Finite Group Databasebeta

← C2×#22578060 #31794253

Subgroups — C2×#22578060 #31794253

Order 17536.

1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 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
1C22×Q8×D137C211yesmaximal, characteristic, complemented
2#22578060C2116yesmaximal, direct product factor
3C2×#17984075C213yesmaximal, complemented
4C2×#17984067C213yesmaximal, complemented
5C2×#17984059C213yesmaximal, complemented
6C2×Q8×Dic137C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊D137C213yesmaximal, complemented
8C22×#41×C137C211yesmaximal, characteristic, complemented
9C22×Dic274C2213yes—
10C2×Q8×D137—28no—
11C2×Q8×D137C2218yescomplemented
12C22×C22⋊D137C2213yes—
13C22×C4×D137C2213yes—
14#17984075C22124yescomplemented
15#17984067C22124yescomplemented
16#17984059C22124yescomplemented
17Q8×Dic137C2218yescomplemented
18C4×C22⋊D137C22124yescomplemented
19C2×C4⋊Dic137C2213yes—
20C2×#14555489C2213yes—
21C2×C22⋊Dic137C2213yes—
22C2×C22⋊Dic137C2216yes—
23C2×#14555487C2211yescharacteristic
24C2×C4×Dic137C2213yes—
25C2×#41×C137—28no—
26C2×#41×C137C2218yescomplemented
27C22×Q8×C137C2211yescharacteristic
28C22×D4×C137C2213yes—
29C23×C4×C137C2213yes—
30C23×D137C2311yescharacteristic
31C2×Dic274—212no—
32C2×Dic274C23112yes—
33Q8×D137—232no—
34C2×C22⋊D137—212no—
35C2×C22⋊D137C23112yes—
36C22×Dic137C2311yescharacteristic
37C22×Dic137C2313yes—
38C22×Dic137C2313yes—
39C2×C4×D137—212no—
40C2×C4×D137C23112yes—
41C4⋊Dic137C23112yes—
42#14555489C23112yes—
43C22⋊Dic137C23112yes—
44C22⋊Dic137C23124yes—
45#14555487C2314yes—
46C4×Dic137C23112yes—
47C24×C137C2313yes—
48#41×C137—232no—
49C2×Q8×C137D418yescomplemented
50C2×Q8×C137C2314yes—
51C2×D4×C137—212no—
52C2×D4×C137C23112yes—
53C22×C4×C137—212no—
54C22×C4×C137C2311yescharacteristic
55C22×C4×C137C23112yes—
56C22×C4×C137C2313yes—
57C22×D137—22no—
58C22×D137—22no—
59C22×D137#4114yescomplemented
60C22×D137C2412yes—
61Dic274—224no—
62C22⋊D137—224no—
63C2×Dic137—26no—
64C2×Dic137—26no—
65C2×Dic137—22no—
66C2×Dic137C2412yes—
67C2×Dic137C2416yes—
68C2×Dic137C2416yes—
69C4×D137—224no—
70C23×C137—26no—
71C23×C137—26no—
72C23×C137—26no—
73C23×C137C2416yes—
74C23×C137C2411yescharacteristic
75Q8×C137C2×D4116yescomplemented
76D4×C137—224no—
77C2×C4×C137—22no—
78C2×C4×C137—224no—
79C2×C4×C137C2×D4112yes—
80C2×C4×C137C2412yes—
81C2×C4×C137C2416yes—
82C22×C137—26no—
83C22×C137—26no—
84C22×C137—26no—
85C22×C137—412no—
86C22×C137C22×D412yes—
87C22×C137C2×#4112yes—
88C22×C1372−1+412yes—
89C22×C137C2511yescharacteristic
90C2×D137—28no—
91C2×D137—22no—
92C2×D137—22no—
93C2×D137C2×#4114yescomplemented
94C4×C137—22no—
95C4×C137C22×D4112yes—
96Dic137—26no—
97Dic137—26no—
98Dic137—22no—
99C2×C137—46no—
100C2×C137#54814yescomplemented
101C2×C137C23×D411yescharacteristic
102C2×C137C22×#4111yescharacteristic
103C2×C137C2×2−1+411yescharacteristic
104D137—24no—
105C137C2×#54811yescharacteristic, complemented, Sylow
106C2×#548—1371nomaximal, Sylow
107C22×#41D13711yescharacteristic, complemented
108C23×Q8—1371no—
109#548—13716no—
110C2×#124—1373no—
111C2×C22⋊Q8—1373no—
