Finite Group Databasebeta

← #28338388

Subgroups — #28338388

Order 13696.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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×D8×D107C211yesmaximal, characteristic
2C2×#16812052C211yesmaximal, characteristic, complemented
3#20292865C211yesmaximal, characteristic, complemented
4#20292429C211yesmaximal, characteristic, complemented
5#20292428C211yesmaximal, characteristic, complemented
6(C2×D107)⋊C16C211yesmaximal, characteristic, complemented
7C2×SD32×C107C211yesmaximal, characteristic, complemented
8C2×D4×D107—21no—
9C2×D856C2211yescharacteristic
10C2×#13133803—21no—
11D8×D107—22no—
12D8×D107—22no—
13C2×C8×D107C2211yescharacteristic
14#16812085—21no—
15#16812052—22no—
16C2×C107⋊C16C2211yescharacteristic, complemented
17#16812043C2211yescharacteristic, complemented
18#16812033—21no—
19C2×Q16×C107C2211yescharacteristic, complemented
20C2×D8×C107C2211yescharacteristic
21SD32×C107—22no—
22C2×C16×C107C2211yescharacteristic, complemented
23C23×D107—41no—
24C2×D428—21no—
25C2×C22⋊D107—41no—
26D4×D107—24no—
27D4×D107—42no—
28C2×C4×D107D411yescharacteristic
29D856—21no—
30D856D412yes—
31#13133803—24no—
32C2×C107⋊C8D411yescharacteristic
33C8×D107—21no—
34C8×D107—21no—
35C4⋊Dic107—21no—
36Dic107⋊C4—41no—
37(C2×D107)⋊C4—41no—
38C107⋊C16—21no—
39C2×Q8×C107—21no—
40C2×D4×C107—21no—
41Q16×C107—21no—
42Q16×C107—21no—
43D8×C107—21no—
44D8×C107D412yes—
45C2×C8×C107C2311yescharacteristic
46C16×C107—21no—
47C23×C107—41no—
48C22×D107—44no—
49C22×D107—41no—
50C22×D107—82no—
51C22×D107—82no—
52C22×D107D811yescharacteristic
53Q8×C107—41no—
54Q8×C107—41no—
55D4×C107—22no—
56D4×C107—41no—
57C2×C4×C107—41no—
58C2×C4×C107C2×D411yescharacteristic
59C22⋊D107—44no—
60C2×Dic107—41no—
61C2×Dic107D811yescharacteristic
62D428—22no—
63D428—41no—
64C4×D107—21no—
65C4×D107—21no—
66C8×C107C2×D411yescharacteristic
67C8×C107C2×D411yescharacteristic
68C107⋊C8—21no—
69C22×C107—42no—
70C22×C107—81no—
71C22×C107—81no—
72C22×C107C2×D811yescharacteristic
73C2×D107—21no—
74C2×D107—21no—
75C2×D107—42no—
76C2×D107—88no—
77C2×D107—81no—
78C2×D107—81no—
79C2×D107SD3212yescomplemented
80C4×C107—81no—
81C4×C107#12011yescharacteristic
82C4×C107AGL(1,ℤ/8)11yescharacteristic
83Dic107—21no—
84Dic107—81no—
85C2×C107—82no—
86C2×C107#44711yescharacteristic
87C2×C107C2×SD3211yescharacteristic
88C2×C107#50911yescharacteristic
89D107—22no—
90D107—82no—
91#2187—1071nomaximal, Sylow
92C107#218711yescharacteristic, complemented, Sylow
93C22×D8—1071no—
94C2×SD32—1071no—
95C2×SD32D10711yescharacteristic, complemented
96#467—1071no—
97#357—1071no—
98#357—1071no—
99C22⋊C16—1071no—
100C22×D4—2141no—
101C2×Q16C2×D10711yescharacteristic, complemented
102C2×D8—1071no—
103C2×D8—2141no—
104C2×D8—2142no—
105C2×D8—2142no—
106C2×D8C2×D10711yescharacteristic
107C22×C8—1071no—
108C22⋊Q8—2141no—
109SD32—22no—
110SD32—2142no—
111C2×C16—1071no—
112C2×C16C2×D10711yescharacteristic, complemented
113#107—1071no—
114#103—2141no—
115C24—4281no—
116C2×Q8—21no—
117C2×D4—21no—
118C2×D4—4281no—
119C2×D4—4282no—
120C2×D4—2144no—
121C2×D4—2141no—
122C22×C4—1071no—
123Q16—21no—
124Q16—21no—
125D8—21no—
126D8—1072no—
127D8—2144no—
128D8—2141no—
129D8C22⋊D10712yes—
130C2×C8—1071no—
131C2×C8—2141no—
132C2×C8—2141no—
133C2×C8C22×D10711yescharacteristic
134C4⋊C4—4281no—
135C4⋊C4—2141no—
136C22⋊C4—4281no—
137C16—21no—
138C16—2141no—
139C23—41no—
140C23—8562no—
141C23—8562no—
142C23—1071no—
143C23—4284no—
144C23—4281no—
145Q8—41no—
146Q8—41no—
147D4—22no—
148D4—41no—
149D4—4284no—
150D4—4281no—
151D4—2142no—
152C2×C4—41no—
153C2×C4—1071no—
154C2×C4—4281no—
155C2×C4—2141no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C8—2141no—
159C8D4×D10711yescharacteristic
160C8C2×C22⋊D10711yescharacteristic
161C22—42no—
162C22—81no—
163C22—81no—
164C22—8568no—
165C22—8561no—
166C22—8561no—
167C22—1072no—
168C22—4282no—
169C22—2141no—
170C22—2141no—
171C22D8×D10711yescharacteristic
172C4—81no—
173C4—8561no—
174C4—2141no—
175C4#1681207711yescharacteristic
176C4#1681213311yescharacteristic
177C2—82no—
178C2—8562no—
179C2—2142no—
180C2#2029275911yescharacteristic
181C2SD32×D10711yescharacteristic
182C2#2029306411yescharacteristic
subgroupG/H
centerC22D8×D107
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
FittingC2×SD32×C107C2
socleC22×C107C2×D8
prime radicalC2×D8×D107C2
hypercenterC2×SD32D107
hypocenterC107#2187
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. C2×#16812052C2semidirect
3. #20292865C2semidirect
4. #20292429C2semidirect
5. #20292428C2semidirect
6. (C2×D107)⋊C16C2semidirect
7. C2×SD32×C107C2semidirect
16. C2×C107⋊C16C22semidirect
17. #16812043C22semidirect
19. C2×Q16×C107C22semidirect
22. C2×C16×C107C22semidirect
79. C2×D107SD32semidirect
92. C107#2187semidirect
95. C2×SD32D107semidirect
101. C2×Q16C2×D107semidirect
112. C2×C16C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1072
C2×#168120522
#202928652
#202924292
#202924282
(C2×D107)⋊C162
C2×SD32×C1072
#2187107