Finite Group Databasebeta

← #34056433

Subgroups — #34056433

Order 19072.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×#18011934C211yesmaximal, characteristic, complemented
2C2×Q16×D149C211yesmaximal, characteristic, complemented
3#23750287C212yesmaximal, complemented
4#23750009C211yesmaximal, characteristic, complemented
5#23749867C211yesmaximal, characteristic, complemented
6#23749862C211yesmaximal, characteristic, complemented
7#23749847C212yesmaximal, complemented
8#23749790C212yesmaximal, complemented
9#23749765C212yesmaximal, complemented
10C8×D596C211yesmaximal, characteristic, complemented
11C4×Q16×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2212yes—
13C2×Q8×D149C2212yes—
14#18011934—24no—
15C2×Dic596C2211yescharacteristic
16C2×#14558726C2212yes—
17Q16×D149—24no—
18C2×#14558725C2212yes—
19C2×#14558723C2212yes—
20C2×D1192C2211yescharacteristic
21C2×C8×D149C2211yescharacteristic
22C2×C8×D149C2211yescharacteristic
23#18011899—22no—
24#18011899C2212yes—
25#18011891—22no—
26#18011891C2212yes—
27#18011886—22no—
28#18011886C2212yes—
29C4×D596—22no—
30C4×D596C2211yescharacteristic
31C4⋊#11924746C2211yescharacteristic
32#18011834C2212yes—
33#18011831C2212yes—
34#18011823C2211yescharacteristic
35#18011823C2211yescharacteristic
36C2×Q16×C149C2211yescharacteristic
37C4×Q8×C149C2212yes—
38#107×C149C2211yescharacteristic
39#103×C149C2212yes—
40C4×C8×C149C2211yescharacteristic
41#14558737—24no—
42#14558737—42no—
43C2×Dic298—22no—
44C2×Dic298C2312yes—
45Q8×D149—24no—
46Q8×D149—42no—
47C2×D596—22no—
48C2×D596—22no—
49C2×D596D411yescharacteristic
50C2×D596C2312yes—
51C2×C4×D149—22no—
52C2×C4×D149—22no—
53C2×C4×D149C2311yescharacteristic
54C2×C4×D149C2311yescharacteristic
55Dic596—22no—
56#14558726—24no—
57#14558725—42no—
58#14558723—24no—
59D1192—22no—
60C2×#11924746C2311yescharacteristic
61C2×#11924746C2311yescharacteristic
62C8×D149—22no—
63C8×D149—22no—
64C4⋊Dic149—22no—
65C4⋊Dic149D411yescharacteristic
66#14558713—22no—
67#14558713—22no—
68#14558713—22no—
69#14558713—22no—
70#14558713D411yescharacteristic
71#14558713D411yescharacteristic
72C2×Q8×C149C2312yes—
73Q16×C149D414yescomplemented
74C2×C8×C149C2311yescharacteristic
75C2×C8×C149C2311yescharacteristic
76C4⋊C4×C149—22no—
77C4⋊C4×C149C2312yes—
78C42×C149—22no—
79C42×C149C2311yescharacteristic
80C22×D149—22no—
81C22×D149C2×D411yescharacteristic
82C22×D149C2×D411yescharacteristic
83Dic298—22no—
84Dic298—22no—
85Dic298—44no—
86D596—22no—
87D596—22no—
88D596—22no—
89D596—44no—
90D596—44no—
91C2×Dic149—22no—
92C2×Dic149C2×D411yescharacteristic
93C2×Dic149C2×D411yescharacteristic
94C4×D149—21no—
95C4×D149—21no—
96C4×D149—21no—
97C4×D149—21no—
98C4×D149—44no—
99C4×D149—44no—
100#11924746—21no—
101#11924746—21no—
102Q8×C149—22no—
103Q8×C149C2×D414yes—
104C2×C4×C149—22no—
105C2×C4×C149—22no—
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2×D411yescharacteristic
108C2×C4×C149C2411yescharacteristic
109C8×C149—21no—
110C8×C149C2×D412yes—
111C22×C149C22×D411yescharacteristic
112C2×D149—21no—
113C2×D149—21no—
114C2×D149—21no—
115C2×D149—21no—
116C2×D149—21no—
117C2×D149—42no—
118C2×D149—42no—
119C2×D149—42no—
120C2×D149#13512yescomplemented
121C4×C149—24no—
122C4×C149—21no—
123C4×C149—21no—
124C4×C149—42no—
125C4×C149C22×D411yescharacteristic
126C4×C149H(5,𝔽2)11yescharacteristic
127Dic149—21no—
128Dic149—21no—
129Dic149—42no—
130C2×C149D4211yescharacteristic
131C2×C149C2×#13511yescharacteristic
132C2×C149#57811yescharacteristic
