Finite Group Databasebeta

← C71×#9993 #31830865

Subgroups — C71×#9993 #31830865

Order 18176.

536 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3193C211yesmaximal, characteristic
2C2×#481×C71C211yesmaximal, characteristic
3C2×#483×C71C211yesmaximal, characteristic, complemented
4C71×#1821C211yesmaximal, characteristic, complemented
5C71×#1472C211yesmaximal, characteristic, complemented
6C71×#1471C211yesmaximal, characteristic
7C71×#1298C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C71C2211yescharacteristic
9C2×#121×C71C2211yescharacteristic
10C22×C4⋊C4×C71C2211yescharacteristic
11#484×C71—21no—
12#483×C71—22no—
13#483×C71—22no—
14#481×C71—22no—
15#481×C71—22no—
16#480×C71—21no—
17#468×C71—22no—
18#450×C71—21no—
19#448×C71—21no—
20C2×#106×C71—21no—
21C2×#106×C71—21no—
22C2×#103×C71—21no—
23C2×#102×C71—21no—
24#407×C71C2211yescharacteristic
25#407×C71C2211yescharacteristic
26C2×C22⋊C8×C71C2211yescharacteristic
27C2×C22⋊C8×C71C2211yescharacteristic
28#343×C71—22no—
29#324×C71—22no—
30#323×C71—22no—
31C22×Q8×C71D411yescharacteristic
32C22×D4×C71—21no—
33C22×D4×C71D411yescharacteristic
34C23×C4×C71—21no—
35C23×C4×C71C2311yescharacteristic
36C2×SD16×C71—22no—
37C2×M16×C71—21no—
38C2×M16×C71D412yes—
39C22×C8×C71—21no—
40C22×C8×C71—21no—
41C22⋊Q8×C71—21no—
42C22⋊Q8×C71—21no—
43C22⋊Q8×C71—21no—
44C22⋊Q8×C71—21no—
45#121×C71—21no—
46#121×C71—21no—
47#121×C71—21no—
48#121×C71—21no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71—21no—
51C2×C4⋊C4×C71—21no—
52C2×C4⋊C4×C71—21no—
53C2×C4⋊C4×C71—21no—
54C2×C4⋊C4×C71—21no—
55C2×C4⋊C4×C71D411yescharacteristic
56C2×C4⋊C4×C71D411yescharacteristic
57C2×C4⋊C4×C71D412yes—
58C2×C22⋊C4×C71—21no—
59C2×C22⋊C4×C71—21no—
60#107×C71—22no—
61#106×C71—24no—
62#106×C71—24no—
63#103×C71—24no—
64#103×C71—22no—
65#102×C71—24no—
66#102×C71—22no—
67C22⋊C8×C71—21no—
68C22⋊C8×C71—21no—
69C22⋊C8×C71—21no—
70C22⋊C8×C71—21no—
71C22⋊C8×C71—21no—
72C22⋊C8×C71—21no—
73C22⋊C8×C71D412yes—
74C22⋊C8×C71D412yes—
75C24×C71—21no—
76C24×C71#4111yescharacteristic
77C2×Q8×C71—21no—
78C2×Q8×C71—21no—
79C2×Q8×C71—42no—
80C2×D4×C71—21no—
81C2×D4×C71—21no—
82C2×D4×C71—42no—
83C2×D4×C71—41no—
84C2×D4×C71—41no—
85C2×D4×C71—82no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—22no—
90C22×C4×C71—42no—
91C22×C4×C71—41no—
92C22×C4×C71—41no—
93C22×C4×C71—82no—
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71C2×D411yescharacteristic
98C22×C4×C71C2×D411yescharacteristic
99C22×C4×C71C2×D411yescharacteristic
100SD16×C71—82no—
101M16×C71—42no—
102M16×C71—42no—
103C2×C8×C71—22no—
104C2×C8×C71—22no—
105C2×C8×C71—22no—
106C2×C8×C71—42no—
107C2×C8×C71—42no—
108C2×C8×C71—41no—
109C4⋊C4×C71—22no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—22no—
112C4⋊C4×C71—21no—
113C4⋊C4×C71—21no—
114C4⋊C4×C71—21no—
115C4⋊C4×C71—41no—
116C4⋊C4×C71—41no—
117C4⋊C4×C71—41no—
118C4⋊C4×C71—41no—
119C22⋊C4×C71—41no—
120C22⋊C4×C71—41no—
121C22⋊C4×C71—41no—
122C22⋊C4×C71—41no—
123C23×C71—22no—
124C23×C71—22no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71—42no—
128C23×C71—81no—
129C23×C71—81no—
130C23×C71#12111yescharacteristic
131C23×C71#12111yescharacteristic
132C23×C71#12111yescharacteristic
133Q8×C71—42no—
134Q8×C71—42no—
135D4×C71—42no—
136D4×C71—42no—
137D4×C71—82no—
138D4×C71—82no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—22no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—42no—
149C2×C4×C71—42no—
150C2×C4×C71—82no—
151C2×C4×C71—82no—
152C2×C4×C71—82no—
153C2×C4×C71—81no—
154C2×C4×C71—81no—
155C2×C4×C71#12011yescharacteristic
156C2×C4×C71#12011yescharacteristic
157C2×C4×C71#12011yescharacteristic
158C2×C4×C71#12711yescharacteristic
159C8×C71—42no—
160C8×C71—42no—
161C8×C71—42no—
162C8×C71—81no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—44no—
171C22×C71—81no—
172C22×C71—81no—
173C22×C71—81no—
