Finite Group Databasebeta

← #34055316

Subgroups — #34055316

Order 19072.

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

normal subgroups only

This group has 362 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#23750236C211yesmaximal, characteristic, complemented
2#23750229C211yesmaximal, characteristic
3#23750027C211yesmaximal, characteristic, complemented
4#23749993C211yesmaximal, characteristic, complemented
5#23749989C211yesmaximal, characteristic, complemented
6#23749868C212yesmaximal, complemented
7#23749851C212yesmaximal, complemented
8#23749726C212yesmaximal, complemented
9SD16×Dic149C212yesmaximal, complemented
10#23749579C211yesmaximal, characteristic, complemented
11#492×C149C211yesmaximal, characteristic, complemented
12C2×#14558719C2211yescharacteristic
13C2×C8×D149C2211yescharacteristic
14#18011906—22no—
15#18011903C2211yescharacteristic
16#18011903C2211yescharacteristic
17#18011901—22no—
18#18011892—22no—
19#18011887C2212yes—
20#18011883—22no—
21#18011882C2212yes—
22Q8×Dic149C2212yes—
23D4×Dic149C2212yes—
24#18011868C2211yescharacteristic
25#18011845C2212yes—
26#18011843C2212yes—
27#18011835C2212yes—
28#18011835C2212yes—
29#18011832C2212yes—
30#18011832C2212yes—
31#18011822C2211yescharacteristic
32#18011821C2211yescharacteristic
33C8×Dic149C2211yescharacteristic
34C2×SD16×C149C2212yes—
35C2×SD16×C149C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37#127×C149C2211yescharacteristic
38C4×C8×C149C2211yescharacteristic
39C2×C22⋊D149—22no—
40C22×Dic149—22no—
41C2×C4×D149C2311yescharacteristic
42#14558719—22no—
43C2×#11924746C2311yescharacteristic
44C2×#11924746C2311yescharacteristic
45C8×D149—22no—
46C4⋊Dic149—22no—
47C4⋊Dic149C2312yes—
48C4⋊Dic149C2312yes—
49#14558715—22no—
50#14558715—22no—
51#14558715—22no—
52#14558715D412yes—
53C22⋊Dic149—22no—
54C22⋊Dic149—22no—
55#14558713—22no—
56#14558713D412yes—
57C4×Dic149—22no—
58C4×Dic149C2311yescharacteristic
59C2×Q8×C149C2312yes—
60C2×D4×C149—22no—
61C2×D4×C149C2312yes—
62SD16×C149—24no—
63SD16×C149—24no—
64C2×C8×C149C2311yescharacteristic
65C2×C8×C149C2311yescharacteristic
66C4⋊C4×C149—22no—
67C42×C149C2311yescharacteristic
68C22×D149C2×D411yescharacteristic
69C22⋊D149—82no—
70C2×Dic149—22no—
71C2×Dic149—22no—
72C2×Dic149—82no—
73C2×Dic149C2×D411yescharacteristic
74C2×Dic149C2×D412yes—
75C4×D149—21no—
76C4×D149—21no—
77#11924746—21no—
78#11924746—21no—
79C23×C149—22no—
80Q8×C149—22no—
81Q8×C149—22no—
82D4×C149—22no—
83D4×C149—22no—
84D4×C149—82no—
85C2×C4×C149—22no—
86C2×C4×C149C2×D412yes—
87C2×C4×C149C2411yescharacteristic
88C8×C149#4112yes—
89C8×C149#4112yes—
90C22×C149—42no—
91C22×C149—42no—
92C22×C149—42no—
93C22×C149C22×D411yescharacteristic
94C2×D149—21no—
95C2×D149—22no—
96C4×C149—22no—
97C4×C149—42no—
98C4×C149C2×#4111yescharacteristic
99C4×C149C2×#4111yescharacteristic
100Dic149—21no—
101Dic149—22no—
102Dic149—42no—
103Dic149—42no—
