Finite Group Databasebeta

← #16828972

Subgroups — #16828972

Order 3712.

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×#11745311C212yesmaximal, complemented
2C2×#11745236C211yesmaximal, characteristic, complemented
3#14271363C214yesmaximal, complemented
4#14271358C214yesmaximal, complemented
5#14271187C211yesmaximal, characteristic, complemented
6#14271167C212yesmaximal, complemented
7C4×#41×C29C211yesmaximal, characteristic, complemented
8C2×#41×C29C2211yescharacteristic
9C22×D116—22no—
10C22×D116C2212yes—
11C4×Q8×C29C2212yescomplemented
12C4×D4×C29—22no—
13C4×D4×C29C2212yescomplemented
14C29×#117—21no—
15C29×#117C2211yescharacteristic
16C2×C42×C29—21no—
17C2×C42×C29C2211yescharacteristic
18#11745311—28no—
19C2×#90784C2214yescomplemented
20C2×#90782C2214yescomplemented
21C2×#90777C2212yes—
22#11745236—21no—
23#11745236—21no—
24#11745236C2214yes—
25#11745157C2218yescomplemented
26C4⋊#6597C2214yescomplemented
27C23×D29—22no—
28#41×C29—22no—
29#41×C29D414yescomplemented
30C2×Q8×C29C2311yescharacteristic
31C2×D4×C29—21no—
32C2×D4×C29C2311yescharacteristic
33C22×C4×C29—21no—
34C22×C4×C29—22no—
35C22×C4×C29D412yes—
36C22×C4×C29C2311yescharacteristic
37C2×D116—24no—
38C2×D116—24no—
39C2×D116—24no—
40C2×D116—24no—
41C2×D116—22no—
42C2×D116—48no—
43C2×D116C2314yes—
44C4⋊C4×C29—22no—
45C4⋊C4×C29C2312yes—
46C22⋊C4×C29—21no—
47C22⋊C4×C29—22no—
48C42×C29—21no—
49C42×C29—22no—
50C42×C29D411yescharacteristic
51C42×C29D411yescharacteristic
52C42×C29C2312yes—
53#90784—28no—
54#90782—28no—
55#90777—24no—
56C2×#6597C2314yes—
57C23×C29—21no—
58C23×C29C2×D411yescharacteristic
59C22×D29—24no—
60C22×D29—48no—
61C22×D29—44no—
62Q8×C29—21no—
63Q8×C29C2×D412yescomplemented
64D4×C29—24no—
65D4×C29—21no—
66D4×C29C2×D412yescomplemented
67C2×C4×C29—21no—
68C2×C4×C29—24no—
69C2×C4×C29—24no—
70C2×C4×C29—22no—
71C2×C4×C29—44no—
72C2×C4×C29C2×D412yes—
73C2×C4×C29C2×D412yes—
74C2×C4×C29C2×D411yescharacteristic
75C2×C4×C29C2×D412yes—
76C2×C4×C29#4112yes—
77C2×C4×C29C2411yescharacteristic
78D116—24no—
79D116—24no—
80D116—48no—
81D116—44no—
82D116—48no—
83#6597—24no—
84#3038—291nomaximal, Sylow
85C22×C29—22no—
86C22×C29—22no—
87C22×C29—42no—
88C22×C29#12112yes—
89C22×C29C22×D411yescharacteristic
90C2×D29—44no—
91C2×D29—48no—
92C2×D29—88no—
93C4×C29—22no—
94C4×C29—22no—
95C4×C29—42no—
96C4×C29#12112yes—
97C4×C29AGL(1,ℤ/8)14yescomplemented
98C4×C29C22×D411yescharacteristic
99C4×C29C2×#4111yescharacteristic
100C2×AGL(1,ℤ/8)—292no—
101C2×#127—291no—
102C4×#41D2911yescharacteristic, complemented
103#460—294no—
104#459—294no—
105#423—291no—
106#418—292no—
107C2×C29—22no—
108C2×C29—42no—
109C2×C29C2×#12111yescharacteristic
110C2×C29C2×AGL(1,ℤ/8)12yes—
111D29—84no—
112C2×#41C2×D2911yescharacteristic
113C22×D4—292no—
114C22×D4—582no—
115AGL(1,ℤ/8)—588no—
116C2×SD16—294no—
