Finite Group Databasebeta

← #123×D379 #37565744

Subgroups — #123×D379 #37565744

Order 24256.

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×D379C211yesmaximal, characteristic
2C22×C4×D379C211yesmaximal, characteristic, complemented
3#27131316C212yesmaximal, complemented
4#27131315C211yesmaximal, characteristic, complemented
5#27131314C211yesmaximal, characteristic, complemented
6#27131313C212yesmaximal, complemented
7#27131312C211yesmaximal, characteristic, complemented
8C4⋊C4×D379C212yesmaximal, complemented
9C22⋊C4×D379C212yesmaximal, complemented
10C22⋊C4×D379C211yesmaximal, characteristic, complemented
11#123×C379C211yesmaximal, characteristic, complemented
12C23×D379C2211yescharacteristic
13C23×D379C2211yescharacteristic
14C2×D1516C2211yescharacteristic
15C2×C22⋊D379C2211yescharacteristic
16C2×C22⋊D379C2211yescharacteristic
17D4×D379—24no—
18C22×Dic379C2211yescharacteristic, complemented
19C2×C4×D379—24no—
20C2×C4×D379C2212yes—
21C2×C4×D379C2212yes—
22C2×C4×D379C2211yescharacteristic
23C2×C4×D379C2212yescomplemented
24C4⋊Dic379C2212yescomplemented
25Dic379⋊C4C2212yescomplemented
26Dic379⋊C4C2212yescomplemented
27C22⋊Dic379C2212yescomplemented
28C22⋊Dic379C2211yescharacteristic, complemented
29(C2×D379)⋊C4C2212yescomplemented
30(C2×D379)⋊C4C2212yescomplemented
31(C2×D379)⋊C4C2212yescomplemented
32C2×D4×C379C2211yescharacteristic
33C22×C4×C379C2211yescharacteristic, complemented
34C4⋊C4×C379C2212yescomplemented
35C22⋊C4×C379C2212yescomplemented
36C22⋊C4×C379C2211yescharacteristic, complemented
37C22×D379—24no—
38C22×D379—22no—
39C22×D379—24no—
40C22×D379D414yes—
41C22×D379C2311yescharacteristic
42C22×D379C2311yescharacteristic
43C22×D379C2311yescharacteristic
44D1516—22no—
45C22⋊D379—22no—
46C22⋊D379—22no—
47C2×Dic379—22no—
48C2×Dic379C2312yes—
49C2×Dic379C2312yes—
50C2×Dic379C2311yescharacteristic
51C4×D379—24no—
52C4×D379—24no—
53C4×D379—22no—
54C4×D379—24no—
55C23×C379C2311yescharacteristic
56C23×C379C2311yescharacteristic
57D4×C379—22no—
58C2×C4×C379—22no—
59C2×C4×C379C2312yes—
60C2×C4×C379C2312yes—
61C2×C4×C379C2311yescharacteristic
62C22×C379—22no—
63C22×C379—21no—
64C22×C379—22no—
65C22×C379C2×D412yes—
66C22×C379C2411yescharacteristic
67C2×D379—22no—
68C2×D379—28no—
69C2×D379—21no—
70C2×D379—22no—
71C2×D379—44no—
72C2×D379C2×D412yes—
73C2×D379C2×D412yes—
74C2×D379#4114yes—
75C4×C379—22no—
76C4×C379—21no—
77C4×C379—22no—
78Dic379—22no—
79Dic379—21no—
80Dic379—22no—
81C2×C379—22no—
82C2×C379—41no—
83C2×C379C22×D411yescharacteristic
84C2×C379C2×#4112yes—
85D379—22no—
86D379—41no—
87D379#12314yesdirect product factor
88C379C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—3791nomaximal, Sylow
90C22×D4—3791no—
91C23×C4—3791no—
92#123—3791no—
93#123—3791no—
94#123—3792no—
95#123—3792no—
96#123—3791no—
