Finite Group Databasebeta

← C71×#8915 #31829787

Subgroups — C71×#8915 #31829787

Order 18176.

396 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2382C211yesmaximal, characteristic
2C71×#1992C211yesmaximal, characteristic
3C71×#1836C212yesmaximal
4C2×#330×C71C211yesmaximal, characteristic
5C71×#1295C212yesmaximal
6C2×#125×C71C2211yescharacteristic
7C2×#97×C71C2211yescharacteristic
8C2×#97×C71C2212yes—
9#389×C71—22no—
10#387×C71—22no—
11#387×C71—22no—
12#384×C71C2211yescharacteristic
13#382×C71C2212yes—
14#340×C71—22no—
15#340×C71—22no—
16#340×C71—22no—
17#330×C71C414yescomplemented
18C22×C8×C71—21no—
19C22×C8×C71—22no—
20#125×C71—22no—
21#125×C71C2×C412yes—
22#125×C71Q812yes—
23C2×C4⋊C4×C71—22no—
24C2×C4⋊C4×C71—21no—
25C2×C4⋊C4×C71—22no—
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71D412yes—
28C2×C4⋊C4×C71D412yes—
29C2×C4⋊C4×C71D412yes—
30C2×C42×C71—22no—
31C2×C42×C71C2311yescharacteristic
32#97×C71—22no—
33#97×C71—22no—
34#97×C71C2×C414yes—
35#95×C71—22no—
36#95×C71—22no—
37#95×C71—22no—
38#95×C71—22no—
39C22×C4×C71—22no—
40C22×C4×C71—22no—
41C22×C4×C71—22no—
42C22×C4×C71—41no—
43C22×C4×C71C2×D411yescharacteristic
44C22×C4×C71C2×D412yes—
45C2×C8×C71—22no—
46C2×C8×C71—24no—
47C2×C8×C71—41no—
48C2×C8×C71—42no—
49C2×C8×C71—42no—
50C2×C8×C71—41no—
51C4⋊C4×C71—24no—
52C4⋊C4×C71—24no—
53C4⋊C4×C71—22no—
54C4⋊C4×C71—42no—
55C4⋊C4×C71—42no—
56C4⋊C4×C71—42no—
57C4⋊C4×C71—44no—
58C4⋊C4×C71—82no—
59C4⋊C4×C71C4⋊C414yes—
60C42×C71—82no—
61C42×C71C22×C411yescharacteristic
62C42×C71C2×Q811yescharacteristic
63C42×C71#4112yes—
64C23×C71#12011yescharacteristic
65C2×C4×C71—21no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—21no—
69C2×C4×C71—24no—
70C2×C4×C71—42no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—81no—
79C2×C4×C71—82no—
80C2×C4×C71C2×C4⋊C411yescharacteristic
81C2×C4×C71C4×D412yes—
82C2×C4×C71C22⋊Q812yes—
83C2×C4×C71#12311yescharacteristic
84C8×C71—42no—
85C8×C71—44no—
86C22×C71#38511yescharacteristic
87C22×C71#45411yescharacteristic
88C22×C71#45412yes—
89C22×C71#45512yes—
90C22×C71#45611yescharacteristic
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—22no—
94C4×C71—22no—
95C4×C71—44no—
96C4×C71—82no—
97C4×C71—82no—
98C4×C71—161no—
99#8915C7111yesmaximal, characteristic, direct product factor, Sylow
100C2×C71#165811yescharacteristic
101C2×C71#166211yescharacteristic
102C2×C71#166412yes—
103C2×C71#185711yescharacteristic
104C2×C71#186012yes—
105#2382C2×C7111yescharacteristic
106#1992C2×C7111yescharacteristic
107#1836C2×C7112yes—
108C2×#330C2×C7111yescharacteristic
109#1295C2×C7112yes—
110C71#891511yescharacteristic, direct product factor, Sylow
111C2×#125C22×C7111yescharacteristic
112C2×#97C22×C7111yescharacteristic
113C2×#97C22×C7112yes—
114#389—22no—
115#387—22no—
116#387—22no—
117#384C22×C7111yescharacteristic
118#382C22×C7112yes—
119#340—22no—
120#340—22no—
121#340—22no—
122#330C4×C7114yescomplemented
123C22×C8—21no—
124C22×C8—22no—
125#125—22no—
126#125C2×C4×C7112yes—
127#125Q8×C7112yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4D4×C7112yes—
133C2×C4⋊C4D4×C7112yes—
134C2×C4⋊C4D4×C7112yes—
135C2×C42—22no—
136C2×C42C23×C7111yescharacteristic
137#97—22no—
138#97—22no—
139#97C2×C4×C7114yes—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4C2×D4×C7111yescharacteristic
149C22×C4C2×D4×C7112yes—
150C2×C8—22no—
151C2×C8—24no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—41no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—82no—
164C4⋊C4C4⋊C4×C7114yes—
165C42—82no—
166C42C22×C4×C7111yescharacteristic
167C42C2×Q8×C7111yescharacteristic
168C42#41×C7112yes—
169C23#120×C7111yescharacteristic
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4—82no—
185C2×C4C2×C4⋊C4×C7111yescharacteristic
186C2×C4C4×D4×C7112yes—
187C2×C4C22⋊Q8×C7112yes—
188C2×C4#123×C7111yescharacteristic
189C8—42no—
190C8—44no—
191C22#385×C7111yescharacteristic
192C22#454×C7111yescharacteristic
193C22#454×C7112yes—
194C22#455×C7112yes—
195C22#456×C7111yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—82no—
202C4—82no—
203C4—161no—
204C2C71×#165811yescharacteristic
205C2C71×#166211yescharacteristic
206C2C71×#166412yes—
207C2C71×#185711yescharacteristic
208C2C71×#186012yes—
subgroupG/H
center, socle, prime radicalC23×C71#120
commutatorC42C22×C4×C71
Frattini, cosocleC2×C42C23×C71
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
17. #330×C71C4semidirect
99. #8915C71direct (N × H)
110. C71#8915direct (N × H)
122. #330C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#23822
C71×#19922
C71×#18362
C2×#330×C712
C71×#12952
#891571