Finite Group Databasebeta

← #28338384

Subgroups — #28338384

Order 13696.

230 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16812052C211yesmaximal, characteristic, complemented
2C2×#16812050C211yesmaximal, characteristic, complemented
3#20292863C211yesmaximal, characteristic, complemented
4#20292431C211yesmaximal, characteristic, complemented
5#20292428C211yesmaximal, characteristic, complemented
6C22⋊(C107⋊C16)C211yesmaximal, characteristic, complemented
7C2×#135×C107C211yesmaximal, characteristic, complemented
8C2×D856C2211yescharacteristic
9#16812083—21no—
10#16812052—22no—
11#16812050—22no—
12C2×C107⋊C16C2211yescharacteristic
13C2×C107⋊C16C2211yescharacteristic
14#16812043C2211yescharacteristic
15#16812031—21no—
16C2×#41×C107—21no—
17#135×C107—22no—
18#135×C107—22no—
19C2×Q16×C107C2211yescharacteristic
20C2×SD16×C107—21no—
21C2×D8×C107C2211yescharacteristic
22C22×C8×C107C2211yescharacteristic
23C2×D428—21no—
24C2×C22⋊D107—41no—
25D856—21no—
26D856—21no—
27C4⋊Dic107—21no—
28(C2×D107)⋊C4—41no—
29C107⋊C16—21no—
30C107⋊C16—21no—
31#41×C107—24no—
32#41×C107—42no—
33C2×Q8×C107—21no—
34C2×D4×C107—21no—
35C2×D4×C107—41no—
36C22×C4×C107—41no—
37C22×C4×C107D411yescharacteristic
38Q16×C107—21no—
39Q16×C107D412yescomplemented
40SD16×C107—24no—
41D8×C107—21no—
42D8×C107D412yescomplemented
43C2×C8×C107—21no—
44C2×C8×C107—21no—
45C2×C8×C107D411yescharacteristic
46C2×C8×C107C2311yescharacteristic
47C23×C107—41no—
48C23×C107D811yescharacteristic
49C22×D107—41no—
50Q8×C107—22no—
51Q8×C107—41no—
52D4×C107—22no—
53D4×C107—44no—
54D4×C107—41no—
55C2×C4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107—44no—
58C2×C4×C107—41no—
59C2×C4×C107D811yescharacteristic
60C2×C4×C107C2×D411yescharacteristic
61C22⋊D107—82no—
62C2×Dic107—41no—
63D428—41no—
64D428—41no—
65C8×C107—21no—
66C8×C107C2×D411yescharacteristic
67C8×C107C2×D411yescharacteristic
68C22×C107—21no—
69C22×C107—21no—
70C22×C107—21no—
71C22×C107—42no—
72C22×C107—81no—
73C22×C107—81no—
74C22×C107C2×D811yescharacteristic
75C2×D107—81no—
76C2×D107—81no—
77C2×D107—81no—
78C4×C107—21no—
79C4×C107—42no—
80C4×C107#12011yescharacteristic
81C4×C107AGL(1,ℤ/8)11yescharacteristic
82Dic107—81no—
83C2×C107—41no—
84C2×C107—82no—
85C2×C107#44711yescharacteristic
86C2×C107#50811yescharacteristic
87C2×C107#50911yescharacteristic
88D107—161no—
89#2186—1071nomaximal, Sylow
90C107#218611yescharacteristic, complemented, Sylow
91C2×#135D10711yescharacteristic, complemented
92C2×SD32—1071no—
93C2×D16—1071no—
94#466—1071no—
95#358—1071no—
96#357—1071no—
97C22⋊C16—1071no—
98C2×#41—21no—
99#135—22no—
100#135—22no—
101C2×Q16C2×D10711yescharacteristic
102C2×SD16—21no—
103C2×D8—1071no—
104C2×D8C2×D10711yescharacteristic
105C22×C8C2×D10711yescharacteristic
106#121—2141no—
107SD32—2142no—
108D16—2142no—
109C2×C16—1071no—
110C2×C16—1071no—
111#107—1071no—
112#102—2141no—
113#41—24no—
114#41—42no—
115C2×Q8—21no—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4—4281no—
119C2×D4—2141no—
120C22×C4—41no—
121C22×C4C22⋊D10711yescharacteristic
122Q16—21no—
123Q16C22⋊D10712yescomplemented
124SD16—24no—
125D8—21no—
126D8—2141no—
127D8—2141no—
128D8C22⋊D10712yescomplemented
129C2×C8—21no—
130C2×C8—21no—
131C2×C8C22⋊D10711yescharacteristic
132C2×C8C22×D10711yescharacteristic
133C4⋊C4—2141no—
134C22⋊C4—4281no—
135C16—2141no—
136C16—2141no—
137C23—41no—
138C23—4281no—
139C23#1313380311yescharacteristic
140Q8—22no—
141Q8—41no—
142D4—22no—
143D4—44no—
144D4—41no—
145D4—8562no—
146D4—4281no—
147D4—4281no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—44no—
151C2×C4—41no—
152C2×C4—4281no—
153C2×C4#1313380311yescharacteristic
154C2×C4C2×C22⋊D10711yescharacteristic
155C8—21no—
156C8C2×C22⋊D10711yescharacteristic
157C8C2×C22⋊D10711yescharacteristic
158C22—21no—
159C22—21no—
160C22—21no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—8561no—
165C22—8561no—
166C22—8561no—
167C22C2×#1313380311yescharacteristic
168C4—21no—
169C4—42no—
170C4—8561no—
171C4#1681207811yescharacteristic
172C4#1681213511yescharacteristic
173C2—41no—
174C2—82no—
175C2—17121no—
176C2#2029276111yescharacteristic
177C2#2029305711yescharacteristic
178C2#2029306211yescharacteristic
subgroupG/H
centerC22C2×#13133803
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
FittingC2×#135×C107C2
socleC22×C107C2×D8
hypercenterC2×#135D107
hypocenterC107#2186
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16812052C2semidirect
2. C2×#16812050C2semidirect
3. #20292863C2semidirect
4. #20292431C2semidirect
5. #20292428C2semidirect
6. C22⋊(C107⋊C16)C2semidirect
7. C2×#135×C107C2semidirect
39. Q16×C107D4semidirect
42. D8×C107D4semidirect
90. C107#2186semidirect
91. C2×#135D107semidirect
123. Q16C22⋊D107semidirect
128. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120522
C2×#168120502
#202928632
#202924312
#202924282
C22⋊(C107⋊C16)2
C2×#135×C1072
#2186107