117C2×D8—294no—
118C2×M16—292no—
119#127—294no—
120#127—581no—
121#127—581no—
122C4×Q8C2×D2912yescomplemented
123C4×D4—22no—
124C4×D4C2×D2912yescomplemented
125#117—21no—
126#117C2×D2911yescharacteristic
127C2×C42—21no—
128C2×C42C2×D2911yescharacteristic
129C4⋊C8—294no—
130#102—298no—
131C29#303811yescharacteristic, complemented, Sylow
132C24—582no—
133#41—22no—
134#41D11614yescomplemented
135C2×Q8C22×D2911yescharacteristic
136C2×D4—21no—
137C2×D4—294no—
138C2×D4—584no—
139C2×D4—584no—
140C2×D4—584no—
141C2×D4—584no—
142C2×D4—582no—
143C2×D4—1168no—
144C2×D4C22×D2911yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4C22⋊D2912yes—
148C22×C4C22×D2911yescharacteristic
149SD16—588no—
150D8—588no—
151M16—584no—
152C2×C8—294no—
153C4⋊C4—22no—
154C4⋊C4C22×D2912yes—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C42—21no—
158C42—22no—
159C42C22⋊D2911yescharacteristic
160C42C22⋊D2911yescharacteristic
161C42C22×D2912yes—
162C23—21no—
163C23—584no—
164C23—1168no—
165C23—1164no—
166C23C2×C22⋊D2911yescharacteristic
167Q8—21no—
168Q8C2×D11612yescomplemented
169D4—24no—
170D4—21no—
171D4—584no—
172D4—584no—
173D4—1168no—
174D4—1164no—
175D4—1168no—
176D4C2×D11612yescomplemented
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4C2×D11612yes—
183C2×C4#9080512yes—
184C2×C4C2×C22⋊D2912yes—
185C2×C4C2×C22⋊D2911yescharacteristic
186C2×C4C2×C22⋊D2912yes—
187C2×C4C23×D2911yescharacteristic
188C8—584no—
189C22—22no—
190C22—22no—
191C22—42no—
192C22—1164no—
193C22—1168no—
194C22—2328no—
195C22#1174529212yes—
196C22C22×C22⋊D2911yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4#1174529212yes—
201C4#1174531114yescomplemented
202C4C22×D11611yescharacteristic
203C4C2×#9080511yescharacteristic
204C2—22no—
205C2—42no—
206C2—2324no—
207C2C2×#1174529211yescharacteristic
208C2C2×#1174531112yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC4×#41×C29C2
socleC22×C29C22×D4
hypercenterC4×#41D29
hypocenterC29#3038
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. C2×#11745236C2semidirect
3. #14271363C2semidirect
4. #14271358C2semidirect
5. #14271187C2semidirect
6. #14271167C2semidirect
7. C4×#41×C29C2semidirect
11. C4×Q8×C29C22semidirect
13. C4×D4×C29C22semidirect
19. C2×#90784C22semidirect
20. C2×#90782C22semidirect
25. #11745157C22semidirect
26. C4⋊#6597C22semidirect
29. #41×C29D4semidirect
63. Q8×C29C2×D4semidirect
66. D4×C29C2×D4semidirect
97. C4×C29AGL(1,ℤ/8)semidirect
102. C4×#41D29semidirect
122. C4×Q8C2×D29semidirect
124. C4×D4C2×D29semidirect
131. C29#3038semidirect
134. #41D116semidirect
168. Q8C2×D116semidirect
176. D4C2×D116semidirect
201. C4#11745311semidirect

Maximal subgroups

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

subgroupindex
C2×#117453112
C2×#117452362
#142713632
#142713582
#142711872
#142711672
C4×#41×C292
#303829