Finite Group Databasebeta

← C71×#12167 #31833039

Subgroups — C71×#12167 #31833039

Order 18176.

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

normal subgroups only

This group has 354 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×#3259C211yesmaximal, characteristic
2C71×#2185C212yesmaximal, complemented
3C71×#2047C211yesmaximal, characteristic, complemented
4C71×#1349C212yesmaximal, complemented
5C71×#1323C211yesmaximal, characteristic, complemented
6C22×Q16×C71C2211yescharacteristic
7#540×C71—21no—
8C2×SD32×C71—22no—
9#478×C71—22no—
10#469×C71—21no—
11#466×C71C2212yes—
12#459×C71—22no—
13#424×C71C2211yescharacteristic
14#419×C71—21no—
15C2×#101×C71—21no—
16#358×C71—22no—
17C22⋊C16×C71C2212yes—
18#341×C71C2211yescharacteristic
19C22×Q8×C71—21no—
20C2×Q16×C71—22no—
21C2×Q16×C71—42no—
22C2×D8×C71—21no—
23C2×D8×C71D412yes—
24C2×M16×C71—21no—
25C2×M16×C71D411yescharacteristic
26C22×C8×C71C2311yescharacteristic
27#127×C71—21no—
28#125×C71—21no—
29#123×C71—42no—
30#121×C71—22no—
31#121×C71—42no—
32C4×D4×C71—22no—
33#117×C71D411yescharacteristic
34SD32×C71—44no—
35C2×C16×C71—22no—
36#107×C71D412yes—
37#106×C71—21no—
38C4⋊C8×C71—22no—
39#103×C71—42no—
40#102×C71—22no—
41#102×C71—22no—
42#101×C71—42no—
43C2×Q8×C71—22no—
44C2×Q8×C71—41no—
45C2×Q8×C71—41no—
46C2×Q8×C71—41no—
47C2×Q8×C71—41no—
48C2×Q8×C71—41no—
49C2×D4×C71—22no—
50C2×D4×C71—42no—
51C2×D4×C71—42no—
52C22×C4×C71—42no—
53C22×C4×C71—41no—
54C22×C4×C71C2×D411yescharacteristic
55Q16×C71—24no—
56Q16×C71—42no—
57D8×C71—42no—
58D8×C71—81no—
59M16×C71—41no—
60M16×C71—41no—
61M16×C71—41no—
62C2×C8×C71—21no—
63C2×C8×C71—41no—
64C2×C8×C71—41no—
65C2×C8×C71C2×D412yes—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71D812yes—
69C22⋊C4×C71—21no—
70C22⋊C4×C71—42no—
71C22⋊C4×C71—42no—
72C42×C71—21no—
73C16×C71—42no—
74C23×C71—42no—
75C23×C71AGL(1,ℤ/8)11yescharacteristic
76Q8×C71—42no—
77Q8×C71—42no—
78Q8×C71—81no—
79Q8×C71—81no—
80D4×C71—42no—
81D4×C71—42no—
82D4×C71—162no—
83D4×C71—162no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71—81no—
90C2×C4×C71—81no—
91C2×C4×C71—162no—
92C2×C4×C71#12011yescharacteristic
93C2×C4×C71C2×D811yescharacteristic
94C8×C71—22no—
95C8×C71—41no—
96C8×C71—81no—
97C22×C71—21no—
98C22×C71—22no—
99C22×C71—82no—
100C22×C71—82no—
101C22×C71—82no—
102C22×C71#44711yescharacteristic
103C4×C71—21no—
104C4×C71—42no—
105C4×C71—82no—
106C4×C71—82no—
107C4×C71#45411yescharacteristic
108C4×C71B(4,𝔽2)11yescharacteristic
109#12167C7111yesmaximal, characteristic, direct product factor, Sylow
110C2×C71—41no—
111C2×C71—162no—
112C2×C71#159811yescharacteristic
113C2×C71#219412yes—
114#3259C2×C7111yescharacteristic
115#2185C2×C7112yescomplemented
116#2047C2×C7111yescharacteristic, complemented
117#1349C2×C7112yescomplemented
118#1323C2×C7111yescharacteristic, complemented
119C71#1216711yescharacteristic, direct product factor, Sylow
120C22×Q16C22×C7111yescharacteristic
121#540—21no—
122C2×SD32—22no—
123#478—22no—
124#469—21no—
125#466C22×C7112yes—
126#459—22no—
127#424C22×C7111yescharacteristic
128#419—21no—
129C2×#101—21no—
130#358—22no—
131C22⋊C16C22×C7112yes—
132#341C22×C7111yescharacteristic
133C22×Q8—21no—
134C2×Q16—22no—
135C2×Q16—42no—
136C2×D8—21no—
137C2×D8D4×C7112yes—
138C2×M16—21no—
139C2×M16D4×C7111yescharacteristic
140C22×C8C23×C7111yescharacteristic
141#127—21no—
142#125—21no—
143#123—42no—
144#121—22no—
145#121—42no—
146C4×D4—22no—
147#117D4×C7111yescharacteristic
148SD32—44no—
149C2×C16—22no—
150#107D4×C7112yes—
151#106—21no—
152C4⋊C8—22no—
153#103—42no—
154#102—22no—
155#102—22no—
156#101—42no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4C2×D4×C7111yescharacteristic
169Q16—24no—
170Q16—42no—
171D8—42no—
172D8—81no—
173M16—41no—
174M16—41no—
175M16—41no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×D4×C7112yes—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4D8×C7112yes—
183C22⋊C4—21no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—21no—
187C16—42no—
188C23—42no—
189C23AGL(1,ℤ/8)×C7111yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—81no—
194D4—42no—
195D4—42no—
196D4—162no—
197D4—162no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—162no—
206C2×C4#120×C7111yescharacteristic
207C2×C4C2×D8×C7111yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—21no—
212C22—22no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22#447×C7111yescharacteristic
217C4—21no—
218C4—42no—
219C4—82no—
220C4—82no—
221C4#454×C7111yescharacteristic
222C4B(4,𝔽2)×C7111yescharacteristic
223C2—41no—
224C2—162no—
225C2C71×#159811yescharacteristic
226C2C71×#219412yes—
subgroupG/H
center, socleC22×C71#447
commutator, Frattini, cosocleC22×C8C23×C71
prime radicalC71×#3259C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ #424×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2185C2semidirect
3. C71×#2047C2semidirect
4. C71×#1349C2semidirect
5. C71×#1323C2semidirect
109. #12167C71direct (N × H)
115. #2185C2×C71semidirect
116. #2047C2×C71semidirect
117. #1349C2×C71semidirect
118. #1323C2×C71semidirect
119. C71#12167direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32592
C71×#21852
C71×#20472
C71×#13492
C71×#13232
#1216771