174C22×C71—162no—
175C22×C71#39211yescharacteristic
176C22×C71#47112yes—
177C22×C71#47112yes—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—21no—
181C4×C71—81no—
182C4×C71—82no—
183C4×C71—81no—
184C4×C71—81no—
185C4×C71#45812yes—
186#9993C7111yesmaximal, characteristic, direct product factor, Sylow
187C2×C71—22no—
188C2×C71—22no—
189C2×C71—41no—
190C2×C71—161no—
191C2×C71#201911yescharacteristic
192C2×C71#201911yescharacteristic
193C2×C71#202211yescharacteristic
194#3193C2×C7111yescharacteristic
195C2×#481C2×C7111yescharacteristic
196C2×#483C2×C7111yescharacteristic, complemented
197#1821C2×C7111yescharacteristic, complemented
198#1472C2×C7111yescharacteristic, complemented
199#1471C2×C7111yescharacteristic
200#1298C2×C7111yescharacteristic, complemented
201C71#999311yescharacteristic, direct product factor, Sylow
202C2×C22⋊Q8C22×C7111yescharacteristic
203C2×#121C22×C7111yescharacteristic
204C22×C4⋊C4C22×C7111yescharacteristic
205#484—21no—
206#483—22no—
207#483—22no—
208#481—22no—
209#481—22no—
210#480—21no—
211#468—22no—
212#450—21no—
213#448—21no—
214C2×#106—21no—
215C2×#106—21no—
216C2×#103—21no—
217C2×#102—21no—
218#407C22×C7111yescharacteristic
219#407C22×C7111yescharacteristic
220C2×C22⋊C8C22×C7111yescharacteristic
221C2×C22⋊C8C22×C7111yescharacteristic
222#343—22no—
223#324—22no—
224#323—22no—
225C22×Q8D4×C7111yescharacteristic
226C22×D4—21no—
227C22×D4D4×C7111yescharacteristic
228C23×C4—21no—
229C23×C4C23×C7111yescharacteristic
230C2×SD16—22no—
231C2×M16—21no—
232C2×M16D4×C7112yes—
233C22×C8—21no—
234C22×C8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239#121—21no—
240#121—21no—
241#121—21no—
242#121—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4D4×C7112yes—
250C2×C4⋊C4D4×C7111yescharacteristic
251C2×C4⋊C4D4×C7111yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254#107—22no—
255#106—24no—
256#106—24no—
257#103—24no—
258#103—22no—
259#102—24no—
260#102—22no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8D4×C7112yes—
268C22⋊C8D4×C7112yes—
269C24—21no—
270C24#41×C7111yescharacteristic
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—82no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4C2×D4×C7111yescharacteristic
289C22×C4C2×D4×C7111yescharacteristic
290C22×C4C2×D4×C7111yescharacteristic
291C22×C4C2×D4×C7111yescharacteristic
292C22×C4C2×D4×C7111yescharacteristic
293C22×C4C2×D4×C7111yescharacteristic
294SD16—82no—
295M16—42no—
296M16—42no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23#121×C7111yescharacteristic
325C23#121×C7111yescharacteristic
326C23#121×C7111yescharacteristic
327Q8—42no—
328Q8—42no—
329D4—42no—
330D4—42no—
331D4—82no—
332D4—82no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4#120×C7111yescharacteristic
350C2×C4#120×C7111yescharacteristic
351C2×C4#120×C7111yescharacteristic
352C2×C4#127×C7111yescharacteristic
353C8—42no—
354C8—42no—
355C8—42no—
356C8—81no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—162no—
369C22#392×C7111yescharacteristic
370C22#471×C7112yes—
371C22#471×C7112yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—82no—
377C4—81no—
378C4—81no—
379C4#458×C7112yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—161no—
384C2C71×#201911yescharacteristic
385C2C71×#201911yescharacteristic
386C2C71×#202211yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC23×C4C23×C71
prime radicalC2×#121×C71C22
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
3. C2×#483×C71C2semidirect
4. C71×#1821C2semidirect
5. C71×#1472C2semidirect
7. C71×#1298C2semidirect
186. #9993C71direct (N × H)
196. C2×#483C2×C71semidirect
197. #1821C2×C71semidirect
198. #1472C2×C71semidirect
200. #1298C2×C71semidirect
201. C71#9993direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#31932
C2×#481×C712
C2×#483×C712
C71×#18212
C71×#14722
C71×#14712
C71×#12982
#999371