Finite Group Databasebeta

← C71×#9977 #31830849

Subgroups — C71×#9977 #31830849

Order 18176.

544 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3186C211yesmaximal, characteristic
2C71×#3094C212yesmaximal, complemented
3C71×#1818C211yesmaximal, characteristic, complemented
4C71×#1471C212yesmaximal, complemented
5C71×#1293C211yesmaximal, characteristic, complemented
6C2×#121×C71C2212yes—
7C2×#117×C71C2211yescharacteristic
8#482×C71—24no—
9#481×C71—22no—
10#480×C71—22no—
11#480×C71—22no—
12#466×C71—22no—
13#447×C71—22no—
14#428×C71—22no—
15#418×C71—22no—
16#407×C71C2212yes—
17C2×C22⋊C8×C71C2211yescharacteristic
18C2×C22⋊C8×C71C2211yescharacteristic
19#336×C71—22no—
20#323×C71—24no—
21C22×D4×C71—22no—
22C22×D4×C71D412yes—
23C23×C4×C71C2311yescharacteristic
24C2×D8×C71—22no—
25C2×M16×C71—22no—
26C22×C8×C71—21no—
27C22×C8×C71D412yes—
28#121×C71—22no—
29#121×C71—22no—
30#121×C71—22no—
31#121×C71—22no—
32#117×C71—22no—
33#117×C71—22no—
34C2×C4⋊C4×C71D412yes—
35C2×C4⋊C4×C71D411yescharacteristic
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C22⋊C4×C71—21no—
38C2×C22⋊C4×C71—22no—
39C2×C42×C71—21no—
40#107×C71—24no—
41#107×C71—22no—
42#106×C71—24no—
43#102×C71—24no—
44#102×C71—24no—
45#102×C71—24no—
46C22⋊C8×C71—22no—
47C22⋊C8×C71—22no—
48C22⋊C8×C71—22no—
49C22⋊C8×C71D414yes—
50C24×C71—22no—
51C24×C71#4111yescharacteristic
52C2×D4×C71—22no—
53C2×D4×C71—22no—
54C2×D4×C71—44no—
55C2×D4×C71—42no—
56C2×D4×C71—42no—
57C2×D4×C71—84no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—22no—
61C22×C4×C71—42no—
62C22×C4×C71C2×D412yes—
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71C2×D411yescharacteristic
65C22×C4×C71C2×D412yes—
66D8×C71—82no—
67M16×C71—44no—
68C2×C8×C71—22no—
69C2×C8×C71—24no—
70C2×C8×C71—42no—
71C2×C8×C71—42no—
72C2×C8×C71—41no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—22no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—22no—
77C22⋊C4×C71—42no—
78C22⋊C4×C71—42no—
79C22⋊C4×C71—42no—
80C42×C71—41no—
81C42×C71—41no—
82C23×C71—22no—
83C23×C71—41no—
84C23×C71—41no—
85C23×C71—44no—
86C23×C71—42no—
87C23×C71—82no—
88C23×C71—82no—
89C23×C71#12112yes—
90C23×C71#12111yescharacteristic
91D4×C71—44no—
92D4×C71—44no—
93D4×C71—84no—
94D4×C71—84no—
95C2×C4×C71—22no—
96C2×C4×C71—24no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—82no—
105C2×C4×C71#12012yes—
106C2×C4×C71#12011yescharacteristic
107C2×C4×C71#12711yescharacteristic
108C8×C71—44no—
109C8×C71—42no—
110C8×C71—81no—
111C22×C71—21no—
112C22×C71—21no—
113C22×C71—22no—
114C22×C71—22no—
115C22×C71—22no—
116C22×C71—82no—
117C22×C71—82no—
118C22×C71—82no—
119C22×C71—82no—
120C22×C71—164no—
121C22×C71#39211yescharacteristic
122C22×C71#47212yes—
123C4×C71—22no—
124C4×C71—21no—
125C4×C71—81no—
126C4×C71—81no—
127C4×C71—82no—
128C4×C71#45812yes—
129#9977C7111yesmaximal, characteristic, direct product factor, Sylow
130C2×C71—22no—
131C2×C71—42no—
132C2×C71—162no—
133C2×C71#201711yescharacteristic
134C2×C71#201811yescharacteristic
135C2×C71#202211yescharacteristic
136#3186C2×C7111yescharacteristic
137#3094C2×C7112yescomplemented
138#1818C2×C7111yescharacteristic, complemented
139#1471C2×C7112yescomplemented
140#1293C2×C7111yescharacteristic, complemented
141C71#997711yescharacteristic, direct product factor, Sylow
142C2×#121C22×C7112yes—
143C2×#117C22×C7111yescharacteristic
144#482—24no—
145#481—22no—
146#480—22no—
147#480—22no—
148#466—22no—
149#447—22no—
150#428—22no—
151#418—22no—
152#407C22×C7112yes—
153C2×C22⋊C8C22×C7111yescharacteristic
154C2×C22⋊C8C22×C7111yescharacteristic
155#336—22no—
156#323—24no—
157C22×D4—22no—
158C22×D4D4×C7112yes—
159C23×C4C23×C7111yescharacteristic
160C2×D8—22no—
161C2×M16—22no—
162C22×C8—21no—
163C22×C8D4×C7112yes—
164#121—22no—
165#121—22no—
166#121—22no—
167#121—22no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4D4×C7111yescharacteristic
171C2×C4⋊C4D4×C7111yescharacteristic
172C2×C4⋊C4D4×C7112yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C42—21no—
176#107—24no—
177#107—22no—
178#106—24no—
179#102—24no—
180#102—24no—
181#102—24no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8D4×C7114yes—
186C24—22no—
187C24#41×C7111yescharacteristic
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—84no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4C2×D4×C7112yes—
199C22×C4C2×D4×C7111yescharacteristic
200C22×C4C2×D4×C7111yescharacteristic
201C22×C4C2×D4×C7112yes—
202D8—82no—
203M16—44no—
204C2×C8—22no—
205C2×C8—24no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—41no—
217C42—41no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—44no—
222C23—42no—
223C23—82no—
224C23—82no—
225C23#121×C7112yes—
226C23#121×C7111yescharacteristic
227D4—44no—
228D4—44no—
229D4—84no—
230D4—84no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4#120×C7112yes—
242C2×C4#120×C7111yescharacteristic
243C2×C4#127×C7111yescharacteristic
244C8—44no—
245C8—42no—
246C8—81no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—164no—
257C22#392×C7111yescharacteristic
258C22#472×C7112yes—
259C4—22no—
260C4—21no—
261C4—81no—
262C4—81no—
263C4—82no—
264C4#458×C7112yes—
265C2—22no—
266C2—42no—
267C2—162no—
268C2C71×#201711yescharacteristic
269C2C71×#201811yescharacteristic
270C2C71×#202211yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC23×C4C23×C71
prime radicalC71×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#3094C2semidirect
3. C71×#1818C2semidirect
4. C71×#1471C2semidirect
5. C71×#1293C2semidirect
129. #9977C71direct (N × H)
137. #3094C2×C71semidirect
138. #1818C2×C71semidirect
139. #1471C2×C71semidirect
140. #1293C2×C71semidirect
141. C71#9977direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#31862
C71×#30942
C71×#18182
C71×#14712
C71×#12932
#997771