Finite Group Databasebeta

← #28345941

Subgroups — #28345941

Order 13696.

584 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16812134C212yesmaximal, complemented
2C2×#16812106C211yesmaximal, characteristic, complemented
3#20292827C214yesmaximal, complemented
4#20292822C214yesmaximal, complemented
5#20292651C211yesmaximal, characteristic, complemented
6#20292631C212yesmaximal, complemented
7C4×#41×C107C211yesmaximal, characteristic, complemented
8C22×D428—22no—
9C22×D428C2212yes—
10#16812134—28no—
11C2×#13133805C2214yescomplemented
12C2×#13133803C2214yescomplemented
13C2×#13133802C2212yes—
14#16812106—21no—
15#16812106—21no—
16#16812106C2214yes—
17C4⋊(C107⋊C8)C2214yescomplemented
18#16812029C2218yescomplemented
19C2×#41×C107C2211yescharacteristic
20C4×Q8×C107C2212yescomplemented
21C4×D4×C107—22no—
22C4×D4×C107C2212yescomplemented
23#117×C107—21no—
24#117×C107C2211yescharacteristic
25C2×C42×C107—21no—
26C2×C42×C107C2211yescharacteristic
27C23×D107—22no—
28C2×D428—24no—
29C2×D428—24no—
30C2×D428—22no—
31C2×D428—24no—
32C2×D428—24no—
33C2×D428—48no—
34C2×D428C2314yes—
35#13133805—28no—
36#13133803—28no—
37#13133802—24no—
38C2×C107⋊C8C2314yes—
39#41×C107—22no—
40#41×C107D414yescomplemented
41C2×Q8×C107C2311yescharacteristic
42C2×D4×C107—21no—
43C2×D4×C107C2311yescharacteristic
44C22×C4×C107—21no—
45C22×C4×C107—22no—
46C22×C4×C107D412yes—
47C22×C4×C107C2311yescharacteristic
48C4⋊C4×C107—22no—
49C4⋊C4×C107C2312yes—
50C22⋊C4×C107—21no—
51C22⋊C4×C107—22no—
52C42×C107—21no—
53C42×C107—22no—
54C42×C107D411yescharacteristic
55C42×C107D411yescharacteristic
56C42×C107C2312yes—
57C23×C107—21no—
58C23×C107C2×D411yescharacteristic
59C22×D107—24no—
60C22×D107—48no—
61C22×D107—44no—
62Q8×C107—21no—
63Q8×C107C2×D412yescomplemented
64D4×C107—24no—
65D4×C107—21no—
66D4×C107C2×D412yescomplemented
67C2×C4×C107—21no—
68C2×C4×C107—24no—
69C2×C4×C107—24no—
70C2×C4×C107—22no—
71C2×C4×C107—44no—
72C2×C4×C107C2×D412yes—
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D412yes—
75C2×C4×C107C2×D412yes—
76C2×C4×C107#4112yes—
77C2×C4×C107C2411yescharacteristic
78D428—24no—
79D428—24no—
80D428—48no—
81D428—44no—
82D428—48no—
83C107⋊C8—24no—
84C22×C107—22no—
85C22×C107—22no—
86C22×C107—42no—
87C22×C107#12112yes—
88C22×C107C22×D411yescharacteristic
89C2×D107—44no—
90C2×D107—48no—
91C2×D107—88no—
92C4×C107—22no—
93C4×C107—22no—
94C4×C107—42no—
95C4×C107#12112yes—
96C4×C107AGL(1,ℤ/8)14yescomplemented
97C4×C107C22×D411yescharacteristic
98C4×C107C2×#4111yescharacteristic
99C2×C107—22no—
100C2×C107—42no—
101C2×C107C2×#12111yescharacteristic
102C2×C107C2×AGL(1,ℤ/8)12yes—
103D107—84no—
104#3038—1071nomaximal, Sylow
105C107#303811yescharacteristic, complemented, Sylow
106C2×AGL(1,ℤ/8)—1072no—
107C2×#127—1071no—
108C4×#41D10711yescharacteristic, complemented
109#460—1074no—
110#459—1074no—
111#423—1071no—
112#418—1072no—
