Finite Group Databasebeta

← #46835698

Subgroups — #46835698

Order 34688.

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

normal subgroups only

This group has 184 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#31774307C211yesmaximal, characteristic, complemented
2#31774267C211yesmaximal, characteristic, complemented
3#31774256C211yesmaximal, characteristic
4#31773639C211yesmaximal, characteristic, complemented
5#31773639C211yesmaximal, characteristic, complemented
6#31773637C211yesmaximal, characteristic
7#563×C271C211yesmaximal, characteristic
8#22576160C2211yescharacteristic
9#22576093—21no—
10#22576091—21no—
11#22576084—21no—
12#22576083—21no—
13#22576083—21no—
14#22576081—21no—
15#22576071C2211yescharacteristic
16#22576071C2211yescharacteristic
17#22576071C2211yescharacteristic
18C4⋊Q8×C271—21no—
19#126×C271—21no—
20#126×C271—21no—
21#125×C271C2211yescharacteristic
22#125×C271C2211yescharacteristic
23#124×C271C2211yescharacteristic
24#123×C271—21no—
25C22⋊Q8×C271—21no—
26C22⋊Q8×C271—21no—
27C2×Dic542—21no—
28C2×C271⋊C8—21no—
29C2×C271⋊C8—21no—
30C2×C271⋊C8—21no—
31C4⋊Dic271—21no—
32C4⋊Dic271—21no—
33C2×Q8×C271—21no—
34C2×Q8×C271D411yescharacteristic
35C2×D4×C271D411yescharacteristic
36C22×C4×C271—21no—
37C4⋊C4×C271—21no—
38C4⋊C4×C271—21no—
39C4⋊C4×C271—21no—
40C4⋊C4×C271—21no—
41C4⋊C4×C271—21no—
42C4⋊C4×C271—21no—
43C4⋊C4×C271D411yescharacteristic
44C4⋊C4×C271D411yescharacteristic
45C4⋊C4×C271D411yescharacteristic
46C4⋊C4×C271D411yescharacteristic
47C22⋊C4×C271—21no—
48C22⋊C4×C271—21no—
49C22⋊C4×C271—21no—
50C22⋊C4×C271—21no—
51C22⋊C4×C271—21no—
52C42×C271—21no—
53C42×C271C2311yescharacteristic
54Dic542—42no—
55C2×Dic271—41no—
56C271⋊C8—22no—
57C271⋊C8—41no—
58C271⋊C8—41no—
59C23×C271—21no—
60Q8×C271—41no—
61Q8×C271—81no—
62D4×C271—41no—
63C2×C4×C271—21no—
64C2×C4×C271—21no—
65C2×C4×C271—21no—
66C2×C4×C271—21no—
67C2×C4×C271—21no—
68C2×C4×C271—81no—
69C2×C4×C271C2×D411yescharacteristic
70C2×C4×C271C2×D411yescharacteristic
71C2×C4×C271C2×D411yescharacteristic
72C22×C271—41no—
73C22×C271—41no—
74C22×C271—41no—
75C22×C271#12011yescharacteristic
76C4×C271—21no—
77C4×C271—21no—
78C4×C271—21no—
79C4×C271—41no—
80C4×C271—41no—
81C4×C271—41no—
82C4×C271—41no—
83C4×C271—41no—
84Dic271—81no—
85C2×C271—81no—
86C2×C271#45411yescharacteristic
87C2×C271#45511yescharacteristic
88C2×C271#45611yescharacteristic
89C271#166411yescharacteristic, complemented, Sylow
90#1664—2711nomaximal, Sylow
91#563D27111yescharacteristic
92#501—2711no—
93#491—2711no—
94#489—2711no—
95#330—2711no—
96#330—2711no—
97#329—2711no—
98C4⋊Q8—21no—
99C4⋊Q8—2711no—
100#126—21no—
101#126—21no—
102#125C2×D27111yescharacteristic
103#125C2×D27111yescharacteristic
104#124C2×D27111yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—5421no—
109#106—5421no—
110#103—5421no—
111#103—5421no—
112#103—5421no—
113#102—5421no—
114#97—2711no—
115#97—2711no—
116#97—2711no—
117C2×Q8—21no—
118C2×Q8—5421no—
119C2×Q8C22⋊D27111yescharacteristic
120C2×D4C22⋊D27111yescharacteristic
121C22×C4—21no—
122C2×C8—5421no—
123C2×C8—5421no—
124C2×C8—5421no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—5421no—
132C4⋊C4—5421no—
133C4⋊C4C22⋊D27111yescharacteristic
134C4⋊C4C22⋊D27111yescharacteristic
135C4⋊C4C22⋊D27111yescharacteristic
136C4⋊C4C22⋊D27111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D27111yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—10842no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—10841no—
156C2×C4C2×C22⋊D27111yescharacteristic
157C2×C4C2×C22⋊D27111yescharacteristic
158C2×C4C2×C22⋊D27111yescharacteristic
159C8—5422no—
160C8—10841no—
161C8—10841no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#2257612911yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—21681no—
175C2—81no—
176C2#3177407011yescharacteristic
177C2#3177407511yescharacteristic
178C2#3177408011yescharacteristic
subgroupG/H
centerC22#22576129
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#563×C271C2
socleC22×C271#120
prime radicalC2×D4×C271D4
hypercenter#563D271
hypocenterC271#1664
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #31774307C2semidirect
2. #31774267C2semidirect
4. #31773639C2semidirect
5. #31773639C2semidirect
89. C271#1664semidirect

Maximal subgroups

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

subgroupindex
#317743072
#317742672
#317742562
#317736392
#317736372
#563×C2712
#1664271