Finite Group Databasebeta

← C71×#12171 #31833043

Subgroups — C71×#12171 #31833043

Order 18176.

332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3261C211yesmaximal, characteristic, complemented
2C71×#2191C212yesmaximal
3C71×#2053C211yesmaximal, characteristic
4C71×#1340C212yesmaximal, complemented
5C71×#1328C211yesmaximal, characteristic, complemented
6#575×C71C2211yescharacteristic
7#541×C71—21no—
8#510×C71—22no—
9#479×C71—22no—
10#470×C71C2212yes—
11#467×C71—21no—
12#462×C71—22no—
13#424×C71C2211yescharacteristic
14#416×C71—21no—
15#410×C71—21no—
16#360×C71—22no—
17#349×C71C2212yes—
18H(3,ℤ/4)×C71C2211yescharacteristic
19C2×#41×C71—21no—
20#137×C71—41no—
21AGL(1,ℤ/8)×C71—41no—
22#135×C71—22no—
23C2×Q16×C71—21no—
24C2×Q16×C71D412yes—
25C2×M16×C71C2311yescharacteristic
26C22×C8×C71D411yescharacteristic
27C4⋊Q8×C71—21no—
28#126×C71—42no—
29#125×C71—21no—
30#125×C71—42no—
31C22⋊Q8×C71—22no—
32C4×Q8×C71—22no—
33#117×C71—21no—
34#117×C71D411yescharacteristic
35Q32×C71—42no—
36SD32×C71—42no—
37M32×C71—22no—
38#107×C71—21no—
39#106×C71D412yes—
40C4⋊C8×C71—22no—
41#103×C71—22no—
42#103×C71—22no—
43#103×C71—41no—
44#102×C71—41no—
45#99×C71—42no—
46#41×C71—41no—
47#41×C71—41no—
48#41×C71—41no—
49#41×C71—41no—
50C2×Q8×C71—21no—
51C2×Q8×C71—22no—
52C2×D4×C71—21no—
53C2×D4×C71—41no—
54C22×C4×C71—41no—
55C22×C4×C71C2×D411yescharacteristic
56Q16×C71—22no—
57Q16×C71—42no—
58Q16×C71—81no—
59SD16×C71—42no—
60D8×C71—22no—
61M16×C71—41no—
62C2×C8×C71—21no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D412yes—
65C4⋊C4×C71—21no—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—42no—
72C22⋊C4×C71—42no—
73C22⋊C4×C71—41no—
74C22⋊C4×C71D812yes—
75C42×C71—21no—
76C42×C71—42no—
77C42×C71—41no—
78C16×C71—42no—
79C23×C71—41no—
80C23×C71C2×D811yescharacteristic
81Q8×C71—41no—
82Q8×C71—42no—
83Q8×C71—41no—
84Q8×C71—42no—
85D4×C71—41no—
86D4×C71—41no—
87D4×C71—81no—
88D4×C71—81no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71—81no—
96C2×C4×C71—81no—
97C2×C4×C71—81no—
98C2×C4×C71#12011yescharacteristic
99C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
100C8×C71—22no—
101C8×C71—41no—
102C22×C71—21no—
103C22×C71—41no—
104C22×C71—81no—
105C22×C71—161no—
106C22×C71#44711yescharacteristic
107C4×C71—21no—
108C4×C71—42no—
109C4×C71—82no—
110C4×C71—81no—
111C4×C71—82no—
112C4×C71—161no—
113C4×C71#45411yescharacteristic
114C4×C71B(4,𝔽2)11yescharacteristic
115#12171C7111yesmaximal, characteristic, direct product factor, Sylow
116C2×C71—21no—
117C2×C71—41no—
118C2×C71—161no—
119C2×C71#159811yescharacteristic
120#3261C2×C7111yescharacteristic, complemented
121#2191C2×C7112yes—
122#2053C2×C7111yescharacteristic
123#1340C2×C7112yescomplemented
124#1328C2×C7111yescharacteristic, complemented
125C71#1217111yescharacteristic, direct product factor, Sylow
126#575C22×C7111yescharacteristic
127#541—21no—
128#510—22no—
129#479—22no—
130#470C22×C7112yes—
131#467—21no—
132#462—22no—
133#424C22×C7111yescharacteristic
134#416—21no—
135#410—21no—
136#360—22no—
137#349C22×C7112yes—
138H(3,ℤ/4)C22×C7111yescharacteristic
139C2×#41—21no—
140#137—41no—
141AGL(1,ℤ/8)—41no—
142#135—22no—
143C2×Q16—21no—
144C2×Q16D4×C7112yes—
145C2×M16C23×C7111yescharacteristic
146C22×C8D4×C7111yescharacteristic
147C4⋊Q8—21no—
148#126—42no—
149#125—21no—
150#125—42no—
151C22⋊Q8—22no—
152C4×Q8—22no—
153#117—21no—
154#117D4×C7111yescharacteristic
155Q32—42no—
156SD32—42no—
157M32—22no—
158#107—21no—
159#106D4×C7112yes—
160C4⋊C8—22no—
161#103—22no—
162#103—22no—
163#103—41no—
164#102—41no—
165#99—42no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×D4—21no—
173C2×D4—41no—
174C22×C4—41no—
175C22×C4C2×D4×C7111yescharacteristic
176Q16—22no—
177Q16—42no—
178Q16—81no—
179SD16—42no—
180D8—22no—
181M16—41no—
182C2×C8—21no—
183C2×C8—41no—
184C2×C8C2×D4×C7112yes—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—41no—
194C22⋊C4D8×C7112yes—
195C42—21no—
196C42—42no—
197C42—41no—
198C16—42no—
199C23—41no—
200C23C2×D8×C7111yescharacteristic
201Q8—41no—
202Q8—42no—
203Q8—41no—
204Q8—42no—
205D4—41no—
206D4—41no—
207D4—81no—
208D4—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4#120×C7111yescharacteristic
219C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
220C8—22no—
221C8—41no—
222C22—21no—
223C22—41no—
224C22—81no—
225C22—161no—
226C22#447×C7111yescharacteristic
227C4—21no—
228C4—42no—
229C4—82no—
230C4—81no—
231C4—82no—
232C4—161no—
233C4#454×C7111yescharacteristic
234C4B(4,𝔽2)×C7111yescharacteristic
235C2—21no—
236C2—41no—
237C2—161no—
238C2C71×#159811yescharacteristic
subgroupG/H
center, socleC2×C71#1598
commutator, Frattini, cosocleC2×M16C23×C71
prime radical#575×C71C22
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C2×C4×C71 ◁ C2×M16×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#3261C2semidirect
4. C71×#1340C2semidirect
5. C71×#1328C2semidirect
115. #12171C71direct (N × H)
120. #3261C2×C71semidirect
123. #1340C2×C71semidirect
124. #1328C2×C71semidirect
125. C71#12171direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32612
C71×#21912
C71×#20532
C71×#13402
C71×#13282
#1217171