Finite Group Databasebeta

← #46838317

Subgroups — #46838317

Order 34688.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#31774697C211yesmaximal, characteristic
2#31774340C212yesmaximal, complemented
3#31774155C211yesmaximal, characteristic, complemented
4#31773708C212yesmaximal, complemented
5#349×C271C211yesmaximal, characteristic, complemented
6C2×#16864654—21no—
7#22576193—22no—
8#22576188—22no—
9#22576182—21no—
10#22576182C2212yes—
11C2×Dic1084C2211yescharacteristic, complemented
12#22576139—21no—
13Dic2168—22no—
14#22576105—22no—
15#22576092C2211yescharacteristic, complemented
16#22576085—21no—
17C2×M16×C271C2211yescharacteristic
18M32×C271C2212yescomplemented
19#16864654—22no—
20#16864654—22no—
21#16864654—41no—
22#16864654—41no—
23C2×Dic542—21no—
24C2×Dic542—21no—
25C2×D1084—21no—
26C2×C22⋊D271—41no—
27C2×C4×D271—41no—
28Dic1084—21no—
29Dic1084—22no—
30Dic1084D412yes—
31#16864643—22no—
32#16864643—22no—
33D2168—21no—
34D2168D412yes—
35C4⋊Dic271—21no—
36Dic271⋊C4—41no—
37C22⋊Dic271—41no—
38C22×C4×C271D411yescharacteristic
39M16×C271—22no—
40C2×C8×C271D411yescharacteristic
41C2×C8×C271C2311yescharacteristic
42C16×C271—22no—
43C22×D271—41no—
44Dic542—22no—
45Dic542—41no—
46Dic542—41no—
47Dic542—41no—
48D1084—22no—
49D1084—41no—
50C22⋊D271—42no—
51C22⋊D271—42no—
52C2×Dic271—41no—
53C2×Dic271—41no—
54C2×Dic271—41no—
55C4×D271—42no—
56C4×D271—42no—
57C23×C271D811yescharacteristic
58C2×C4×C271—21no—
59C2×C4×C271—21no—
60C2×C4×C271D811yescharacteristic
61C2×C4×C271C2×D411yescharacteristic
62C8×C271—21no—
63C8×C271C2×D412yes—
64C22×C271—21no—
65C22×C271—41no—
66C22×C271C2×D811yescharacteristic
67C2×D271—42no—
68C2×D271—82no—
69C4×C271—21no—
70C4×C271#12011yescharacteristic
71C4×C271AGL(1,ℤ/8)11yescharacteristic
72Dic271—42no—
73Dic271—81no—
74Dic271—81no—
75C2×C271—21no—
76C2×C271—41no—
77C2×C271#44711yescharacteristic
78D271—82no—
79C271#219111yescharacteristic, complemented, Sylow
80#2191—2711nomaximal, Sylow
81#575—2711no—
82#510—2712no—
83#470—2711no—
84#360—2712no—
85#349D27111yescharacteristic, complemented
86C2×#41—5421no—
87#137—5422no—
88AGL(1,ℤ/8)—5422no—
89#135—2712no—
90#135—5421no—
91C2×Q16—2711no—
92C2×M16C2×D27111yescharacteristic
93C22⋊Q8—5421no—
94Q32—5422no—
95SD32—5422no—
96M32C2×D27112yescomplemented
97#106—2711no—
98#103—5421no—
99#41—5422no—
100#41—5422no—
101#41—10841no—
102#41—10841no—
103C2×Q8—5421no—
104C2×Q8—5421no—
105C2×D4—5421no—
106C2×D4—10841no—
107C22×C4—10841no—
108C22×C4D108411yescharacteristic
109Q16—2712no—
110Q16—5421no—
111Q16—5422no—
112SD16—5422no—
113SD16—5422no—
114D8—2712no—
115D8—5421no—
116M16—22no—
117C2×C8D108411yescharacteristic
118C2×C8C22×D27111yescharacteristic
119C4⋊C4—5421no—
120C4⋊C4—10841no—
121C22⋊C4—10841no—
122C16—22no—
123C23—10841no—
124C23D216811yescharacteristic
125Q8—5422no—
126Q8—10841no—
127Q8—10841no—
128Q8—10841no—
129D4—5422no—
130D4—10841no—
131D4—10842no—
132D4—10842no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—10842no—
136C2×C4—10841no—
137C2×C4—10842no—
138C2×C4—10841no—
139C2×C4—10841no—
140C2×C4D216811yescharacteristic
141C2×C4C2×D108411yescharacteristic
142C8—21no—
143C8D4×D27112yes—
144C22—21no—
145C22—41no—
146C22—10842no—
147C22—21682no—
148C22C2×D216811yescharacteristic
149C4—21no—
150C4—10842no—
151C4—21681no—
152C4—21681no—
153C4#2257613011yescharacteristic
154C4#2257618811yescharacteristic
155C2—21no—
156C2—41no—
157C2—21682no—
158C2#3177403311yescharacteristic
subgroupG/H
centerC2#31774033
commutator, cosocleC2×C8×C271C23
FrattiniC2×C8C22×D271
Fitting#349×C271C2
socleC2×C271#447
prime radical#31774697C2
hypercenter#349D271
hypocenterC271#2191
LCSG ▷ C2×C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #31774340C2semidirect
3. #31774155C2semidirect
4. #31773708C2semidirect
5. #349×C271C2semidirect
11. C2×Dic1084C22semidirect
15. #22576092C22semidirect
18. M32×C271C22semidirect
79. C271#2191semidirect
85. #349D271semidirect
96. M32C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746972
#317743402
#317741552
#317737082
#349×C2712
#2191271