Finite Group Databasebeta

← C2×#14271336 #16828828

Subgroups — C2×#14271336 #16828828

Order 3712.

728 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11745309C211yesmaximal, characteristic, complemented
3#14271336C218yesmaximal, direct product factor
4C2×#11745185C212yesmaximal, complemented
5C2×#11745184C211yesmaximal, characteristic, complemented
6C2×C29×2−1+4C211yesmaximal, characteristic, complemented
7C29×2−1+4—24no—
8C29×2−1+4—22no—
9C29×2−1+4C2214yescomplemented
10C2×#41×C29—22no—
11C2×#41×C29—22no—
12C2×#41×C29C2212yes—
13C22×Q8×C29—22no—
14C22×Q8×C29C2211yescharacteristic
15C22×D116C2211yescharacteristic
16#11745311—24no—
17#11745311C2218yescomplemented
18#11745309—22no—
19#11745309C2214yescomplemented
20C2×#90784—22no—
21C2×#90782—22no—
22C2×#90781—22no—
23C2×C4×Dic29C2211yescharacteristic
24C2×#90777C2212yes—
25#11745185C2218yescomplemented
26#11745184C2214yescomplemented
27C23×D29—21no—
28#41×C29—28no—
29#41×C29—28no—
30#41×C29—28no—
31#41×C29—28no—
32#41×C29—22no—
33#41×C29—22no—
34#41×C29D414yes—
35#41×C29C2314yes—
36C2×Q8×C29—22no—
37C2×Q8×C29—28no—
38C2×Q8×C29—28no—
39C2×Q8×C29—22no—
40C2×Q8×C29—22no—
41C2×Q8×C29D412yes—
42C2×Q8×C29D412yes—
43C2×Q8×C29C2312yes—
44C2×D4×C29—22no—
45C2×D4×C29—22no—
46C2×D4×C29D412yes—
47C22×C4×C29—24no—
48C22×C4×C29—22no—
49C22×C4×C29D412yes—
50C22×C4×C29C2311yescharacteristic
51C22×Dic29—21no—
52C2×D116—22no—
53C2×D116—22no—
54C2×D116—21no—
55C2×D116C2312yes—
56#90784—28no—
57#90782—28no—
58#90781—28no—
59C4×Dic29—21no—
60C4×Dic29C2312yes—
61#90777—22no—
62#90777C2314yes—
63C2×#6597—22no—
64C23×C29—22no—
65C23×C29C2×D411yescharacteristic
66C22×D29—22no—
67C22×D29—44no—
68C22×D29—41no—
69C22×D29—41no—
70Q8×C29—28no—
71Q8×C29—22no—
72Q8×C29—24no—
73Q8×C29—24no—
74Q8×C29C2×D414yes—
75D4×C29—28no—
76D4×C29—24no—
77D4×C29—24no—
78D4×C29—22no—
79D4×C29C2×D414yes—
80C2×C4×C29—216no—
81C2×C4×C29—22no—
82C2×C4×C29—22no—
83C2×C4×C29—22no—
84C2×C4×C29—24no—
85C2×C4×C29—24no—
86C2×C4×C29C2×D414yes—
87C2×C4×C29C2×D411yescharacteristic
88C2×C4×C29C2×D412yes—
89C2×C4×C29C2×D412yes—
90C2×C4×C29C2411yescharacteristic
91C2×Dic29—22no—
92C2×Dic29—42no—
93D116—24no—
94D116—42no—
95#6597—24no—
96C2×#454—291nomaximal, Sylow
97C22×C29—22no—
98C22×C29—24no—
99C22×C29—44no—
100C22×C29#12011yescharacteristic
101C22×C29#12011yescharacteristic
102C22×C29C22×D411yescharacteristic
103C2×D29—44no—
104C2×D29—42no—
105C2×D29—82no—
106C2×D29—82no—
107C4×C29—24no—
108C4×C29—24no—
109C4×C29#12012yes—
110C4×C29C22×D412yes—
111Dic29—42no—
112C2×2−1+4D2911yescharacteristic, complemented
113C2×AGL(1,ℤ/8)—292no—
114C2×#124—291no—
115#454—298no—
116C2×#104—292no—
117C2×#101—291no—
118C2×C29—21no—
119C2×C29—21no—
120C2×C29—44no—
121C2×C29#45412yescomplemented
122C2×C29C2×#12011yescharacteristic
