Finite Group Databasebeta

← #28336425

Subgroups — #28336425

Order 13696.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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#20292644C212yesmaximal, complemented
2#20292623C212yesmaximal, complemented
3#20292601C211yesmaximal, characteristic, complemented
4#20292594C211yesmaximal, characteristic, complemented
5#585×C107C211yesmaximal, characteristic
6C2×#13133802C2211yescharacteristic, complemented
7C22×C107⋊C8C2211yescharacteristic, complemented
8#16812068C2212yescomplemented
9#16812036—24no—
10#16812033—22no—
11#16812030—22no—
12#16812024—22no—
13C22⋊(C107⋊C8)—22no—
142−1+4×C107—22no—
152−1+4×C107C412yescomplemented
16H(5,𝔽2)×C107—22no—
17H(5,𝔽2)×C107—22no—
18H(5,𝔽2)×C107C412yescomplemented
19C2×#41×C107—24no—
20C2×#41×C107—22no—
21C2×#41×C107C2212yes—
22C2×#41×C107C2211yescharacteristic
23#13133802—22no—
24C2×C107⋊C8—22no—
25C2×C107⋊C8—21no—
26C2×C107⋊C8—21no—
27C4⋊Dic107—22no—
28C22⋊Dic107—22no—
29C4×Dic107—22no—
30#41×C107—24no—
31#41×C107—28no—
32#41×C107—24no—
33#41×C107—24no—
34#41×C107—24no—
35#41×C107—24no—
36#41×C107—22no—
37#41×C107—22no—
38#41×C107—22no—
39#41×C107—22no—
40#41×C107C2×C414yes—
41#41×C107D414yes—
42C2×Q8×C107—22no—
43C2×Q8×C107—24no—
44C2×Q8×C107C2×C411yescharacteristic
45C2×Q8×C107D412yes—
46C2×D4×C107—28no—
47C2×D4×C107—22no—
48C2×D4×C107—24no—
49C2×D4×C107—22no—
50C2×D4×C107—24no—
51C2×D4×C107C2×C411yescharacteristic
52C2×D4×C107D412yes—
53C2×D4×C107D412yes—
54C22×C4×C107—24no—
55C22×C4×C107—22no—
56C22×C4×C107D412yes—
57C22×C4×C107C2311yescharacteristic
58C23×C107—24no—
59C23×C107—22no—
60C23×C107C22⋊C412yes—
61C23×C107C2×D411yescharacteristic
62Q8×C107—24no—
63Q8×C107—22no—
64Q8×C107—22no—
65Q8×C107C22⋊C414yes—
66D4×C107—28no—
67D4×C107—24no—
68D4×C107—24no—
69D4×C107—24no—
70D4×C107—24no—
71D4×C107—22no—
72D4×C107—22no—
73D4×C107C22⋊C414yes—
74C2×C4×C107—24no—
75C2×C4×C107—28no—
76C2×C4×C107—22no—
77C2×C4×C107—24no—
78C2×C4×C107—24no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107C22⋊C412yes—
82C2×C4×C107C22×C411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D411yescharacteristic
86C2×Dic107—42no—
87C107⋊C8—41no—
88C107⋊C8—41no—
89C22×C107—22no—
90C22×C107—24no—
91C22×C107—41no—
92C22×C107—42no—
93C22×C107—82no—
94C22×C107—84no—
95C22×C107C2×C22⋊C411yescharacteristic
96C22×C107#12012yes—
97C4×C107—24no—
98C4×C107—22no—
99C4×C107C2×C22⋊C412yes—
100C4×C107#12011yescharacteristic
101C4×C107#12011yescharacteristic
102Dic107—82no—
103C2×C107—22no—
104C2×C107—21no—
105C2×C107—42no—
106C2×C107—44no—
107C2×C107#37911yescharacteristic
108#1794—1071nomaximal, Sylow
109C107#179411yescharacteristic, complemented, Sylow
110#585D10711yescharacteristic
111#421—1072no—
112#416—1072no—
113#410—1071no—
114#408—1071no—
1152−1+4—22no—
1162−1+4Dic10712yescomplemented
117H(5,𝔽2)—22no—
118H(5,𝔽2)—22no—
119H(5,𝔽2)Dic10712yescomplemented
120C2×#41—24no—
121C2×#41—22no—
122C2×#41C2×D10712yes—
123C2×#41C2×D10711yescharacteristic
124C2×M16—1071no—
125C22×C8—1071no—
126#117—1072no—
127#104—2144no—
128#103—2142no—
129#102—2142no—
130#99—2142no—
131C22⋊C8—2142no—
132#41—24no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—24no—
137#41—24no—
138#41—22no—
139#41—22no—
140#41—22no—
141#41—22no—
142#41C2×Dic10714yes—
143#41C22⋊D10714yes—
144C2×Q8—22no—
145C2×Q8—24no—
146C2×Q8C2×Dic10711yescharacteristic
147C2×Q8C22⋊D10712yes—
148C2×D4—28no—
149C2×D4—22no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—24no—
153C2×D4C2×Dic10711yescharacteristic
154C2×D4C22⋊D10712yes—
155C2×D4C22⋊D10712yes—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4C22⋊D10712yes—
159C22×C4C22×D10711yescharacteristic
160M16—2142no—
161C2×C8—2141no—
162C2×C8—2141no—
163C2×C8—2142no—
164C4⋊C4—2142no—
165C22⋊C4—2142no—
166C42—2142no—
167C23—24no—
168C23—22no—
169C23C22⋊Dic10712yes—
170C23C2×C22⋊D10711yescharacteristic
171Q8—24no—
172Q8—22no—
173Q8—22no—
174Q8C22⋊Dic10714yes—
175D4—28no—
176D4—24no—
177D4—24no—
178D4—24no—
179D4—24no—
180D4—22no—
181D4—22no—
182D4C22⋊Dic10714yes—
183C2×C4—24no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—4282no—
191C2×C4C22⋊Dic10712yes—
192C2×C4C22×Dic10711yescharacteristic
193C2×C4C2×C22⋊D10712yes—
194C2×C4C2×C22⋊D10712yes—
195C2×C4C2×C22⋊D10711yescharacteristic
196C8—4281no—
197C8—4281no—
198C22—22no—
199C22—24no—
200C22—41no—
201C22—42no—
202C22—82no—
203C22—84no—
204C22C2×C22⋊Dic10711yescharacteristic
205C22#1681207812yes—
206C4—24no—
207C4—22no—
208C4—8562no—
209C4C2×C22⋊Dic10712yes—
210C4#1681207811yescharacteristic
211C4#1681207811yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—44no—
216C2#2029248311yescharacteristic
subgroupG/H
centerC4#16812078
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting, prime radical#585×C107C2
socleC2×C107#379
cosocleC22×C4×C107C23
hypercenter#585D107
hypocenterC107#1794
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20292644C2semidirect
2. #20292623C2semidirect
3. #20292601C2semidirect
4. #20292594C2semidirect
6. C2×#13133802C22semidirect
7. C22×C107⋊C8C22semidirect
8. #16812068C22semidirect
15. 2−1+4×C107C4semidirect
18. H(5,𝔽2)×C107C4semidirect
109. C107#1794semidirect
116. 2−1+4Dic107semidirect
119. H(5,𝔽2)Dic107semidirect

Maximal subgroups

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

subgroupindex
#202926442
#202926232
#202926012
#202925942
#585×C1072
#1794107