Finite Group Databasebeta

← #28336424

Subgroups — #28336424

Order 13696.

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

normal subgroups only

This group has 548 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
1#20293463C211yesmaximal, characteristic
2#20292643C212yesmaximal, complemented
3#20292624C212yesmaximal, complemented
4#20292601C211yesmaximal, characteristic, complemented
5#408×C107C211yesmaximal, characteristic, complemented
6#16812157C412yescomplemented
7#16812156—22no—
8#16812155—22no—
9#16812154—22no—
10#16812154C412yescomplemented
11C2×#13133818—22no—
12C2×#13133818C2212yes—
13#41×D107—24no—
14#16812068C2212yescomplemented
15#16812035—24no—
16#16812034—22no—
17#16812031—22no—
18#16812024—22no—
19C2×#41×C107C2211yescharacteristic
20C2×M16×C107C2211yescharacteristic, complemented
21C22×C8×C107C2211yescharacteristic, complemented
22C22⋊C8×C107—22no—
23#13133819—24no—
24#13133818—24no—
25#13133818—24no—
26#13133818—24no—
27#13133818—22no—
28#13133818—22no—
29#13133818C2×C414yes—
30#13133818D414yes—
31#13133817—28no—
32#13133817—24no—
33C2×Dic214—22no—
34C2×Dic214D412yes—
35Q8×D107—24no—
36C2×D428—22no—
37C2×D428D412yes—
38C2×C22⋊D107—24no—
39C2×C22⋊D107—22no—
40D4×D107—28no—
41D4×D107—24no—
42C2×C4×D107—24no—
43C2×C4×D107—22no—
44C4⋊Dic107—22no—
45C22⋊Dic107—22no—
46C4×Dic107—22no—
47#41×C107—22no—
48#41×C107—22no—
49C2×Q8×C107C2×C411yescharacteristic
50C2×D4×C107C2×C411yescharacteristic
51C2×D4×C107D412yes—
52C22×C4×C107D412yes—
53C22×C4×C107C2311yescharacteristic
54M16×C107—22no—
55C2×C8×C107—22no—
56C2×C8×C107—21no—
57C2×C8×C107—21no—
58C23×C107C22⋊C412yes—
59C23×C107C2×D411yescharacteristic
60C22×D107—24no—
61C22×D107—22no—
62Q8×C107—22no—
63D4×C107—24no—
64D4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—22no—
67C2×C4×C107C22⋊C412yes—
68C2×C4×C107C22×C411yescharacteristic
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D411yescharacteristic
72C22⋊D107—28no—
73C22⋊D107—24no—
74C22⋊D107—24no—
75C2×Dic107—24no—
76C2×Dic107—22no—
77C2×Dic107—42no—
78D428—24no—
79D428—22no—
80D428C22⋊C414yes—
81C4×D107—28no—
82C4×D107—24no—
83C4×D107—24no—
84Dic214—24no—
85Dic214—22no—
86Dic214C22⋊C414yes—
87C8×C107—41no—
88C8×C107—41no—
89C22×C107—22no—
90C22×C107—41no—
91C22×C107—42no—
92C22×C107C2×C22⋊C411yescharacteristic
93C22×C107#12012yes—
94C2×D107—24no—
95C2×D107—82no—
96C2×D107—84no—
97C4×C107—22no—
98C4×C107C2×C22⋊C412yes—
99C4×C107#12011yescharacteristic
100C4×C107#12011yescharacteristic
101Dic107—24no—
102Dic107—82no—
103C2×C107—22no—
104C2×C107—21no—
105C2×C107—42no—
106C2×C107#37911yescharacteristic
107D107—44no—
108#1794—1071nomaximal, Sylow
109C107#179411yescharacteristic, complemented, Sylow
110#585—1071no—
111#421—1072no—
112#416—1072no—
113#410—1071no—
114#408D10711yescharacteristic, complemented
1152−1+4—1072no—
1162−1+4—2142no—
117H(5,𝔽2)—1072no—
118H(5,𝔽2)—2142no—
119H(5,𝔽2)—2142no—
120C2×#41—1072no—
121C2×#41—2144no—
122C2×#41—2142no—
123C2×#41C2×D10711yescharacteristic
124C2×M16C2×D10711yescharacteristic, complemented
125C22×C8C2×D10711yescharacteristic, complemented
126#117—1072no—
127#104—2144no—
128#103—2142no—
129#102—2142no—
130#99—2142no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—1074no—
135#41—1074no—
136#41—2144no—
137#41—2144no—
138#41—2144no—
139#41—2148no—
140#41—2144no—
141#41—2144no—
142#41—2142no—
143#41—2142no—
144C2×Q8—1072no—
145C2×Q8—2142no—
146C2×Q8—2144no—
147C2×Q8C4×D10711yescharacteristic
148C2×D4—1072no—
149C2×D4—2142no—
150C2×D4—2144no—
151C2×D4—2148no—
152C2×D4—2144no—
153C2×D4—2142no—
154C2×D4C4×D10711yescharacteristic
155C2×D4D42812yes—
156C22×C4—2144no—
157C22×C4—2142no—
158C22×C4C22⋊D10712yes—
159C22×C4C22×D10711yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—2142no—
165C22⋊C4—2142no—
166C42—2142no—
167C23—2144no—
168C23—2142no—
169C23(C2×D107)⋊C412yes—
170C23C2×D42811yescharacteristic
171Q8—22no—
172Q8—1074no—
173Q8—2144no—
174Q8—2142no—
175D4—24no—
176D4—22no—
177D4—1074no—
178D4—2144no—
179D4—2148no—
180D4—2144no—
181D4—2144no—
182D4—2142no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—4282no—
186C2×C4—2148no—
187C2×C4—2144no—
188C2×C4—2142no—
189C2×C4—2144no—
190C2×C4—2144no—
191C2×C4(C2×D107)⋊C412yes—
192C2×C4C2×C4×D10711yescharacteristic
193C2×C4D4×D10712yes—
194C2×C4D4×D10712yes—
195C2×C4C2×C22⋊D10711yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—8562no—
202C22—8564no—
203C22—2144no—
204C22C2×(C2×D107)⋊C411yescharacteristic
205C22#1681207912yes—
206C4—22no—
207C4—8562no—
208C4—2144no—
209C4C22⋊C4×D10712yes—
210C4#1681207711yescharacteristic
211C4#1681207711yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—4284no—
216C2#2029248211yescharacteristic
subgroupG/H
centerC4#16812077
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#408×C107C2
socleC2×C107#379
prime radical#20293463C2
cosocleC22×C4×C107C23
hypercenter#408D107
hypocenterC107#1794
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20292643C2semidirect
3. #20292624C2semidirect
4. #20292601C2semidirect
5. #408×C107C2semidirect
6. #16812157C4semidirect
10. #16812154C4semidirect
14. #16812068C22semidirect
20. C2×M16×C107C22semidirect
21. C22×C8×C107C22semidirect
109. C107#1794semidirect
114. #408D107semidirect
124. C2×M16C2×D107semidirect
125. C22×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
#202926432
#202926242
#202926012
#408×C1072
#1794107