123D29—82no—
1242−1+4—24no—
1252−1+4—22no—
1262−1+4C2×D2914yescomplemented
127C2×#41—22no—
128C2×#41—22no—
129C2×#41C2×D2912yes—
130C22×Q8—22no—
131C22×Q8C2×D2911yescharacteristic
132C22×D4—291no—
133AGL(1,ℤ/8)—298no—
134AGL(1,ℤ/8)—584no—
135C2×SD16—582no—
136C2×D8—582no—
137C2×M16—292no—
138#124—294no—
139#124—582no—
140C2×C22⋊C4—582no—
141C2×C42—291no—
142#104—298no—
143#101—294no—
144C29C2×#45411yescharacteristic, complemented, Sylow
145C24—581no—
146#41—28no—
147#41—28no—
148#41—28no—
149#41—28no—
150#41—22no—
151#41—22no—
152#41C22⋊D2914yes—
153#41C22×D2914yes—
154C2×Q8—22no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8C22⋊D2912yes—
160C2×Q8C22⋊D2912yes—
161C2×Q8C22×D2912yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—292no—
165C2×D4—582no—
166C2×D4—582no—
167C2×D4—581no—
168C2×D4C22⋊D2912yes—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—581no—
172C22×C4C22⋊D2912yes—
173C22×C4C22×D2911yescharacteristic
174SD16—588no—
175D8—588no—
176M16—294no—
177M16—582no—
178C2×C8—582no—
179C22⋊C4—588no—
180C42—292no—
181C42—581no—
182C23—22no—
183C23—582no—
184C23—1164no—
185C23—1161no—
186C23—1161no—
187C23C2×C22⋊D2911yescharacteristic
188Q8—28no—
189Q8—22no—
190Q8—24no—
191Q8—24no—
192Q8C2×C22⋊D2914yes—
193D4—28no—
194D4—24no—
195D4—24no—
196D4—22no—
197D4—584no—
198D4—1162no—
199D4C2×C22⋊D2914yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—582no—
207C2×C4—1162no—
208C2×C4C2×C22⋊D2914yes—
209C2×C4C2×C22⋊D2911yescharacteristic
210C2×C4C2×C22⋊D2912yes—
211C2×C4C2×C22⋊D2912yes—
212C2×C4C23×D2911yescharacteristic
213C8—584no—
214C22—22no—
215C22—24no—
216C22—44no—
217C22—1164no—
218C22—1162no—
219C22—2322no—
220C22—2322no—
221C22#1174531511yescharacteristic
222C22#1174531511yescharacteristic
223C22C22×C22⋊D2911yescharacteristic
224C4—24no—
225C4—24no—
226C4—1162no—
227C4#1174531512yes—
228C4C22×C22⋊D2912yes—
229C2—21no—
230C2—21no—
231C2—44no—
232C2—2322no—
233C2#1427133612yesdirect product factor
234C2C2×#1174531511yescharacteristic
subgroupG/H
centerC22#11745315
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×2−1+4C2
socleC22×C29#120
hypercenterC2×2−1+4D29
hypocenterC29C2×#454
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. C2×#11745309C2semidirect
3. #14271336C2direct (N × H)
4. C2×#11745185C2semidirect
5. C2×#11745184C2semidirect
6. C2×C29×2−1+4C2semidirect
9. C29×2−1+4C22semidirect
17. #11745311C22semidirect
19. #11745309C22semidirect
25. #11745185C22semidirect
26. #11745184C22semidirect
112. C2×2−1+4D29semidirect
121. C2×C29#454semidirect
126. 2−1+4C2×D29semidirect
144. C29C2×#454semidirect
233. C2#14271336direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117453112
C2×#117453092
#142713362
C2×#117451852
C2×#117451842
C2×C29×2−1+42
C2×#45429