Finite Group Databasebeta

← #28337502

Subgroups — #28337502

Order 13696.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16812135C212yesmaximal, complemented
2#20293208C211yesmaximal, characteristic, complemented
3C2×#16812037C212yesmaximal, complemented
4#20292396C211yesmaximal, characteristic, complemented
5#390×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2212yes—
7#16812135—24no—
8#16812135—24no—
9C2×#13133806—22no—
10C2×#13133803—22no—
11C2×#13133802C2212yes—
12#16812089—22no—
13#16812084—22no—
14#16812083—22no—
15#16812037—24no—
16C22×D4×C107C2211yescharacteristic
17C2×C4⋊C4×C107C2211yescharacteristic
18C2×C22⋊C4×C107—22no—
19C2×C42×C107C2211yescharacteristic
20#13133818—22no—
21#13133818—22no—
22#13133818—22no—
23#13133818—22no—
24C2×Dic214D412yes—
25C2×D428—22no—
26C2×D428D412yes—
27C2×C22⋊D107—22no—
28C2×C4×D107—22no—
29#13133806—28no—
30#13133803—28no—
31#13133802—22no—
32#13133802D414yescomplemented
33C2×C107⋊C8—22no—
34C4⋊Dic107—22no—
35(C2×D107)⋊C4—24no—
36C24×C107—21no—
37C2×D4×C107—22no—
38C2×D4×C107—22no—
39C2×D4×C107—21no—
40C2×D4×C107D412yescomplemented
41C22×C4×C107—21no—
42C22×C4×C107D412yes—
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—22no—
45C22⋊C4×C107—44no—
46C42×C107—22no—
47C23×C107—22no—
48C23×C107—42no—
49C23×C107—42no—
50C23×C107—41no—
51C23×C107—41no—
52C23×C107C2×D411yescharacteristic
53C22×D107—22no—
54D4×C107—24no—
55D4×C107—42no—
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—41no—
59C2×C4×C107—41no—
60C2×C4×C107—42no—
61C2×C4×C107C2×D412yes—
62C2×C4×C107C2×D412yes—
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107#4111yescharacteristic
65C22⋊D107—42no—
66C22⋊D107—42no—
67C2×Dic107—22no—
68D428—22no—
69D428—22no—
70D428—82no—
71C4×D107—42no—
72C4×D107—42no—
73Dic214—22no—
74Dic214—22no—
75C107⋊C8—24no—
76C22×C107—21no—
77C22×C107—21no—
78C22×C107—44no—
79C22×C107—42no—
80C22×C107—82no—
81C22×C107—81no—
82C22×C107—81no—
83C22×C107#12011yescharacteristic
84C22×C107#12111yescharacteristic
85C22×C107#12711yescharacteristic
86C2×D107—42no—
87C2×D107—42no—
88C2×D107—42no—
89C4×C107—42no—
90C4×C107—42no—
91C4×C107#12012yes—
92C4×C107#12112yes—
93Dic107—42no—
94C2×C107—21no—
95C2×C107—21no—
96C2×C107—82no—
97C2×C107#39211yescharacteristic
98C2×C107#45311yescharacteristic
99C2×C107#45411yescharacteristic
100D107—82no—
101#2006—1071nomaximal, Sylow
102C107#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1072no—
104#537—1071no—
105C2×#104—1072no—
106#390D10711yescharacteristic, complemented
107#343—1071no—
108C2×#41—1072no—
109C22×D4C2×D10711yescharacteristic
110AGL(1,ℤ/8)—2144no—
111AGL(1,ℤ/8)—2144no—
112C2×SD16—2142no—
113C2×D8—2142no—
114C2×M16—1072no—
115C22⋊Q8—2142no—
116#121—2142no—
117#121—2142no—
118C2×C4⋊C4C2×D10711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D10711yescharacteristic
121#104—2144no—
122C24—21no—
123#41—2142no—
124#41—2142no—
125#41—2142no—
126#41—2142no—
127C2×Q8—1072no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—1072no—
132C2×D4—2142no—
133C2×D4—2142no—
134C2×D4D42812yescomplemented
135C22×C4—21no—
136C22×C4—2142no—
137C22×C4C22⋊D10712yes—
138C22×C4C22×D10711yescharacteristic
139SD16—2148no—
140D8—2148no—
141M16—1074no—
142M16—2142no—
143C2×C8—2142no—
144C4⋊C4—22no—
145C4⋊C4—2142no—
146C22⋊C4—44no—
147C22⋊C4—2144no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—2142no—
155C23C2×C22⋊D10711yescharacteristic
156Q8—2142no—
157Q8—2142no—
158D4—24no—
159D4—42no—
160D4—8562no—
161D4—4282no—
162D4—4282no—
163D4—2142no—
164D4—2142no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—4282no—
171C2×C4—4282no—
172C2×C4—2142no—
173C2×C4D4×D10712yes—
174C2×C4D4×D10712yes—
175C2×C4C2×D42811yescharacteristic
176C2×C4#1313381811yescharacteristic
177C8—2144no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—4282no—
186C22—4282no—
187C22—4282no—
188C22#1681207711yescharacteristic
189C22#1681208311yescharacteristic
190C22#1681210511yescharacteristic
191C4—42no—
192C4—42no—
193C4—4282no—
194C4#1681207912yes—
195C4#1681208212yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—8562no—
200C2#2029253611yescharacteristic
201C2#2029279411yescharacteristic
202C2#2029279911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#390×C107C2
socleC22×C107#120
hypercenter#390D107
hypocenterC107#2006
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293208C2semidirect
3. C2×#16812037C2semidirect
4. #20292396C2semidirect
5. #390×C107C2semidirect
32. #13133802D4semidirect
40. C2×D4×C107D4semidirect
102. C107#2006semidirect
106. #390D107semidirect
134. C2×D4D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202932082
C2×#168120372
#202923962
#390×C1072
#2006107