Finite Group Databasebeta

← #123×D317 #35203723

Subgroups — #123×D317 #35203723

Order 20288.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×D4×D317C211yesmaximal, characteristic
2C22×C4×D317C211yesmaximal, characteristic, complemented
3#23778765C212yesmaximal, complemented
4#23778764C211yesmaximal, characteristic, complemented
5#23778763C211yesmaximal, characteristic, complemented
6#23778762C212yesmaximal, complemented
7#23778761C211yesmaximal, characteristic, complemented
8C4⋊C4×D317C212yesmaximal, complemented
9C22⋊C4×D317C212yesmaximal, complemented
10C22⋊C4×D317C211yesmaximal, characteristic, complemented
11#123×C317C211yesmaximal, characteristic, complemented
12C23×D317C2211yescharacteristic
13C23×D317C2211yescharacteristic
14C2×D1268C2211yescharacteristic
15C2×C22⋊D317C2211yescharacteristic
16C2×C22⋊D317C2211yescharacteristic
17D4×D317—24no—
18C22×Dic317C2211yescharacteristic, complemented
19C2×C4×D317—24no—
20C2×C4×D317C2211yescharacteristic
21C2×C4×D317C2212yes—
22C2×C4×D317C2212yes—
23C2×C4×D317C2212yescomplemented
24C4⋊Dic317C2212yescomplemented
25#19099553C2212yescomplemented
26#19099553C2212yescomplemented
27C22⋊Dic317C2212yescomplemented
28C22⋊Dic317C2211yescharacteristic, complemented
29#19099551C2212yescomplemented
30#19099551C2212yescomplemented
31#19099551C2212yescomplemented
32C2×D4×C317C2211yescharacteristic
33C22×C4×C317C2211yescharacteristic, complemented
34C4⋊C4×C317C2212yescomplemented
35C22⋊C4×C317C2212yescomplemented
36C22⋊C4×C317C2211yescharacteristic, complemented
37C22×D317—24no—
38C22×D317—22no—
39C22×D317—24no—
40C22×D317D414yes—
41C22×D317C2311yescharacteristic
42C22×D317C2311yescharacteristic
43C22×D317C2311yescharacteristic
44D1268—22no—
45C22⋊D317—22no—
46C22⋊D317—22no—
47C2×Dic317—22no—
48C2×Dic317C2312yes—
49C2×Dic317C2312yes—
50C2×Dic317C2311yescharacteristic
51C4×D317—24no—
52C4×D317—24no—
53C4×D317—22no—
54C4×D317—24no—
55C23×C317C2311yescharacteristic
56C23×C317C2311yescharacteristic
57D4×C317—22no—
58C2×C4×C317—22no—
59C2×C4×C317C2312yes—
60C2×C4×C317C2312yes—
61C2×C4×C317C2311yescharacteristic
62C22×C317—22no—
63C22×C317—21no—
64C22×C317—22no—
65C22×C317C2×D412yes—
66C22×C317C2411yescharacteristic
67C2×D317—22no—
68C2×D317—28no—
69C2×D317—21no—
70C2×D317—22no—
71C2×D317—44no—
72C2×D317C2×D412yes—
73C2×D317C2×D412yes—
74C2×D317#4114yes—
75C4×C317—22no—
76C4×C317—21no—
77C4×C317—22no—
78Dic317—22no—
79Dic317—21no—
80Dic317—22no—
81C2×C317—22no—
82C2×C317—41no—
83C2×C317C22×D411yescharacteristic
84C2×C317C2×#4112yes—
85D317—22no—
86D317—41no—
87D317#12314yesdirect product factor
88C317C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—3171nomaximal, Sylow
90C22×D4—3171no—
91C23×C4—3171no—
92#123—3171no—
93#123—3171no—
94#123—3172no—
95#123—3172no—
96#123—3171no—
