Finite Group Databasebeta

← C71×#7833 #31828705

Subgroups — C71×#7833 #31828705

Order 18176.

438 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2804C211yesmaximal, characteristic
2C71×#2095C212yesmaximal, complemented
3C71×#2094C211yesmaximal, characteristic
4C71×#1272C212yesmaximal, complemented
5C71×#1270C211yesmaximal, characteristic
6C2×#107×C71—22no—
7C2×#106×C71—21no—
8C2×#103×C71—22no—
9C2×#102×C71—21no—
10#400×C71—21no—
11#393×C71—22no—
12#391×C71C2212yes—
13#390×C71C2211yescharacteristic
14#386×C71—21no—
15#385×C71—22no—
16#384×C71C2211yescharacteristic
17#336×C71C2212yes—
18#336×C71C2211yescharacteristic
19C22×Q8×C71D412yes—
20C22×D4×C71D411yescharacteristic
21C23×C4×C71—21no—
22C22×C8×C71—21no—
23C22×C8×C71—22no—
24C2×C4⋊C4×C71—21no—
25C2×C4⋊C4×C71—22no—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71D411yescharacteristic
28C2×C4⋊C4×C71D412yes—
29C2×C22⋊C4×C71—22no—
30C2×C22⋊C4×C71—22no—
31C2×C22⋊C4×C71—21no—
32C2×C42×C71C2311yescharacteristic
33#107×C71—44no—
34#106×C71—42no—
35#103×C71—44no—
36#102×C71—42no—
37#95×C71—22no—
38#95×C71—21no—
39#95×C71—22no—
40#95×C71—22no—
41C24×C71—21no—
42C2×Q8×C71—22no—
43C2×Q8×C71—22no—
44C2×Q8×C71—42no—
45C2×Q8×C71—42no—
46C2×D4×C71—21no—
47C2×D4×C71—41no—
48C2×D4×C71—41no—
49C2×D4×C71Q1612yes—
50C22×C4×C71—22no—
51C22×C4×C71—21no—
52C22×C4×C71—22no—
53C22×C4×C71—41no—
54C22×C4×C71—82no—
55C22×C4×C71—81no—
56C22×C4×C71C2×D411yescharacteristic
57C22×C4×C71C2×D412yes—
58C2×C8×C71—22no—
59C2×C8×C71—24no—
60C2×C8×C71—41no—
61C2×C8×C71—42no—
62C2×C8×C71—42no—
63C2×C8×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—81no—
71C22⋊C4×C71—42no—
72C22⋊C4×C71—42no—
73C22⋊C4×C71—82no—
74C22⋊C4×C71—81no—
75C42×C71—41no—
76C23×C71—22no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71—41no—
80C23×C71—41no—
81C23×C71#12011yescharacteristic
82Q8×C71—44no—
83Q8×C71—82no—
84D4×C71—42no—
85D4×C71—81no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—84no—
99C2×C4×C71—81no—
100C2×C4×C71—82no—
101C2×C4×C71C2×Q1611yescharacteristic
102C2×C4×C71#13512yes—
103C2×C4×C71#13711yescharacteristic
104C2×C4×C71#13712yes—
105C8×C71—42no—
106C8×C71—44no—
107C22×C71—42no—
108C22×C71—44no—
109C22×C71—82no—
110C22×C71—81no—
111C22×C71—81no—
112C22×C71#45111yescharacteristic
113C22×C71#45212yes—
114C22×C71#45311yescharacteristic
115C22×C71#45312yes—
116C22×C71#45811yescharacteristic
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—82no—
122C4×C71—81no—
123C4×C71—82no—
124C4×C71—161no—
125#7833C7111yesmaximal, characteristic, direct product factor, Sylow
126C2×C71—82no—
127C2×C71#161411yescharacteristic
128C2×C71#161812yes—
129C2×C71#163011yescharacteristic
130C2×C71#163212yes—
131C2×C71#165511yescharacteristic
132#2804C2×C7111yescharacteristic
133#2095C2×C7112yescomplemented
134#2094C2×C7111yescharacteristic
135#1272C2×C7112yescomplemented
136#1270C2×C7111yescharacteristic
137C71#783311yescharacteristic, direct product factor, Sylow
138C2×#107—22no—
139C2×#106—21no—
140C2×#103—22no—
141C2×#102—21no—
142#400—21no—
143#393—22no—
144#391C22×C7112yes—
145#390C22×C7111yescharacteristic
146#386—21no—
147#385—22no—
148#384C22×C7111yescharacteristic
149#336C22×C7111yescharacteristic
150#336C22×C7112yes—
151C22×Q8D4×C7112yes—
152C22×D4D4×C7111yescharacteristic
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4D4×C7112yes—
160C2×C4⋊C4D4×C7111yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42C23×C7111yescharacteristic
165#107—44no—
166#106—42no—
167#103—44no—
168#102—42no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95—22no—
173C24—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4Q16×C7112yes—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×D4×C7111yescharacteristic
189C22×C4C2×D4×C7112yes—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#120×C7111yescharacteristic
214Q8—44no—
215Q8—82no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4C2×Q16×C7111yescharacteristic
234C2×C4#135×C7112yes—
235C2×C4#137×C7111yescharacteristic
236C2×C4#137×C7112yes—
237C8—42no—
238C8—44no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#451×C7111yescharacteristic
245C22#452×C7112yes—
246C22#453×C7111yescharacteristic
247C22#453×C7112yes—
248C22#458×C7111yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—81no—
255C4—82no—
256C4—161no—
257C2—82no—
258C2C71×#161411yescharacteristic
259C2C71×#161812yes—
260C2C71×#163011yescharacteristic
261C2C71×#163212yes—
262C2C71×#165511yescharacteristic
subgroupG/H
center, socleC23×C71#120
commutator, Frattini, cosocleC2×C42C23×C71
prime radicalC22×D4×C71D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2095C2semidirect
4. C71×#1272C2semidirect
125. #7833C71direct (N × H)
133. #2095C2×C71semidirect
135. #1272C2×C71semidirect
137. C71#7833direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#28042
C71×#20952
C71×#20942
C71×#12722
C71×#12702
#783371