104C2×C149—82no—
105C2×C149#53211yescharacteristic
106C2×C149#57712yes—
107D149—41no—
108C149#317411yescharacteristic, complemented, Sylow
109#3174—1491nomaximal, Sylow
110#541—1491no—
111#540—1491no—
112#499—1491no—
113#492D14911yescharacteristic, complemented
114#491—1491no—
115#490—1491no—
116#468—1492no—
117#465—1492no—
118#440—1492no—
119C4×SD16—1492no—
120#405—1491no—
121C2×SD16C2×D14912yes—
122C2×SD16C2×D14912yes—
123C2×M16—1491no—
124C22×C8—1491no—
125C4⋊Q8C2×D14911yescharacteristic
126#127C2×D14911yescharacteristic
127#126—2982no—
128#125—2982no—
129#125—1491no—
130#125—1491no—
131#123—2982no—
132C22⋊Q8—1492no—
133#121—2982no—
134#121—1492no—
135C4×Q8—1492no—
136C4×D4—1492no—
137#117—1491no—
138#107—1492no—
139#106—1492no—
140#103—1492no—
141#103—1492no—
142#102—1492no—
143#102—1492no—
144#97—1491no—
145#97—1491no—
146C4×C8—1491no—
147C4×C8C2×D14911yescharacteristic
148C2×Q8C22×D14912yes—
149C2×D4—22no—
150C2×D4—2982no—
151C2×D4C22×D14912yes—
152C22×C4—2982no—
153C22×C4—1491no—
154SD16—24no—
155SD16—24no—
156M16—2982no—
157C2×C8—2982no—
158C2×C8—1491no—
159C2×C8—1491no—
160C2×C8C22×D14911yescharacteristic
161C2×C8C22×D14911yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—2982no—
164C4⋊C4—2982no—
165C4⋊C4—2982no—
166C4⋊C4—2982no—
167C4⋊C4—1492no—
168C4⋊C4—1492no—
169C4⋊C4—1492no—
170C22⋊C4—2982no—
171C22⋊C4—2982no—
172C22⋊C4—2982no—
173C22⋊C4—1492no—
174C42—2982no—
175C42—1491no—
176C42C22×D14911yescharacteristic
177C23—22no—
178C23—1491no—
179Q8—22no—
180Q8—22no—
181D4—22no—
182D4—22no—
183D4—82no—
184D4—11922no—
185C2×C4—22no—
186C2×C4—2981no—
187C2×C4—2981no—
188C2×C4—2982no—
189C2×C4—2982no—
190C2×C4—1492no—
191C2×C4—1491no—
192C2×C4—11922no—
193C2×C4D4×D14912yes—
194C2×C4C23×D14911yescharacteristic
195C8—2981no—
196C8—2981no—
197C8#1455873512yes—
198C8#1455873512yes—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—2981no—
203C22—2982no—
204C22C2×D4×D14911yescharacteristic
205C4—22no—
206C4—42no—
207C4—2981no—
208C4—2982no—
209C4—5962no—
210C4—5962no—
211C4C2×#1455873511yescharacteristic
212C4C2×#1455873511yescharacteristic
213C2—82no—
214C2—5961no—
215C2#2375017311yescharacteristic
216C2#2375043112yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#492×C149C2
socleC22×C149C22×D4
prime radical#23750229C2
hypercenter#492D149
hypocenterC149#3174
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #23750236C2semidirect
3. #23750027C2semidirect
4. #23749993C2semidirect
5. #23749989C2semidirect
6. #23749868C2semidirect
7. #23749851C2semidirect
8. #23749726C2semidirect
9. SD16×Dic149C2semidirect
10. #23749579C2semidirect
11. #492×C149C2semidirect
108. C149#3174semidirect
113. #492D149semidirect

Maximal subgroups

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

subgroupindex
#237502362
#237502292
#237500272
#237499932
#237499892
#237498682
#237498512
#237497262
SD16×Dic1492
#237495792
#492×C1492
#3174149