Finite Group Databasebeta

← C71×#9967 #31830839

Subgroups — C71×#9967 #31830839

Order 18176.

804 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#451×C71C212yesmaximal
2C71×#2659C211yesmaximal, characteristic
3C2×#328×C71C212yesmaximal
4C71×#1446C211yesmaximal, characteristic
5C71×#1280C211yesmaximal, characteristic
6C23×Q8×C71C2211yescharacteristic
7C22×Q16×C71—22no—
8C2×C22⋊Q8×C71C2212yes—
9#451×C71—24no—
10#451×C71—24no—
11C2×#103×C71—22no—
12C2×C22⋊C8×C71C2212yes—
13#398×C71—22no—
14#395×C71—22no—
15#380×C71—22no—
16C2×#95×C71—22no—
17C2×#95×C71C2211yescharacteristic
18C2×#95×C71C2211yescharacteristic
19#328×C71—24no—
20C22×Q8×C71—21no—
21C22×Q8×C71—22no—
22C22×Q8×C71—22no—
23C22×Q8×C71—44no—
24C22×Q8×C71D412yes—
25C22×Q8×C71D412yes—
26C23×C4×C71—21no—
27C23×C4×C71—21no—
28C23×C4×C71D412yes—
29C23×C4×C71C2311yescharacteristic
30C2×Q16×C71—28no—
31C2×Q16×C71—48no—
32C22×C8×C71—22no—
33C22⋊Q8×C71—22no—
34C22⋊Q8×C71—22no—
35C22⋊Q8×C71—22no—
36C22⋊Q8×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71D412yes—
41C2×C22⋊C4×C71—22no—
42#103×C71—28no—
43C22⋊C8×C71—22no—
44C22⋊C8×C71D414yes—
45#95×C71—24no—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71—44no—
49C24×C71C2×D411yescharacteristic
50C2×Q8×C71—28no—
51C2×Q8×C71—24no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71—44no—
55C2×Q8×C71—44no—
56C2×Q8×C71—44no—
57C2×Q8×C71—44no—
58C2×Q8×C71—48no—
59C2×Q8×C71—42no—
60C22×C4×C71—24no—
61C22×C4×C71—22no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—41no—
70C22×C4×C71—41no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—42no—
77C22×C4×C71Q1614yescomplemented
78C22×C4×C71C2×D412yes—
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D412yes—
81C22×C4×C71#4111yescharacteristic
82Q16×C71—416no—
83C2×C8×C71—24no—
84C2×C8×C71—44no—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—82no—
90C4⋊C4×C71—82no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—22no—
96C23×C71#12011yescharacteristic
97C23×C71#12111yescharacteristic
98C23×C71#12711yescharacteristic
99C23×C71C2×Q1612yes—
100C23×C71#13712yes—
101Q8×C71—28no—
102Q8×C71—44no—
103Q8×C71—48no—
104Q8×C71—44no—
105Q8×C71—82no—
106C2×C4×C71—24no—
107C2×C4×C71—24no—
108C2×C4×C71—24no—
109C2×C4×C71—41no—
110C2×C4×C71—42no—
111C2×C4×C71—41no—
112C2×C4×C71—42no—
113C2×C4×C71—44no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—44no—
118C2×C4×C71—44no—
119C2×C4×C71—44no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—44no—
126C2×C4×C71—82no—
127C2×C4×C71—82no—
128C2×C4×C71—81no—
129C2×C4×C71—81no—
130C2×C4×C71#12012yes—
131C2×C4×C71#12112yes—
132C8×C71—44no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—22no—
136C22×C71—22no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71#39211yescharacteristic
142C22×C71#45112yes—
143C22×C71#45511yescharacteristic
144C22×C71#45611yescharacteristic
145C22×C71B(4,𝔽2)12yes—
146C22×C71#46212yes—
147C22×C71#49411yescharacteristic
148C22×C71#49711yescharacteristic
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—44no—
152C4×C71—44no—
153C4×C71—82no—
154C4×C71—82no—
155C4×C71—82no—
156#9967C7111yesmaximal, characteristic, direct product factor, Sylow
157C2×C71—21no—
158C2×C71—22no—
159C2×C71—21no—
160C2×C71#160212yes—
161C2×C71#160512yes—
162C2×C71#200711yescharacteristic
163C2×C71#201611yescharacteristic
164C2×C71#202111yescharacteristic
165C2×#451C2×C7112yes—
166#2659C2×C7111yescharacteristic
167C2×#328C2×C7112yes—
168#1446C2×C7111yescharacteristic
169#1280C2×C7111yescharacteristic
170C71#996711yescharacteristic, direct product factor, Sylow
171C23×Q8C22×C7111yescharacteristic
172C22×Q16—22no—
173C2×C22⋊Q8C22×C7112yes—
174#451—24no—
175#451—24no—
176C2×#103—22no—
177C2×C22⋊C8C22×C7112yes—
178#398—22no—
179#395—22no—
180#380—22no—
181C2×#95—22no—
182C2×#95C22×C7111yescharacteristic
183C2×#95C22×C7111yescharacteristic
184#328—24no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8D4×C7112yes—
190C22×Q8D4×C7112yes—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4D4×C7112yes—
194C23×C4C23×C7111yescharacteristic
195C2×Q16—28no—
196C2×Q16—48no—
197C22×C8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4D4×C7112yes—
206C2×C22⋊C4—22no—
207#103—28no—
208C22⋊C8—22no—
209C22⋊C8D4×C7114yes—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—44no—
214C24C2×D4×C7111yescharacteristic
215C2×Q8—28no—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4Q16×C7114yescomplemented
243C22×C4C2×D4×C7112yes—
244C22×C4C2×D4×C7111yescharacteristic
245C22×C4C2×D4×C7112yes—
246C22×C4#41×C7111yescharacteristic
247Q16—416no—
248C2×C8—24no—
249C2×C8—44no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#120×C7111yescharacteristic
262C23#121×C7111yescharacteristic
263C23#127×C7111yescharacteristic
264C23C2×Q16×C7112yes—
265C23#137×C7112yes—
266Q8—28no—
267Q8—44no—
268Q8—48no—
269Q8—44no—
270Q8—82no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4#120×C7112yes—
296C2×C4#121×C7112yes—
297C8—44no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22#392×C7111yescharacteristic
307C22#451×C7112yes—
308C22#455×C7111yescharacteristic
309C22#456×C7111yescharacteristic
310C22B(4,𝔽2)×C7112yes—
311C22#462×C7112yes—
312C22#494×C7111yescharacteristic
313C22#497×C7111yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—82no—
319C4—82no—
320C4—82no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2C71×#160212yes—
325C2C71×#160512yes—
326C2C71×#200711yescharacteristic
327C2C71×#201611yescharacteristic
328C2C71×#202111yescharacteristic
subgroupG/H
center, socleC23×C71#120
commutator, Frattini, cosocleC23×C4C23×C71
prime radicalC24×C71C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
77. C22×C4×C71Q16semidirect
156. #9967C71direct (N × H)
170. C71#9967direct (N × H)
242. C22×C4Q16×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#451×C712
C71×#26592
C2×#328×C712
C71×#14462
C71×#12802
#996771