Finite Group Databasebeta

← #34056993

Subgroups — #34056993

Order 19072.

420 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#23750279C211yesmaximal, characteristic, complemented
2#23750278C211yesmaximal, characteristic, complemented
3#23749977C211yesmaximal, characteristic, complemented
4#23749962C212yesmaximal, complemented
5#23749872C212yesmaximal, complemented
6#23749831C211yesmaximal, characteristic, complemented
7#23749771C212yesmaximal, complemented
8C4×#14558721C211yesmaximal, characteristic, complemented
9#23749702C211yesmaximal, characteristic, complemented
10#23749623C212yesmaximal, complemented
11#474×C149C211yesmaximal, characteristic, complemented
12C2×#14558737—21no—
13C2×#14558735C2212yes—
14C2×#14558725C2212yes—
15C2×#14558721C2211yescharacteristic
16C2×#14558719C2212yes—
17#18011911C2211yescharacteristic
18#18011896—22no—
19#18011895—22no—
20#18011891C2212yes—
21#18011887—21no—
22#18011886C2212yes—
23#18011885—21no—
24C4×D596C2211yescharacteristic
25C4×D596C2211yescharacteristic
26C2×C4⋊Dic149—21no—
27C4⋊C4×D149C2212yes—
28#18011844C2211yescharacteristic
29#18011836C2212yes—
30#18011834C2212yes—
31#18011834C2212yes—
32#18011832C2211yescharacteristic
33#18011831C2211yescharacteristic
34#18011823C2212yes—
35C4×#11924746C2211yescharacteristic
36C4⋊Q8×C149C2211yescharacteristic
37C4×D4×C149C2211yescharacteristic
38#107×C149C2212yes—
39C4⋊C8×C149C2211yescharacteristic
40#102×C149C2212yes—
41#14558737—44no—
42#14558735—24no—
43#14558735—42no—
44C2×Dic298C2312yes—
45C2×D596—22no—
46C2×D596D411yescharacteristic
47C2×D596C2311yescharacteristic
48C2×C22⋊D149—22no—
49C22×Dic149—22no—
50C2×C4×D149—22no—
51C2×C4×D149—21no—
52C2×C4×D149C2312yes—
53#14558725—42no—
54#14558721—22no—
55#14558719—42no—
56C2×#11924746C2311yescharacteristic
57C2×#11924746C2311yescharacteristic
58C4⋊Dic149—22no—
59C4⋊Dic149—21no—
60C4⋊Dic149—21no—
61C4⋊Dic149D411yescharacteristic
62C4⋊Dic149C2311yescharacteristic
63#14558715—22no—
64#14558713—22no—
65#14558713—22no—
66#14558713—21no—
67#14558713D412yes—
68C2×Q8×C149—21no—
69C2×D4×C149C2311yescharacteristic
70C22×C4×C149—21no—
71C2×C8×C149C2312yes—
72C4⋊C4×C149—21no—
73C4⋊C4×C149C2312yes—
74C4⋊C4×C149C2311yescharacteristic
75C22⋊C4×C149—21no—
76C42×C149C2311yescharacteristic
77C22×D149—21no—
78C22×D149C2×D412yes—
79Dic298—22no—
80Dic298—22no—
81D596—22no—
82D596—21no—
83D596—21no—
84D596—44no—
85C22⋊D149—44no—
86C2×Dic149—21no—
87C2×Dic149—22no—
88C2×Dic149—44no—
89C2×Dic149C2×D412yes—
90C4×D149—22no—
91C4×D149—22no—
92C4×D149—44no—
93C4×D149—42no—
94#11924746—21no—
95#11924746—21no—
96C23×C149—21no—
97Q8×C149—42no—
98D4×C149—21no—
99D4×C149#4112yescomplemented
100C2×C4×C149—22no—
101C2×C4×C149—21no—
102C2×C4×C149—42no—
103C2×C4×C149C2×D411yescharacteristic
104C2×C4×C149C2×D411yescharacteristic
105C2×C4×C149C2411yescharacteristic
106C8×C149—22no—
107C22×C149—22no—
108C22×C149—41no—
109C22×C149—41no—
110C22×C149C22×D411yescharacteristic
111C2×D149—22no—
112C2×D149—22no—
113C2×D149—22no—
114C2×D149—41no—
115C2×D149—41no—
116C2×D149—41no—
117C4×C149—21no—
118C4×C149—41no—
119C4×C149—42no—
120C4×C149AGL(1,ℤ/8)12yes—
121C4×C149C2×#4111yescharacteristic
122C4×C149H(5,𝔽2)11yescharacteristic
123Dic149—22no—
124Dic149—42no—
125Dic149—41no—
126C2×C149—42no—
127C2×C149#54611yescharacteristic
128C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C149#57811yescharacteristic
