Finite Group Databasebeta

← #34058103

Subgroups — #34058103

Order 19072.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#23750224C216yesmaximal, complemented
2#23750215C216yesmaximal, complemented
3#23750187C213yesmaximal, complemented
4#23750161C216yesmaximal, complemented
5#23750138C216yesmaximal, complemented
6C2×#18011871C213yesmaximal, complemented
7#543×C149C211yesmaximal, characteristic, complemented
8C2×#14558737C2212yescomplemented
9C22×C22⋊D149C2213yes—
10C23×Dic149C2211yescharacteristic
11C22×C4×D149C2213yescomplemented
12#18011906C22112yescomplemented
13#18011902C2216yescomplemented
14#18011900C2216yescomplemented
15#18011896C22112yescomplemented
16#18011895C22112yescomplemented
17#18011887C2216yescomplemented
18#18011886C2216yescomplemented
19#18011885C2216yes—
20#18011885C2216yes—
21#18011881C22112yescomplemented
22Q8×Dic149C2212yescomplemented
23#18011873C2216yescomplemented
24#18011871C22112yescomplemented
25#18011871C22112yescomplemented
26#18011869C2216yescomplemented
27C2×#14558715C2213yescomplemented
28C2×#14558713C2213yescomplemented
29C2×C4×Dic149C2213yescomplemented
30C22⋊Q8×C149C2216yescomplemented
31C22⋊Q8×C149C2216yescomplemented
32C2×C22⋊C4×C149C2213yes—
33C23×D149C2313yes—
34#14558737—44no—
35C2×D596C2316yes—
36C2×D596C2316yes—
37C2×C22⋊D149—212no—
38C2×C22⋊D149C23112yes—
39C22×Dic149—42no—
40C22×Dic149C2313yes—
41C22×Dic149C2313yes—
42C22×Dic149C2313yes—
43C2×C4×D149—46no—
44C2×C4×D149C2316yes—
45C2×C4×D149C2316yes—
46C4⋊Dic149C2316yes—
47C4⋊Dic149C2316yes—
48#14558715C23112yes—
49#14558713C23112yes—
50#14558713C23112yes—
51#14558713C23112yes—
52C4×Dic149C2316yes—
53C4×Dic149C2316yes—
54C24×C149C2311yescharacteristic
55C2×Q8×C149C2312yes—
56C2×Q8×C149C2312yes—
57C22×C4×C149C2313yes—
58C22×C4×C149C2313yes—
59C4⋊C4×C149C2316yes—
60C4⋊C4×C149C2316yes—
61C22⋊C4×C149C23112yes—
62C22×D149—26no—
63C22×D149—43no—
64C22×D149—43no—
65C22×D149C2416yes—
66D596—46no—
67D596—46no—
68C22⋊D149—224no—
69C2×Dic149—26no—
70C2×Dic149—26no—
71C2×Dic149—43no—
72C2×Dic149C2413yes—
73C2×Dic149C2416yes—
74C2×Dic149C2411yescharacteristic
75C4×D149—212no—
76C4×D149—46no—
77C23×C149—41no—
78C23×C149—41no—
79C23×C149#4114yescomplemented
80C23×C149C2413yes—
81Q8×C149—42no—
82Q8×C149—42no—
83C2×C4×C149—43no—
84C2×C4×C149—43no—
85C2×C4×C149C2416yes—
86C2×C4×C149C2416yes—
87C22×C149—23no—
88C22×C149—23no—
89C22×C149—44no—
90C22×C149C2×#4116yes—
91C22×C149C2511yescharacteristic
92C2×D149—26no—
93C2×D149—26no—
94C2×D149—26no—
95C2×D149—412no—
96C4×C149—26no—
97C4×C149—26no—
98Dic149—26no—
99Dic149—23no—
100Dic149H(5,𝔽2)12yes—
101C2×C149—26no—
102C2×C149C22×#4111yescharacteristic
103C2×C149C2×H(5,𝔽2)11yescharacteristic
104C2×C149#58511yescharacteristic
105D149—46no—
106C149#345911yescharacteristic, complemented, Sylow
107#3459—1491nomaximal, Sylow
108#543D14911yescharacteristic, complemented
109#539—1496no—
110#538—1496no—
111#534—1493no—
112#529—1496no—
113#525—1496no—
114C2×C4×D4—1493no—
115C2×#41—1492no—
116C22×D4—1493no—
117C23×C4—1491no—
