Finite Group Databasebeta

← C71×#10359 #31831231

Subgroups — C71×#10359 #31831231

Order 18176.

400 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 106 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
2C71×#1993C212yesmaximal
3C71×#1960C211yesmaximal, characteristic
4C71×#1295C211yesmaximal, characteristic
5C71×#1295C212yesmaximal
6C2×C4⋊Q8×C71C2211yescharacteristic
7C2×#103×C71—21no—
8C2×#103×C71—21no—
9C2×#97×C71C2211yescharacteristic
10C2×#97×C71C2212yes—
11#400×C71—21no—
12#400×C71—22no—
13#395×C71—21no—
14#395×C71—22no—
15#382×C71C2211yescharacteristic
16#382×C71C2212yes—
17#340×C71—22no—
18#340×C71—22no—
19C22×Q8×C71—21no—
20C22×Q8×C71—21no—
21C22×C8×C71—21no—
22C22×C8×C71—22no—
23C4⋊Q8×C71—22no—
24C4⋊Q8×C71—21no—
25C4⋊Q8×C71Q812yescomplemented
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71—22no—
29C2×C4⋊C4×C71D411yescharacteristic
30C2×C4⋊C4×C71D411yescharacteristic
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D412yes—
33C2×C42×C71C2311yescharacteristic
34#103×C71—42no—
35#103×C71—42no—
36#97×C71—22no—
37#97×C71—22no—
38#97×C71—22no—
39#95×C71—22no—
40#95×C71—21no—
41#95×C71—21no—
42#95×C71—22no—
43#95×C71—22no—
44#95×C71—22no—
45C2×Q8×C71—22no—
46C2×Q8×C71—41no—
47C2×Q8×C71—41no—
48C2×Q8×C71—41no—
49C2×Q8×C71—41no—
50C2×Q8×C71—41no—
51C2×Q8×C71—81no—
52C2×Q8×C71—82no—
53C22×C4×C71—21no—
54C22×C4×C71—22no—
55C22×C4×C71—21no—
56C22×C4×C71—22no—
57C22×C4×C71—41no—
58C22×C4×C71C2×D412yes—
59C22×C4×C71C2×D411yescharacteristic
60C2×C8×C71—22no—
61C2×C8×C71—24no—
62C2×C8×C71—41no—
63C2×C8×C71—42no—
64C2×C8×C71—42no—
65C2×C8×C71—41no—
66C4⋊C4×C71—24no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—81no—
73C4⋊C4×C71—82no—
74C42×C71C2×Q811yescharacteristic
75C42×C71#4112yes—
76C42×C71#4111yescharacteristic
77C23×C71#12011yescharacteristic
78Q8×C71—42no—
79Q8×C71—42no—
80Q8×C71—81no—
81Q8×C71—81no—
82Q8×C71—84no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—42no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
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—81no—
99C2×C4×C71—81no—
100C2×C4×C71C22⋊Q812yes—
101C2×C4×C71C22⋊Q811yescharacteristic
102C2×C4×C71#12412yes—
103C2×C4×C71#12411yescharacteristic
104C8×C71—42no—
105C8×C71—44no—
106C22×C71#39311yescharacteristic
107C22×C71#45311yescharacteristic
108C22×C71#45411yescharacteristic
109C22×C71#45512yes—
110C22×C71#45612yes—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—81no—
116C4×C71—82no—
117C4×C71—81no—
118C4×C71—82no—
119C4×C71—82no—
120#10359C7111yesmaximal, characteristic, direct product factor, Sylow
121C2×C71#165911yescharacteristic
122C2×C71#166011yescharacteristic
123C2×C71#166412yes—
124C2×C71#202711yescharacteristic
125C2×C71#202812yes—
126#2844C2×C7111yescharacteristic
127#1993C2×C7112yes—
128#1960C2×C7111yescharacteristic
129#1295C2×C7111yescharacteristic
130#1295C2×C7112yes—
131C71#1035911yescharacteristic, direct product factor, Sylow
132C2×C4⋊Q8C22×C7111yescharacteristic
133C2×#103—21no—
134C2×#103—21no—
135C2×#97C22×C7111yescharacteristic
136C2×#97C22×C7112yes—
137#400—22no—
138#400—21no—
139#395—22no—
140#395—21no—
141#382C22×C7111yescharacteristic
142#382C22×C7112yes—
143#340—22no—
144#340—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8Q8×C7112yescomplemented
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4D4×C7112yes—
156C2×C4⋊C4D4×C7112yes—
157C2×C4⋊C4D4×C7111yescharacteristic
158C2×C4⋊C4D4×C7111yescharacteristic
159C2×C42C23×C7111yescharacteristic
160#103—42no—
161#103—42no—
162#97—22no—
163#97—22no—
164#97—22no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—82no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4C2×D4×C7112yes—
185C22×C4C2×D4×C7111yescharacteristic
186C2×C8—22no—
187C2×C8—24no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C4⋊C4—24no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—82no—
200C42C2×Q8×C7111yescharacteristic
201C42#41×C7112yes—
202C42#41×C7111yescharacteristic
203C23#120×C7111yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—81no—
207Q8—81no—
208Q8—84no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4C22⋊Q8×C7112yes—
227C2×C4C22⋊Q8×C7111yescharacteristic
228C2×C4#124×C7112yes—
229C2×C4#124×C7111yescharacteristic
230C8—42no—
231C8—44no—
232C22#393×C7111yescharacteristic
233C22#453×C7111yescharacteristic
234C22#454×C7111yescharacteristic
235C22#455×C7112yes—
236C22#456×C7112yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—81no—
245C4—82no—
246C2C71×#165911yescharacteristic
247C2C71×#166011yescharacteristic
248C2C71×#166412yes—
249C2C71×#202711yescharacteristic
250C2C71×#202812yes—
subgroupG/H
center, socle, prime radicalC23×C71#120
commutator, Frattini, cosocleC2×C42C23×C71
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
25. C4⋊Q8×C71Q8semidirect
120. #10359C71direct (N × H)
131. C71#10359direct (N × H)
151. C4⋊Q8Q8×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#28442
C71×#19932
C71×#19602
C71×#12952
#1035971