Finite Group Databasebeta

← C2×#21422261 #28508997

Subgroups — C2×#21422261 #28508997

Order 15104.

474 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16837195C212yesmaximal
2#21422261C214yesmaximal, direct product factor
3C2×C59×#492C211yesmaximal, characteristic
4C2×#14273642—22no—
5#16837195—22no—
6#16837195C2214yes—
7C22×SD16×C59—22no—
8C2×C4⋊Q8×C59C411yescharacteristic, complemented
9C2×#127×C59C2211yescharacteristic
10C59×#492—22no—
11C59×#492C412yescomplemented
12C59×#492C2212yes—
13C2×C4×C8×C59C411yescharacteristic, complemented
14C2×C22⋊Dic59—42no—
15#14273642—24no—
16#14273642—42no—
17C22×Q8×C59—21no—
18C22×D4×C59—41no—
19C22×D4×C59D412yes—
20C2×SD16×C59—24no—
21C2×SD16×C59—24no—
22C2×SD16×C59—44no—
23C2×SD16×C59—44no—
24C22×C8×C59—21no—
25C4⋊Q8×C59—21no—
26C4⋊Q8×C59—41no—
27C4⋊Q8×C59C2×C412yescomplemented
28#127×C59—41no—
29#127×C59C2×C411yescharacteristic
30#127×C59D412yes—
31#127×C59C2311yescharacteristic
32C2×C4⋊C4×C59—41no—
33C2×C42×C59C2×C411yescharacteristic
34C4×C8×C59—21no—
35C4×C8×C59C2×C412yescomplemented
36C24×C59—22no—
37C2×Q8×C59—22no—
38C2×Q8×C59—42no—
39C2×Q8×C59—42no—
40C2×Q8×C59—41no—
41C2×D4×C59—22no—
42C2×D4×C59—42no—
43C2×D4×C59—42no—
44C2×D4×C59—41no—
45C2×D4×C59—42no—
46C2×D4×C59—41no—
47C2×D4×C59—84no—
48C2×D4×C59C22⋊C412yes—
49C2×D4×C59C2×D412yes—
50C22×C4×C59—21no—
51C22×C4×C59—41no—
52C22×C4×C59C22⋊C411yescharacteristic
53C22×Dic59—82no—
54SD16×C59—48no—
55SD16×C59—48no—
56C2×C8×C59—22no—
57C2×C8×C59—22no—
58C2×C8×C59—22no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—42no—
61C42×C59—21no—
62C42×C59C22⋊C411yescharacteristic
63C42×C59C22×C411yescharacteristic
64C22⋊Dic59—44no—
65C22⋊Dic59—82no—
66C23×C59—22no—
67C23×C59—22no—
68C23×C59—42no—
69C23×C59—42no—
70C23×C59—84no—
71C23×C59#9911yescharacteristic
72Q8×C59—44no—
73Q8×C59—82no—
74D4×C59—42no—
75D4×C59—42no—
76D4×C59—82no—
77D4×C59—82no—
78D4×C59—84no—
79D4×C59—82no—
80C2×C4×C59—24no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—42no—
86C2×C4×C59—81no—
87C2×C4×C59—81no—
88C2×C4×C59#9911yescharacteristic
89C2×C4×C59C2×C22⋊C411yescharacteristic
90C2×Dic59—84no—
91C2×Dic59—164no—
92C8×C59—24no—
93C2×#1409—591nomaximal, Sylow
94C22×C59—22no—
95C22×C59—42no—
96C22×C59—42no—
97C22×C59—82no—
98C22×C59—82no—
99C22×C59—84no—
100C22×C59—162no—
101C22×C59#35311yescharacteristic
102C22×C59#35311yescharacteristic
103C22×C59C2×#9911yescharacteristic
104C4×C59—22no—
105C4×C59—22no—
106C4×C59—21no—
107C4×C59—21no—
108C4×C59—82no—
109Dic59—164no—
110C2×#492D5911yescharacteristic
111C2×#353—592no—
112#1409—594no—
113C2×C59—21no—
114C2×C59—21no—
115C2×C59—82no—
116C2×C59—82no—
117C2×C59#140912yescomplemented
118C2×C59C2×#35311yescharacteristic
119C22×SD16—22no—
120C2×C4⋊Q8Dic5911yescharacteristic, complemented
