Finite Group Databasebeta

← #34055321

Subgroups — #34055321

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#23750238C211yesmaximal, characteristic, complemented
2#23750231C211yesmaximal, characteristic
3#23749998C211yesmaximal, characteristic, complemented
4#23749995C211yesmaximal, characteristic, complemented
5#23749991C211yesmaximal, characteristic, complemented
6#23749872C212yesmaximal, complemented
7#23749855C212yesmaximal, complemented
8#23749724C212yesmaximal, complemented
9#23749712C212yesmaximal, complemented
10#23749579C211yesmaximal, characteristic, complemented
11#499×C149C211yesmaximal, characteristic, complemented
12C2×#14558725C2212yes—
13C2×#14558725C2212yes—
14C2×#14558719C2211yescharacteristic
15C2×C8×D149C2211yescharacteristic
16#18011911C2211yescharacteristic
17#18011908C2211yescharacteristic
18#18011906—22no—
19#18011902—22no—
20#18011896—22no—
21#18011891C2212yes—
22#18011886—22no—
23#18011886C2212yes—
24#18011876C2212yes—
25#18011873C2212yes—
26#18011868C2211yescharacteristic
27#18011834C2212yes—
28#18011834C2212yes—
29#18011831C2212yes—
30#18011831C2212yes—
31#18011822C2211yescharacteristic
32#18011821C2211yescharacteristic
33C8×Dic149C2211yescharacteristic
34#125×C149C2211yescharacteristic
35#125×C149C2211yescharacteristic
36#107×C149C2212yes—
37#106×C149C2212yes—
38C4×C8×C149C2211yescharacteristic
39C2×Dic298C2312yes—
40C2×D596—22no—
41C2×D596—22no—
42C2×D596C2312yes—
43C2×C4×D149—22no—
44C2×C4×D149C2311yescharacteristic
45#14558725—24no—
46#14558725—24no—
47#14558719—22no—
48C2×#11924746C2311yescharacteristic
49C2×#11924746C2311yescharacteristic
50C8×D149—22no—
51C4⋊Dic149—22no—
52C4⋊Dic149—22no—
53#14558715—22no—
54#14558715D412yes—
55#14558713—22no—
56#14558713—22no—
57#14558713—22no—
58#14558713D412yes—
59C4×Dic149—22no—
60C4×Dic149C2311yescharacteristic
61C2×C8×C149C2311yescharacteristic
62C2×C8×C149C2311yescharacteristic
63C4⋊C4×C149—22no—
64C4⋊C4×C149—22no—
65C4⋊C4×C149C2312yes—
66C4⋊C4×C149C2312yes—
67C42×C149C2311yescharacteristic
68C22×D149—22no—
69C22×D149C2×D411yescharacteristic
70Dic298—22no—
71Dic298—22no—
72D596—22no—
73D596—22no—
74D596—82no—
75D596—82no—
76C2×Dic149—22no—
77C2×Dic149C2×D412yes—
78C2×Dic149C2×D411yescharacteristic
79C4×D149—21no—
80C4×D149—21no—
81C4×D149—82no—
82#11924746—21no—
83#11924746—21no—
84C2×C4×C149—22no—
85C2×C4×C149—22no—
86C2×C4×C149C2×D412yes—
87C2×C4×C149C2411yescharacteristic
88C8×C149#4112yes—
89C8×C149#4112yes—
90C22×C149C22×D411yescharacteristic
91C2×D149—21no—
92C2×D149—22no—
93C2×D149—42no—
94C2×D149—42no—
95C2×D149—42no—
96C4×C149—22no—
97C4×C149—42no—
98C4×C149—42no—
99C4×C149C2×#4111yescharacteristic
100C4×C149C2×#4111yescharacteristic
101Dic149—21no—
102Dic149—22no—
103Dic149—42no—
104C2×C149#53211yescharacteristic
105C2×C149#57712yes—
106D149—41no—
107D149—82no—
108C149#317411yescharacteristic, complemented, Sylow
109#3174—1491nomaximal, Sylow
110#541—1491no—
111#540—1491no—
112#499D14911yescharacteristic, complemented
113#492—1491no—
114#491—1491no—
115#490—1491no—
116#468—1492no—
117#465—1492no—
118#440—1492no—
119C4×SD16—1492no—
120#405—1491no—
121C2×SD16—1492no—
122C2×SD16—1492no—
123C2×M16—1491no—
124C22×C8—1491no—
125C4⋊Q8—1491no—
126#127—1491no—
127#126—2982no—
128#125—2982no—
129#125C2×D14911yescharacteristic
130#125C2×D14911yescharacteristic
131#123—2982no—
132C22⋊Q8—1492no—
133#121—2982no—
134#121—1492no—
135C4×Q8—1492no—
136C4×D4—1492no—
137#117—1491no—
138#107C2×D14912yes—
139#106C2×D14912yes—
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×Q8—1492no—
149C2×D4—2982no—
150C2×D4—2982no—
151C2×D4—1492no—
152C22×C4—2982no—
153C22×C4—1491no—
154SD16—2984no—
155SD16—2984no—
156M16—2982no—
157C2×C8—2982no—
158C2×C8—1491no—
159C2×C8—1491no—
160C2×C8C22×D14911yescharacteristic
161C2×C8C22×D14911yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—2982no—
165C4⋊C4—2982no—
166C4⋊C4—2982no—
167C4⋊C4—1492no—
168C4⋊C4C22×D14912yes—
169C4⋊C4C22×D14912yes—
170C22⋊C4—2982no—
171C22⋊C4—2982no—
172C22⋊C4—2982no—
173C22⋊C4—1492no—
174C42—2982no—
175C42—1491no—
176C42C22×D14911yescharacteristic
177C23—2982no—
178C23—1491no—
179Q8—2982no—
180Q8—2982no—
181D4—2982no—
182D4—2982no—
183D4—11922no—
184D4—11922no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—2981no—
188C2×C4—2981no—
189C2×C4—2982no—
190C2×C4—1492no—
191C2×C4—1491no—
192C2×C4—11922no—
193C2×C4D4×D14912yes—
194C2×C4C23×D14911yescharacteristic
195C8—2981no—
196C8—2981no—
197C8#1455873712yes—
198C8#1455873712yes—
199C22—2981no—
200C22—2982no—
201C22—5962no—
202C22—5962no—
203C22—5962no—
204C22C2×D4×D14911yescharacteristic
205C4—22no—
206C4—42no—
207C4—42no—
208C4—2981no—
209C4—2982no—
210C4—5962no—
211C4C2×#1455873711yescharacteristic
212C4C2×#1455873711yescharacteristic
213C2—5961no—
214C2—11922no—
215C2#2375017611yescharacteristic
216C2#2375043212yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#499×C149C2
socleC22×C149C22×D4
prime radical#23750231C2
hypercenter#499D149
hypocenterC149#3174
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #23750238C2semidirect
3. #23749998C2semidirect
4. #23749995C2semidirect
5. #23749991C2semidirect
6. #23749872C2semidirect
7. #23749855C2semidirect
8. #23749724C2semidirect
9. #23749712C2semidirect
10. #23749579C2semidirect
11. #499×C149C2semidirect
108. C149#3174semidirect
112. #499D149semidirect

Maximal subgroups

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

subgroupindex
#237502382
#237502312
#237499982
#237499952
#237499912
#237498722
#237498552
#237497242
#237497122
#237495792
#499×C1492
#3174149