130D149—42no—
131D149—81no—
132C149#333011yescharacteristic, complemented, Sylow
133#3330—1491nomaximal, Sylow
134#547—1491no—
135#547—1491no—
136#487—1491no—
137#484—1492no—
138#474D14911yescharacteristic, complemented
139#468—1492no—
140#461—1491no—
141#449—1492no—
142C4×D8—1491no—
143#436—1491no—
144#417—1492no—
145C2×#41—2981no—
146C2×#41—1492no—
147C2×SD16—1492no—
148C2×D8—1491no—
149C2×M16—1492no—
150C4⋊Q8—1491no—
151C4⋊Q8C2×D14911yescharacteristic
152#123—2982no—
153#123—2982no—
154C22⋊Q8—2981no—
155C22⋊Q8—1492no—
156#121—2981no—
157#121—1492no—
158C4×D4—1491no—
159C4×D4—1491no—
160C4×D4C2×D14911yescharacteristic
161C2×C4⋊C4—2981no—
162C2×C4⋊C4—1492no—
163#107—1491no—
164#107C2×D14912yes—
165C4⋊C8C2×D14911yescharacteristic
166#103—1492no—
167#103—1492no—
168#103—1492no—
169#102—1491no—
170#102—1491no—
171#102C2×D14912yes—
172C22⋊C8—1492no—
173C4×C8—1491no—
174#41—2984no—
175#41—5962no—
176#41—5964no—
177C2×Q8—21no—
178C2×Q8—1492no—
179C2×D4—2982no—
180C2×D4—2982no—
181C2×D4—1491no—
182C2×D4—1491no—
183C2×D4C22×D14911yescharacteristic
184C22×C4—21no—
185C22×C4—2982no—
186C22×C4—2982no—
187C22×C4—2981no—
188C22×C4—1492no—
189SD16—5962no—
190D8—2982no—
191M16—5962no—
192C2×C8—1491no—
193C2×C8—1491no—
194C2×C8C22×D14912yes—
195C4⋊C4—21no—
196C4⋊C4—2982no—
197C4⋊C4—2982no—
198C4⋊C4—2981no—
199C4⋊C4—2981no—
200C4⋊C4—1491no—
201C4⋊C4—1491no—
202C4⋊C4C22×D14911yescharacteristic
203C4⋊C4C22×D14912yes—
204C22⋊C4—21no—
205C22⋊C4—2982no—
206C22⋊C4—2982no—
207C22⋊C4—2981no—
208C22⋊C4—1492no—
209C42C22×D14911yescharacteristic
210C23—21no—
211C23—2981no—
212C23—1492no—
213Q8—42no—
214Q8—2982no—
215Q8—2982no—
216D4—21no—
217D4—2982no—
218D4—2981no—
219D4—2981no—
220D4—5964no—
221D4—5964no—
222D4#1455873712yescomplemented
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—2982no—
227C2×C4—2982no—
228C2×C4—2982no—
229C2×C4—2981no—
230C2×C4—5964no—
231C2×C4—5964no—
232C2×C4—5962no—
233C2×C4—1492no—
234C2×C4D4×D14911yescharacteristic
235C2×C4D4×D14911yescharacteristic
236C2×C4C23×D14911yescharacteristic
237C8—22no—
238C8—2981no—
239C8—2981no—
240C22—22no—
241C22—41no—
242C22—41no—
243C22—2982no—
244C22—2982no—
245C22—2982no—
246C22—5961no—
247C22—5961no—
248C22—5961no—
249C22C2×D4×D14911yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—2982no—
254C4—5961no—
255C4—5962no—
256C4#1801193512yes—
257C4C2×#1455873711yescharacteristic
258C4#1801195711yescharacteristic
259C2—42no—
260C2—5962no—
261C2—11921no—
262C2#2375026611yescharacteristic
263C2C2×#1801193511yescharacteristic
264C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#474×C149C2
socleC22×C149C22×D4
hypercenter#474D149
hypocenterC149#3330
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23750279C2semidirect
2. #23750278C2semidirect
3. #23749977C2semidirect
4. #23749962C2semidirect
5. #23749872C2semidirect
6. #23749831C2semidirect
7. #23749771C2semidirect
8. C4×#14558721C2semidirect
9. #23749702C2semidirect
10. #23749623C2semidirect
11. #474×C149C2semidirect
99. D4×C149#41semidirect
132. C149#3330semidirect
138. #474D149semidirect
222. D4#14558737semidirect

Maximal subgroups

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

subgroupindex
#237502792
#237502782
#237499772
#237499622
#237498722
#237498312
#237497712
C4×#145587212
#237497022
#237496232
#474×C1492
#3330149