121C2×#127C2×D5911yescharacteristic
122#492—22no—
123#492Dic5912yescomplemented
124#492C2×D5912yes—
125C2×#99—1182no—
126C2×C4×C8Dic5911yescharacteristic, complemented
127#353—594no—
128#353—1182no—
129C59C2×#140911yescharacteristic, complemented, Sylow
130C22×Q8—21no—
131C22×D4—41no—
132C22×D4C22⋊D5912yes—
133C2×SD16—24no—
134C2×SD16—24no—
135C2×SD16—44no—
136C2×SD16—44no—
137C22×C8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—41no—
140C4⋊Q8C2×Dic5912yescomplemented
141#127—41no—
142#127C2×Dic5911yescharacteristic
143#127C22⋊D5912yes—
144#127C22×D5911yescharacteristic
145C2×C4⋊C4—41no—
146C2×C22⋊C4—2362no—
147C2×C42C2×Dic5911yescharacteristic
148#99—1184no—
149#99—2362no—
150C4×C8—21no—
151C4×C8C2×Dic5912yescomplemented
152C24—22no—
153C2×Q8—22no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—41no—
157C2×D4—22no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—84no—
164C2×D4C22⋊Dic5912yes—
165C2×D4C2×C22⋊D5912yes—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—4722no—
169C22×C4C22⋊Dic5911yescharacteristic
170SD16—48no—
171SD16—48no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—4722no—
178C22⋊C4—2364no—
179C42—21no—
180C42C22⋊Dic5911yescharacteristic
181C42C22×Dic5911yescharacteristic
182C23—22no—
183C23—22no—
184C23—42no—
185C23—42no—
186C23—84no—
187C23#1427364211yescharacteristic
188Q8—44no—
189Q8—82no—
190D4—42no—
191D4—42no—
192D4—82no—
193D4—82no—
194D4—84no—
195D4—82no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—9444no—
205C2×C4—4724no—
206C2×C4#1427364211yescharacteristic
207C2×C4C2×C22⋊Dic5911yescharacteristic
208C8—24no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—82no—
213C22—82no—
214C22—84no—
215C22—162no—
216C22#1683719511yescharacteristic
217C22#1683719511yescharacteristic
218C22C2×#1427364211yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—9444no—
225C2—21no—
226C2—21no—
227C2—82no—
228C2—82no—
229C2#2142226112yesdirect product factor
230C2C2×#1683719511yescharacteristic
subgroupG/H
centerC22#16837195
commutatorC42×C59C22×C4
Frattini#127C22×D59
FittingC2×C59×#492C2
socleC22×C59#353
prime radicalC2×#127×C59C22
cosocle#127×C59C23
hypercenterC2×#492D59
hypocenterC59C2×#1409
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
2. #21422261C2direct (N × H)
8. C2×C4⋊Q8×C59C4semidirect
11. C59×#492C4semidirect
13. C2×C4×C8×C59C4semidirect
27. C4⋊Q8×C59C2×C4semidirect
35. C4×C8×C59C2×C4semidirect
117. C2×C59#1409semidirect
120. C2×C4⋊Q8Dic59semidirect
123. #492Dic59semidirect
126. C2×C4×C8Dic59semidirect
129. C59C2×#1409semidirect
140. C4⋊Q8C2×Dic59semidirect
151. C4×C8C2×Dic59semidirect
229. C2#21422261direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168371952
#214222612
C2×C59×#4922
C2×#140959