Finite Group Databasebeta

← C2×#17986336 #22598247

Subgroups — C2×#17986336 #22598247

Order 9088.

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

normal subgroups only

This group has 460 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×#14556108C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic
3#17986336C218yesmaximal, direct product factor
4C2×#14556061C211yesmaximal, characteristic, complemented
5C2×#14556053C211yesmaximal, characteristic, complemented
6C2×#14556051C211yesmaximal, characteristic, complemented
7C2×(C2×D71)⋊C8C211yesmaximal, characteristic
8C2×#107×C71C211yesmaximal, characteristic, complemented
9C22×Dic142C2211yescharacteristic
10C22×C4×D71—21no—
11C22×C4×D71C2211yescharacteristic
12C22×C71⋊C8C2211yescharacteristic
13#14556108—22no—
14#14556108C2214yes—
15C2×C4⋊Dic71—21no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×Dic71⋊C4—21no—
18C4⋊C4×D71—22no—
19C4⋊C4×D71C2214yes—
20C2×(C2×D71)⋊C4—21no—
21#14556061C2214yescomplemented
22#14556053C2214yescomplemented
23#14556051C2214yescomplemented
24(C2×D71)⋊C8C2214yes—
25C22×C8×C71C2211yescharacteristic
26C2×C4⋊C4×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28#107×C71C2214yescomplemented
29C23×D71D411yescharacteristic
30C2×Dic142—22no—
31C2×Dic142—22no—
32C2×Dic142—21no—
33C2×Dic142C2312yes—
34C22×Dic71—21no—
35C22×Dic71—21no—
36C22×Dic71D411yescharacteristic
37C2×C4×D71—22no—
38C2×C4×D71—22no—
39C2×C4×D71—24no—
40C2×C4×D71—44no—
41C2×C4×D71D412yes—
42C2×C4×D71C2312yes—
43C2×C71⋊C8—22no—
44C2×C71⋊C8C2312yes—
45C4⋊Dic71—21no—
46C4⋊Dic71—24no—
47C4⋊Dic71C2312yes—
48Dic71⋊C4—24no—
49(C2×D71)⋊C4—24no—
50C22×C4×C71—21no—
51C22×C4×C71—21no—
52C22×C4×C71C2311yescharacteristic
53C2×C8×C71—22no—
54C2×C8×C71C2312yes—
55C4⋊C4×C71—21no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71C2312yes—
58C4⋊C4×C71C2312yes—
59C23×C71C2×D411yescharacteristic
60C22×D71—22no—
61C22×D71—22no—
62C22×D71Q1614yes—
63C22×D71C2×D412yes—
64C2×C4×C71—22no—
65C2×C4×C71—22no—
66C2×C4×C71—42no—
67C2×C4×C71—42no—
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71#4112yes—
70C2×C4×C71#4112yes—
71C2×C4×C71C2411yescharacteristic
72C2×Dic71—22no—
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71—42no—
76C2×Dic71—42no—
77C2×Dic71C2×D412yes—
78C4×D71—24no—
79C4×D71—22no—
80C4×D71—22no—
81C4×D71—48no—
82Dic142—22no—
83Dic142—22no—
84Dic142—42no—
85C8×C71—22no—
86C71⋊C8—22no—
87C22×C71#12312yes—
88C22×C71C2×Q1612yes—
89C22×C71AGL(1,ℤ/8)12yes—
90C22×C71C22×D411yescharacteristic
91C2×D71—22no—
92C2×D71—22no—
93C2×D71—28no—
94C2×D71C2×Q1614yes—
95C4×C71—42no—
96C4×C71—42no—
97C4×C71#12312yes—
98C4×C71C2×#4111yescharacteristic
99C4×C71C2×#4111yescharacteristic
100Dic71—22no—
101Dic71—42no—
102Dic71—42no—
103C2×C71#48414yescomplemented
104C2×C71C2×#12311yescharacteristic
105C2×C71C22×Q1611yescharacteristic
