Finite Group Databasebeta

← #46840011

Subgroups — #46840011

Order 34688.

1088 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C2×#22576147C212yesmaximal, complemented
2C2×#22576139C211yesmaximal, characteristic, complemented
3#31773811C214yesmaximal, complemented
4#31773800C212yesmaximal, complemented
5#31773753C214yesmaximal, complemented
6C4×C22⋊Dic271C211yesmaximal, characteristic, complemented
7C23×D4×C271C211yesmaximal, characteristic
8C22×Dic542C2211yescharacteristic, complemented
9#22576147—28no—
10#22576139—24no—
11C2×C4⋊Dic271C2211yescharacteristic, complemented
12C2×Dic271⋊C4C2212yescomplemented
13C2×C22⋊Dic271C2212yes—
14C2×C22⋊Dic271C2212yes—
15C2×C22⋊Dic271C2218yescomplemented
16C2×C22⋊Dic271C2218yescomplemented
17C2×C4×Dic271C2212yescomplemented
18C271⋊#95C2212yescomplemented
19C25×C271C2212yes—
20C22×D4×C271—28no—
21C22×D4×C271—24no—
22C22×D4×C271C2212yes—
23C22×D4×C271C2212yes—
24C23×C4×C271C2211yescharacteristic
25C2×Dic542—22no—
26C2×Dic542—24no—
27C22×Dic271C2312yes—
28C22×Dic271C2314yes—
29C22×Dic271C2312yes—
30C4⋊Dic271—22no—
31Dic271⋊C4—24no—
32C22⋊Dic271—24no—
33C22⋊Dic271—216no—
34C22⋊Dic271—24no—
35C22⋊Dic271—48no—
36C4×Dic271—24no—
37C24×C271—28no—
38C24×C271—24no—
39C24×C271—28no—
40C24×C271—44no—
41C24×C271C2314yes—
42C24×C271C2311yescharacteristic
43C2×D4×C271—232no—
44C2×D4×C271—28no—
45C2×D4×C271—24no—
46C2×D4×C271—24no—
47C2×D4×C271—24no—
48C2×D4×C271D418yes—
49C22×C4×C271—24no—
50C22×C4×C271D414yes—
51C22×C4×C271C2311yescharacteristic
52C22×C4×C271C2311yescharacteristic
53Dic542—44no—
54C2×Dic271—24no—
55C2×Dic271—24no—
56C2×Dic271—22no—
57C2×Dic271—22no—
58C2×Dic271—24no—
59C2×Dic271—24no—
60C2×Dic271—24no—
61C23×C271—22no—
62C23×C271—22no—
63C23×C271—216no—
64C23×C271—24no—
65C23×C271—24no—
66C23×C271—28no—
67C23×C271—28no—
68C23×C271—432no—
69C23×C271—48no—
70C23×C271—48no—
71C23×C271C2×D418yes—
72C23×C271C2×D412yes—
73C23×C271#4114yes—
74C23×C271C2411yescharacteristic
75D4×C271—216no—
76D4×C271—216no—
77C2×C4×C271—22no—
78C2×C4×C271—28no—
79C2×C4×C271C2×D412yes—
80C2×C4×C271C2×D414yes—
81C2×C4×C271C2×D412yes—
82C22×C271—22no—
83C22×C271—22no—
84C22×C271—28no—
85C22×C271—28no—
86C22×C271—216no—
87C22×C271—432no—
88C22×C271—44no—
89C22×C271—44no—
90C22×C271—48no—
91C22×C271#12414yescomplemented
92C22×C271C22×D411yescharacteristic
93C22×C271C22×D412yes—
94C22×C271C2×#4112yes—
95C22×C271H(5,𝔽2)12yes—
96C4×C271—22no—
97C4×C271#12014yes—
98Dic271—42no—
99Dic271—42no—
100Dic271—44no—
101C2×C271—24no—
102C2×C271—48no—
103C2×C271C2×#12011yescharacteristic
104C2×C271C2×#12411yescharacteristic
105C2×C271#53511yescharacteristic
106C2×C271#54614yes—
107C271#243511yescharacteristic, complemented, Sylow
108#2435—2711nomaximal, Sylow
109C23×D4D27111yescharacteristic
110C2×#124—2712no—