133D149—22no—
134D149—41no—
135D149—82no—
136C149#328511yescharacteristic, complemented, Sylow
137#3285—1491nomaximal, Sylow
138C2×#135—1491no—
139C22×Q16—1491no—
140#548—1492no—
141#495—1491no—
142#467—1491no—
143#466—1491no—
144#464—1492no—
145#452—1492no—
146#448—1492no—
147C4×Q16D14911yescharacteristic, complemented
148C8×D4—1491no—
149C2×#41—1492no—
150C22×Q8—1492no—
151#135—2984no—
152C2×Q16—2984no—
153C2×Q16—1492no—
154C2×Q16—1491no—
155C2×Q16C2×D14911yescharacteristic
156C2×SD16—1492no—
157C2×SD16—1492no—
158C2×D8—1491no—
159C22×C8—1491no—
160C22×C8—1491no—
161#124—2982no—
162#124—1492no—
163C22⋊Q8—2982no—
164C22⋊Q8—1492no—
165#121—2982no—
166#121—1492no—
167C4×Q8C2×D14912yes—
168C4×D4—2982no—
169C4×D4—1491no—
170#107C2×D14911yescharacteristic
171C4⋊C8—1491no—
172#103—1492no—
173#103C2×D14912yes—
174#102—1492no—
175C22⋊C8—1491no—
176C22⋊C8—1491no—
177C4×C8C2×D14911yescharacteristic
178#41—2984no—
179#41—5962no—
180C2×Q8—2982no—
181C2×Q8—2984no—
182C2×Q8—5962no—
183C2×Q8—1492no—
184C2×Q8C22×D14912yes—
185C2×D4—2982no—
186C2×D4—2982no—
187C2×D4—1491no—
188C2×D4—1492no—
189C22×C4—2982no—
190C22×C4—2982no—
191C22×C4—1491no—
192C22×C4—1491no—
193Q16—2984no—
194Q16—2982no—
195Q16D59614yescomplemented
196SD16—2984no—
197SD16—5962no—
198D8—2982no—
199C2×C8—2982no—
200C2×C8—2982no—
201C2×C8—1491no—
202C2×C8—1491no—
203C2×C8C22×D14911yescharacteristic
204C2×C8C22×D14911yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—2982no—
207C4⋊C4—1491no—
208C4⋊C4C22×D14912yes—
209C22⋊C4—2982no—
210C22⋊C4—2982no—
211C22⋊C4—2982no—
212C22⋊C4—2982no—
213C22⋊C4—1491no—
214C22⋊C4—1491no—
215C42—22no—
216C42C22×D14911yescharacteristic
217C23—2982no—
218C23—1491no—
219C23—1491no—
220Q8—22no—
221Q8—2982no—
222Q8—2982no—
223Q8—5964no—
224Q8C2×D59614yes—
225D4—2982no—
226D4—2982no—
227D4—2982no—
228D4—5964no—
229D4—5964no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—2981no—
233C2×C4—2981no—
234C2×C4—2981no—
235C2×C4—2981no—
236C2×C4—2982no—
237C2×C4—5964no—
238C2×C4—5964no—
239C2×C4—1491no—
240C2×C4—1491no—
241C2×C4D4×D14911yescharacteristic
242C2×C4D4×D14911yescharacteristic
243C2×C4C23×D14911yescharacteristic
244C8—21no—
245C8—2981no—
246C8—2981no—
247C8C2×D59612yes—
248C22—2981no—
249C22—2981no—
250C22—2981no—
251C22—2981no—
252C22—2981no—
253C22—5962no—
254C22—5962no—
255C22—5962no—
256C22—1492no—
257C22C2×D4×D14911yescharacteristic
258C4—21no—
259C4—21no—
260C4—24no—
261C4—42no—
262C4—2981no—
263C4—2981no—
264C4—5962no—
265C4C22×D59611yescharacteristic
266C4#1801195711yescharacteristic
267C2—2982no—
268C2—5961no—
269C2—11922no—
270C2D4×D59611yescharacteristic
271C2#135×D14911yescharacteristic
272C2#2375043811yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×Q16×C149C2
socleC22×C149C22×D4
hypercenterC4×Q16D149
hypocenterC149#3285
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#18011934C2semidirect
2. C2×Q16×D149C2semidirect
3. #23750287C2semidirect
4. #23750009C2semidirect
5. #23749867C2semidirect
6. #23749862C2semidirect
7. #23749847C2semidirect
8. #23749790C2semidirect
9. #23749765C2semidirect
10. C8×D596C2semidirect
11. C4×Q16×C149C2semidirect
73. Q16×C149D4semidirect
120. C2×D149#135semidirect
136. C149#3285semidirect
147. C4×Q16D149semidirect
195. Q16D596semidirect

Maximal subgroups

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

subgroupindex
C2×#180119342
C2×Q16×D1492
#237502872
#237500092
#237498672
#237498622
#237498472
#237497902
#237497652
C8×D5962
C4×Q16×C1492
#3285149