Finite Group Databasebeta

← #28336429

Subgroups — #28336429

Order 13696.

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×#16812155C211yesmaximal, characteristic
2#20292645C212yesmaximal, complemented
3#20292631C212yesmaximal, complemented
4#20292611C211yesmaximal, characteristic, complemented
5#410×C107C211yesmaximal, characteristic, complemented
6#16812155—24no—
7#16812155—24no—
8C2×#13133819—21no—
9C2×#13133818C412yescomplemented
10C2×#13133818C2211yescharacteristic
11C22×D428C412yescomplemented
12C22×D428C2211yescharacteristic
13C2×D4×D107—21no—
14C2×D4×D107—22no—
15C2×#13133802C2212yescomplemented
16#16812037—24no—
17#16812029—24no—
18#16812028—21no—
19#16812026—21no—
20C2×#41×C107C2211yescharacteristic
21#117×C107C2212yescomplemented
22#99×C107—22no—
23C23×D107—22no—
24C23×D107—21no—
25#13133819—28no—
26#13133818—22no—
27#13133818—24no—
28#13133818—24no—
29#13133818D414yes—
30C2×Dic214C2×C411yescharacteristic
31C2×D428—22no—
32C2×D428—21no—
33C2×D428—22no—
34C2×D428—24no—
35C2×D428—24no—
36C2×D428C2×C411yescharacteristic
37C2×D428C2×C412yes—
38C2×D428D414yes—
39C2×C22⋊D107—22no—
40C2×C22⋊D107—21no—
41D4×D107—216no—
42D4×D107—28no—
43C2×C4×D107—22no—
44C2×C4×D107—21no—
45#13133802—22no—
46#13133802—22no—
47C2×C107⋊C8—22no—
48#41×C107—22no—
49#41×C107—22no—
50C2×Q8×C107D411yescharacteristic
51C2×D4×C107D411yescharacteristic
52C2×D4×C107D412yes—
53C22×C4×C107C2×C412yes—
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—22no—
56C22⋊C4×C107—22no—
57C42×C107—22no—
58C23×C107C22⋊C412yes—
59C23×C107C2×D411yescharacteristic
60C22×D107—22no—
61C22×D107—21no—
62C22×D107—28no—
63C22×D107—24no—
64C22×D107—82no—
65C22×D107—84no—
66Q8×C107—22no—
67D4×C107—24no—
68D4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—42no—
72C2×C4×C107C22⋊C412yes—
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D412yes—
75C2×C4×C107C2×D412yes—
76C2×C4×C107C2×D411yescharacteristic
77C22⋊D107—28no—
78C22⋊D107—24no—
79C2×Dic107—21no—
80D428—28no—
81D428—21no—
82D428—22no—
83D428—24no—
84D428C22⋊C412yes—
85D428C22⋊C414yes—
86C4×D107—28no—
87C4×D107—24no—
88Dic214—21no—
89Dic214C22⋊C412yes—
90C107⋊C8—42no—
91C22×C107—22no—
92C22×C107—41no—
93C22×C107—42no—
94C22×C107C2×C22⋊C411yescharacteristic
95C22×C107#12012yes—
96C2×D107—24no—
97C2×D107—22no—
98C2×D107—44no—
99C2×D107—42no—
100C2×D107—84no—
101C2×D107—88no—
102C4×C107—22no—
103C4×C107—82no—
104C4×C107C2×C22⋊C411yescharacteristic
105C4×C107C2×C22⋊C411yescharacteristic
106C4×C107#12012yes—
107Dic107—22no—
108C2×C107—22no—
109C2×C107—21no—
110C2×C107—42no—
111C2×C107#37911yescharacteristic
112D107—44no—
113D107—42no—
114#1795—1071nomaximal, Sylow
115C107#179511yescharacteristic, complemented, Sylow
116C2×H(5,𝔽2)—1071no—
117#421—1072no—
118#418—1072no—
119#413—1071no—
120#410D10711yescharacteristic, complemented
121H(5,𝔽2)—2144no—
122H(5,𝔽2)—2144no—
123C2×#41—1072no—
124C2×#41—1071no—
125C2×#41—2141no—
126C2×#41C2×D10711yescharacteristic
127C22×D4—1072no—
128C22×D4—1071no—
129C22×D4—2141no—
130C22×D4—2142no—
131C2×M16—1072no—
132#117C2×D10712yescomplemented
133#104—2144no—
134#102—2144no—
135#101—2141no—
136#100—2141no—
137#99—22no—
138C24—2142no—
139C24—2141no—
140#41—22no—
141#41—22no—
142#41—1074no—
143#41—2144no—
144#41—2144no—
145#41—2148no—
146#41—2142no—
147C2×Q8—1071no—
148C2×Q8D42811yescharacteristic
149C2×D4—1072no—
150C2×D4—1071no—
151C2×D4—1074no—
152C2×D4—2144no—
153C2×D4—2144no—
154C2×D4—21416no—
155C2×D4—2148no—
156C2×D4—2142no—
157C2×D4—2141no—
158C2×D4—2142no—
159C2×D4—2142no—
160C2×D4—2141no—
161C2×D4D42811yescharacteristic
162C2×D4C22⋊D10712yes—
163C22×C4—2142no—
164C22×C4—2141no—
165C22×C4C4×D10712yes—
166C22×C4C22×D10711yescharacteristic
167M16—2142no—
168M16—2142no—
169C2×C8—2142no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—8562no—
174C23—8564no—
175C23—2148no—
176C23—2144no—
177C23—2142no—
178C23—2141no—
179C23(C2×D107)⋊C412yes—
180C23C2×C22⋊D10711yescharacteristic
181Q8—22no—
182Q8—1072no—
183Q8—2141no—
184D4—24no—
185D4—22no—
186D4—1074no—
187D4—1072no—
188D4—2148no—
189D4—2148no—
190D4—2144no—
191D4—2141no—
192D4—2142no—
193D4—2144no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—2148no—
198C2×C4—2141no—
199C2×C4—2144no—
200C2×C4(C2×D107)⋊C412yes—
201C2×C4C2×C4×D10711yescharacteristic
202C2×C4D4×D10712yes—
203C2×C4D4×D10712yes—
204C2×C4C2×D42811yescharacteristic
205C8—4282no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—8564no—
210C22—8568no—
211C22—4284no—
212C22—4282no—
213C22—2144no—
214C22—2142no—
215C22C2×(C2×D107)⋊C411yescharacteristic
216C22#1681207712yes—
217C4—22no—
218C4—82no—
219C4—2142no—
220C4C22⋊C4×D10711yescharacteristic
221C4C22⋊C4×D10711yescharacteristic
222C4#1681207912yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—4284no—
227C2—4282no—
228C2#2029248211yescharacteristic
subgroupG/H
centerC2#20292482
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#410×C107C2
socleC2×C107#379
prime radicalC2×#16812155C2
cosocleC22×C4×C107C23
hypercenter#410D107
hypocenterC107#1795
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20292645C2semidirect
3. #20292631C2semidirect
4. #20292611C2semidirect
5. #410×C107C2semidirect
9. C2×#13133818C4semidirect
11. C22×D428C4semidirect
15. C2×#13133802C22semidirect
21. #117×C107C22semidirect
115. C107#1795semidirect
120. #410D107semidirect
132. #117C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
#202926452
#202926312
#202926112
#410×C1072
#1795107