Finite Group Databasebeta

← C2×#20265360 #27180474

Subgroups — C2×#20265360 #27180474

Order 12928.

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

normal subgroups only

This group has 708 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×#15692285C212yesmaximal, complemented
2C2×#15692253C211yesmaximal, characteristic, complemented
3#20265360C218yesmaximal, direct product factor
4C2×#15692182C212yesmaximal, complemented
5C2×#15692174C211yesmaximal, characteristic, complemented
6C2×2−1+4×C101C211yesmaximal, characteristic
7C22×Dic202C2211yescharacteristic
8#15692285—24no—
9#15692285C2218yescomplemented
10C2×#13108064—22no—
11C2×#13108062—22no—
12C2×#13108058C2212yes—
13#15692253—22no—
14#15692253C2214yescomplemented
15C2×#13108053—22no—
16C2×C4×Dic101C2211yescharacteristic
17#15692182C2218yescomplemented
18#15692174C2214yescomplemented
192−1+4×C101—24no—
202−1+4×C101—22no—
212−1+4×C101C2214yes—
22C2×#41×C101—22no—
23C2×#41×C101—22no—
24C2×#41×C101C2212yes—
25C22×Q8×C101—22no—
26C22×Q8×C101C2211yescharacteristic
27C2×Dic202—22no—
28C2×Dic202—22no—
29C2×Dic202—21no—
30C2×Dic202C2312yes—
31C22×Dic101—21no—
32C22×Dic101—21no—
33#13108064—28no—
34#13108062—28no—
35#13108058—22no—
36#13108058C2314yes—
37C2×#11717579—22no—
38#13108053—28no—
39C4×Dic101—21no—
40C4×Dic101C2312yes—
41#41×C101—28no—
42#41×C101—28no—
43#41×C101—28no—
44#41×C101—28no—
45#41×C101—22no—
46#41×C101—22no—
47#41×C101D414yes—
48#41×C101C2314yes—
49C2×Q8×C101—22no—
50C2×Q8×C101—28no—
51C2×Q8×C101—28no—
52C2×Q8×C101—22no—
53C2×Q8×C101—22no—
54C2×Q8×C101D412yes—
55C2×Q8×C101D412yes—
56C2×Q8×C101C2312yes—
57C2×D4×C101—22no—
58C2×D4×C101—22no—
59C2×D4×C101D412yes—
60C22×C4×C101—24no—
61C22×C4×C101—22no—
62C22×C4×C101D412yes—
63C22×C4×C101C2311yescharacteristic
64C23×C101—22no—
65C23×C101C2×D411yescharacteristic
66Q8×C101—28no—
67Q8×C101—22no—
68Q8×C101—24no—
69Q8×C101—24no—
70Q8×C101C2×D414yes—
71D4×C101—28no—
72D4×C101—24no—
73D4×C101—24no—
74D4×C101—22no—
75D4×C101C2×D414yes—
76C2×C4×C101—216no—
77C2×C4×C101—22no—
78C2×C4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—24no—
81C2×C4×C101—24no—
82C2×C4×C101C2×D414yes—
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×D412yes—
85C2×C4×C101C2×D412yes—
86C2×C4×C101C2411yescharacteristic
87C2×Dic101—24no—
88C2×Dic101—22no—
89C2×Dic101—22no—
90C2×Dic101—41no—
91C2×Dic101—41no—
92Dic202—24no—
93Dic202—42no—
94#11717579—24no—
95C22×C101—22no—
96C22×C101—24no—
97C22×C101—44no—
98C22×C101#12011yescharacteristic
99C22×C101#12011yescharacteristic
100C22×C101C22×D411yescharacteristic
101C4×C101—24no—
102C4×C101—24no—
103C4×C101#12012yes—
104C4×C101C22×D412yes—
105Dic101—24no—
106Dic101—42no—
107C2×C101—21no—
108C2×C101—21no—
109C2×C101—44no—
110C2×C101#45612yescomplemented
111C2×C101C2×#12011yescharacteristic
