Finite Group Databasebeta

← #46782715

Subgroups — #46782715

Order 33664.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22554516C211yesmaximal, characteristic
2#30681956C212yesmaximal, complemented
3#30681942C212yesmaximal, complemented
4#30681922C211yesmaximal, characteristic, complemented
5#410×C263C211yesmaximal, characteristic, complemented
6#22554516—24no—
7#22554516—24no—
8C2×#16862377—21no—
9C2×#16862376C412yescomplemented
10C2×#16862376C2211yescharacteristic
11C22×D1052C412yescomplemented
12C22×D1052C2211yescharacteristic
13C2×D4×D263—22no—
14C2×D4×D263—21no—
15C2×#16862360C2212yescomplemented
16#22554398—24no—
17#22554390—24no—
18#22554389—21no—
19#22554387—21no—
20C2×#41×C263C2211yescharacteristic
21#117×C263C2212yescomplemented
22#99×C263—22no—
23C23×D263—22no—
24C23×D263—21no—
25#16862377—28no—
26#16862376—24no—
27#16862376—24no—
28#16862376—22no—
29#16862376D414yes—
30C2×Dic526C2×C411yescharacteristic
31C2×D1052—24no—
32C2×D1052—24no—
33C2×D1052—22no—
34C2×D1052—21no—
35C2×D1052—22no—
36C2×D1052C2×C411yescharacteristic
37C2×D1052C2×C412yes—
38C2×D1052D414yes—
39C2×C22⋊D263—22no—
40C2×C22⋊D263—21no—
41D4×D263—216no—
42D4×D263—28no—
43C2×C4×D263—22no—
44C2×C4×D263—21no—
45#16862360—22no—
46#16862360—22no—
47C2×C263⋊C8—22no—
48#41×C263—22no—
49#41×C263—22no—
50C2×Q8×C263D411yescharacteristic
51C2×D4×C263D412yes—
52C2×D4×C263D411yescharacteristic
53C22×C4×C263C2×C412yes—
54C22×C4×C263C2311yescharacteristic
55C4⋊C4×C263—22no—
56C22⋊C4×C263—22no—
57C42×C263—22no—
58C22×D263—22no—
59C22×D263—21no—
60C22×D263—28no—
61C22×D263—24no—
62C22×D263—82no—
63C22×D263—84no—
64Dic526—21no—
65Dic526C22⋊C412yes—
66D1052—28no—
67D1052—22no—
68D1052—21no—
69D1052—24no—
70D1052C22⋊C412yes—
71D1052C22⋊C414yes—
72C22⋊D263—28no—
73C22⋊D263—24no—
74C2×Dic263—21no—
75C4×D263—28no—
76C4×D263—24no—
77C263⋊C8—42no—
78C23×C263C22⋊C412yes—
79C23×C263C2×D411yescharacteristic
80Q8×C263—22no—
81D4×C263—24no—
82D4×C263—22no—
83C2×C4×C263—22no—
84C2×C4×C263—22no—
85C2×C4×C263—42no—
86C2×C4×C263C22⋊C412yes—
87C2×C4×C263C22×C411yescharacteristic
88C2×C4×C263C2×D412yes—
89C2×C4×C263C2×D412yes—
90C2×C4×C263C2×D411yescharacteristic
91C22×C263—22no—
92C22×C263—41no—
93C22×C263—42no—
94C22×C263C2×C22⋊C411yescharacteristic
95C22×C263#12012yes—
96C2×D263—24no—
97C2×D263—22no—
98C2×D263—44no—
99C2×D263—42no—
100C2×D263—84no—
101C2×D263—88no—
102C4×C263—22no—
103C4×C263—82no—
104C4×C263C2×C22⋊C411yescharacteristic
105C4×C263C2×C22⋊C411yescharacteristic
106C4×C263#12012yes—
107Dic263—22no—
108C2×C263—22no—
109C2×C263—21no—
110C2×C263—42no—
111C2×C263#37911yescharacteristic
112D263—44no—
113D263—42no—
114C263#179511yescharacteristic, complemented, Sylow
115#1795—2631nomaximal, Sylow
116C2×H(5,𝔽2)—2631no—
117#421—2632no—
118#418—2632no—
