Finite Group Databasebeta

← #46849776

Subgroups — #46849776

Order 34688.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#31774496C216yesmaximal, complemented
2#31774487C216yesmaximal, complemented
3#31774459C213yesmaximal, complemented
4#31774433C216yesmaximal, complemented
5#31774410C216yesmaximal, complemented
6C2×(C22⋊D271)⋊C4C213yesmaximal, complemented
7#543×C271C211yesmaximal, characteristic, complemented
8C2×#16864655C2212yescomplemented
9C22×C22⋊D271C2213yes—
10C23×Dic271C2211yescharacteristic
11C22×C4×D271C2213yescomplemented
12#22576155C22112yescomplemented
13#22576151C2216yescomplemented
14#22576149C2216yescomplemented
15#22576145C22112yescomplemented
16#22576144C22112yescomplemented
17#22576136C2216yescomplemented
18#22576135C2216yescomplemented
19#22576134C2216yes—
20#22576134C2216yes—
21#22576130C22112yescomplemented
22Q8×Dic271C2212yescomplemented
23#22576122C2216yescomplemented
24(C22⋊D271)⋊C4C22112yescomplemented
25(C22⋊D271)⋊C4C22112yescomplemented
26#22576118C2216yescomplemented
27C2×Dic271⋊C4C2213yescomplemented
28C2×(C2×D271)⋊C4C2213yescomplemented
29C2×C4×Dic271C2213yescomplemented
30C22⋊Q8×C271C2216yescomplemented
31C22⋊Q8×C271C2216yescomplemented
32C2×C22⋊C4×C271C2213yes—
33C23×D271C2313yes—
34#16864655—44no—
35C2×D1084C2316yes—
36C2×D1084C2316yes—
37C2×C22⋊D271—212no—
38C2×C22⋊D271C23112yes—
39C22×Dic271—42no—
40C22×Dic271C2313yes—
41C22×Dic271C2313yes—
42C22×Dic271C2313yes—
43C2×C4×D271—46no—
44C2×C4×D271C2316yes—
45C2×C4×D271C2316yes—
46C4⋊Dic271C2316yes—
47C4⋊Dic271C2316yes—
48Dic271⋊C4C23112yes—
49(C2×D271)⋊C4C23112yes—
50(C2×D271)⋊C4C23112yes—
51(C2×D271)⋊C4C23112yes—
52C4×Dic271C2316yes—
53C4×Dic271C2316yes—
54C24×C271C2311yescharacteristic
55C2×Q8×C271C2312yes—
56C2×Q8×C271C2312yes—
57C22×C4×C271C2313yes—
58C22×C4×C271C2313yes—
59C4⋊C4×C271C2316yes—
60C4⋊C4×C271C2316yes—
61C22⋊C4×C271C23112yes—
62C22×D271—26no—
63C22×D271—43no—
64C22×D271—43no—
65C22×D271C2416yes—
66D1084—46no—
67D1084—46no—
68C22⋊D271—224no—
69C2×Dic271—26no—
70C2×Dic271—26no—
71C2×Dic271—43no—
72C2×Dic271C2413yes—
73C2×Dic271C2416yes—
74C2×Dic271C2411yescharacteristic
75C4×D271—212no—
76C4×D271—46no—
77C23×C271—41no—
78C23×C271—41no—
79C23×C271#4114yescomplemented
80C23×C271C2413yes—
81Q8×C271—42no—
82Q8×C271—42no—
83C2×C4×C271—43no—
84C2×C4×C271—43no—
85C2×C4×C271C2416yes—
86C2×C4×C271C2416yes—
87C22×C271—23no—
88C22×C271—23no—
89C22×C271—44no—
90C22×C271C2×#4116yes—
91C22×C271C2511yescharacteristic
92C2×D271—26no—
93C2×D271—26no—
94C2×D271—26no—
95C2×D271—412no—
96C4×C271—26no—
97C4×C271—26no—
98Dic271—26no—
99Dic271—23no—
100Dic271H(5,𝔽2)12yes—
101C2×C271—26no—
102C2×C271C22×#4111yescharacteristic
103C2×C271C2×H(5,𝔽2)11yescharacteristic
104C2×C271#58511yescharacteristic
105D271—46no—
106C271#345911yescharacteristic, complemented, Sylow
107#3459—2711nomaximal, Sylow
108#543D27111yescharacteristic, complemented
109#539—2716no—
110#538—2716no—
111#534—2713no—
112#529—2716no—
113#525—2716no—
114C2×C4×D4—2713no—
115C2×#41—2712no—
116C22×D4—2713no—
117C23×C4—2711no—
