Finite Group Databasebeta

← C71×#8757 #31829629

Subgroups — C71×#8757 #31829629

Order 18176.

438 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2316C211yesmaximal, characteristic
2C71×#1463C212yesmaximal
3C71×#1305C214yesmaximal
4C2×#117×C71C411yescharacteristic
5C2×#117×C71C411yescharacteristic
6C22×C4⋊C4×C71C2211yescharacteristic
7#407×C71C2214yes—
8#389×C71—22no—
9#389×C71—22no—
10#380×C71C412yes—
11#380×C71C2212yes—
12#324×C71—22no—
13#324×C71—22no—
14C23×C4×C71C2×C411yescharacteristic
15C23×C4×C71C2×C411yescharacteristic
16C23×C4×C71C2311yescharacteristic
17C2×M16×C71—24no—
18#117×C71—42no—
19#117×C71—42no—
20C2×C4⋊C4×C71—24no—
21C2×C4⋊C4×C71—21no—
22C2×C4⋊C4×C71—21no—
23C2×C4⋊C4×C71—22no—
24C2×C4⋊C4×C71—22no—
25C2×C4⋊C4×C71D412yes—
26C2×C4⋊C4×C71D412yes—
27C2×C22⋊C4×C71C2×C412yes—
28C2×C22⋊C4×C71C2×C412yes—
29C2×C42×C71—21no—
30C2×C42×C71—21no—
31C22⋊C8×C71—24no—
32C22⋊C8×C71—24no—
33#95×C71—22no—
34#95×C71—22no—
35C24×C71C22×C411yescharacteristic
36C22×C4×C71—22no—
37C22×C4×C71—22no—
38C22×C4×C71—22no—
39C22×C4×C71—22no—
40C22×C4×C71—22no—
41C22×C4×C71—22no—
42C22×C4×C71—81no—
43C22×C4×C71—81no—
44C22×C4×C71C22⋊C411yescharacteristic
45C22×C4×C71C22⋊C411yescharacteristic
46C22×C4×C71C22⋊C412yes—
47C22×C4×C71C2×D412yes—
48C22×C4×C71#4114yes—
49M16×C71—84no—
50C2×C8×C71—44no—
51C4⋊C4×C71—24no—
52C4⋊C4×C71—24no—
53C4⋊C4×C71—42no—
54C4⋊C4×C71—42no—
55C4⋊C4×C71—82no—
56C4⋊C4×C71—82no—
57C22⋊C4×C71—42no—
58C22⋊C4×C71—42no—
59C42×C71—42no—
60C42×C71—42no—
61C23×C71—21no—
62C23×C71—21no—
63C23×C71—41no—
64C23×C71—41no—
65C23×C71C2×C22⋊C411yescharacteristic
66C23×C71#11712yes—
67C2×C4×C71—28no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—42no—
74C2×C4×C71—44no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—82no—
83C2×C4×C71#10012yes—
84C2×C4×C71#10112yes—
85C2×C4×C71#12314yes—
86C8×C71—84no—
87C22×C71—21no—
88C22×C71—21no—
89C22×C71—22no—
90C22×C71—22no—
91C22×C71—22no—
92C22×C71—42no—
93C22×C71—42no—
94C22×C71#38111yescharacteristic
95C22×C71C2×#10011yescharacteristic
96C22×C71C2×#10111yescharacteristic
97C4×C71—22no—
98C4×C71—22no—
99C4×C71—24no—
100C4×C71—42no—
101C4×C71—82no—
102C4×C71—82no—
103#8757C7111yesmaximal, characteristic, direct product factor, Sylow
104C2×C71—21no—
105C2×C71—21no—
106C2×C71—42no—
107C2×C71#182111yescharacteristic
108C2×C71#182811yescharacteristic
109C2×C71#183011yescharacteristic
110#2316C2×C7111yescharacteristic
111#1463C2×C7112yes—
112#1305C2×C7114yes—
113C71#875711yescharacteristic, direct product factor, Sylow
114C2×#117C4×C7111yescharacteristic
115C2×#117C4×C7111yescharacteristic
116C22×C4⋊C4C22×C7111yescharacteristic
117#407C22×C7114yes—
118#389—22no—
119#389—22no—
120#380C4×C7112yes—
121#380C22×C7112yes—
122#324—22no—
123#324—22no—
124C23×C4C2×C4×C7111yescharacteristic
125C23×C4C2×C4×C7111yescharacteristic
126C23×C4C23×C7111yescharacteristic
127C2×M16—24no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4D4×C7112yes—
136C2×C4⋊C4D4×C7112yes—
137C2×C22⋊C4C2×C4×C7112yes—
138C2×C22⋊C4C2×C4×C7112yes—
139C2×C42—21no—
140C2×C42—21no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143#95—22no—
144#95—22no—
145C24C22×C4×C7111yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C22⋊C4×C7111yescharacteristic
155C22×C4C22⋊C4×C7111yescharacteristic
156C22×C4C22⋊C4×C7112yes—
157C22×C4C2×D4×C7112yes—
158C22×C4#41×C7114yes—
159M16—84no—
160C2×C8—44no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—42no—
170C42—42no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊C4×C7111yescharacteristic
176C23#117×C7112yes—
177C2×C4—28no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4#100×C7112yes—
194C2×C4#101×C7112yes—
195C2×C4#123×C7114yes—
196C8—84no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22#381×C7111yescharacteristic
205C22C2×#100×C7111yescharacteristic
206C22C2×#101×C7111yescharacteristic
207C4—22no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—82no—
212C4—82no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C71×#182111yescharacteristic
217C2C71×#182811yescharacteristic
218C2C71×#183011yescharacteristic
subgroupG/H
center, socleC22×C71#381
commutatorC24C22×C4×C71
Frattini, cosocleC23×C4C23×C71
prime radicalC24×C71C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
103. #8757C71direct (N × H)
113. C71#8757direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#23162
C71×#14632
C71×#13052
#875771