112C2×#456—1011nomaximal, Sylow
113C101C2×#45611yescharacteristic, complemented, Sylow
114C2×2−1+4D10111yescharacteristic
115C2×#137—1012no—
116C2×C4⋊Q8—1011no—
117#456—1018no—
118C2×#104—1012no—
119C2×#101—1011no—
1202−1+4—24no—
1212−1+4—22no—
1222−1+4C2×D10114yes—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41C2×D10112yes—
126C22×Q8—22no—
127C22×Q8—1011no—
128C22×Q8C2×D10111yescharacteristic
129#137—1018no—
130#137—2024no—
131C2×Q16—2022no—
132C2×SD16—2022no—
133C2×M16—1012no—
134C4⋊Q8—1014no—
135C4⋊Q8—2022no—
136C2×C4⋊C4—2022no—
137C2×C42—1011no—
138#104—1018no—
139#101—1014no—
140#41—28no—
141#41—28no—
142#41—28no—
143#41—28no—
144#41—22no—
145#41—22no—
146#41C22⋊D10114yes—
147#41C22×D10114yes—
148C2×Q8—22no—
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1012no—
154C2×Q8—2022no—
155C2×Q8—2022no—
156C2×Q8—2021no—
157C2×Q8C22⋊D10112yes—
158C2×Q8C22⋊D10112yes—
159C2×Q8C22×D10112yes—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4C22⋊D10112yes—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—2021no—
166C22×C4—2021no—
167C22×C4C22⋊D10112yes—
168C22×C4C22×D10111yescharacteristic
169Q16—2028no—
170SD16—2028no—
171M16—1014no—
172M16—2022no—
173C2×C8—2022no—
174C4⋊C4—2028no—
175C42—1012no—
176C42—2021no—
177C23—22no—
178C23C2×C22⋊D10111yescharacteristic
179Q8—28no—
180Q8—24no—
181Q8—22no—
182Q8—24no—
183Q8—4042no—
184Q8—2024no—
185Q8C2×C22⋊D10114yes—
186D4—28no—
187D4—24no—
188D4—24no—
189D4—22no—
190D4C2×C22⋊D10114yes—
191C2×C4—216no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—4041no—
198C2×C4—4041no—
199C2×C4—2022no—
200C2×C4—2024no—
201C2×C4—2022no—
202C2×C4C2×C22⋊D10114yes—
203C2×C4C2×C22⋊D10112yes—
204C2×C4C2×C22⋊D10112yes—
205C2×C4C2×C22⋊D10111yescharacteristic
206C2×C4C23×D10111yescharacteristic
207C8—2024no—
208C22—22no—
209C22—24no—
210C22—44no—
211C22#1569222411yescharacteristic
212C22#1569222411yescharacteristic
213C22C22×C22⋊D10111yescharacteristic
214C4—24no—
215C4—24no—
216C4—4042no—
217C4—2024no—
218C4#1569222412yes—
219C4C22×C22⋊D10112yes—
220C2—21no—
221C2—21no—
222C2—44no—
223C2#2026536012yesdirect product factor
224C2C2×#1569222411yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting, prime radicalC2×2−1+4×C101C2
socleC22×C101#120
hypercenterC2×2−1+4D101
hypocenterC101C2×#456
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#15692285C2semidirect
2. C2×#15692253C2semidirect
3. #20265360C2direct (N × H)
4. C2×#15692182C2semidirect
5. C2×#15692174C2semidirect
9. #15692285C22semidirect
14. #15692253C22semidirect
17. #15692182C22semidirect
18. #15692174C22semidirect
110. C2×C101#456semidirect
113. C101C2×#456semidirect
223. C2#20265360direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156922852
C2×#156922532
#202653602
C2×#156921822
C2×#156921742
C2×2−1+4×C1012
C2×#456101