Finite Group Databasebeta

← C71×#13667 #31834539

Subgroups — C71×#13667 #31834539

Order 18176.

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

normal subgroups only

This group has 294 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×#3414C211yesmaximal, characteristic, complemented
2C71×#2263C212yesmaximal, complemented
3C71×#2218C211yesmaximal, characteristic, complemented
4C71×#1418C212yesmaximal, complemented
5C71×#1400C211yesmaximal, characteristic, complemented
6C2×#135×C71—21no—
7#510×C71—22no—
8#509×C71—22no—
9#508×C71—21no—
10#508×C71C2212yes—
11C2×D16×C71C2211yescharacteristic, complemented
12C2×M32×C71C2211yescharacteristic
13#466×C71—21no—
14SD64×C71—22no—
15D32×C71—22no—
16M64×C71C2212yescomplemented
17#367×C71C2211yescharacteristic, complemented
18#357×C71—21no—
19C2×#41×C71—41no—
20#135×C71—22no—
21#135×C71—22no—
22#135×C71—41no—
23#135×C71—41no—
24C2×Q16×C71—21no—
25C2×SD16×C71—41no—
26C2×D8×C71—21no—
27C2×D8×C71—21no—
28C22×C8×C71D411yescharacteristic
29#121×C71—41no—
30Q32×C71—21no—
31Q32×C71D412yescomplemented
32SD32×C71—22no—
33SD32×C71—22no—
34D16×C71—21no—
35D16×C71—22no—
36D16×C71D412yescomplemented
37M32×C71—22no—
38C2×C16×C71D411yescharacteristic
39C2×C16×C71C2311yescharacteristic
40#107×C71—21no—
41#102×C71—41no—
42C32×C71—22no—
43#41×C71—42no—
44#41×C71—42no—
45#41×C71—81no—
46#41×C71—81no—
47C2×Q8×C71—41no—
48C2×D4×C71—41no—
49C2×D4×C71—41no—
50C2×D4×C71—81no—
51C2×D4×C71—81no—
52C22×C4×C71—81no—
53C22×C4×C71D811yescharacteristic
54Q16×C71—22no—
55Q16×C71—41no—
56SD16×C71—42no—
57SD16×C71—42no—
58D8×C71—22no—
59D8×C71—41no—
60D8×C71—41no—
61D8×C71—41no—
62C2×C8×C71—21no—
63C2×C8×C71—21no—
64C2×C8×C71D811yescharacteristic
65C2×C8×C71C2×D411yescharacteristic
66C4⋊C4×C71—41no—
67C22⋊C4×C71—81no—
68C16×C71—21no—
69C16×C71C2×D412yes—
70C23×C71—81no—
71C23×C71—81no—
72C23×C71D1611yescharacteristic
73Q8×C71—42no—
74Q8×C71—81no—
75D4×C71—42no—
76D4×C71—81no—
77D4×C71—82no—
78D4×C71—82no—
79D4×C71—81no—
80D4×C71—81no—
81D4×C71—162no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—82no—
85C2×C4×C71—81no—
86C2×C4×C71—82no—
87C2×C4×C71—81no—
88C2×C4×C71D1611yescharacteristic
89C2×C4×C71C2×D811yescharacteristic
90C8×C71—21no—
91C8×C71#12011yescharacteristic
92C8×C71AGL(1,ℤ/8)11yescharacteristic
93C22×C71—21no—
94C22×C71—41no—
95C22×C71—82no—
96C22×C71—161no—
97C22×C71—161no—
98C22×C71—162no—
99C22×C71—161no—
100C22×C71C2×D1611yescharacteristic
101C4×C71—21no—
102C4×C71—82no—
103C4×C71—161no—
104C4×C71#44711yescharacteristic
105C4×C71#50911yescharacteristic
106#13667C7111yesmaximal, characteristic, direct product factor, Sylow
107C2×C71—21no—
108C2×C71—41no—
109C2×C71—162no—
110C2×C71—321no—
111C2×C71#218411yescharacteristic
112#3414C2×C7111yescharacteristic, complemented
113#2263C2×C7112yescomplemented
114#2218C2×C7111yescharacteristic, complemented
115#1418C2×C7112yescomplemented