97#123D31711yescharacteristic, direct product factor
98C2×C4⋊C4—3172no—
99C2×C22⋊C4—3171no—
100C2×C22⋊C4—3172no—
101C24—3171no—
102C24—3171no—
103C2×D4—3171no—
104C2×D4—3171no—
105C2×D4—3171no—
106C2×D4—6344no—
107C2×D4C2×D31711yescharacteristic
108C22×C4—3171no—
109C22×C4—3172no—
110C22×C4—3172no—
111C22×C4—3171no—
112C22×C4—3172no—
113C22×C4—6344no—
114C22×C4C2×D31711yescharacteristic, complemented
115C4⋊C4—3172no—
116C4⋊C4—3172no—
117C4⋊C4—3172no—
118C4⋊C4C2×D31712yescomplemented
119C22⋊C4—3171no—
120C22⋊C4—3172no—
121C22⋊C4—3172no—
122C22⋊C4—3172no—
123C22⋊C4—3172no—
124C22⋊C4C2×D31711yescharacteristic, complemented
125C22⋊C4C2×D31712yescomplemented
126C23—3171no—
127C23—3174no—
128C23—3171no—
129C23—3171no—
130C23—6344no—
131C23—6342no—
132C23—6344no—
133C23C22×D31711yescharacteristic
134C23C22×D31711yescharacteristic
135D4—22no—
136D4—6342no—
137D4—6342no—
138D4—6342no—
139C2×C4—22no—
140C2×C4—3172no—
141C2×C4—3172no—
142C2×C4—3171no—
143C2×C4—6344no—
144C2×C4—6344no—
145C2×C4—6342no—
146C2×C4—6344no—
147C2×C4—6342no—
148C2×C4C22×D31712yes—
149C2×C4C22×D31712yes—
150C2×C4C22×D31711yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—3172no—
155C22—3174no—
156C22—3172no—
157C22—6348no—
158C22—6342no—
159C22—6341no—
160C22—6342no—
161C22—12684no—
162C22D4×D31712yes—
163C22C23×D31711yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—6342no—
168C4—6341no—
169C4—6342no—
170C2—22no—
171C2—41no—
172C2—3174no—
173C2—6342no—
174C2—12681no—
175C2C2×D4×D31711yescharacteristic
176C2#41×D31712yes—
subgroupG/H
center, FrattiniC22C23×D317
commutator, socle, cosocleC22×C317C24
Fitting#123×C317C2
prime radicalC2×D4×D317C2
hypercenter#123D317
hypocenterC317C2×#123
LCSG ▷ C22×C317 ▷ C317
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D317C2semidirect
3. #23778765C2semidirect
4. #23778764C2semidirect
5. #23778763C2semidirect
6. #23778762C2semidirect
7. #23778761C2semidirect
8. C4⋊C4×D317C2semidirect
9. C22⋊C4×D317C2semidirect
10. C22⋊C4×D317C2semidirect
11. #123×C317C2semidirect
18. C22×Dic317C22semidirect
23. C2×C4×D317C22semidirect
24. C4⋊Dic317C22semidirect
25. #19099553C22semidirect
26. #19099553C22semidirect
27. C22⋊Dic317C22semidirect
28. C22⋊Dic317C22semidirect
29. #19099551C22semidirect
30. #19099551C22semidirect
31. #19099551C22semidirect
33. C22×C4×C317C22semidirect
34. C4⋊C4×C317C22semidirect
35. C22⋊C4×C317C22semidirect
36. C22⋊C4×C317C22semidirect
87. D317#123direct (N × H)
88. C317C2×#123semidirect
97. #123D317direct (N × H)
114. C22×C4C2×D317semidirect
118. C4⋊C4C2×D317semidirect
124. C22⋊C4C2×D317semidirect
125. C22⋊C4C2×D317semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3172
C22×C4×D3172
#237787652
#237787642
#237787632
#237787622
#237787612
C4⋊C4×D3172
C22⋊C4×D3172
#123×C3172
C2×#123317