Finite Group Databasebeta

← #28338434

Subgroups — #28338434

Order 13696.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293432C211yesmaximal, characteristic
2#20293070C211yesmaximal, characteristic, complemented
3C2×#16812054C211yesmaximal, characteristic, complemented
4#20292902C211yesmaximal, characteristic, complemented
5#20292441C211yesmaximal, characteristic, complemented
6#20292435C211yesmaximal, characteristic, complemented
7#350×C107C211yesmaximal, characteristic, complemented
8#16812155—21no—
9#16812139—21no—
10#16812137—22no—
11#16812137—21no—
12#16812131—21no—
13#16812131C2211yescharacteristic
14C2×Dic428C2211yescharacteristic, complemented
15C2×D856—21no—
16C2×D856C2211yescharacteristic
17Dic856—21no—
18#16812054—22no—
19#16812054—21no—
20#16812045C2211yescharacteristic, complemented
21#16812027—21no—
22#131×C107C2211yescharacteristic
23M32×C107C2211yescharacteristic, complemented
24C2×C16×C107C2211yescharacteristic, complemented
25#13133819—41no—
26#13133818—22no—
27#13133818—21no—
28C2×Dic214—21no—
29C2×D428—22no—
30C2×D428—21no—
31D4×D107—42no—
32D4×D107—41no—
33Dic428—21no—
34Dic428—21no—
35Dic428D411yescharacteristic
36#13133808—41no—
37#13133807—22no—
38#13133807—21no—
39#13133806—41no—
40D856—22no—
41D856—21no—
42D856D411yescharacteristic
43D856D412yes—
44#13133802—21no—
45#41×C107D411yescharacteristic
46M16×C107—21no—
47M16×C107D411yescharacteristic
48C2×C8×C107—21no—
49C2×C8×C107C2311yescharacteristic
50C16×C107—21no—
51C16×C107—21no—
52C22×D107—42no—
53C22×D107—41no—
54Q8×C107D811yescharacteristic
55D4×C107—21no—
56D4×C107D811yescharacteristic
57C2×C4×C107—21no—
58C2×C4×C107C2×D411yescharacteristic
59C22⋊D107—42no—
60C22⋊D107—41no—
61C2×Dic107—41no—
62D428—21no—
63D428—22no—
64D428—42no—
65D428—41no—
66C4×D107—42no—
67C4×D107—41no—
68Dic214—21no—
69Dic214—41no—
70Dic214—41no—
71C8×C107—21no—
72C8×C107C2×D411yescharacteristic
73C8×C107C2×D411yescharacteristic
74C107⋊C8—41no—
75C22×C107—21no—
76C22×C107AGL(1,ℤ/8)11yescharacteristic
77C2×D107—42no—
78C2×D107—41no—
79C2×D107—161no—
80C2×D107—162no—
81C4×C107—21no—
82C4×C107#12011yescharacteristic
83C4×C107C2×D811yescharacteristic
84Dic107—41no—
85Dic107—81no—
86C2×C107—21no—
87C2×C107—41no—
88C2×C107#44711yescharacteristic
89D107—82no—
90D107—81no—
91#2194—1071nomaximal, Sylow
92C107#219411yescharacteristic, complemented, Sylow
93#576—1071no—
94#510—1071no—
95C2×SD32—1071no—
96#473—1071no—
97#361—1071no—
98#359—1071no—
99#350D10711yescharacteristic, complemented
100H(5,𝔽2)—2141no—
101#137—2141no—
102AGL(1,ℤ/8)—2142no—
103AGL(1,ℤ/8)—2141no—
104#135—1071no—
105#135—2141no—
106C2×Q16—1071no—
107C2×D8—1071no—
108C2×D8—2141no—
109#131C2×D10711yescharacteristic
110Q32—2141no—
111SD32—2142no—
112SD32—2141no—
113M32C2×D10711yescharacteristic, complemented
114C2×C16C2×D10711yescharacteristic, complemented
115#108—1071no—
116#101—2141no—
117#41—4281no—
118#41—2142no—
119#41—2141no—
120#41D42811yescharacteristic
121C2×Q8—2141no—
122C2×D4—4282no—
123C2×D4—4281no—
124C2×D4—2142no—
125C2×D4—2141no—
126Q16—1071no—
127Q16—4281no—
128Q16—2141no—
129Q16—2141no—
130SD16—4281no—
131SD16—2142no—
132SD16—2141no—
133D8—1072no—
134D8—1071no—
135D8—2142no—
136D8—2141no—
137M16—21no—
138M16—2141no—
139M16D42811yescharacteristic
140C2×C8—21no—
141C2×C8C22×D10711yescharacteristic
142C16—21no—
143C16—21no—
144C23—4282no—
145C23—4281no—
146Q8—4281no—
147Q8—4281no—
148Q8—2141no—
149Q8D85611yescharacteristic
150D4—21no—
151D4—4282no—
152D4—4282no—
153D4—4281no—
154D4—4281no—
155D4—2142no—
156D4—2141no—
157D4D85611yescharacteristic
158C2×C4—21no—
159C2×C4—4282no—
160C2×C4—4281no—
161C2×C4—4281no—
162C2×C4C2×D42811yescharacteristic
163C8—21no—
164C8—4281no—
165C8D4×D10711yescharacteristic
166C8D4×D10711yescharacteristic
167C22—21no—
168C22—4282no—
169C22—4281no—
170C22—17121no—
171C22—17122no—
172C22#1681213711yescharacteristic
173C4—21no—
174C4—8561no—
175C4—4281no—
176C4#1681207911yescharacteristic
177C4C2×D85611yescharacteristic
178C2—21no—
179C2—41no—
180C2—8562no—
181C2—8561no—
182C2#2029276311yescharacteristic
subgroupG/H
centerC2#20292763
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#350×C107C2
socleC2×C107#447
prime radical#20293432C2
hypercenter#350D107
hypocenterC107#2194
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #20293070C2semidirect
3. C2×#16812054C2semidirect
4. #20292902C2semidirect
5. #20292441C2semidirect
6. #20292435C2semidirect
7. #350×C107C2semidirect
14. C2×Dic428C22semidirect
20. #16812045C22semidirect
23. M32×C107C22semidirect
24. C2×C16×C107C22semidirect
92. C107#2194semidirect
99. #350D107semidirect
113. M32C2×D107semidirect
114. C2×C16C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934322
#202930702
C2×#168120542
#202929022
#202924412
#202924352
#350×C1072
#2194107