112C2×#121—1373no—
113C2×C4×Q8—1371no—
114C2×C4×D4—1373no—
115C2×#41—28no—
116C2×#41C2×D13718yescomplemented
117C22×Q8—2748no—
118C22×Q8—1378no—
119C22×Q8—1373no—
120C22×Q8C2×D13711yescharacteristic
121C22×D4—1373no—
122C22×D4C2×D13713yes—
123C23×C4—1373no—
124C23×C4C2×D13713yes—
125#124—13724no—
126C22⋊Q8—13724no—
127#121—13724no—
128C4×Q8—1378no—
129C4×D4—13724no—
130C2×C4⋊C4—1373no—
131C2×C4⋊C4—1373no—
132C2×C22⋊C4—1373no—
133C2×C22⋊C4—1376no—
134C2×C22⋊C4—1371no—
135C2×C42—1373no—
136C24—1371no—
137C24C22×D13713yes—
138#41—232no—
139C2×Q8—27432no—
140C2×Q8—27412no—
141C2×Q8—13712no—
142C2×Q8C22⋊D13718yescomplemented
143C2×Q8C22×D13714yes—
144C2×D4—212no—
145C2×D4—27412no—
146C2×D4—13712no—
147C2×D4C22×D137112yes—
148C22×C4—212no—
149C22×C4—27412no—
150C22×C4—1371no—
151C22×C4—1373no—
152C22×C4—1373no—
153C22×C4—13712no—
154C22×C4C22×D137112yes—
155C22×C4C22×D13711yescharacteristic
156C22×C4C22×D13713yes—
157C4⋊C4—13712no—
158C4⋊C4—13712no—
159C22⋊C4—13712no—
160C22⋊C4—13724no—
161C22⋊C4—1374no—
162C42—13712no—
163C23—26no—
164C23—26no—
165C23—26no—
166C23—2742no—
167C23—2742no—
168C23—1374no—
169C23—1372no—
170C23C23×D13716yes—
171C23C23×D13711yescharacteristic
172Q8—27424no—
173Q8C2×C22⋊D137116yescomplemented
174D4—224no—
175D4—27424no—
176C2×C4—22no—
177C2×C4—224no—
178C2×C4—2746no—
179C2×C4—2746no—
180C2×C4—2742no—
181C2×C4—27424no—
182C2×C4—1372no—
183C2×C4—1376no—
184C2×C4—1376no—
185C2×C4C2×C22⋊D137112yes—
186C2×C4C23×D13712yes—
187C2×C4C23×D13716yes—
188C22—26no—
189C22—26no—
190C22—26no—
191C22—412no—
192C22—2742no—
193C22—2742no—
194C22—2748no—
195C22—1374no—
196C22C22×C22⋊D13712yes—
197C22#41×D13712yes—
198C22#1798413512yes—
199C22C24×D13711yescharacteristic
200C4—22no—
201C4—2746no—
202C4—2746no—
203C4—2742no—
204C4C22×C22⋊D137112yes—
205C2—46no—
206C2—2744no—
207C2#2257806014yesdirect product factor
208C2C23×C22⋊D13711yescharacteristic
209C2C2×#41×D13711yescharacteristic
210C2C2×#1798413511yescharacteristic
subgroupG/H
centerC23C23×D137
commutator, cosocleC22×C137C25
FrattiniC22C24×D137
FittingC22×#41×C137C2
socleC23×C137C24
hypercenterC22×#41D137
hypocenterC137C2×#548
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D137C2semidirect
2. #22578060C2direct (N × H)
3. C2×#17984075C2semidirect
4. C2×#17984067C2semidirect
5. C2×#17984059C2semidirect
6. C2×Q8×Dic137C2semidirect
7. C2×C4×C22⋊D137C2semidirect
8. C22×#41×C137C2semidirect
11. C2×Q8×D137C22semidirect
14. #17984075C22semidirect
15. #17984067C22semidirect
16. #17984059C22semidirect
17. Q8×Dic137C22semidirect
18. C4×C22⋊D137C22semidirect
26. C2×#41×C137C22semidirect
49. C2×Q8×C137D4semidirect
59. C22×D137#41semidirect
75. Q8×C137C2×D4semidirect
93. C2×D137C2×#41semidirect
100. C2×C137#548semidirect
105. C137C2×#548semidirect
107. C22×#41D137semidirect
116. C2×#41C2×D137semidirect
142. C2×Q8C22⋊D137semidirect
173. Q8C2×C22⋊D137semidirect
207. C2#22578060direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1372
#225780602
C2×#179840752
C2×#179840672
C2×#179840592
C2×Q8×Dic1372
C2×C4×C22⋊D1372
C22×#41×C1372
C2×#548137