118C23×C4—1493no—
119#126—14912no—
120#125—1496no—
121#124—1496no—
122#123—14912no—
123#123—14912no—
124C22⋊Q8—1496no—
125C22⋊Q8C2×D14916yescomplemented
126C22⋊Q8C2×D14916yescomplemented
127#121—1496no—
128#121—1496no—
129#121—1496no—
130#120—14912no—
131C4×Q8—1492no—
132C4×D4—1496no—
133C4×D4—14912no—
134C4×D4—14912no—
135#117—1496no—
136C2×C4⋊C4—1493no—
137C2×C22⋊C4—1493no—
138C2×C22⋊C4C2×D14913yes—
139C2×C42—1493no—
140C24—1493no—
141C24C22×D14911yescharacteristic
142#41—5964no—
143C2×Q8C22×D14912yes—
144C2×Q8C22×D14912yes—
145C2×D4—29812no—
146C2×D4—14912no—
147C2×D4—1496no—
148C2×D4—1496no—
149C22×C4—5966no—
150C22×C4—5962no—
151C22×C4—1493no—
152C22×C4—1493no—
153C22×C4—1496no—
154C22×C4—1493no—
155C22×C4—1496no—
156C22×C4C22×D14913yes—
157C22×C4C22×D14913yes—
158C4⋊C4—14912no—
159C4⋊C4—1496no—
160C4⋊C4—1496no—
161C4⋊C4C22×D14916yes—
162C4⋊C4C22×D14916yes—
163C22⋊C4—14912no—
164C22⋊C4—14912no—
165C22⋊C4—14912no—
166C22⋊C4C22×D149112yes—
167C42—1496no—
168C42—1496no—
169C23—41no—
170C23—41no—
171C23—2986no—
172C23—5963no—
173C23—5963no—
174C23—1496no—
175C23#1455873714yescomplemented
176C23C23×D14913yes—
177Q8—42no—
178Q8—42no—
179D4—29824no—
180D4—5966no—
181D4—5966no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—29812no—
185C2×C4—2986no—
186C2×C4—2986no—
187C2×C4—5966no—
188C2×C4—5963no—
189C2×C4—1496no—
190C2×C4—1493no—
191C2×C4—1491no—
192C2×C4C23×D14916yes—
193C2×C4C23×D14916yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—2986no—
198C22—2986no—
199C22—2986no—
200C22—59612no—
201C22C2×#1455873716yes—
202C22C24×D14911yescharacteristic
203C4—26no—
204C4—26no—
205C4—2986no—
206C4—2983no—
207C4—1492no—
208C2—26no—
209C2—5966no—
210C2C22×#1455873711yescharacteristic
211C2H(5,𝔽2)×D14911yescharacteristic
212C2#2375046011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#543×C149C2
hypercenter#543D149
hypocenterC149#3459
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #23750224C2semidirect
2. #23750215C2semidirect
3. #23750187C2semidirect
4. #23750161C2semidirect
5. #23750138C2semidirect
6. C2×#18011871C2semidirect
7. #543×C149C2semidirect
8. C2×#14558737C22semidirect
11. C22×C4×D149C22semidirect
12. #18011906C22semidirect
13. #18011902C22semidirect
14. #18011900C22semidirect
15. #18011896C22semidirect
16. #18011895C22semidirect
17. #18011887C22semidirect
18. #18011886C22semidirect
21. #18011881C22semidirect
22. Q8×Dic149C22semidirect
23. #18011873C22semidirect
24. #18011871C22semidirect
25. #18011871C22semidirect
26. #18011869C22semidirect
27. C2×#14558715C22semidirect
28. C2×#14558713C22semidirect
29. C2×C4×Dic149C22semidirect
30. C22⋊Q8×C149C22semidirect
31. C22⋊Q8×C149C22semidirect
79. C23×C149#41semidirect
106. C149#3459semidirect
108. #543D149semidirect
125. C22⋊Q8C2×D149semidirect
126. C22⋊Q8C2×D149semidirect
175. C23#14558737semidirect

Maximal subgroups

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

subgroupindex
#237502242
#237502152
#237501872
#237501612
#237501382
C2×#180118712
#543×C1492
#3459149