Finite Group Databasebeta

← #46846748

Subgroups — #46846748

Order 34688.

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

normal subgroups only

This group has 400 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#31774450C211yesmaximal, characteristic, complemented
2C2×#22576147C211yesmaximal, characteristic
3#31774255C214yesmaximal, complemented
4#31774250C214yesmaximal, complemented
5#31773907C211yesmaximal, characteristic, complemented
6#31773905C211yesmaximal, characteristic, complemented
7#31773902C211yesmaximal, characteristic
8M16×Dic271C211yesmaximal, characteristic, complemented
9#418×C271C211yesmaximal, characteristic
10C22×Dic542C2211yescharacteristic
11C2×#16864638C2211yescharacteristic
12#22576159C2212yescomplemented
13#22576151C2212yescomplemented
14#22576147—22no—
15#22576147C2212yes—
16#22576147C2212yes—
17#22576137—22no—
18C4×Dic542—22no—
19#22576119C2211yescharacteristic
20C2×C22⋊Dic271—22no—
21C2×C4×Dic271C2211yescharacteristic
22#22576085C2214yescomplemented
23#22576083C2214yescomplemented
24#22576081C2214yes—
25#22576070C2212yescomplemented
26C8×Dic271C2212yescomplemented
27C22×D4×C271C2211yescharacteristic
28C2×M16×C271C2211yescharacteristic
29#117×C271C2211yescharacteristic
30#102×C271C2214yes—
31C2×Dic542—22no—
32C2×Dic542—21no—
33C2×Dic542—22no—
34C2×Dic542—22no—
35C2×Dic542C2312yes—
36C22×Dic271—21no—
37C22×Dic271D412yes—
38#16864638—22no—
39C2×C271⋊C8C2312yes—
40C4⋊Dic271—22no—
41C4⋊Dic271C2312yes—
42Dic271⋊C4—22no—
43Dic271⋊C4—22no—
44C22⋊Dic271—24no—
45C22⋊Dic271—24no—
46C22⋊Dic271—21no—
47C4×Dic271—21no—
48C4×Dic271D412yes—
49C4×Dic271C2311yescharacteristic
50C4×Dic271C2311yescharacteristic
51C24×C271—21no—
52C2×D4×C271—21no—
53C2×D4×C271—22no—
54C2×D4×C271—22no—
55C2×D4×C271C2312yes—
56C22×C4×C271C2311yescharacteristic
57M16×C271—22no—
58C2×C8×C271C2312yes—
59C4⋊C4×C271C2312yes—
60C22⋊C4×C271—21no—
61C42×C271—21no—
62Dic542—22no—
63Dic542—22no—
64Dic542—44no—
65Dic542—42no—
66C2×Dic271—22no—
67C2×Dic271—22no—
68C2×Dic271—42no—
69C2×Dic271—42no—
70C2×Dic271C2×D412yes—
71C2×Dic271C2×D412yes—
72C271⋊C8—22no—
73C23×C271—22no—
74C23×C271—44no—
75C23×C271—42no—
76C23×C271C2×D411yescharacteristic
77D4×C271—22no—
78D4×C271—22no—
79D4×C271—42no—
80C2×C4×C271—22no—
81C2×C4×C271C2×D411yescharacteristic
82C2×C4×C271#4112yes—
83C2×C4×C271#4112yes—
84C2×C4×C271C2411yescharacteristic
85C8×C271—22no—
86C22×C271—22no—
87C22×C271—42no—
88C22×C271—44no—
89C22×C271—84no—
90C22×C271#12412yes—
91C22×C271C22×D411yescharacteristic
92C4×C271—42no—
93C4×C271#12412yes—
94C4×C271C2×#4111yescharacteristic
95C4×C271C2×#4111yescharacteristic
96Dic271—22no—
97Dic271—22no—
98Dic271—42no—
99Dic271—42no—
100C2×C271—22no—
101C2×C271—82no—
102C2×C271C2×#12411yescharacteristic
103C2×C271#57512yes—
104C271#314011yescharacteristic, complemented, Sylow
105#3140—2711nomaximal, Sylow
106#533—2711no—
107C2×#124—2711no—
108#489—2714no—
109#488—2714no—
110#419—2711no—
111#419—2711no—
112#418—2711no—
113#418D27111yescharacteristic
114C4×M16—2711no—
115C22×Q8—2711no—
116C22×D4C2×D27111yescharacteristic
117C2×M16—2711no—
118C2×M16C2×D27111yescharacteristic
119C4⋊Q8—2712no—
120#125—2712no—
121#124—2712no—
122#124—2712no—
123#124—5422no—
124C22⋊Q8—5422no—
125C4×Q8—5422no—
126#117—2711no—
127#117C2×D27111yescharacteristic
128C2×C22⋊C4—5422no—
129C2×C42—2711no—
130#103—2714no—
131#103—2714no—
132#102—2714no—
133#102C2×D27114yes—
134#97—2712no—
135C4×C8—2712no—
136C24—21no—
137C2×Q8—2712no—
138C2×Q8—5422no—
139C2×Q8—5421no—
140C2×Q8—5422no—
141C2×Q8—5422no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22×D27112yes—
146C22×C4—2712no—
147C22×C4—5421no—
148C22×C4C22×D27111yescharacteristic
149M16—22no—
150M16—5422no—
151C2×C8—2712no—
152C2×C8C22×D27112yes—
153C4⋊C4—2712no—
154C4⋊C4—5422no—
155C4⋊C4—5422no—
156C4⋊C4—5422no—
157C4⋊C4C22×D27112yes—
158C22⋊C4—21no—
159C22⋊C4—5424no—
160C22⋊C4—5424no—
161C22⋊C4—5421no—
162C42—21no—
163C42—2712no—
164C42—2711no—
165C42—2711no—
166C42—5421no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D27111yescharacteristic
171Q8—5422no—
172Q8—5422no—
173Q8—10844no—
174Q8—10842no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—2712no—
180C2×C4—2712no—
181C2×C4—5422no—
182C2×C4—5422no—
183C2×C4—10842no—
184C2×C4—10842no—
185C2×C4D4×D27111yescharacteristic
186C2×C4#1686465312yes—
187C2×C4#1686465412yes—
188C2×C4C23×D27111yescharacteristic
189C8—22no—
190C8—5422no—
191C22—22no—
192C22—42no—
193C22—44no—
194C22—84no—
195C22#2257614612yes—
196C22C2×D4×D27111yescharacteristic
197C4—42no—
198C4—5422no—
199C4—5422no—
200C4—10842no—
201C4—10842no—
202C4#2257614612yes—
203C4C2×#1686465311yescharacteristic
204C4C2×#1686465411yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#2257614611yescharacteristic
208C2#3177469312yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#418×C271C2
socleC22×C271C22×D4
prime radicalC22×D4×C271C22
hypercenter#418D271
hypocenterC271#3140
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #31774450C2semidirect
3. #31774255C2semidirect
4. #31774250C2semidirect
5. #31773907C2semidirect
6. #31773905C2semidirect
8. M16×Dic271C2semidirect
12. #22576159C22semidirect
13. #22576151C22semidirect
22. #22576085C22semidirect
23. #22576083C22semidirect
25. #22576070C22semidirect
26. C8×Dic271C22semidirect
104. C271#3140semidirect

Maximal subgroups

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

subgroupindex
#317744502
C2×#225761472
#317742552
#317742502
#317739072
#317739052
#317739022
M16×Dic2712
#418×C2712
#3140271