Finite Group Databasebeta

← #46834986

Subgroups — #46834986

Order 34688.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D271C211yesmaximal, characteristic
2#31773952C212yesmaximal, complemented
3#31773637C211yesmaximal, characteristic, complemented
4#31773636C212yesmaximal, complemented
5#329×C271C211yesmaximal, characteristic, complemented
6C2×Q8×D271C411yescharacteristic
7C2×D4×D271C411yescharacteristic
8#22576149C411yescharacteristic
9#22576148C2211yescharacteristic
10#22576147C411yescharacteristic
11#22576146—22no—
12C22⋊C4×D271—22no—
13C42×D271C2211yescharacteristic
14Dic271⋊C8—22no—
15(C2×D271)⋊C8—22no—
16#22576071C2212yescomplemented
17#124×C271C2211yescharacteristic
18#97×C271C2212yescomplemented
19C23×D271—21no—
20C2×Dic542—21no—
21C2×Dic542C2×C411yescharacteristic
22Q8×D271—42no—
23C2×D1084C2×C411yescharacteristic
24C2×C22⋊D271—21no—
25D4×D271—42no—
26C2×C4×D271—21no—
27C2×C4×D271C2×C411yescharacteristic
28C2×C4×D271D412yes—
29C2×C271⋊C8—22no—
30C22⋊Dic271—22no—
31(C2×D271)⋊C4—22no—
32(C2×D271)⋊C4—22no—
33C4×Dic271C2×C411yescharacteristic
34C4×Dic271D412yes—
35C2×Q8×C271C2×C411yescharacteristic
36C2×D4×C271C2×C411yescharacteristic
37C2×C8×C271—22no—
38C22⋊C4×C271—22no—
39C42×C271C2311yescharacteristic
40C22×D271—21no—
41C22×D271—44no—
42C22×D271—42no—
43C22×D271C22⋊C411yescharacteristic
44Dic542—42no—
45Dic542—41no—
46D1084—41no—
47C22⋊D271—42no—
48C2×Dic271—21no—
49C2×Dic271C22⋊C412yes—
50C2×Dic271C22⋊C411yescharacteristic
51C4×D271—22no—
52C4×D271—42no—
53C4×D271—42no—
54C271⋊C8—42no—
55C23×C271—21no—
56Q8×C271—41no—
57D4×C271—41no—
58C2×C4×C271—21no—
59C2×C4×C271C22×C411yescharacteristic
60C2×C4×C271C2×D412yes—
61C8×C271—42no—
62C22×C271—41no—
63C22×C271—42no—
64C22×C271C2×C22⋊C411yescharacteristic
65C2×D271—22no—
66C2×D271—41no—
67C2×D271—42no—
68C2×D271—84no—
69C2×D271#10012yes—
70C4×C271—21no—
71C4×C271—22no—
72C4×C271—41no—
73Dic271—21no—
74Dic271—22no—
75Dic271—41no—
76C2×C271—81no—
77C2×C271C2×#10011yescharacteristic
78C2×C271#42112yes—
79D271—22no—
80D271—81no—
81C271#153311yescharacteristic, complemented, Sylow
82#1533—2711nomaximal, Sylow
83C2×#124—2711no—
84#432—2712no—
85#329—2711no—
86#329—2712no—
87#329D27111yescharacteristic, complemented
88C22×Q8—2711no—
89C22×D4—2711no—
90#124—2711no—
91#124—2711no—
92#124—2711no—
93#124—5422no—
94#124C2×D27111yescharacteristic
95C2×C22⋊C4—5422no—
96C2×C42—2711no—
97C4⋊C8—5422no—
98C22⋊C8—5422no—
99#97—2712no—
100#97C2×D27112yescomplemented
101C24—5421no—
102C2×Q8—2711no—
103C2×Q8—5421no—
104C2×Q8—10842no—
105C2×Q8C4×D27111yescharacteristic
106C2×D4—2711no—
107C2×D4—5421no—
108C2×D4—10842no—
109C2×D4C4×D27111yescharacteristic
110C22×C4—2711no—
111C22×C4—2712no—
112C22×C4—5421no—
113C2×C8—22no—
114C2×C8—5422no—
115C22⋊C4—22no—
116C22⋊C4—5422no—
117C22⋊C4—5422no—
118C22⋊C4—5422no—
119C42—2711no—
120C42—2712no—
121C42C22×D27111yescharacteristic
122C23—21no—
123C23—2711no—
124C23—5421no—
125C23—10842no—
126C23—10844no—
127Q8—41no—
128Q8—10841no—
129Q8—10842no—
130D4—41no—
131D4—10841no—
132D4—10842no—
133C2×C4—21no—
134C2×C4—2712no—
135C2×C4—2711no—
136C2×C4—5422no—
137C2×C4—5421no—
138C2×C4—10842no—
139C2×C4—10842no—
140C2×C4C2×C4×D27111yescharacteristic
141C2×C4D4×D27112yes—
142C8—42no—
143C8—10842no—
144C22—41no—
145C22—42no—
146C22—2712no—
147C22—5422no—
148C22—10841no—
149C22—10842no—
150C22—21684no—
151C22C22⋊C4×D27111yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—5421no—
156C4—5422no—
157C4—10841no—
158C2—81no—
159C2—5422no—
160C2—21681no—
161C2#100×D27111yescharacteristic
162C2#3177391212yes—
subgroupG/H
centerC22C22⋊C4×D271
commutatorC2×C4×C271C22×C4
FrattiniC42C22×D271
Fitting#329×C271C2
socleC22×C271C2×C22⋊C4
prime radicalC2×D4×D271C4
cosocleC42×C271C23
hypercenter#329D271
hypocenterC271#1533
LCSG ▷ C2×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #31773952C2semidirect
3. #31773637C2semidirect
4. #31773636C2semidirect
5. #329×C271C2semidirect
16. #22576071C22semidirect
18. #97×C271C22semidirect
81. C271#1533semidirect
87. #329D271semidirect
100. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#124×D2712
#317739522
#317736372
#317736362
#329×C2712
#1533271