Finite Group Databasebeta

← #34057824

Subgroups — #34057824

Order 19072.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D149C212yesmaximal, complemented
2#23750067C212yesmaximal, complemented
3#23750064C212yesmaximal, complemented
4#23750062C212yesmaximal, complemented
5#23750059C212yesmaximal, complemented
6#23750057C211yesmaximal, characteristic, complemented
7#23750055C212yesmaximal, complemented
8C2×#18011850C211yesmaximal, characteristic, complemented
9#508×C149C211yesmaximal, characteristic, complemented
10#41×D149—22no—
11#18011934C2212yes—
12#18011933C2212yescomplemented
13#18011932—22no—
14#18011931—22no—
15#18011930C2212yes—
16Q16×D149C2212yescomplemented
17SD16×D149—22no—
18D8×D149C2212yescomplemented
19C2×C8×D149C2211yescharacteristic
20Dic1192C2211yescharacteristic, complemented
21#18011857C2212yescomplemented
22#18011856C2212yescomplemented
23#18011855C2212yescomplemented
24#18011854C2212yescomplemented
25D2384C2211yescharacteristic, complemented
26#18011852C2212yescomplemented
27#18011850C2212yescomplemented
28#18011850C2212yescomplemented
29C2×#14558711C2211yescharacteristic, complemented
30#135×C149C2212yescomplemented
31Q32×C149C2211yescharacteristic, complemented
32SD32×C149C2212yescomplemented
33D16×C149C2211yescharacteristic, complemented
34C2×C16×C149C2211yescharacteristic, complemented
35#14558737—22no—
36#14558736—22no—
37#14558736—42no—
38#14558735—22no—
39#14558735—42no—
40Q8×D149—22no—
41D4×D149—22no—
42D4×D149—42no—
43C2×C4×D149—42no—
44C2×C4×D149D411yescharacteristic
45Dic596C2312yes—
46#14558726—22no—
47#14558725—22no—
48#14558724—22no—
49#14558723—22no—
50D1192C2312yes—
51#14558721—22no—
52C2×#11924746D411yescharacteristic
53C8×D149D412yes—
54C8×D149C2312yes—
55#14558711C2312yes—
56#41×C149—22no—
57Q16×C149C2312yes—
58SD16×C149—22no—
59D8×C149C2312yes—
60C2×C8×C149C2311yescharacteristic
61C16×C149C2311yescharacteristic
62C16×C149C2311yescharacteristic
63C22×D149—42no—
64C22×D149D811yescharacteristic
65Dic298—22no—
66Dic298—42no—
67D596—22no—
68D596—42no—
69C22⋊D149—42no—
70C22⋊D149—42no—
71C22⋊D149—42no—
72C2×Dic149—42no—
73C2×Dic149D811yescharacteristic
74C4×D149—42no—
75C4×D149—42no—
76C4×D149—42no—
77C4×D149D812yes—
78C4×D149C2×D412yes—
79#11924746C2×D412yes—
80Q8×C149—22no—
81D4×C149—22no—
82D4×C149—42no—
83C2×C4×C149—42no—
84C2×C4×C149C2×D411yescharacteristic
85C8×C149C2×D411yescharacteristic
86C8×C149C2411yescharacteristic
87C22×C149—42no—
88C22×C149C2×D811yescharacteristic
89C2×D149—41no—
90C2×D149—42no—
91C2×D149—84no—
92C2×D149C2×D812yes—
93C4×C149—42no—
94C4×C149C2×D811yescharacteristic
95C4×C149C22×D411yescharacteristic
96Dic149—42no—
97Dic149C2×D812yes—
98C2×C149—21no—
99C2×C149—82no—
100C2×C149C22×D811yescharacteristic
101D149—22no—
102D149—82no—
103C149#341411yescharacteristic, complemented, Sylow
104#3414—1491nomaximal, Sylow
105C2×#135—1492no—
106#510—1492no—
107#510—1492no—
108#509—1492no—
109#509—1492no—
110#508—1492no—
111#508—1491no—
112#508D14911yescharacteristic, complemented
113C2×M32—1491no—
114C2×#41—2982no—
115#135—2982no—