118C23×C4—2713no—
119#126—27112no—
120#125—2716no—
121#124—2716no—
122#123—27112no—
123#123—27112no—
124C22⋊Q8—2716no—
125C22⋊Q8C2×D27116yescomplemented
126C22⋊Q8C2×D27116yescomplemented
127#121—2716no—
128#121—2716no—
129#121—2716no—
130#120—27112no—
131C4×Q8—2712no—
132C4×D4—2716no—
133C4×D4—27112no—
134C4×D4—27112no—
135#117—2716no—
136C2×C4⋊C4—2713no—
137C2×C22⋊C4—2713no—
138C2×C22⋊C4C2×D27113yes—
139C2×C42—2713no—
140C24—2713no—
141C24C22×D27111yescharacteristic
142#41—10844no—
143C2×Q8C22×D27112yes—
144C2×Q8C22×D27112yes—
145C2×D4—27112no—
146C2×D4—2716no—
147C2×D4—2716no—
148C2×D4—54212no—
149C22×C4—2713no—
150C22×C4—2713no—
151C22×C4—2713no—
152C22×C4—2716no—
153C22×C4—2716no—
154C22×C4—10846no—
155C22×C4—10842no—
156C22×C4C22×D27113yes—
157C22×C4C22×D27113yes—
158C4⋊C4—27112no—
159C4⋊C4—2716no—
160C4⋊C4—2716no—
161C4⋊C4C22×D27116yes—
162C4⋊C4C22×D27116yes—
163C22⋊C4—27112no—
164C22⋊C4—27112no—
165C22⋊C4—27112no—
166C22⋊C4C22×D271112yes—
167C42—2716no—
168C42—2716no—
169C23—41no—
170C23—41no—
171C23—2716no—
172C23—5426no—
173C23—10843no—
174C23—10843no—
175C23#1686465514yescomplemented
176C23C23×D27113yes—
177Q8—42no—
178Q8—42no—
179D4—54224no—
180D4—10846no—
181D4—10846no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—2711no—
185C2×C4—2716no—
186C2×C4—2713no—
187C2×C4—54212no—
188C2×C4—5426no—
189C2×C4—5426no—
190C2×C4—10846no—
191C2×C4—10843no—
192C2×C4C23×D27116yes—
193C2×C4C23×D27116yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—5426no—
198C22—5426no—
199C22—5426no—
200C22—108412no—
201C22C2×#1686465516yes—
202C22C24×D27111yescharacteristic
203C4—26no—
204C4—26no—
205C4—2712no—
206C4—5426no—
207C4—5423no—
208C2—26no—
209C2—10846no—
210C2C22×#1686465511yescharacteristic
211C2H(5,𝔽2)×D27111yescharacteristic
212C2#3177473211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#543×C271C2
hypercenter#543D271
hypocenterC271#3459
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #31774496C2semidirect
2. #31774487C2semidirect
3. #31774459C2semidirect
4. #31774433C2semidirect
5. #31774410C2semidirect
6. C2×(C22⋊D271)⋊C4C2semidirect
7. #543×C271C2semidirect
8. C2×#16864655C22semidirect
11. C22×C4×D271C22semidirect
12. #22576155C22semidirect
13. #22576151C22semidirect
14. #22576149C22semidirect
15. #22576145C22semidirect
16. #22576144C22semidirect
17. #22576136C22semidirect
18. #22576135C22semidirect
21. #22576130C22semidirect
22. Q8×Dic271C22semidirect
23. #22576122C22semidirect
24. (C22⋊D271)⋊C4C22semidirect
25. (C22⋊D271)⋊C4C22semidirect
26. #22576118C22semidirect
27. C2×Dic271⋊C4C22semidirect
28. C2×(C2×D271)⋊C4C22semidirect
29. C2×C4×Dic271C22semidirect
30. C22⋊Q8×C271C22semidirect
31. C22⋊Q8×C271C22semidirect
79. C23×C271#41semidirect
106. C271#3459semidirect
108. #543D271semidirect
125. C22⋊Q8C2×D271semidirect
126. C22⋊Q8C2×D271semidirect
175. C23#16864655semidirect

Maximal subgroups

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

subgroupindex
#317744962
#317744872
#317744592
#317744332
#317744102
C2×(C22⋊D271)⋊C42
#543×C2712
#3459271