Finite Group Databasebeta

← #46835289

Subgroups — #46835289

Order 34688.

316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 53 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#31774523C211yesmaximal, characteristic, complemented
2#31774050C212yesmaximal, complemented
3#31773806C211yesmaximal, characteristic
4#31773635C212yesmaximal, complemented
5#323×C271C211yesmaximal, characteristic, complemented
6C22×D1084C2211yescharacteristic
7C22×C22⋊D271—21no—
8C2×#16864643—22no—
9#22576139—21no—
10#22576139C2212yes—
11#22576137—22no—
12#22576137—22no—
13C2×C22⋊Dic271—21no—
14C2×(C2×D271)⋊C4—21no—
15#22576082—22no—
16C271⋊#95C2211yescharacteristic
17C2×C22⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19C23×D271—21no—
20C2×Dic542—21no—
21C2×Dic542D412yescomplemented
22C2×D1084—22no—
23C2×D1084—42no—
24C2×C22⋊D271—42no—
25C2×C22⋊D271—42no—
26C2×C22⋊D271—42no—
27C22×Dic271—21no—
28C22×Dic271—21no—
29C22×Dic271—21no—
30#16864643—44no—
31C4⋊Dic271—21no—
32C4⋊Dic271D412yescomplemented
33Dic271⋊C4—22no—
34Dic271⋊C4—22no—
35C22⋊Dic271—22no—
36C22⋊Dic271—22no—
37(C2×D271)⋊C4—42no—
38C24×C271D411yescharacteristic
39C22×C4×C271D411yescharacteristic
40C22×C4×C271C2311yescharacteristic
41C2×C8×C271—22no—
42C22⋊C4×C271—22no—
43C22×D271—22no—
44C22×D271—41no—
45C22×D271—41no—
46C22×D271—41no—
47C22×D271—41no—
48C22×D271—41no—
49C22×D271—41no—
50Dic542—42no—
51Dic542—81no—
52D1084—24no—
53D1084—42no—
54C22⋊D271—48no—
55C2×Dic271—22no—
56C2×Dic271—22no—
57C2×Dic271—22no—
58C2×Dic271—41no—
59C2×Dic271—41no—
60C2×Dic271—81no—
61C2×Dic271—81no—
62C23×C271—21no—
63C23×C271—21no—
64C23×C271—41no—
65C23×C271—41no—
66C23×C271SD1612yescomplemented
67C23×C271C2×D411yescharacteristic
68C2×C4×C271—21no—
69C2×C4×C271—41no—
70C2×C4×C271—41no—
71C2×C4×C271C2×D412yes—
72C8×C271—42no—
73C22×C271—21no—
74C22×C271—21no—
75C22×C271—22no—
76C22×C271—42no—
77C22×C271—41no—
78C22×C271—41no—
79C22×C271—81no—
80C22×C271#12011yescharacteristic
81C22×C271C2×SD1611yescharacteristic
82C22×C271#13711yescharacteristic
83C2×D271—42no—
84C2×D271—42no—
85C2×D271—42no—
86C2×D271—81no—
87C2×D271—81no—
88C2×D271—81no—
89C2×D271—81no—
90C4×C271—22no—
91C4×C271—41no—
92Dic271—42no—
93Dic271—42no—
94Dic271—42no—
95C2×C271—21no—
96C2×C271—21no—
97C2×C271—42no—
98C2×C271#44811yescharacteristic
99C2×C271#45311yescharacteristic
100C2×C271B(4,𝔽2)11yescharacteristic
101D271—82no—
102C271#159611yescharacteristic, complemented, Sylow
103#1596—2711nomaximal, Sylow
104#543—2711no—
105#450—2712no—
106#392—2711no—
107#328—2712no—
108#323D27111yescharacteristic, complemented
109C22×D4—2711no—
110C22×D4—5421no—
111C2×SD16—5422no—
112C22⋊Q8—2712no—
113C22⋊Q8—5422no—
114C22⋊Q8—5422no—
115C22⋊Q8—5421no—
116C2×C22⋊C4—5421no—
117C2×C22⋊C4—5421no—
118C2×C22⋊C4C2×D27111yescharacteristic
119#102—5422no—
120C22⋊C8C2×D27112yes—
121#95—2711no—
122C24—5421no—
123C24D108411yescharacteristic
124C2×Q8—2712no—
125C2×Q8—5421no—
126C2×D4—5422no—
127C2×D4—10842no—
128C2×D4—10842no—
129C2×D4—10842no—
130C2×D4—10842no—
131C22×C4—5421no—
132C22×C4—5421no—
133C22×C4—5421no—
134C22×C4D108411yescharacteristic
135C22×C4C22×D27111yescharacteristic
136SD16—10844no—
137C2×C8—22no—
138C4⋊C4—2712no—
139C4⋊C4—5422no—
140C4⋊C4—5421no—
141C4⋊C4—5422no—
142C22⋊C4—22no—
143C22⋊C4—5422no—
144C22⋊C4—5422no—
145C22⋊C4—10842no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—5422no—
151C23—10841no—
152C23—10841no—
153C23—10841no—
154C23—10841no—
155C23—10841no—
156C23—10841no—
157C23#1686464312yescomplemented
158C23C2×D108411yescharacteristic
159Q8—10842no—
160Q8—21681no—
161D4—5424no—
162D4—10842no—
163D4—10848no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—5422no—
168C2×C4—5422no—
169C2×C4—5422no—
170C2×C4—10841no—
171C2×C4—10841no—
172C2×C4—21681no—
173C2×C4—21681no—
174C2×C4D4×D27112yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—10842no—
184C22—10842no—
185C22—10842no—
186C22—21681no—
187C22—21681no—
188C22—21681no—
189C22—21681no—
190C22#2257613011yescharacteristic
191C22C2×#1686464311yescharacteristic
192C22#2257619311yescharacteristic
193C4—22no—
194C4—41no—
195C4—10842no—
196C4—10842no—
197C4—10842no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—21682no—
202C2#3177403811yescharacteristic
203C2#3177406711yescharacteristic
204C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#323×C271C2
socleC22×C271#120
prime radical#31773806C2
hypercenter#323D271
hypocenterC271#1596
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #31774523C2semidirect
2. #31774050C2semidirect
4. #31773635C2semidirect
5. #323×C271C2semidirect
21. C2×Dic542D4semidirect
32. C4⋊Dic271D4semidirect
66. C23×C271SD16semidirect
102. C271#1596semidirect
108. #323D271semidirect
157. C23#16864643semidirect

Maximal subgroups

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

subgroupindex
#317745232
#317740502
#317738062
#317736352
#323×C2712
#1596271