
112C8×C149—22no—
113C22×C149—22no—
114C22×C149—41no—
115C22×C149—41no—
116C22×C149C22×D411yescharacteristic
117C2×D149—22no—
118C2×D149—22no—
119C2×D149—22no—
120C2×D149—22no—
121C2×D149—41no—
122C2×D149—41no—
123C4×C149—21no—
124C4×C149—42no—
125C4×C149—41no—
126C4×C149#13512yescomplemented
127C4×C149C22×D411yescharacteristic
128C4×C149H(5,𝔽2)11yescharacteristic
129Dic149—22no—
130Dic149—42no—
131Dic149—41no—
132C2×C149—42no—
133C2×C149D4211yescharacteristic
134C2×C149C2×#13511yescharacteristic
135C2×C149#57811yescharacteristic
136D149—42no—
137D149—42no—
138C149#328311yescharacteristic, complemented, Sylow
139#3283—1491nomaximal, Sylow
140C2×#135—1492no—
141#547—1491no—
142#547—1491no—
143#494—1491no—
144#467—1492no—
145#461—1491no—
146#461D14911yescharacteristic, complemented
147#452—1492no—
148#452—1492no—
149C4×D8—1491no—
150C8×D4—1491no—
151C2×#41—1492no—
152C2×#41—1492no—
153#135—2988no—
154C2×Q16—1492no—
155C2×Q16—1492no—
156C2×SD16—1492no—
157C2×SD16C2×D14912yes—
158C2×D8—1491no—
159C22×C8—1492no—
160C4⋊Q8—1491no—
161C4⋊Q8C2×D14911yescharacteristic
162#123—2982no—
163#123—2982no—
164C22⋊Q8—1492no—
165C22⋊Q8—1492no—
166#121—2981no—
167#121—2981no—
168C4×D4—1491no—
169C4×D4—1491no—
170C4×D4C2×D14911yescharacteristic
171C2×C4⋊C4—2981no—
172C2×C4⋊C4—2981no—
173#107—1491no—
174C4⋊C8C2×D14911yescharacteristic
175#103—1492no—
176#103C2×D14912yes—
177#102—1491no—
178#102—1491no—
179C22⋊C8—1492no—
180C4×C8—1491no—
181#41—2984no—
182#41—2984no—
183#41—5962no—
184#41—5962no—
185C2×Q8—1492no—
186C2×Q8C22×D14912yes—
187C2×D4—2982no—
188C2×D4—2982no—
189C2×D4—1491no—
190C2×D4—1491no—
191C2×D4C22×D14911yescharacteristic
192C22×C4—21no—
193C22×C4—2982no—
194C22×C4—2982no—
195C22×C4—2981no—
196C22×C4—1492no—
197Q16—2984no—
198Q16—5962no—
199SD16—24no—
200SD16—2984no—
201D8—1494no—
202C2×C8—2984no—
203C2×C8—1491no—
204C2×C8—1491no—
205C2×C8C22×D14912yes—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—2982no—
209C4⋊C4—2982no—
210C4⋊C4—2981no—
211C4⋊C4—2981no—
212C4⋊C4—1491no—
213C4⋊C4—1491no—
214C4⋊C4C22×D14911yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—2982no—
217C22⋊C4—2982no—
218C22⋊C4—2981no—
219C22⋊C4—1492no—
220C42C22×D14911yescharacteristic
221C23—21no—
222C23—2981no—
223C23—1492no—
224Q8—22no—
225Q8—22no—
226Q8—2982no—
227Q8—2982no—
228D4—21no—
229D4—2982no—
230D4—2981no—
231D4—5964no—
232D4—5964no—
233D4—1492no—
234D4D4×D14912yes—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—2982no—
239C2×C4—2982no—
240C2×C4—2981no—
241C2×C4—2982no—
242C2×C4—5964no—
243C2×C4—5964no—
244C2×C4—5962no—
245C2×C4—1492no—
246C2×C4D4×D14911yescharacteristic
247C2×C4D4×D14911yescharacteristic
248C2×C4C23×D14911yescharacteristic
249C8—22no—
250C8—2981no—
251C8—1492no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—2982no—
256C22—2982no—
257C22—2982no—
258C22—2982no—
259C22—5961no—
260C22—5961no—
261C22C2×D4×D14911yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—2982no—
266C4—5961no—
267C4—5962no—
268C4#1801193212yescomplemented
269C4C2×D4×D14911yescharacteristic
270C4#1801195711yescharacteristic
271C2—42no—
272C2—5962no—
273C2—5962no—
274C2#2375025911yescharacteristic
275C2C2×#1801193211yescharacteristic
276C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#461×C149C2
socleC22×C149C22×D4
hypercenter#461D149
hypocenterC149#3283
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#18011932C2semidirect
2. #23750279C2semidirect
3. #23750278C2semidirect
4. #23750004C2semidirect
5. #23749865C2semidirect
6. #23749831C2semidirect
7. #23749790C2semidirect
8. #23749784C2semidirect
9. C4×#14558721C2semidirect
10. #23749690C2semidirect
11. #461×C149C2semidirect
58. #14558721D4semidirect
126. C4×C149#135semidirect
138. C149#3283semidirect
146. #461D149semidirect
268. C4#18011932semidirect

Maximal subgroups

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

subgroupindex
C2×#180119322
#237502792
#237502782
#237500042
#237498652
#237498312
#237497902
#237497842
C4×#145587212
#237496902
#461×C1492
#3283149