111C2×C22⋊Q8—2711no—
112#394—2714no—
113#390—2712no—
114#379—2714no—
115C4×C22⋊C4—2711no—
116C25C2×D27112yes—
117C22×Q8—2711no—
118C22×D4—28no—
119C22×D4—24no—
120C22×D4C2×D27112yes—
121C22×D4C2×D27112yes—
122C23×C4C2×D27111yescharacteristic
123#124—5428no—
124C22⋊Q8—5424no—
125C2×C4⋊C4—2711no—
126C2×C4⋊C4—2712no—
127C2×C22⋊C4—2712no—
128C2×C22⋊C4—2718no—
129C2×C22⋊C4—2712no—
130C2×C22⋊C4—2718no—
131C2×C42—2712no—
132#95—2712no—
133C24—28no—
134C24—24no—
135C24—28no—
136C24—44no—
137C24C22×D27114yes—
138C24C22×D27111yescharacteristic
139C2×Q8—5422no—
140C2×Q8—5424no—
141C2×D4—232no—
142C2×D4—28no—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4C22⋊D27118yes—
147C22×C4—24no—
148C22×C4—2714no—
149C22×C4—2712no—
150C22×C4—2712no—
151C22×C4C22⋊D27114yes—
152C22×C4C22×D27111yescharacteristic
153C22×C4C22×D27111yescharacteristic
154C4⋊C4—5424no—
155C4⋊C4—5422no—
156C22⋊C4—5424no—
157C22⋊C4—54216no—
158C22⋊C4—5424no—
159C22⋊C4—10848no—
160C42—5424no—
161C23—22no—
162C23—22no—
163C23—216no—
164C23—24no—
165C23—24no—
166C23—28no—
167C23—28no—
168C23—432no—
169C23—48no—
170C23—48no—
171C23C2×C22⋊D27118yes—
172C23C2×C22⋊D27112yes—
173C23#1686465314yes—
174C23C23×D27111yescharacteristic
175Q8—10844no—
176D4—216no—
177D4—216no—
178C2×C4—22no—
179C2×C4—28no—
180C2×C4—5424no—
181C2×C4—5424no—
182C2×C4—5422no—
183C2×C4—5422no—
184C2×C4—5424no—
185C2×C4—5424no—
186C2×C4—5424no—
187C2×C4C2×C22⋊D27112yes—
188C2×C4C2×C22⋊D27114yes—
189C2×C4C2×C22⋊D27112yes—
190C22—22no—
191C22—28no—
192C22—22no—
193C22—216no—
194C22—28no—
195C22—432no—
196C22—44no—
197C22—48no—
198C22—44no—
199C22#2257614714yescomplemented
200C22C22×C22⋊D27111yescharacteristic
201C22C22×C22⋊D27112yes—
202C22C2×#1686465312yes—
203C22#2257620512yes—
204C4—22no—
205C4—10842no—
206C4—10844no—
207C4—10842no—
208C4#2257612914yes—
209C2—24no—
210C2—48no—
211C2C2×#2257612911yescharacteristic
212C2C2×#2257614711yescharacteristic
213C2#3177446411yescharacteristic
214C2#3177454114yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting, prime radicalC23×D4×C271C2
hypercenterC23×D4D271
hypocenterC271#2435
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22576147C2semidirect
2. C2×#22576139C2semidirect
3. #31773811C2semidirect
4. #31773800C2semidirect
5. #31773753C2semidirect
6. C4×C22⋊Dic271C2semidirect
8. C22×Dic542C22semidirect
11. C2×C4⋊Dic271C22semidirect
12. C2×Dic271⋊C4C22semidirect
15. C2×C22⋊Dic271C22semidirect
16. C2×C22⋊Dic271C22semidirect
17. C2×C4×Dic271C22semidirect
18. C271⋊#95C22semidirect
91. C22×C271#124semidirect
107. C271#2435semidirect
199. C22#22576147semidirect

Maximal subgroups

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

subgroupindex
C2×#225761472
C2×#225761392
#317738112
#317738002
#317737532
C4×C22⋊Dic2712
C23×D4×C2712
#2435271