113C2×#41C2×D10711yescharacteristic
114C22×D4—1072no—
115C22×D4—2142no—
116AGL(1,ℤ/8)—2148no—
117C2×SD16—1074no—
118C2×D8—1074no—
119C2×M16—1072no—
120#127—1074no—
121#127—2141no—
122#127—2141no—
123C4×Q8C2×D10712yescomplemented
124C4×D4—22no—
125C4×D4C2×D10712yescomplemented
126#117—21no—
127#117C2×D10711yescharacteristic
128C2×C42—21no—
129C2×C42C2×D10711yescharacteristic
130C4⋊C8—1074no—
131#102—1078no—
132C24—2142no—
133#41—22no—
134#41D42814yescomplemented
135C2×Q8C22×D10711yescharacteristic
136C2×D4—21no—
137C2×D4—1074no—
138C2×D4—4288no—
139C2×D4—2144no—
140C2×D4—2144no—
141C2×D4—2142no—
142C2×D4—2144no—
143C2×D4—2144no—
144C2×D4C22×D10711yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4C22⋊D10712yes—
148C22×C4C22×D10711yescharacteristic
149SD16—2148no—
150D8—2148no—
151M16—2144no—
152C2×C8—1074no—
153C4⋊C4—22no—
154C4⋊C4C22×D10712yes—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C42—21no—
158C42—22no—
159C42C22⋊D10711yescharacteristic
160C42C22⋊D10711yescharacteristic
161C42C22×D10712yes—
162C23—21no—
163C23—4288no—
164C23—4284no—
165C23—2144no—
166C23C2×C22⋊D10711yescharacteristic
167Q8—21no—
168Q8C2×D42812yescomplemented
169D4—24no—
170D4—21no—
171D4—4288no—
172D4—4284no—
173D4—4288no—
174D4—2144no—
175D4—2144no—
176D4C2×D42812yescomplemented
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4C2×C22⋊D10712yes—
183C2×C4C2×C22⋊D10711yescharacteristic
184C2×C4C2×C22⋊D10712yes—
185C2×C4C2×D42812yes—
186C2×C4#1313381812yes—
187C2×C4C23×D10711yescharacteristic
188C8—2144no—
189C22—22no—
190C22—22no—
191C22—42no—
192C22—8568no—
193C22—4288no—
194C22—4284no—
195C22#1681208312yes—
196C22C22×C22⋊D10711yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4#1681208312yes—
201C4#1681213414yescomplemented
202C4C22×D42811yescharacteristic
203C4C2×#1313381811yescharacteristic
204C2—22no—
205C2—42no—
206C2—8564no—
207C2C2×#1681208311yescharacteristic
208C2C2×#1681213412yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×#41×C107C2
socleC22×C107C22×D4
hypercenterC4×#41D107
hypocenterC107#3038
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C2×#16812106C2semidirect
3. #20292827C2semidirect
4. #20292822C2semidirect
5. #20292651C2semidirect
6. #20292631C2semidirect
7. C4×#41×C107C2semidirect
11. C2×#13133805C22semidirect
12. C2×#13133803C22semidirect
17. C4⋊(C107⋊C8)C22semidirect
18. #16812029C22semidirect
20. C4×Q8×C107C22semidirect
22. C4×D4×C107C22semidirect
40. #41×C107D4semidirect
63. Q8×C107C2×D4semidirect
66. D4×C107C2×D4semidirect
96. C4×C107AGL(1,ℤ/8)semidirect
105. C107#3038semidirect
108. C4×#41D107semidirect
123. C4×Q8C2×D107semidirect
125. C4×D4C2×D107semidirect
134. #41D428semidirect
168. Q8C2×D428semidirect
176. D4C2×D428semidirect
201. C4#16812134semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×#168121062
#202928272
#202928222
#202926512
#202926312
C4×#41×C1072
#3038107