Finite Group Databasebeta

← #46839644

Subgroups — #46839644

Order 34688.

840 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 163 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×#22576122C213yesmaximal, complemented
2C2×(C22⋊D271)⋊C4C213yesmaximal, complemented
3#31773832C213yesmaximal, complemented
4#31773807C211yesmaximal, characteristic, complemented
5#31773806C213yesmaximal, complemented
6#31773742C211yesmaximal, characteristic, complemented
7#399×C271C211yesmaximal, characteristic, complemented
8C22×D1084C2213yes—
9C22×C22⋊D271C2213yes—
10C23×Dic271C2211yescharacteristic
11C22×C4×D271C2213yes—
12#22576122—212no—
13(C22⋊D271)⋊C4—212no—
14C2×Dic271⋊C4C2213yes—
15C2×(C2×D271)⋊C4C2216yes—
16C2×(C2×D271)⋊C4C2213yes—
17C2×C4×Dic271C2213yes—
18C271⋊#95C2213yes—
19C2×C4⋊C4×C271C2213yes—
20C2×C22⋊C4×C271C2213yes—
21#95×C271C2211yescharacteristic
22C23×D271C2313yes—
23C2×D1084—212no—
24C2×D1084—26no—
25C2×C22⋊D271—26no—
26C2×C22⋊D271—26no—
27C2×C22⋊D271—26no—
28C22×Dic271—26no—
29C22×Dic271C2311yescharacteristic
30C22×Dic271C2313yes—
31C22×Dic271C2311yescharacteristic
32C2×C4×D271—26no—
33C2×C4×D271—212no—
34Dic271⋊C4—26no—
35(C2×D271)⋊C4—212no—
36(C2×D271)⋊C4—26no—
37C4×Dic271D4112yescomplemented
38C24×C271C2311yescharacteristic
39C22×C4×C271C2313yes—
40C22×C4×C271C2313yes—
41C4⋊C4×C271—26no—
42C22⋊C4×C271—26no—
43C22×D271—23no—
44C22×D271—26no—
45C22×D271—23no—
46C22×D271—26no—
47C22×D271#4116yes—
48D1084—224no—
49C22⋊D271—224no—
50C2×Dic271—23no—
51C2×Dic271—23no—
52C2×Dic271—23no—
53C2×Dic271—44no—
54C2×Dic271C2×D416yes—
55C2×Dic271C2×D416yes—
56C4×D271—412no—
57C23×C271—23no—
58C23×C271—23no—
59C23×C271#4112yes—
60C23×C271C2411yescharacteristic
61C2×C4×C271—26no—
62C2×C4×C271—23no—
63C2×C4×C271—26no—
64C2×C4×C271C2×D416yes—
65C22×C271—26no—
66C22×C271—43no—
67C22×C271—41no—
68C22×C271C22×D413yes—
69C22×C271C2×#4113yes—
70C22×C271C2×#4111yescharacteristic
71C2×D271—26no—
72C2×D271—212no—
73C2×D271—43no—
74C2×D271—46no—
75C2×D271—43no—
76C4×C271—26no—
77C4×C271—43no—
78Dic271—26no—
79Dic271—41no—
80Dic271#12714yes—
81C2×C271—42no—
82C2×C271#52513yes—
83C2×C271#52513yes—
84C2×C271C2×#12711yescharacteristic
85D271—46no—
86C271#239711yescharacteristic, complemented, Sylow
87#2397—2711nomaximal, Sylow
88C2×C4×D4—2713no—
89C2×C4×D4—2713no—
90#399—2713no—
91#399D27111yescharacteristic, complemented
92#392—2713no—
93#392—2711no—
94#376—2711no—
95C22×D4—2713no—
96C22×D4—2713no—
97C23×C4—2711no—
98C23×C4—2713no—
99C4×D4—54212no—
100C4×D4—54212no—
101C2×C4⋊C4—2713no—
102C2×C4⋊C4C2×D27113yes—
103C2×C22⋊C4—2716no—
104C2×C22⋊C4—2713no—
105C2×C22⋊C4C2×D27113yes—
106C2×C42—2713no—
107#95—2713no—
108#95C2×D27111yescharacteristic
109C24—2713no—
110C24C22×D27111yescharacteristic
111C2×D4—54212no—
112C2×D4—5426no—
113C2×D4—5426no—
114C2×D4—5426no—
115C2×D4—5426no—
116C22×C4—2711no—
117C22×C4—2713no—
118C22×C4—2711no—
119C22×C4—5426no—
120C22×C4—54212no—
121C22×C4—5426no—
122C22×C4C22×D27113yes—
123C22×C4C22×D27113yes—
124C4⋊C4—26no—
125C4⋊C4—5426no—
126C22⋊C4—26no—
127C22⋊C4—54212no—
128C22⋊C4—5426no—
129C42—27112no—
130C23—23no—
131C23—23no—
132C23—2716no—
133C23—5426no—
134C23—5423no—
135C23—5426no—
136C23—5423no—
137C23#1686465512yes—
138C23C23×D27111yescharacteristic
139D4—54224no—
140D4—54224no—
141C2×C4—26no—
142C2×C4—23no—
143C2×C4—26no—
144C2×C4—2716no—
145C2×C4—2716no—
146C2×C4—5423no—
147C2×C4—5423no—
148C2×C4—5423no—
149C2×C4—108412no—
150C2×C4—10844no—
151C2×C4D4×D27116yes—
152C22—26no—
153C22—43no—
154C22—41no—
155C22—54212no—
156C22—5426no—
157C22—10843no—
158C22—10846no—
159C22—10843no—
160C22C2×D4×D27113yes—
161C22#41×D27113yes—
162C22C2×#1686465511yescharacteristic
163C4—26no—
164C4—43no—
165C4—2714no—
166C4—5426no—
167C4—10841no—
168C2—42no—
169C2—10846no—
170C2#3177440513yes—
171C2#3177441013yes—
172C2#127×D27111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#399×C271C2
hypercenter#399D271
hypocenterC271#2397
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#22576122C2semidirect
2. C2×(C22⋊D271)⋊C4C2semidirect
3. #31773832C2semidirect
4. #31773807C2semidirect
5. #31773806C2semidirect
6. #31773742C2semidirect
7. #399×C271C2semidirect
37. C4×Dic271D4semidirect
86. C271#2397semidirect
91. #399D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761222
C2×(C22⋊D271)⋊C42
#317738322
#317738072
#317738062
#317737422
#399×C2712
#2397271