Finite Group Databasebeta

← #34057319

Subgroups — #34057319

Order 19072.

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

normal subgroups only

This group has 354 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#23750340C211yesmaximal, characteristic
2#23750335C211yesmaximal, characteristic, complemented
3#23750030C211yesmaximal, characteristic, complemented
4#23749965C212yesmaximal, complemented
5#23749937C212yesmaximal, complemented
6#23749920C211yesmaximal, characteristic, complemented
7#23749915C211yesmaximal, characteristic, complemented
8#23749868C212yesmaximal, complemented
9#23749704C211yesmaximal, characteristic, complemented
10#23749664C212yesmaximal, complemented
11C4×SD16×C149C211yesmaximal, characteristic, complemented
12C2×#14558719C2212yes—
13#18011911C2211yescharacteristic
14#18011911C2211yescharacteristic
15#18011903—21no—
16#18011897—22no—
17#18011889—22no—
18#18011887C2212yes—
19#18011885—21no—
20#18011882C2212yes—
21C4×D596—21no—
22C4×D596C2211yescharacteristic
23#18011870—21no—
24#18011869C2212yes—
25C4⋊C4×D149C2212yes—
26#18011845C2212yes—
27#18011845C2212yes—
28#18011844C2212yes—
29#18011842C2212yes—
30C4⋊#11924746C2211yescharacteristic
31#18011835C2212yes—
32#18011832C2212yes—
33#18011823C2212yes—
34C2×SD16×C149C2211yescharacteristic
35C4×Q8×C149C2211yescharacteristic
36C4×D4×C149C2211yescharacteristic
37#106×C149C2211yescharacteristic
38#103×C149C2211yescharacteristic
39#102×C149C2211yescharacteristic
40C4×C8×C149C2211yescharacteristic
41C2×Dic298—21no—
42C2×Dic298—21no—
43C2×D596D411yescharacteristic
44C2×C22⋊D149—22no—
45C2×C4×D149—22no—
46C2×C4×D149C2312yes—
47#14558719—24no—
48C2×#11924746C2312yes—
49C4⋊Dic149—21no—
50C4⋊Dic149—21no—
51C4⋊Dic149D411yescharacteristic
52C4⋊Dic149C2312yes—
53C4⋊Dic149C2312yes—
54#14558715—22no—
55#14558715—22no—
56C22⋊Dic149—22no—
57#14558713—22no—
58#14558713—22no—
59#14558713D412yes—
60C4×Dic149—22no—
61C2×Q8×C149C2311yescharacteristic
62C2×D4×C149C2311yescharacteristic
63C22×C4×C149—21no—
64SD16×C149—22no—
65C2×C8×C149C2311yescharacteristic
66C2×C8×C149C2311yescharacteristic
67C4⋊C4×C149—21no—
68C4⋊C4×C149C2311yescharacteristic
69C4⋊C4×C149C2311yescharacteristic
70C22⋊C4×C149—21no—
71C42×C149—21no—
72C42×C149C2311yescharacteristic
73C22×D149C2×D412yes—
74Dic298—81no—
75Dic298—81no—
76D596—22no—
77C22⋊D149—82no—
78C2×Dic149—22no—
79C2×Dic149—22no—
80C2×Dic149C2×D412yes—
81C4×D149—22no—
82C4×D149—22no—
83C4×D149—82no—
84#11924746—22no—
85C23×C149—21no—
86Q8×C149—21no—
87Q8×C149—21no—
88D4×C149—21no—
89D4×C149—21no—
90C2×C4×C149—21no—
91C2×C4×C149—21no—
92C2×C4×C149—21no—
93C2×C4×C149—42no—
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2×D411yescharacteristic
96C2×C4×C149C2411yescharacteristic
97C8×C149—21no—
98C8×C149#4112yes—
99C22×C149—41no—
100C22×C149—41no—
101C22×C149—41no—
102C22×C149C22×D411yescharacteristic
103C2×D149—22no—
104C2×D149—22no—
105C2×D149—22no—
106C4×C149—21no—
107C4×C149—41no—
108C4×C149—41no—
109C4×C149—41no—
110C4×C149#13712yes—
111C4×C149C2×#4111yescharacteristic
112C4×C1492−1+411yescharacteristic
113Dic149—22no—
114Dic149—42no—
115Dic149—42no—
116C2×C149—81no—
117C2×C149#54711yescharacteristic
118C2×C149C2×#13711yescharacteristic
