Finite Group Databasebeta

← C71×#12173 #31833045

Subgroups — C71×#12173 #31833045

Order 18176.

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

normal subgroups only

This group has 384 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×#3253C211yesmaximal, characteristic, complemented
2C71×#2190C212yesmaximal, complemented
3C71×#2050C211yesmaximal, characteristic, complemented
4C71×#1341C212yesmaximal, complemented
5C71×#1327C211yesmaximal, characteristic, complemented
6#575×C71C2211yescharacteristic
7#532×C71—21no—
8#509×C71—22no—
9#476×C71—22no—
10#469×C71—21no—
11#469×C71C2212yes—
12#459×C71—22no—
13#423×C71C2211yescharacteristic
14C2×#104×C71—21no—
15#413×C71—21no—
16#360×C71—22no—
17#349×C71C2212yes—
18#338×C71C2211yescharacteristic
19C2×#41×C71—21no—
20C2×#41×C71—41no—
21#137×C71—41no—
22AGL(1,ℤ/8)×C71—41no—
23#135×C71—22no—
24C2×D8×C71—21no—
25C2×D8×C71D412yes—
26C2×M16×C71—21no—
27C2×M16×C71D411yescharacteristic
28C2×M16×C71C2311yescharacteristic
29C4⋊Q8×C71—21no—
30#127×C71—21no—
31#124×C71—41no—
32#121×C71—22no—
33#121×C71—41no—
34C4×D4×C71—22no—
35C4×D4×C71—41no—
36C2×C42×C71D411yescharacteristic
37SD32×C71—42no—
38D16×C71—42no—
39M32×C71—22no—
40#106×C71—21no—
41#106×C71D412yes—
42C4⋊C8×C71—22no—
43#104×C71—41no—
44#104×C71—41no—
45#102×C71—22no—
46#102×C71—22no—
47#101×C71—41no—
48#100×C71—41no—
49#41×C71—41no—
50#41×C71—41no—
51#41×C71—41no—
52#41×C71—41no—
53#41×C71—84no—
54C2×Q8×C71—21no—
55C2×Q8×C71—41no—
56C2×D4×C71—21no—
57C2×D4×C71—22no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71—42no—
61C2×D4×C71—41no—
62C22×C4×C71—21no—
63C22×C4×C71—42no—
64C22×C4×C71—41no—
65C22×C4×C71C2×D411yescharacteristic
66Q16×C71—22no—
67SD16×C71—42no—
68D8×C71—22no—
69D8×C71—42no—
70D8×C71—81no—
71M16×C71—41no—
72M16×C71—41no—
73M16×C71—41no—
74M16×C71—41no—
75C2×C8×C71—21no—
76C2×C8×C71—41no—
77C2×C8×C71C2×D412yes—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—41no—
80C22⋊C4×C71—42no—
81C22⋊C4×C71—42no—
82C42×C71—21no—
83C42×C71D811yescharacteristic
84C42×C71D811yescharacteristic
85C16×C71—42no—
86C23×C71—41no—
87C23×C71—42no—
88C23×C71AGL(1,ℤ/8)11yescharacteristic
89Q8×C71—41no—
90Q8×C71—41no—
91Q8×C71—82no—
92D4×C71—41no—
93D4×C71—42no—
94D4×C71—41no—
95D4×C71—42no—
96D4×C71—82no—
97D4×C71—82no—
98D4×C71—81no—
99D4×C71—84no—
100D4×C71—81no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—42no—
106C2×C4×C71—41no—
107C2×C4×C71—42no—
108C2×C4×C71—81no—
109C2×C4×C71—84no—
110C2×C4×C71—81no—
111C2×C4×C71#12011yescharacteristic
112C2×C4×C71C2×D811yescharacteristic
113C8×C71—22no—
114C8×C71—41no—
115C8×C71—81no—
116C22×C71—21no—
117C22×C71—41no—
118C22×C71—82no—
119C22×C71—82no—
120C22×C71—81no—
121C22×C71—82no—
122C22×C71—161no—
123C22×C71#44711yescharacteristic
124C4×C71—22no—
125C4×C71—21no—
126C4×C71—41no—
127C4×C71—82no—
128C4×C71—81no—
129C4×C71#45411yescharacteristic
130C4×C71B(4,𝔽2)11yescharacteristic
131#12173C7111yesmaximal, characteristic, direct product factor, Sylow