116#135—2982no—
117#135—1492no—
118#135—1492no—
119#135—1492no—
120#135C2×D14912yescomplemented
121C2×Q16—1492no—
122C2×SD16—2982no—
123C2×D8—1492no—
124C22×C8—1491no—
125Q32—1492no—
126Q32—1491no—
127Q32C2×D14911yescharacteristic, complemented
128SD32—1492no—
129SD32—1492no—
130SD32—1492no—
131SD32C2×D14912yescomplemented
132D16—1492no—
133D16—1491no—
134D16C2×D14911yescharacteristic, complemented
135M32—1492no—
136M32—1492no—
137C2×C16—1491no—
138C2×C16C2×D14911yescharacteristic, complemented
139#41—22no—
140#41—2982no—
141#41—2982no—
142#41—2982no—
143#41—5962no—
144#41—5962no—
145C2×Q8—2982no—
146C2×D4—2982no—
147C2×D4—5962no—
148C22×C4—5962no—
149C22×C4—1491no—
150Q16—2982no—
151Q16—1492no—
152Q16C22×D14912yes—
153SD16—22no—
154SD16—2982no—
155SD16—2982no—
156SD16—2982no—
157D8—2982no—
158D8—1492no—
159D8C22×D14912yes—
160C2×C8—1491no—
161C2×C8—1492no—
162C2×C8—1492no—
163C2×C8C22×D14911yescharacteristic
164C16—1492no—
165C16C22×D14911yescharacteristic
166C16C22×D14911yescharacteristic
167C23—5962no—
168C23—1491no—
169Q8—22no—
170Q8—2982no—
171Q8—5962no—
172D4—22no—
173D4—42no—
174D4—2982no—
175D4—5962no—
176D4—5962no—
177D4—5962no—
178D4—5962no—
179C2×C4—42no—
180C2×C4—5962no—
181C2×C4—5962no—
182C2×C4—5962no—
183C2×C4—5962no—
184C2×C4—1491no—
185C2×C4—1492no—
186C2×C4—1492no—
187C2×C4D4×D14911yescharacteristic
188C8—1492no—
189C8D4×D14911yescharacteristic
190C8C23×D14911yescharacteristic
191C22—42no—
192C22—5961no—
193C22—5962no—
194C22—1492no—
195C22—11924no—
196C22D8×D14911yescharacteristic
197C4—42no—
198C4—5962no—
199C4—1492no—
200C4D8×D14911yescharacteristic
201C4C2×D4×D14911yescharacteristic
202C2—21no—
203C2—82no—
204C2—2982no—
205C2—11922no—
206C2C2×D8×D14911yescharacteristic
subgroupG/H
centerC4D8×D149
commutator, cosocleC8×C149C24
FrattiniC8C23×D149
Fitting#508×C149C2
socleC2×C149C22×D8
hypercenter#508D149
hypocenterC149#3414
LCSG ▷ C8×C149 ▷ C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D149C2semidirect
2. #23750067C2semidirect
3. #23750064C2semidirect
4. #23750062C2semidirect
5. #23750059C2semidirect
6. #23750057C2semidirect
7. #23750055C2semidirect
8. C2×#18011850C2semidirect
9. #508×C149C2semidirect
12. #18011933C22semidirect
16. Q16×D149C22semidirect
18. D8×D149C22semidirect
20. Dic1192C22semidirect
21. #18011857C22semidirect
22. #18011856C22semidirect
23. #18011855C22semidirect
24. #18011854C22semidirect
25. D2384C22semidirect
26. #18011852C22semidirect
27. #18011850C22semidirect
28. #18011850C22semidirect
29. C2×#14558711C22semidirect
30. #135×C149C22semidirect
31. Q32×C149C22semidirect
32. SD32×C149C22semidirect
33. D16×C149C22semidirect
34. C2×C16×C149C22semidirect
103. C149#3414semidirect
112. #508D149semidirect
120. #135C2×D149semidirect
127. Q32C2×D149semidirect
131. SD32C2×D149semidirect
134. D16C2×D149semidirect
138. C2×C16C2×D149semidirect

Maximal subgroups

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

subgroupindex
#135×D1492
#237500672
#237500642
#237500622
#237500592
#237500572
#237500552
C2×#180118502
#508×C1492
#3414149