Finite Group Databasebeta

← #28346752

Subgroups — #28346752

Order 13696.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293232C211yesmaximal, characteristic
2#20292951C212yesmaximal, complemented
3#20292934C212yesmaximal, complemented
4#20292827C212yesmaximal, complemented
5#20292766C212yesmaximal, complemented
6#20292725C212yesmaximal, complemented
7C4×#13133805C212yesmaximal, complemented
8#20292688C211yesmaximal, characteristic, complemented
9#519×C107C211yesmaximal, characteristic, complemented
10C2×#13133805C2212yes—
11C2×#13133805C2212yes—
12#16812106C2211yescharacteristic
13#16812101C2211yescharacteristic
14#16812094—22no—
15#16812083C2212yes—
16#16812082—22no—
17C4×D428C2212yes—
18#16812042C2212yes—
19#16812040C2212yes—
20C4⋊(C107⋊C8)C2212yes—
21#16812033C2212yes—
22#16812033C2212yes—
23#16812029C2212yes—
24#16812029C2212yes—
25C22⋊(C107⋊C8)C2212yes—
26#16812020C2211yescharacteristic
27C4×C107⋊C8C2211yescharacteristic
28C22×Q8×C107C2211yescharacteristic
29C4×Q8×C107—22no—
30C4×Q8×C107C2212yes—
31C4×Q8×C107C2212yes—
32#117×C107—22no—
33#117×C107C2211yescharacteristic
34#117×C107C2211yescharacteristic
35C2×D428—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133805—24no—
40#13133805—24no—
41C2×C107⋊C8C2312yes—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—22no—
45(C2×D107)⋊C4—22no—
46(C2×D107)⋊C4—22no—
47C2×Q8×C107—24no—
48C2×Q8×C107—21no—
49C2×Q8×C107C2312yes—
50C22×C4×C107—21no—
51C22×C4×C107C2311yescharacteristic
52C4⋊C4×C107—22no—
53C4⋊C4×C107—22no—
54C4⋊C4×C107D412yes—
55C4⋊C4×C107C2312yes—
56C22⋊C4×C107—21no—
57C22⋊C4×C107D412yes—
58C42×C107—22no—
59C42×C107—22no—
60C42×C107—21no—
61C42×C107C2311yescharacteristic
62C42×C107C2311yescharacteristic
63C23×C107C2×D411yescharacteristic
64C22×D107—22no—
65Q8×C107—24no—
66Q8×C107—22no—
67Q8×C107#4114yes—
68C2×C4×C107—24no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107C2×D412yes—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2411yescharacteristic
76C22⋊D107—82no—
77C2×Dic107—22no—
78D428—22no—
79D428—22no—
80D428—82no—
81C4×D107—82no—
82C107⋊C8—22no—
83C107⋊C8—22no—
84C22×C107—21no—
85C22×C107—22no—
86C22×C107C22×D411yescharacteristic
87C2×D107—42no—
88C2×D107—42no—
89C2×D107—42no—
90C4×C107—24no—
91C4×C107—21no—
92C4×C107—22no—
93C4×C107—22no—
94C4×C107—42no—
95C4×C107C2×#4112yes—
96Dic107—42no—
97C2×C107—41no—
98C2×C107#52511yescharacteristic
99C2×C107#57712yes—
100D107—82no—
101#3126—1071nomaximal, Sylow
102C107#312611yescharacteristic, complemented, Sylow
103#540—1071no—
104#519D10711yescharacteristic, complemented
105#482—1072no—
106#479—1072no—
107#460—1072no—
108#448—1072no—
109#440—1072no—
110C4×SD16—1072no—
111#433—1071no—
112C22×Q8C2×D10711yescharacteristic
113C2×SD16—1072no—
114C2×SD16—1072no—
115#127—1071no—
116#125—1071no—
117#123—2142no—
118#121—1072no—
119#121—2142no—
120C4×Q8—22no—
121C4×Q8C2×D10712yes—
122C4×Q8C2×D10712yes—
123C4×D4—1072no—
124#117—22no—
125#117C2×D10711yescharacteristic
126#117C2×D10711yescharacteristic
127#107—1072no—
128#106—1072no—
129C4⋊C8—1072no—
130#103—1072no—
131#103—1072no—
132#102—1072no—
133#102—1072no—
134C22⋊C8—1072no—
135#97—1071no—
136C4×C8—1071no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D10712yes—
140C2×D4—1072no—
141C2×D4—2142no—
142C2×D4—2142no—
143C22×C4—21no—
144C22×C4—2142no—
145C22×C4C22×D10711yescharacteristic
146SD16—2144no—
147SD16—2144no—
148C2×C8—1072no—
149C2×C8—1072no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1072no—
153C4⋊C4—2142no—
154C4⋊C4C22⋊D10712yes—
155C4⋊C4C22×D10712yes—
156C22⋊C4—21no—
157C22⋊C4—2142no—
158C22⋊C4—2142no—
159C22⋊C4C22⋊D10712yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D10711yescharacteristic
164C42C22×D10711yescharacteristic
165C23—2142no—
166C23C2×C22⋊D10711yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1313381814yes—
170D4—8562no—
171D4—8562no—
172D4—2142no—
173D4—2142no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—8562no—
179C2×C4—2142no—
180C2×C4C2×C22⋊D10712yes—
181C2×C4C2×C22⋊D10712yes—
182C2×C4C2×C22⋊D10711yescharacteristic
183C2×C4C23×D10711yescharacteristic
184C8—2142no—
185C8—2142no—
186C22—21no—
187C22—22no—
188C22—4282no—
189C22—4282no—
190C22—4282no—
191C22C22×C22⋊D10711yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—4282no—
198C4C2×#1313381812yes—
199C2—41no—
200C2—8562no—
201C2#2029313811yescharacteristic
202C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#519×C107C2
socleC22×C107C22×D4
prime radical#20293232C2
hypercenter#519D107
hypocenterC107#3126
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #20292951C2semidirect
3. #20292934C2semidirect
4. #20292827C2semidirect
5. #20292766C2semidirect
6. #20292725C2semidirect
7. C4×#13133805C2semidirect
8. #20292688C2semidirect
9. #519×C107C2semidirect
102. C107#3126semidirect
104. #519D107semidirect

Maximal subgroups

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

subgroupindex
#202932322
#202929512
#202929342
#202928272
#202927662
#202927252
C4×#131338052
#202926882
#519×C1072
#3126107