Finite Group Databasebeta

← C2×#20265359 #27180473

Subgroups — C2×#20265359 #27180473

Order 12928.

708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692303C211yesmaximal, characteristic
2C2×#15692286C212yesmaximal, complemented
3#20265359C218yesmaximal, direct product factor
4C2×#15692183C212yesmaximal, complemented
5C2×#15692174C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C101C211yesmaximal, characteristic, complemented
7#15692303—24no—
8#15692303—22no—
9#15692303C2214yes—
10C2×#13108075—22no—
11C2×#13108074—22no—
12C2×#13108074C2212yes—
13C2×Q8×D101—22no—
14#15692286—24no—
15#15692286C2218yescomplemented
16C2×#13108064—22no—
17C2×#13108061—22no—
18C2×#13108058C2212yes—
19#15692183C2218yescomplemented
20#15692174C2214yescomplemented
21C22×Q8×C101C2211yescharacteristic
22C22×Q8×C101C2211yescharacteristic
23C4⋊Q8×C101—22no—
24C4⋊Q8×C101C2214yescomplemented
25C2×C4⋊C4×C101—22no—
26C2×C42×C101C2211yescharacteristic
27#13108075—28no—
28#13108075—28no—
29#13108074—28no—
30#13108074—28no—
31#13108074—22no—
32#13108074—22no—
33#13108074D414yes—
34#13108074C2314yes—
35C2×Dic202—22no—
36C2×Dic202D412yes—
37Q8×D101—28no—
38Q8×D101—28no—
39C2×D404—22no—
40C2×D404D412yes—
41C2×C22⋊D101—22no—
42C2×C4×D101—24no—
43C2×C4×D101—22no—
44#13108064—28no—
45#13108061—28no—
46#13108058—22no—
47#13108058C2314yes—
48C2×#11717579—22no—
49C2×Q8×C101—22no—
50C2×Q8×C101—22no—
51C2×Q8×C101—21no—
52C2×Q8×C101—22no—
53C2×Q8×C101—22no—
54C2×Q8×C101D412yes—
55C2×Q8×C101C2312yes—
56C2×Q8×C101C2312yes—
57C22×C4×C101—21no—
58C22×C4×C101—21no—
59C22×C4×C101D412yes—
60C22×C4×C101C2311yescharacteristic
61C4⋊C4×C101—28no—
62C42×C101—21no—
63C42×C101C2312yes—
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66Q8×C101—24no—
67Q8×C101—24no—
68Q8×C101—24no—
69Q8×C101—42no—
70C2×C4×C101—24no—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—22no—
74C2×C4×C101—22no—
75C2×C4×C101—41no—
76C2×C4×C101—41no—
77C2×C4×C101C2×D412yes—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×D414yes—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2411yescharacteristic
82C22⋊D101—24no—
83C22⋊D101—24no—
84C2×Dic101—22no—
85D404—28no—
86D404—22no—
87D404C2×D414yes—
88C4×D101—216no—
89C4×D101—24no—
90C4×D101—24no—
91Dic202—28no—
92Dic202—22no—
93Dic202C2×D414yes—
94#11717579—24no—
95C22×C101—22no—
96C22×C101#12011yescharacteristic
97C22×C101#12011yescharacteristic
98C22×C101C22×D411yescharacteristic
99C2×D101—24no—
100C2×D101—44no—
101C4×C101—24no—
102C4×C101—24no—
103C4×C101—42no—
104C4×C101#12012yes—
105C4×C101C22×D412yes—
106Dic101—24no—
107C2×C101—21no—
108C2×C101—21no—
109C2×C101#45612yescomplemented
110C2×C101C2×#12011yescharacteristic
111D101—44no—
112C2×#456—1011nomaximal, Sylow
113C101C2×#45611yescharacteristic, complemented, Sylow
114C2×2−1+4—1011no—
115C2×#137—1012no—
116C2×C4⋊Q8D10111yescharacteristic, complemented
117#456—1018no—
118C2×#104—1012no—
119C2×#101—1011no—
1202−1+4—1014no—
1212−1+4—2024no—
1222−1+4—2022no—
123C2×#41—1012no—
124C2×#41—2022no—
125C2×#41—2022no—
126C22×Q8—2022no—
127C22×Q8C2×D10111yescharacteristic
128C22×Q8C2×D10111yescharacteristic
129#137—1018no—
130#137—2024no—
131C2×Q16—2022no—
132C2×SD16—2022no—
133C2×M16—1012no—
134C4⋊Q8—22no—
135C4⋊Q8C2×D10114yescomplemented
136C2×C4⋊C4—22no—
137C2×C42C2×D10111yescharacteristic
138#104—1018no—
139#101—1014no—
140#41—1014no—
141#41—1014no—
142#41—2028no—
143#41—2028no—
144#41—2028no—
145#41—2022no—
146#41—2022no—
147#41—2028no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1012no—
154C2×Q8—2022no—
155C2×Q8—2028no—
156C2×Q8—2028no—
157C2×Q8C22⋊D10112yes—
158C2×Q8C22×D10112yes—
159C2×Q8C22×D10112yes—
160C2×D4—1012no—
161C2×D4—2022no—
162C2×D4—2022no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—2024no—
166C22×C4—2022no—
167C22×C4C22⋊D10112yes—
168C22×C4C22×D10111yescharacteristic
169Q16—2028no—
170SD16—2028no—
171M16—1014no—
172M16—2022no—
173C2×C8—2022no—
174C4⋊C4—28no—
175C42—21no—
176C42C22×D10112yes—
177C23—2022no—
178C23C2×C22⋊D10111yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—42no—
183Q8—1014no—
184Q8—2028no—
185Q8—2022no—
186D4—1014no—
187D4—2028no—
188D4—2024no—
189D4—2024no—
190D4—2022no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—20216no—
199C2×C4—2022no—
200C2×C4—2024no—
201C2×C4—2024no—
202C2×C4D4×D10112yes—
203C2×C4D4×D10112yes—
204C2×C4C2×C22⋊D10114yes—
205C2×C4C2×C22⋊D10111yescharacteristic
206C2×C4C23×D10111yescharacteristic
207C8—2024no—
208C22—22no—
209C22—4044no—
210C22—2024no—
211C22#1569222311yescharacteristic
212C22#1569222311yescharacteristic
213C22C22×C22⋊D10111yescharacteristic
214C4—24no—
215C4—24no—
216C4—42no—
217C4—2024no—
218C4#1569222312yes—
219C4C2×D4×D10112yes—
220C2—21no—
221C2—21no—
222C2—4044no—
223C2#2026535912yesdirect product factor
224C2C2×#1569222311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×C4⋊Q8×C101C2
socleC22×C101#120
prime radicalC2×#15692303C2
hypercenterC2×C4⋊Q8D101
hypocenterC101C2×#456
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15692286C2semidirect
3. #20265359C2direct (N × H)
4. C2×#15692183C2semidirect
5. C2×#15692174C2semidirect
6. C2×C4⋊Q8×C101C2semidirect
15. #15692286C22semidirect
19. #15692183C22semidirect
20. #15692174C22semidirect
24. C4⋊Q8×C101C22semidirect
109. C2×C101#456semidirect
113. C101C2×#456semidirect
116. C2×C4⋊Q8D101semidirect
135. C4⋊Q8C2×D101semidirect
223. C2#20265359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923032
C2×#156922862
#202653592
C2×#156921832
C2×#156921742
C2×C4⋊Q8×C1012
C2×#456101