Finite Group Databasebeta

← #46846655

Subgroups — #46846655

Order 34688.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#31774502C211yesmaximal, characteristic
2#31774221C212yesmaximal, complemented
3#31774204C212yesmaximal, complemented
4#31774097C212yesmaximal, complemented
5#31774036C212yesmaximal, complemented
6#31773995C212yesmaximal, complemented
7C4×#16864641C212yesmaximal, complemented
8#31773958C211yesmaximal, characteristic, complemented
9#519×C271C211yesmaximal, characteristic, complemented
10C2×#16864641C2212yes—
11C2×#16864641C2212yes—
12#22576157C2211yescharacteristic
13#22576152C2211yescharacteristic
14#22576145—22no—
15#22576134C2212yes—
16#22576133—22no—
17C4×D1084C2212yes—
18#22576093C2212yes—
19#22576091C2212yes—
20C4⋊(C271⋊C8)C2212yes—
21#22576084C2212yes—
22#22576084C2212yes—
23#22576080C2212yes—
24#22576080C2212yes—
25C22⋊(C271⋊C8)C2212yes—
26#22576071C2211yescharacteristic
27C4×C271⋊C8C2211yescharacteristic
28C22×Q8×C271C2211yescharacteristic
29C4×Q8×C271—22no—
30C4×Q8×C271C2212yes—
31C4×Q8×C271C2212yes—
32#117×C271—22no—
33#117×C271C2211yescharacteristic
34#117×C271C2211yescharacteristic
35C2×D1084—22no—
36C2×D1084C2312yes—
37C2×C22⋊D271—22no—
38C2×C4×D271—22no—
39#16864641—24no—
40#16864641—24no—
41C2×C271⋊C8C2312yes—
42C2×C271⋊C8C2312yes—
43C4⋊Dic271C2312yes—
44Dic271⋊C4—22no—
45(C2×D271)⋊C4—22no—
46(C2×D271)⋊C4—22no—
47C2×Q8×C271—24no—
48C2×Q8×C271—21no—
49C2×Q8×C271C2312yes—
50C22×C4×C271—21no—
51C22×C4×C271C2311yescharacteristic
52C4⋊C4×C271—22no—
53C4⋊C4×C271—22no—
54C4⋊C4×C271D412yes—
55C4⋊C4×C271C2312yes—
56C22⋊C4×C271—21no—
57C22⋊C4×C271D412yes—
58C42×C271—22no—
59C42×C271—22no—
60C42×C271—21no—
61C42×C271C2311yescharacteristic
62C42×C271C2311yescharacteristic
63C22×D271—22no—
64D1084—22no—
65D1084—22no—
66D1084—82no—
67C22⋊D271—82no—
68C2×Dic271—22no—
69C4×D271—82no—
70C271⋊C8—22no—
71C271⋊C8—22no—
72C23×C271C2×D411yescharacteristic
73Q8×C271—24no—
74Q8×C271—22no—
75Q8×C271#4114yes—
76C2×C4×C271—24no—
77C2×C4×C271—22no—
78C2×C4×C271—22no—
79C2×C4×C271—22no—
80C2×C4×C271C2×D412yes—
81C2×C4×C271C2×D412yes—
82C2×C4×C271C2×D411yescharacteristic
83C2×C4×C271C2411yescharacteristic
84C22×C271—21no—
85C22×C271—22no—
86C22×C271C22×D411yescharacteristic
87C2×D271—42no—
88C2×D271—42no—
89C2×D271—42no—
90C4×C271—24no—
91C4×C271—21no—
92C4×C271—22no—
93C4×C271—22no—
94C4×C271—42no—
95C4×C271C2×#4112yes—
96Dic271—42no—
97C2×C271—41no—
98C2×C271#52511yescharacteristic
99C2×C271#57712yes—
100D271—82no—
101C271#312611yescharacteristic, complemented, Sylow
102#3126—2711nomaximal, Sylow
103#540—2711no—
104#519D27111yescharacteristic, complemented
105#482—2712no—
106#479—2712no—
107#460—2712no—
108#448—2712no—
109#440—2712no—
110C4×SD16—2712no—
111#433—2711no—
112C22×Q8C2×D27111yescharacteristic
113C2×SD16—2712no—
114C2×SD16—2712no—
115#127—2711no—
116#125—2711no—
117#123—5422no—
118#121—2712no—
119#121—5422no—
120C4×Q8—22no—
121C4×Q8C2×D27112yes—
122C4×Q8C2×D27112yes—
123C4×D4—2712no—
124#117—22no—
125#117C2×D27111yescharacteristic
126#117C2×D27111yescharacteristic
127#107—2712no—
128#106—2712no—
129C4⋊C8—2712no—
130#103—2712no—
131#103—2712no—
132#102—2712no—
133#102—2712no—
134C22⋊C8—2712no—
135#97—2711no—
136C4×C8—2711no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D27112yes—
140C2×D4—2712no—
141C2×D4—5422no—
142C2×D4—5422no—
143C22×C4—21no—
144C22×C4—5422no—
145C22×C4C22×D27111yescharacteristic
146SD16—5424no—
147SD16—5424no—
148C2×C8—2712no—
149C2×C8—2712no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—2712no—
153C4⋊C4—5422no—
154C4⋊C4C22⋊D27112yes—
155C4⋊C4C22×D27112yes—
156C22⋊C4—21no—
157C22⋊C4—5422no—
158C22⋊C4—5422no—
159C22⋊C4C22⋊D27112yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D27111yescharacteristic
164C42C22×D27111yescharacteristic
165C23—5422no—
166C23C2×C22⋊D27111yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1686465414yes—
170D4—5422no—
171D4—5422no—
172D4—21682no—
173D4—21682no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—5422no—
179C2×C4—21682no—
180C2×C4C2×C22⋊D27112yes—
181C2×C4C2×C22⋊D27112yes—
182C2×C4C2×C22⋊D27111yescharacteristic
183C2×C4C23×D27111yescharacteristic
184C8—5422no—
185C8—5422no—
186C22—21no—
187C22—22no—
188C22—10842no—
189C22—10842no—
190C22—10842no—
191C22C22×C22⋊D27111yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—10842no—
198C4C2×#1686465412yes—
199C2—41no—
200C2—21682no—
201C2#3177440811yescharacteristic
202C2#3177470512yes—
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#519×C271C2
socleC22×C271C22×D4
prime radical#31774502C2
hypercenter#519D271
hypocenterC271#3126
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #31774221C2semidirect
3. #31774204C2semidirect
4. #31774097C2semidirect
5. #31774036C2semidirect
6. #31773995C2semidirect
7. C4×#16864641C2semidirect
8. #31773958C2semidirect
9. #519×C271C2semidirect
101. C271#3126semidirect
104. #519D271semidirect

Maximal subgroups

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

subgroupindex
#317745022
#317742212
#317742042
#317740972
#317740362
#317739952
C4×#168646412
#317739582
#519×C2712
#3126271