106C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
107D71—24no—
108C2×#484—711nomaximal, Sylow
109C71C2×#48411yescharacteristic, complemented, Sylow
110C2×C22⋊Q8—711no—
111C22×C4⋊C4—711no—
112#484—718no—
113C2×#107—711no—
114C2×#107D7111yescharacteristic, complemented
115C2×#103—711no—
116C2×#103—711no—
117C2×C22⋊C8—711no—
118C22×Q8—711no—
119C23×C4—711no—
120C23×C4—1421no—
121C22×C8—711no—
122C22×C8C2×D7111yescharacteristic
123C22⋊Q8—714no—
124C22⋊Q8—1422no—
125C2×C4⋊C4—711no—
126C2×C4⋊C4—714no—
127C2×C4⋊C4—1422no—
128C2×C4⋊C4—1421no—
129C2×C4⋊C4—1421no—
130C2×C4⋊C4C2×D7111yescharacteristic
131C2×C4⋊C4C2×D7111yescharacteristic
132C2×C22⋊C4—1421no—
133#107—714no—
134#107C2×D7114yescomplemented
135#103—714no—
136#103—714no—
137C22⋊C8—714no—
138C24—711no—
139C2×Q8—712no—
140C2×Q8—1422no—
141C2×Q8—1422no—
142C2×Q8—1421no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2844no—
146C22×C4—711no—
147C22×C4—712no—
148C22×C4—712no—
149C22×C4—1424no—
150C22×C4—1421no—
151C22×C4—1421no—
152C22×C4—1422no—
153C22×C4—1422no—
154C22×C4C22×D7111yescharacteristic
155C2×C8—22no—
156C2×C8—712no—
157C2×C8—1422no—
158C2×C8C22×D7112yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—712no—
162C4⋊C4—1424no—
163C4⋊C4—1424no—
164C4⋊C4—1421no—
165C4⋊C4C22×D7112yes—
166C4⋊C4C22×D7112yes—
167C22⋊C4—1424no—
168C23—712no—
169C23—714no—
170C23—1422no—
171C23—1422no—
172C23D4×D7111yescharacteristic
173Q8—2842no—
174Q8—1422no—
175Q8—1422no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—2848no—
181C2×C4—2842no—
182C2×C4—2842no—
183C2×C4—712no—
184C2×C4—1422no—
185C2×C4—1422no—
186C2×C4—1422no—
187C2×C4—1422no—
188C2×C4—1422no—
189C2×C4—1424no—
190C2×C4D4×D7111yescharacteristic
191C2×C4#1176586512yes—
192C2×C4#1176586612yes—
193C2×C4C23×D7111yescharacteristic
194C8—22no—
195C8—1422no—
196C22—714no—
197C22—1428no—
198C22—1422no—
199C22—1422no—
200C22#1455611312yes—
201C22Q16×D7112yes—
202C22#1455615212yes—
203C22C2×D4×D7111yescharacteristic
204C4—42no—
205C4—42no—
206C4—2842no—
207C4—2842no—
208C4—1422no—
209C4#1455611312yes—
210C4C2×#1176586511yescharacteristic
211C4C2×#1176586611yescharacteristic
212C2—1424no—
213C2#1798633614yesdirect product factor
214C2C2×#1455611311yescharacteristic
215C2C2×Q16×D7111yescharacteristic
216C2C2×#1455615211yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#107×C71C2
socleC23×C71C2×D4
hypercenterC2×#107D71
hypocenterC71C2×#484
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
3. #17986336C2direct (N × H)
4. C2×#14556061C2semidirect
5. C2×#14556053C2semidirect
6. C2×#14556051C2semidirect
8. C2×#107×C71C2semidirect
21. #14556061C22semidirect
22. #14556053C22semidirect
23. #14556051C22semidirect
28. #107×C71C22semidirect
103. C2×C71#484semidirect
109. C71C2×#484semidirect
114. C2×#107D71semidirect
134. #107C2×D71semidirect
213. C2#17986336direct (N × H)