Finite Group Databasebeta

← C71×#12185 #31833057

Subgroups — C71×#12185 #31833057

Order 18176.

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

normal subgroups only

This group has 360 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×#2844C211yesmaximal, characteristic, complemented
2C71×#2040C211yesmaximal, characteristic
3C71×#1616C212yesmaximal
4C71×#1353C212yesmaximal, complemented
5C71×#1324C211yesmaximal, characteristic, complemented
6#537×C71C2211yescharacteristic
7#481×C71—22no—
8C2×#104×C71—21no—
9#417×C71—21no—
10#400×C71—22no—
11#400×C71C2212yes—
12#395×C71—22no—
13#395×C71—21no—
14#382×C71—21no—
15#382×C71C2211yescharacteristic
16#343×C71C2211yescharacteristic
17#327×C71—22no—
18#324×C71C2212yes—
19C2×#41×C71—21no—
20C22×Q8×C71—21no—
21C2×M16×C71—21no—
22C22⋊Q8×C71—41no—
23#121×C71—22no—
24#121×C71—41no—
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71D411yescharacteristic
29C2×C4⋊C4×C71D412yes—
30C2×C4⋊C4×C71C2311yescharacteristic
31C2×C42×C71D411yescharacteristic
32#106×C71—42no—
33#104×C71—42no—
34#103×C71—41no—
35#102×C71—41no—
36#102×C71—42no—
37C22⋊C8×C71—22no—
38#95×C71—22no—
39#95×C71—21no—
40#95×C71—22no—
41#95×C71—22no—
42#95×C71—22no—
43#95×C71D412yes—
44#41×C71—41no—
45#41×C71—41no—
46#41×C71—41no—
47#41×C71—41no—
48C2×Q8×C71—21no—
49C2×Q8×C71—82no—
50C2×Q8×C71—81no—
51C2×D4×C71—21no—
52C2×D4×C71—22no—
53C2×D4×C71—41no—
54C22×C4×C71—21no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71—21no—
58C22×C4×C71—41no—
59C22×C4×C71C2×D411yescharacteristic
60C22×C4×C71C2×D412yes—
61M16×C71—41no—
62M16×C71—41no—
63C2×C8×C71—42no—
64C2×C8×C71—41no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—81no—
70C4⋊C4×C71—81no—
71C4⋊C4×C71—82no—
72C22⋊C4×C71—42no—
73C42×C71—21no—
74C42×C71SD1612yes—
75C23×C71—41no—
76C23×C71#12011yescharacteristic
77Q8×C71—41no—
78Q8×C71—41no—
79Q8×C71—84no—
80D4×C71—41no—
81D4×C71—41no—
82D4×C71—82no—
83D4×C71—81no—
84D4×C71—81no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71—81no—
102C2×C4×C71—81no—
103C2×C4×C71C2×SD1611yescharacteristic
104C2×C4×C71#13711yescharacteristic
105C8×C71—81no—
106C8×C71—82no—
107C22×C71—21no—
108C22×C71—21no—
109C22×C71—81no—
110C22×C71—81no—
111C22×C71—81no—
112C22×C71#44811yescharacteristic
113C22×C71#45411yescharacteristic
114C22×C71B(4,𝔽2)11yescharacteristic
115C4×C71—21no—
116C4×C71—21no—
117C4×C71—42no—
118C4×C71—42no—
119C4×C71—82no—
120C4×C71—81no—
121C4×C71—81no—
122C4×C71—82no—
123#12185C7111yesmaximal, characteristic, direct product factor, Sylow
124C2×C71—21no—
125C2×C71—21no—
126C2×C71—161no—
127C2×C71#159911yescharacteristic
128C2×C71#220111yescharacteristic
129C2×C71#220211yescharacteristic
130#2844C2×C7111yescharacteristic, complemented
131#2040C2×C7111yescharacteristic
132#1616C2×C7112yes—
133#1353C2×C7112yescomplemented
134#1324C2×C7111yescharacteristic, complemented
135C71#1218511yescharacteristic, direct product factor, Sylow
136#537C22×C7111yescharacteristic
137#481—22no—
138C2×#104—21no—
139#417—21no—
140#400—22no—
141#400C22×C7112yes—
142#395—22no—
143#395—21no—
144#382—21no—
145#382C22×C7111yescharacteristic
146#343C22×C7111yescharacteristic
147#327—22no—
148#324C22×C7112yes—
149C2×#41—21no—
150C22×Q8—21no—
151C2×M16—21no—
152C22⋊Q8—41no—
153#121—22no—
154#121—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4D4×C7111yescharacteristic
159C2×C4⋊C4D4×C7112yes—
160C2×C4⋊C4C23×C7111yescharacteristic
161C2×C42D4×C7111yescharacteristic
162#106—42no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—42no—
167C22⋊C8—22no—
168#95—22no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95D4×C7112yes—
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—82no—
180C2×Q8—81no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4C2×D4×C7111yescharacteristic
190C22×C4C2×D4×C7112yes—
191M16—41no—
192M16—41no—
193C2×C8—42no—
194C2×C8—41no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—82no—
202C22⋊C4—42no—
203C42—21no—
204C42SD16×C7112yes—
205C23—41no—
206C23#120×C7111yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—84no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4C2×SD16×C7111yescharacteristic
234C2×C4#137×C7111yescharacteristic
235C8—81no—
236C8—82no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#448×C7111yescharacteristic
243C22#454×C7111yescharacteristic
244C22B(4,𝔽2)×C7111yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—81no—
252C4—82no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2C71×#159911yescharacteristic
257C2C71×#220111yescharacteristic
258C2C71×#220211yescharacteristic
subgroupG/H
center, socleC22×C71B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C71
prime radical#537×C71C22
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
1. C71×#2844C2semidirect
4. C71×#1353C2semidirect
5. C71×#1324C2semidirect
123. #12185C71direct (N × H)
130. #2844C2×C71semidirect
133. #1353C2×C71semidirect
134. #1324C2×C71semidirect
135. C71#12185direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#28442
C71×#20402
C71×#16162
C71×#13532
C71×#13242
#1218571