Finite Group Databasebeta

← C2×#22586505 #31917760

Subgroups — C2×#22586505 #31917760

Order 18176.

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×#17985792C212yesmaximal
2#22586505C214yesmaximal, direct product factor
3C2×#492×C71C211yesmaximal, characteristic
4C2×#14556043—22no—
5#17985792—22no—
6#17985792C2214yes—
7C22×SD16×C71—22no—
8C2×C4⋊Q8×C71C411yescharacteristic, complemented
9C2×#127×C71C2211yescharacteristic
10#492×C71—22no—
11#492×C71C412yescomplemented
12#492×C71C2212yes—
13C2×C4×C8×C71C411yescharacteristic, complemented
14C2×C22⋊Dic71—42no—
15#14556043—24no—
16#14556043—42no—
17C22×Q8×C71—21no—
18C22×D4×C71—41no—
19C22×D4×C71D412yes—
20C2×SD16×C71—24no—
21C2×SD16×C71—24no—
22C2×SD16×C71—44no—
23C2×SD16×C71—44no—
24C22×C8×C71—21no—
25C4⋊Q8×C71—21no—
26C4⋊Q8×C71—41no—
27C4⋊Q8×C71C2×C412yescomplemented
28#127×C71—41no—
29#127×C71C2×C411yescharacteristic
30#127×C71D412yes—
31#127×C71C2311yescharacteristic
32C2×C4⋊C4×C71—41no—
33C2×C42×C71C2×C411yescharacteristic
34C4×C8×C71—21no—
35C4×C8×C71C2×C412yescomplemented
36C22×Dic71—82no—
37C22⋊Dic71—44no—
38C22⋊Dic71—82no—
39C24×C71—22no—
40C2×Q8×C71—22no—
41C2×Q8×C71—42no—
42C2×Q8×C71—42no—
43C2×Q8×C71—41no—
44C2×D4×C71—22no—
45C2×D4×C71—42no—
46C2×D4×C71—42no—
47C2×D4×C71—41no—
48C2×D4×C71—42no—
49C2×D4×C71—41no—
50C2×D4×C71—84no—
51C2×D4×C71C22⋊C412yes—
52C2×D4×C71C2×D412yes—
53C22×C4×C71—21no—
54C22×C4×C71—41no—
55C22×C4×C71C22⋊C411yescharacteristic
56SD16×C71—48no—
57SD16×C71—48no—
58C2×C8×C71—22no—
59C2×C8×C71—22no—
60C2×C8×C71—22no—
61C4⋊C4×C71—42no—
62C4⋊C4×C71—42no—
63C42×C71—21no—
64C42×C71C22⋊C411yescharacteristic
65C42×C71C22×C411yescharacteristic
66C23×C71—22no—
67C23×C71—22no—
68C23×C71—42no—
69C23×C71—42no—
70C23×C71—84no—
71C23×C71#9911yescharacteristic
72Q8×C71—44no—
73Q8×C71—82no—
74D4×C71—42no—
75D4×C71—42no—
76D4×C71—82no—
77D4×C71—82no—
78D4×C71—84no—
79D4×C71—82no—
80C2×C4×C71—24no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—42no—
86C2×C4×C71—81no—
87C2×C4×C71—81no—
88C2×C4×C71#9911yescharacteristic
89C2×C4×C71C2×C22⋊C411yescharacteristic
90C2×Dic71—84no—
91C2×Dic71—164no—
92C8×C71—24no—
93C22×C71—22no—
94C22×C71—42no—
95C22×C71—42no—
96C22×C71—82no—
97C22×C71—82no—
98C22×C71—84no—
99C22×C71—162no—
100C22×C71#35311yescharacteristic
101C22×C71#35311yescharacteristic
102C22×C71C2×#9911yescharacteristic
103C4×C71—22no—
104C4×C71—22no—
105C4×C71—21no—
106C4×C71—21no—
107C4×C71—82no—
108Dic71—164no—
109C2×#1409—711nomaximal, Sylow
110C2×C71—21no—
111C2×C71—21no—
112C2×C71—82no—
113C2×C71—82no—
114C2×C71#140912yescomplemented
115C2×C71C2×#35311yescharacteristic
116C2×#492D7111yescharacteristic
117C2×#353—712no—
118#1409—714no—
119C71C2×#140911yescharacteristic, complemented, Sylow
120C22×SD16—22no—
121C2×C4⋊Q8Dic7111yescharacteristic, complemented
122C2×#127C2×D7111yescharacteristic
123#492—22no—
124#492Dic7112yescomplemented
125#492C2×D7112yes—
126C2×#99—1422no—
127C2×C4×C8Dic7111yescharacteristic, complemented
128#353—714no—
129#353—1422no—
130C22×Q8—21no—
131C22×D4—41no—
132C22×D4C22⋊D7112yes—
133C2×SD16—24no—
134C2×SD16—24no—
135C2×SD16—44no—
136C2×SD16—44no—
137C22×C8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—41no—
140C4⋊Q8C2×Dic7112yescomplemented
141#127—41no—
142#127C2×Dic7111yescharacteristic
143#127C22⋊D7112yes—
144#127C22×D7111yescharacteristic
145C2×C4⋊C4—41no—
146C2×C22⋊C4—2842no—
147C2×C42C2×Dic7111yescharacteristic
148#99—2842no—
149#99—1424no—
150C4×C8—21no—
151C4×C8C2×Dic7112yescomplemented
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⋊Dic7112yes—
165C2×D4C2×C22⋊D7112yes—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—5682no—
169C22×C4C22⋊Dic7111yescharacteristic
170SD16—48no—
171SD16—48no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—2844no—
178C22⋊C4—5682no—
179C42—21no—
180C42C22⋊Dic7111yescharacteristic
181C42C22×Dic7111yescharacteristic
182C23—22no—
183C23—22no—
184C23—42no—
185C23—42no—
186C23—84no—
187C23#1455604311yescharacteristic
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—5684no—
205C2×C4—11364no—
206C2×C4#1455604311yescharacteristic
207C2×C4C2×C22⋊Dic7111yescharacteristic
208C8—24no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—82no—
213C22—82no—
214C22—84no—
215C22—162no—
216C22#1798579211yescharacteristic
217C22#1798579211yescharacteristic
218C22C2×#1455604311yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—11364no—
225C2—21no—
226C2—21no—
227C2—82no—
228C2—82no—
229C2#2258650512yesdirect product factor
230C2C2×#1798579211yescharacteristic
subgroupG/H
centerC22#17985792
commutatorC42×C71C22×C4
Frattini#127C22×D71
FittingC2×#492×C71C2
socleC22×C71#353
prime radicalC2×#127×C71C22
cosocle#127×C71C23
hypercenterC2×#492D71
hypocenterC71C2×#1409
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
2. #22586505C2direct (N × H)
8. C2×C4⋊Q8×C71C4semidirect
11. #492×C71C4semidirect
13. C2×C4×C8×C71C4semidirect
27. C4⋊Q8×C71C2×C4semidirect
35. C4×C8×C71C2×C4semidirect
114. C2×C71#1409semidirect
119. C71C2×#1409semidirect
121. C2×C4⋊Q8Dic71semidirect
124. #492Dic71semidirect
127. C2×C4×C8Dic71semidirect
140. C4⋊Q8C2×Dic71semidirect
151. C4×C8C2×Dic71semidirect
229. C2#22586505direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179857922
#225865052
C2×#492×C712
C2×#140971