132C2×C71—21no—
133C2×C71—41no—
134C2×C71—162no—
135C2×C71—161no—
136C2×C71#159811yescharacteristic
137#3253C2×C7111yescharacteristic, complemented
138#2190C2×C7112yescomplemented
139#2050C2×C7111yescharacteristic, complemented
140#1341C2×C7112yescomplemented
141#1327C2×C7111yescharacteristic, complemented
142C71#1217311yescharacteristic, direct product factor, Sylow
143#575C22×C7111yescharacteristic
144#532—21no—
145#509—22no—
146#476—22no—
147#469—21no—
148#469C22×C7112yes—
149#459—22no—
150#423C22×C7111yescharacteristic
151C2×#104—21no—
152#413—21no—
153#360—22no—
154#349C22×C7112yes—
155#338C22×C7111yescharacteristic
156C2×#41—21no—
157C2×#41—41no—
158#137—41no—
159AGL(1,ℤ/8)—41no—
160#135—22no—
161C2×D8—21no—
162C2×D8D4×C7112yes—
163C2×M16—21no—
164C2×M16D4×C7111yescharacteristic
165C2×M16C23×C7111yescharacteristic
166C4⋊Q8—21no—
167#127—21no—
168#124—41no—
169#121—22no—
170#121—41no—
171C4×D4—22no—
172C4×D4—41no—
173C2×C42D4×C7111yescharacteristic
174SD32—42no—
175D16—42no—
176M32—22no—
177#106—21no—
178#106D4×C7112yes—
179C4⋊C8—22no—
180#104—41no—
181#104—41no—
182#102—22no—
183#102—22no—
184#101—41no—
185#100—41no—
186#41—41no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—84no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—41no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4C2×D4×C7111yescharacteristic
203Q16—22no—
204SD16—42no—
205D8—22no—
206D8—42no—
207D8—81no—
208M16—41no—
209M16—41no—
210M16—41no—
211M16—41no—
212C2×C8—21no—
213C2×C8—41no—
214C2×C8C2×D4×C7112yes—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C42—21no—
220C42D8×C7111yescharacteristic
221C42D8×C7111yescharacteristic
222C16—42no—
223C23—41no—
224C23—42no—
225C23AGL(1,ℤ/8)×C7111yescharacteristic
226Q8—41no—
227Q8—41no—
228Q8—82no—
229D4—41no—
230D4—42no—
231D4—41no—
232D4—42no—
233D4—82no—
234D4—82no—
235D4—81no—
236D4—84no—
237D4—81no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—84no—
247C2×C4—81no—
248C2×C4#120×C7111yescharacteristic
249C2×C4C2×D8×C7111yescharacteristic
250C8—22no—
251C8—41no—
252C8—81no—
253C22—21no—
254C22—41no—
255C22—82no—
256C22—82no—
257C22—81no—
258C22—82no—
259C22—161no—
260C22#447×C7111yescharacteristic
261C4—22no—
262C4—21no—
263C4—41no—
264C4—82no—
265C4—81no—
266C4#454×C7111yescharacteristic
267C4B(4,𝔽2)×C7111yescharacteristic
268C2—21no—
269C2—41no—
270C2—162no—
271C2—161no—
272C2C71×#159811yescharacteristic
subgroupG/H
center, socleC2×C71#1598
commutator, Frattini, cosocleC2×M16C23×C71
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×#3253C2semidirect
2. C71×#2190C2semidirect
3. C71×#2050C2semidirect
4. C71×#1341C2semidirect
5. C71×#1327C2semidirect
131. #12173C71direct (N × H)
137. #3253C2×C71semidirect
138. #2190C2×C71semidirect
139. #2050C2×C71semidirect
140. #1341C2×C71semidirect
141. #1327C2×C71semidirect
142. C71#12173direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32532
C71×#21902
C71×#20502
C71×#13412
C71×#13272
#1217371