Finite Group Databasebeta

← #28349551

Subgroups — #28349551

Order 13696.

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

normal subgroups only

This group has 380 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#20293362C212yesmaximal, complemented
2#20293016C212yesmaximal, complemented
3#20293011C211yesmaximal, characteristic, complemented
4#20292850C212yesmaximal, complemented
5#20292843C212yesmaximal, complemented
6#20292823C212yesmaximal, complemented
7#20292729C212yesmaximal, complemented
8C4×D856C211yesmaximal, characteristic, complemented
9#446×C107C211yesmaximal, characteristic, complemented
10C2×Dic428C2211yescharacteristic
11C2×#13133807C2212yes—
12C2×#13133807C2212yes—
13C2×D856C2211yescharacteristic
14C2×D856C2211yescharacteristic
15C2×D856C2212yes—
16#16812106C2212yes—
17#16812098—22no—
18#16812097C2212yes—
19#16812097C2212yes—
20#16812094—22no—
21#16812094—22no—
22C4×Dic214C2212yes—
23C4×D428C2212yes—
24C4×D428C2212yes—
25#16812071—22no—
26#16812043C2211yescharacteristic
27#16812034C2212yes—
28#16812031C2212yes—
29#16812031C2212yes—
30C4×Q8×C107C2211yescharacteristic
31C4⋊C8×C107C2212yes—
32C4⋊C8×C107C2211yescharacteristic
33#97×C107C2212yes—
34C4×C8×C107C2211yescharacteristic
35C2×Dic214C2312yes—
36C2×D428—22no—
37C2×D428—22no—
38C2×D428C2312yes—
39C2×D428C2312yes—
40C2×C4×D107—22no—
41C2×C4×D107—22no—
42Dic428—41no—
43#13133807—24no—
44#13133807—42no—
45D856—22no—
46D856—42no—
47D856—41no—
48C4⋊Dic107C2312yes—
49Dic107⋊C4—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—22no—
56C4×Dic107—22no—
57C2×Q8×C107D411yescharacteristic
58C2×C8×C107C2312yes—
59C2×C8×C107C2311yescharacteristic
60C2×C8×C107C2311yescharacteristic
61C4⋊C4×C107D412yes—
62C4⋊C4×C107D411yescharacteristic
63C42×C107C2311yescharacteristic
64C42×C107C2312yes—
65C22×D107—22no—
66C22×D107—22no—
67Q8×C107—41no—
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2411yescharacteristic
73C2×Dic107—22no—
74C2×Dic107—22no—
75D428—22no—
76D428—22no—
77D428—22no—
78D428—22no—
79D428—44no—
80D428—82no—
81C4×D107—82no—
82C4×D107—82no—
83Dic214—22no—
84Dic214—22no—
85C8×C107—22no—
86C8×C107—21no—
87C8×C107#4112yes—
88C22×C107C22×D411yescharacteristic
89C2×D107—42no—
90C2×D107—42no—
91C2×D107—42no—
92C2×D107—42no—
93C2×D107—42no—
94C2×D107—42no—
95C4×C107—22no—
96C4×C107—21no—
97C4×C107—21no—
98C4×C107—22no—
99C4×C107C2×#4111yescharacteristic
100C4×C107H(5,𝔽2)11yescharacteristic
101Dic107—42no—
102Dic107—42no—
103C2×C107#55011yescharacteristic
104C2×C107#57511yescharacteristic
105C2×C107#57611yescharacteristic
106D107—82no—
107D107—82no—
108#3403—1071nomaximal, Sylow
109C107#340311yescharacteristic, complemented, Sylow
110#559—1072no—
111#496—1072no—
112#495—1071no—
113#464—1072no—
114#463—1072no—
115#459—1072no—
116#446D10711yescharacteristic, complemented
117#440—1072no—
118C4×D8—1071no—
119C2×Q16—1071no—
120C2×SD16—1072no—
121C2×SD16—1072no—
122C2×D8—1071no—
123C2×D8—1072no—
124C2×D8—1071no—
125#127—1072no—
126#124—1072no—
127#124—1072no—
128#124—2142no—
129#123—2142no—
130#123—2142no—
131C4×Q8—1072no—
132C4×Q8C2×D10711yescharacteristic
133C4×D4—1072no—
134C4×D4—1072no—
135C4×D4—2142no—
136#107—1071no—
137C4⋊C8C2×D10712yes—
138C4⋊C8C2×D10711yescharacteristic
139#103—1072no—
140#102—1072no—
141#102—1072no—
142#97C2×D10712yes—
143C4×C8C2×D10711yescharacteristic
144C2×Q8—1072no—
145C2×Q8D42811yescharacteristic
146C2×D4—1072no—
147C2×D4—1072no—
148C2×D4—2142no—
149C2×D4—2142no—
150C22×C4—2142no—
151C22×C4—2142no—
152Q16—4281no—
153SD16—4282no—
154SD16—2144no—
155D8—4282no—
156D8—4281no—
157D8—2142no—
158C2×C8C22×D10711yescharacteristic
159C2×C8C22×D10712yes—
160C2×C8C22×D10711yescharacteristic
161C4⋊C4—1072no—
162C4⋊C4—2142no—
163C4⋊C4D42812yes—
164C4⋊C4D42811yescharacteristic
165C22⋊C4—2142no—
166C22⋊C4—2142no—
167C22⋊C4—2142no—
168C22⋊C4—2142no—
169C22⋊C4—2142no—
170C22⋊C4—2142no—
171C42—2142no—
172C42C22×D10711yescharacteristic
173C42C22×D10712yes—
174C23—2142no—
175C23—2142no—
176Q8—41no—
177Q8—2142no—
178Q8—2142no—
179D4—8562no—
180D4—4284no—
181D4—2142no—
182D4—2142no—
183D4—2142no—
184D4—2142no—
185C2×C4—8562no—
186C2×C4—8562no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4C2×D42812yes—
190C2×C4C2×D42811yescharacteristic
191C2×C4C2×D42811yescharacteristic
192C2×C4C2×D42812yes—
193C2×C4C23×D10711yescharacteristic
194C8—22no—
195C8—21no—
196C8#1313381912yes—
197C22—4282no—
198C22—4282no—
199C22—4282no—
200C22—4282no—
201C22—4282no—
202C22—4282no—
203C22C22×D42811yescharacteristic
204C4—22no—
205C4—21no—
206C4—21no—
207C4—22no—
208C4—4282no—
209C4—4282no—
210C4C2×#1313381911yescharacteristic
211C4#1681215511yescharacteristic
212C2—8562no—
213C2—8562no—
214C2#2029329911yescharacteristic
215C2#2029342711yescharacteristic
216C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#446×C107C2
socleC22×C107C22×D4
hypercenter#446D107
hypocenterC107#3403
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #20293362C2semidirect
2. #20293016C2semidirect
3. #20293011C2semidirect
4. #20292850C2semidirect
5. #20292843C2semidirect
6. #20292823C2semidirect
7. #20292729C2semidirect
8. C4×D856C2semidirect
9. #446×C107C2semidirect
109. C107#3403semidirect
116. #446D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202930162
#202930112
#202928502
#202928432
#202928232
#202927292
C4×D8562
#446×C1072
#3403107