Finite Group Databasebeta

← #28338429

Subgroups — #28338429

Order 13696.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#20293429C211yesmaximal, characteristic
2C2×#16812052C211yesmaximal, characteristic, complemented
3#20292899C211yesmaximal, characteristic, complemented
4#20292441C211yesmaximal, characteristic, complemented
5#20292434C211yesmaximal, characteristic, complemented
6#20292412C211yesmaximal, characteristic, complemented
7#510×C107C211yesmaximal, characteristic, complemented
8#16812155—21no—
9#16812140—21no—
10#16812136—22no—
11#16812134—21no—
12#16812132—21no—
13C2×D856C2211yescharacteristic
14D8×D107—21no—
15#16812115C2211yescharacteristic
16#16812052—22no—
17C2×C107⋊C16C2211yescharacteristic, complemented
18#16812045C2211yescharacteristic, complemented
19#16812028—21no—
20#135×C107C2211yescharacteristic
21C2×Q16×C107C2211yescharacteristic, complemented
22Q32×C107—21no—
23SD32×C107—21no—
24M32×C107C2211yescharacteristic, complemented
25#13133819—22no—
26#13133818—41no—
27#13133818D411yescharacteristic
28C2×D428—21no—
29C2×D428—41no—
30D4×D107—22no—
31D4×D107—42no—
32#13133808—41no—
33#13133805—22no—
34#13133805—41no—
35D856—21no—
36D856D412yes—
37#13133803—22no—
38#13133802—21no—
39#13133802D411yescharacteristic
40#13133801—21no—
41C8×D107—21no—
42C107⋊C16—21no—
43#41×C107—21no—
44C2×Q8×C107—21no—
45Q16×C107—21no—
46Q16×C107—21no—
47Q16×C107D411yescharacteristic
48SD16×C107—21no—
49D8×C107D411yescharacteristic
50C2×C8×C107C2311yescharacteristic
51C16×C107—21no—
52C22×D107—42no—
53C22×D107—41no—
54Q8×C107—21no—
55Q8×C107—41no—
56Q8×C107—41no—
57D4×C107—21no—
58D4×C107—41no—
59C2×C4×C107—41no—
60C2×C4×C107—41no—
61C2×C4×C107C2×D411yescharacteristic
62C22⋊D107—21no—
63C22⋊D107—42no—
64D428—22no—
65D428—42no—
66D428—41no—
67D428D811yescharacteristic
68C4×D107—21no—
69C4×D107—42no—
70Dic214D811yescharacteristic
71C8×C107C2×D411yescharacteristic
72C8×C107C2×D411yescharacteristic
73C107⋊C8—21no—
74C107⋊C8—41no—
75C22×C107—41no—
76C22×C107AGL(1,ℤ/8)11yescharacteristic
77C2×D107—21no—
78C2×D107—42no—
79C2×D107—162no—
80C2×D107—161no—
81C4×C107—41no—
82C4×C107—81no—
83C4×C107#12011yescharacteristic
84C4×C107C2×D811yescharacteristic
85Dic107—21no—
86C2×C107—21no—
87C2×C107—81no—
88C2×C107#44711yescharacteristic
89D107—41no—
90D107—82no—
91#2194—1071nomaximal, Sylow
92C107#219411yescharacteristic, complemented, Sylow
93#576—1071no—
94#510D10711yescharacteristic, complemented
95C2×SD32—1071no—
96#473—1071no—
97#361—1071no—
98#359—1071no—
99#350—1071no—
100H(5,𝔽2)—2141no—
101#137—2141no—
102AGL(1,ℤ/8)—2141no—
103AGL(1,ℤ/8)—2142no—
104#135—2141no—
105#135C2×D10711yescharacteristic
106C2×Q16C2×D10711yescharacteristic, complemented
107C2×D8—1071no—
108C2×D8—2141no—
109#131—1071no—
110Q32—21no—
111SD32—21no—
112SD32—2142no—
113M32C2×D10711yescharacteristic, complemented
114C2×C16—1071no—
115#108—1071no—
116#101—2141no—
117#41—21no—
118#41—1071no—
119#41—4281no—
120#41—2142no—
121C2×Q8—21no—
122C2×D4—4281no—
123C2×D4—4282no—
124C2×D4—2142no—
125C2×D4—2141no—
126Q16—21no—
127Q16—21no—
128Q16—4281no—
129Q16C22⋊D10711yescharacteristic
130SD16—21no—
131SD16—4281no—
132SD16—2142no—
133D8—1072no—
134D8—2142no—
135D8—2141no—
136D8C22⋊D10711yescharacteristic
137M16—1071no—
138M16—2141no—
139M16—2141no—
140C2×C8—2141no—
141C2×C8C22×D10711yescharacteristic
142C16—21no—
143C16—2141no—
144C23—4282no—
145C23—4281no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—1071no—
150D4—21no—
151D4—41no—
152D4—1071no—
153D4—4282no—
154D4—4282no—
155D4—4281no—
156D4—2141no—
157D4—2142no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—4282no—
161C2×C4—2141no—
162C2×C4D4×D10711yescharacteristic
163C8—4281no—
164C8—2141no—
165C8D4×D10711yescharacteristic
166C8C2×C22⋊D10711yescharacteristic
167C22—41no—
168C22—4282no—
169C22—17121no—
170C22—17122no—
171C22—2141no—
172C22#1681213311yescharacteristic
173C4—41no—
174C4—81no—
175C4—2141no—
176C4#1681207711yescharacteristic
177C4D8×D10711yescharacteristic
178C2—21no—
179C2—81no—
180C2—8562no—
181C2—4281no—
182C2#2029275911yescharacteristic
subgroupG/H
centerC2#20292759
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#510×C107C2
socleC2×C107#447
prime radical#20293429C2
hypercenter#510D107
hypocenterC107#2194
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#16812052C2semidirect
3. #20292899C2semidirect
4. #20292441C2semidirect
5. #20292434C2semidirect
6. #20292412C2semidirect
7. #510×C107C2semidirect
17. C2×C107⋊C16C22semidirect
18. #16812045C22semidirect
21. C2×Q16×C107C22semidirect
24. M32×C107C22semidirect
92. C107#2194semidirect
94. #510D107semidirect
106. C2×Q16C2×D107semidirect
113. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934292
C2×#168120522
#202928992
#202924412
#202924342
#202924122
#510×C1072
#2194107