97#123D37911yescharacteristic, direct product factor
98C2×C4⋊C4—3792no—
99C2×C22⋊C4—3791no—
100C2×C22⋊C4—3792no—
101C24—3791no—
102C24—3791no—
103C2×D4—3791no—
104C2×D4—3791no—
105C2×D4—3791no—
106C2×D4—7584no—
107C2×D4C2×D37911yescharacteristic
108C22×C4—3791no—
109C22×C4—3792no—
110C22×C4—3792no—
111C22×C4—3791no—
112C22×C4—3792no—
113C22×C4—7584no—
114C22×C4C2×D37911yescharacteristic, complemented
115C4⋊C4—3792no—
116C4⋊C4—3792no—
117C4⋊C4—3792no—
118C4⋊C4C2×D37912yescomplemented
119C22⋊C4—3791no—
120C22⋊C4—3792no—
121C22⋊C4—3792no—
122C22⋊C4—3792no—
123C22⋊C4—3792no—
124C22⋊C4C2×D37911yescharacteristic, complemented
125C22⋊C4C2×D37912yescomplemented
126C23—3791no—
127C23—3794no—
128C23—3791no—
129C23—3791no—
130C23—7582no—
131C23—7584no—
132C23—7584no—
133C23C22×D37911yescharacteristic
134C23C22×D37911yescharacteristic
135D4—22no—
136D4—7582no—
137D4—7582no—
138D4—7582no—
139C2×C4—22no—
140C2×C4—3792no—
141C2×C4—3792no—
142C2×C4—3791no—
143C2×C4—7584no—
144C2×C4—7584no—
145C2×C4—7582no—
146C2×C4—7584no—
147C2×C4—7582no—
148C2×C4C22×D37912yes—
149C2×C4C22×D37912yes—
150C2×C4C22×D37911yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—3792no—
155C22—3794no—
156C22—3792no—
157C22—15164no—
158C22—7588no—
159C22—7581no—
160C22—7582no—
161C22—7582no—
162C22D4×D37912yes—
163C22C23×D37911yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—7582no—
168C4—7581no—
169C4—7582no—
170C2—22no—
171C2—41no—
172C2—3794no—
173C2—15161no—
174C2—7582no—
175C2C2×D4×D37911yescharacteristic
176C2#41×D37912yes—
subgroupG/H
center, FrattiniC22C23×D379
commutator, socle, cosocleC22×C379C24
Fitting#123×C379C2
prime radicalC2×D4×D379C2
hypercenter#123D379
hypocenterC379C2×#123
LCSG ▷ C22×C379 ▷ C379
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D379C2semidirect
3. #27131316C2semidirect
4. #27131315C2semidirect
5. #27131314C2semidirect
6. #27131313C2semidirect
7. #27131312C2semidirect
8. C4⋊C4×D379C2semidirect
9. C22⋊C4×D379C2semidirect
10. C22⋊C4×D379C2semidirect
11. #123×C379C2semidirect
18. C22×Dic379C22semidirect
23. C2×C4×D379C22semidirect
24. C4⋊Dic379C22semidirect
25. Dic379⋊C4C22semidirect
26. Dic379⋊C4C22semidirect
27. C22⋊Dic379C22semidirect
28. C22⋊Dic379C22semidirect
29. (C2×D379)⋊C4C22semidirect
30. (C2×D379)⋊C4C22semidirect
31. (C2×D379)⋊C4C22semidirect
33. C22×C4×C379C22semidirect
34. C4⋊C4×C379C22semidirect
35. C22⋊C4×C379C22semidirect
36. C22⋊C4×C379C22semidirect
87. D379#123direct (N × H)
88. C379C2×#123semidirect
97. #123D379direct (N × H)
114. C22×C4C2×D379semidirect
118. C4⋊C4C2×D379semidirect
124. C22⋊C4C2×D379semidirect
125. C22⋊C4C2×D379semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3792
C22×C4×D3792
#271313162
#271313152
#271313142
#271313132
#271313122
C4⋊C4×D3792
C22⋊C4×D3792
#123×C3792
C2×#123379