116#1400C2×C7111yescharacteristic, complemented
117C71#1366711yescharacteristic, direct product factor, Sylow
118C2×#135—21no—
119#510—22no—
120#509—22no—
121#508—21no—
122#508C22×C7112yes—
123C2×D16C22×C7111yescharacteristic, complemented
124C2×M32C22×C7111yescharacteristic
125#466—21no—
126SD64—22no—
127D32—22no—
128M64C22×C7112yescomplemented
129#367C22×C7111yescharacteristic, complemented
130#357—21no—
131C2×#41—41no—
132#135—22no—
133#135—22no—
134#135—41no—
135#135—41no—
136C2×Q16—21no—
137C2×SD16—41no—
138C2×D8—21no—
139C2×D8—21no—
140C22×C8D4×C7111yescharacteristic
141#121—41no—
142Q32—21no—
143Q32D4×C7112yescomplemented
144SD32—22no—
145SD32—22no—
146D16—21no—
147D16—22no—
148D16D4×C7112yescomplemented
149M32—22no—
150C2×C16D4×C7111yescharacteristic
151C2×C16C23×C7111yescharacteristic
152#107—21no—
153#102—41no—
154C32—22no—
155#41—42no—
156#41—42no—
157#41—81no—
158#41—81no—
159C2×Q8—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—81no—
163C2×D4—81no—
164C22×C4—81no—
165C22×C4D8×C7111yescharacteristic
166Q16—22no—
167Q16—41no—
168SD16—42no—
169SD16—42no—
170D8—22no—
171D8—41no—
172D8—41no—
173D8—41no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8D8×C7111yescharacteristic
177C2×C8C2×D4×C7111yescharacteristic
178C4⋊C4—41no—
179C22⋊C4—81no—
180C16—21no—
181C16C2×D4×C7112yes—
182C23—81no—
183C23—81no—
184C23D16×C7111yescharacteristic
185Q8—42no—
186Q8—81no—
187D4—42no—
188D4—81no—
189D4—82no—
190D4—82no—
191D4—81no—
192D4—81no—
193D4—162no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—81no—
200C2×C4D16×C7111yescharacteristic
201C2×C4C2×D8×C7111yescharacteristic
202C8—21no—
203C8#120×C7111yescharacteristic
204C8AGL(1,ℤ/8)×C7111yescharacteristic
205C22—21no—
206C22—41no—
207C22—82no—
208C22—161no—
209C22—161no—
210C22—162no—
211C22—161no—
212C22C2×D16×C7111yescharacteristic
213C4—21no—
214C4—82no—
215C4—161no—
216C4#447×C7111yescharacteristic
217C4#509×C7111yescharacteristic
218C2—21no—
219C2—41no—
220C2—162no—
221C2—321no—
222C2C71×#218411yescharacteristic
subgroupG/H
center, socleC2×C71#2184
commutator, Frattini, cosocleC2×C16C23×C71
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C2×C4×C71 ◁ C22×C8×C71 ◁ C2×M32×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#3414C2semidirect
2. C71×#2263C2semidirect
3. C71×#2218C2semidirect
4. C71×#1418C2semidirect
5. C71×#1400C2semidirect
11. C2×D16×C71C22semidirect
16. M64×C71C22semidirect
17. #367×C71C22semidirect
31. Q32×C71D4semidirect
36. D16×C71D4semidirect
106. #13667C71direct (N × H)
112. #3414C2×C71semidirect
113. #2263C2×C71semidirect
114. #2218C2×C71semidirect
115. #1418C2×C71semidirect
116. #1400C2×C71semidirect
117. C71#13667direct (N × H)
123. C2×D16C22×C71semidirect
128. M64C22×C71semidirect
129. #367C22×C71semidirect
143. Q32D4×C71semidirect
148. D16D4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#34142
C71×#22632
C71×#22182
C71×#14182
C71×#14002
#1366771