119C2×C149#57611yescharacteristic
120D149—42no—
121C149#335711yescharacteristic, complemented, Sylow
122#3357—1491nomaximal, Sylow
123#555—1491no—
124#554—1491no—
125#500—1491no—
126#485—1492no—
127#480—1492no—
128#477—1491no—
129#476—1491no—
130#468—1492no—
131C4×SD16D14911yescharacteristic, complemented
132#436—1491no—
133#428—1492no—
134C2×SD16C2×D14911yescharacteristic
135C2×M16—1492no—
136C4⋊Q8—1491no—
137C4⋊Q8—1491no—
138#125—2981no—
139#124—2982no—
140C22⋊Q8—2982no—
141C22⋊Q8—1492no—
142#121—2981no—
143#121—1492no—
144C4×Q8C2×D14911yescharacteristic
145C4×D4—2981no—
146C4×D4—1491no—
147C4×D4C2×D14911yescharacteristic
148#117—2981no—
149#117—1492no—
150C2×C4⋊C4—1492no—
151#107—1492no—
152#107—1492no—
153#107—1492no—
154#106—1492no—
155#106C2×D14911yescharacteristic
156C4⋊C8—1491no—
157#103—1492no—
158#103C2×D14911yescharacteristic
159#102—1492no—
160#102C2×D14911yescharacteristic
161C22⋊C8—1492no—
162C4×C8C2×D14911yescharacteristic
163C2×Q8—2981no—
164C2×Q8—2981no—
165C2×Q8C22×D14911yescharacteristic
166C2×D4—2982no—
167C2×D4—1491no—
168C2×D4C22×D14911yescharacteristic
169C22×C4—21no—
170C22×C4—2982no—
171C22×C4—1492no—
172SD16—22no—
173M16—2984no—
174C2×C8—1492no—
175C2×C8C22×D14911yescharacteristic
176C2×C8C22×D14911yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—2982no—
179C4⋊C4—2982no—
180C4⋊C4—2981no—
181C4⋊C4—2981no—
182C4⋊C4—1492no—
183C4⋊C4—1492no—
184C4⋊C4—1491no—
185C4⋊C4C22×D14911yescharacteristic
186C4⋊C4C22×D14911yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—2982no—
189C22⋊C4—2982no—
190C22⋊C4—2982no—
191C22⋊C4—1492no—
192C42—21no—
193C42—2982no—
194C42C22×D14911yescharacteristic
195C23—21no—
196C23—1492no—
197Q8—21no—
198Q8—21no—
199Q8—11921no—
200Q8—11921no—
201D4—21no—
202D4—21no—
203D4—2982no—
204D4—11922no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—2982no—
210C2×C4—2982no—
211C2×C4—2982no—
212C2×C4—2982no—
213C2×C4—1492no—
214C2×C4—11922no—
215C2×C4D4×D14911yescharacteristic
216C2×C4D4×D14911yescharacteristic
217C2×C4C23×D14911yescharacteristic
218C8—21no—
219C8—2982no—
220C8#1455873612yes—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—2982no—
225C22—2982no—
226C22—2982no—
227C22C2×D4×D14911yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—2982no—
233C4—5962no—
234C4—5962no—
235C4#1801194012yes—
236C4C2×#1455873611yescharacteristic
237C4#1801195811yescharacteristic
238C2—81no—
239C2—5962no—
240C2#2375028011yescharacteristic
241C2C2×#1801194011yescharacteristic
242C2#2375042711yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×SD16×C149C2
socleC22×C149C22×D4
prime radical#23750340C2
hypercenterC4×SD16D149
hypocenterC149#3357
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #23750335C2semidirect
3. #23750030C2semidirect
4. #23749965C2semidirect
5. #23749937C2semidirect
6. #23749920C2semidirect
7. #23749915C2semidirect
8. #23749868C2semidirect
9. #23749704C2semidirect
10. #23749664C2semidirect
11. C4×SD16×C149C2semidirect
121. C149#3357semidirect
131. C4×SD16D149semidirect

Maximal subgroups

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

subgroupindex
#237503402
#237503352
#237500302
#237499652
#237499372
#237499202
#237499152
#237498682
#237497042
#237496642
C4×SD16×C1492
#3357149