Finite Group Databasebeta

← C71×#12256 #31833128

Subgroups — C71×#12256 #31833128

Order 18176.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C71×#2551C211yesmaximal, characteristic
2C71×#2060C211yesmaximal, characteristic
3C71×#1605C212yesmaximal
4C71×#1354C212yesmaximal
5C71×#1325C211yesmaximal, characteristic
6#544×C71C2211yescharacteristic
7C2×C4⋊Q8×C71—22no—
8#451×C71—22no—
9#428×C71—21no—
10#395×C71—21no—
11#395×C71C2212yes—
12#389×C71—22no—
13#384×C71C2211yescharacteristic
14C4×C4⋊C4×C71—21no—
15#343×C71C2211yescharacteristic
16#339×C71—21no—
17#328×C71—22no—
18#324×C71C2212yes—
19C22×Q8×C71—21no—
20C22×Q8×C71D412yes—
21C2×Q16×C71—42no—
22C2×M16×C71—21no—
23C4⋊Q8×C71—41no—
24C4⋊Q8×C71—48no—
25#125×C71—41no—
26C22⋊Q8×C71—22no—
27#117×C71—21no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—22no—
33C2×C4⋊C4×C71D411yescharacteristic
34C2×C4⋊C4×C71C2311yescharacteristic
35C2×C42×C71—22no—
36C2×C42×C71D411yescharacteristic
37#107×C71—41no—
38#106×C71—41no—
39#103×C71—42no—
40C22⋊C8×C71—22no—
41#95×C71—21no—
42#95×C71—22no—
43#95×C71D412yes—
44C2×Q8×C71—22no—
45C2×Q8×C71—44no—
46C2×Q8×C71—44no—
47C2×Q8×C71—42no—
48C2×Q8×C71—81no—
49C22×C4×C71—21no—
50C22×C4×C71—22no—
51C22×C4×C71—22no—
52C22×C4×C71—21no—
53C22×C4×C71C2×D411yescharacteristic
54C22×C4×C71C2×D412yes—
55Q16×C71—84no—
56M16×C71—22no—
57M16×C71—41no—
58C2×C8×C71—41no—
59C2×C8×C71—42no—
60C4⋊C4×C71—24no—
61C4⋊C4×C71—24no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71—21no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—44no—
68C4⋊C4×C71—44no—
69C4⋊C4×C71—44no—
70C4⋊C4×C71—42no—
71C22⋊C4×C71—41no—
72C42×C71—21no—
73C42×C71—44no—
74C42×C71—41no—
75C42×C71Q1612yescomplemented
76C23×C71#12011yescharacteristic
77Q8×C71—44no—
78Q8×C71—82no—
79Q8×C71—84no—
80Q8×C71—82no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—24no—
87C2×C4×C71—24no—
88C2×C4×C71—22no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71C2×Q1611yescharacteristic
99C2×C4×C71#13711yescharacteristic
100C8×C71—42no—
101C8×C71—82no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71#45111yescharacteristic
105C22×C71#45611yescharacteristic
106C22×C71B(4,𝔽2)11yescharacteristic
107C4×C71—24no—
108C4×C71—21no—
109C4×C71—21no—
110C4×C71—44no—
111C4×C71—42no—
112C4×C71—44no—
113C4×C71—81no—
114C4×C71—81no—
115C4×C71—81no—
116#12256C7111yesmaximal, characteristic, direct product factor, Sylow
117C2×C71—21no—
118C2×C71—21no—
119C2×C71#160511yescharacteristic
120C2×C71#219611yescharacteristic
121C2×C71#220511yescharacteristic
122#2551C2×C7111yescharacteristic
123#2060C2×C7111yescharacteristic
124#1605C2×C7112yes—
125#1354C2×C7112yes—
126#1325C2×C7111yescharacteristic
127C71#1225611yescharacteristic, direct product factor, Sylow
128#544C22×C7111yescharacteristic
129C2×C4⋊Q8—22no—
130#451—22no—
131#428—21no—
132#395—21no—
133#395C22×C7112yes—
134#389—22no—
135#384C22×C7111yescharacteristic
136C4×C4⋊C4—21no—
137#343C22×C7111yescharacteristic
138#339—21no—
139#328—22no—
140#324C22×C7112yes—
141C22×Q8—21no—
142C22×Q8D4×C7112yes—
143C2×Q16—42no—
144C2×M16—21no—
145C4⋊Q8—41no—
146C4⋊Q8—48no—
147#125—41no—
148C22⋊Q8—22no—
149#117—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4D4×C7111yescharacteristic
156C2×C4⋊C4C23×C7111yescharacteristic
157C2×C42—22no—
158C2×C42D4×C7111yescharacteristic
159#107—41no—
160#106—41no—
161#103—42no—
162C22⋊C8—22no—
163#95—21no—
164#95—22no—
165#95D4×C7112yes—
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—42no—
170C2×Q8—81no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×D4×C7111yescharacteristic
176C22×C4C2×D4×C7112yes—
177Q16—84no—
178M16—22no—
179M16—41no—
180C2×C8—41no—
181C2×C8—42no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C22⋊C4—41no—
194C42—21no—
195C42—44no—
196C42—41no—
197C42Q16×C7112yescomplemented
198C23#120×C7111yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—82no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4C2×Q16×C7111yescharacteristic
221C2×C4#137×C7111yescharacteristic
222C8—42no—
223C8—82no—
224C22—21no—
225C22—21no—
226C22#451×C7111yescharacteristic
227C22#456×C7111yescharacteristic
228C22B(4,𝔽2)×C7111yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2C71×#160511yescharacteristic
241C2C71×#219611yescharacteristic
242C2C71×#220511yescharacteristic
subgroupG/H
center, socleC22×C71B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C71
prime radicalC23×C71#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C71 ◁ C2×C4⋊C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
75. C42×C71Q16semidirect
116. #12256C71direct (N × H)
127. C71#12256direct (N × H)
197. C42Q16×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#25512
C71×#20602
C71×#16052
C71×#13542
C71×#13252
#1225671