119#413—2631no—
120#410D26311yescharacteristic, complemented
121H(5,𝔽2)—5264no—
122H(5,𝔽2)—5264no—
123C2×#41—2632no—
124C2×#41—2631no—
125C2×#41—5261no—
126C2×#41C2×D26311yescharacteristic
127C22×D4—2632no—
128C22×D4—2631no—
129C22×D4—5262no—
130C22×D4—5261no—
131C2×M16—2632no—
132#117C2×D26312yescomplemented
133#104—5264no—
134#102—5264no—
135#101—5261no—
136#100—5261no—
137#99—22no—
138C24—5262no—
139C24—5261no—
140#41—22no—
141#41—22no—
142#41—2634no—
143#41—5264no—
144#41—5264no—
145#41—5268no—
146#41—5262no—
147C2×Q8—2631no—
148C2×Q8D105211yescharacteristic
149C2×D4—2632no—
150C2×D4—2631no—
151C2×D4—2634no—
152C2×D4—5264no—
153C2×D4—5264no—
154C2×D4—52616no—
155C2×D4—5268no—
156C2×D4—5262no—
157C2×D4—5261no—
158C2×D4—5262no—
159C2×D4—5262no—
160C2×D4—5261no—
161C2×D4C22⋊D26312yes—
162C2×D4D105211yescharacteristic
163C22×C4—5262no—
164C22×C4—5261no—
165C22×C4C4×D26312yes—
166C22×C4C22×D26311yescharacteristic
167M16—5262no—
168M16—5262no—
169C2×C8—5262no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—5268no—
174C23—5262no—
175C23—5261no—
176C23—5264no—
177C23—21042no—
178C23—21044no—
179C23(C2×D263)⋊C412yes—
180C23C2×C22⋊D26311yescharacteristic
181Q8—22no—
182Q8—2632no—
183Q8—5261no—
184D4—24no—
185D4—22no—
186D4—2634no—
187D4—2632no—
188D4—5268no—
189D4—5268no—
190D4—5264no—
191D4—5261no—
192D4—5262no—
193D4—5264no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—5268no—
198C2×C4—5261no—
199C2×C4—5264no—
200C2×C4(C2×D263)⋊C412yes—
201C2×C4C2×C4×D26311yescharacteristic
202C2×C4D4×D26312yes—
203C2×C4D4×D26312yes—
204C2×C4C2×D105211yescharacteristic
205C8—10522no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—5264no—
210C22—5262no—
211C22—10524no—
212C22—10522no—
213C22—21044no—
214C22—21048no—
215C22C2×(C2×D263)⋊C411yescharacteristic
216C22#2255443812yes—
217C4—22no—
218C4—82no—
219C4—5262no—
220C4C22⋊C4×D26311yescharacteristic
221C4C22⋊C4×D26311yescharacteristic
222C4#2255444012yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—10524no—
227C2—10522no—
228C2#3068179311yescharacteristic
subgroupG/H
centerC2#30681793
commutatorC2×C4×C263C22×C4
FrattiniC22×C4C22×D263
Fitting#410×C263C2
socleC2×C263#379
prime radicalC2×#22554516C2
cosocleC22×C4×C263C23
hypercenter#410D263
hypocenterC263#1795
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #30681956C2semidirect
3. #30681942C2semidirect
4. #30681922C2semidirect
5. #410×C263C2semidirect
9. C2×#16862376C4semidirect
11. C22×D1052C4semidirect
15. C2×#16862360C22semidirect
21. #117×C263C22semidirect
114. C263#1795semidirect
120. #410D263semidirect
132. #117C2×D263semidirect

Maximal subgroups

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

subgroupindex
C2×#225545162
#306819562
#306819422
#306819222
#410×C2632
#1795263