Finite Group Databasebeta

← C71×#9995 #31830867

Subgroups — C71×#9995 #31830867

Order 18176.

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

normal subgroups only

This group has 510 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×#3195C211yesmaximal, characteristic
2C2×#484×C71C211yesmaximal, characteristic
3C2×#483×C71C211yesmaximal, characteristic
4C71×#1821C211yesmaximal, characteristic
5C71×#1472C211yesmaximal, characteristic
6C71×#1472C211yesmaximal, characteristic
7C71×#1298C211yesmaximal, characteristic
8C2×C22⋊Q8×C71C2211yescharacteristic
9C2×C22⋊Q8×C71C2211yescharacteristic
10C22×C4⋊C4×C71C2211yescharacteristic
11#484×C71—22no—
12#484×C71—22no—
13#483×C71—22no—
14#483×C71—21no—
15#483×C71—21no—
16#483×C71—22no—
17#470×C71—22no—
18#451×C71—21no—
19#451×C71—21no—
20C2×#107×C71—21no—
21C2×#106×C71—21no—
22C2×#103×C71—21no—
23C2×#103×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#324×C71—22no—
31C22×Q8×C71D411yescharacteristic
32C22×Q8×C71D411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71C2311yescharacteristic
35C2×Q16×C71—22no—
36C2×M16×C71—21no—
37C2×M16×C71D412yes—
38C22×C8×C71—21no—
39C22×C8×C71—21no—
40C22⋊Q8×C71—21no—
41C22⋊Q8×C71—21no—
42C22⋊Q8×C71—21no—
43C22⋊Q8×C71—21no—
44C22⋊Q8×C71—21no—
45C22⋊Q8×C71—21no—
46C22⋊Q8×C71—21no—
47C22⋊Q8×C71—21no—
48C2×C4⋊C4×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—24no—
61#106×C71—24no—
62#106×C71—22no—
63#103×C71—24no—
64#103×C71—24no—
65#103×C71—22no—
66#103×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#4111yescharacteristic
76C2×Q8×C71—21no—
77C2×Q8×C71—21no—
78C2×Q8×C71—21no—
79C2×Q8×C71—21no—
80C2×Q8×C71—42no—
81C2×Q8×C71—42no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—22no—
87C22×C4×C71—42no—
88C22×C4×C71—41no—
89C22×C4×C71—41no—
90C22×C4×C71—82no—
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97Q16×C71—82no—
98M16×C71—42no—
99M16×C71—42no—
100C2×C8×C71—22no—
101C2×C8×C71—22no—
102C2×C8×C71—22no—
103C2×C8×C71—42no—
104C2×C8×C71—42no—
105C2×C8×C71—41no—
106C4⋊C4×C71—22no—
107C4⋊C4×C71—21no—
108C4⋊C4×C71—21no—
109C4⋊C4×C71—22no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—21no—
112C4⋊C4×C71—41no—
113C4⋊C4×C71—41no—
114C4⋊C4×C71—41no—
115C4⋊C4×C71—41no—
116C4⋊C4×C71—41no—
117C4⋊C4×C71—41no—
118C22⋊C4×C71—41no—
119C22⋊C4×C71—41no—
120C22⋊C4×C71—41no—
121C22⋊C4×C71—41no—
122C23×C71—22no—
123C23×C71—22no—
124C23×C71—41no—
125C23×C71#12111yescharacteristic
126C23×C71#12111yescharacteristic
127C23×C71#12111yescharacteristic
128Q8×C71—42no—
129Q8×C71—42no—
130Q8×C71—42no—
131Q8×C71—42no—
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—22no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—42no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—82no—
146C2×C4×C71—82no—
147C2×C4×C71—82no—
148C2×C4×C71—81no—
149C2×C4×C71—81no—
150C2×C4×C71#12011yescharacteristic
151C2×C4×C71#12011yescharacteristic
152C2×C4×C71#12011yescharacteristic
153C2×C4×C71#12711yescharacteristic
154C8×C71—42no—
155C8×C71—42no—
156C8×C71—42no—
157C8×C71—81no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—44no—
166C22×C71#39211yescharacteristic
167C22×C71#47112yes—
168C22×C71#47312yes—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—81no—
173C4×C71—82no—
174C4×C71—81no—
175C4×C71—81no—
176C4×C71—81no—
177C4×C71#45812yes—
178#9995C7111yesmaximal, characteristic, direct product factor, Sylow
179C2×C71—22no—
180C2×C71—22no—
181C2×C71—41no—
182C2×C71#201911yescharacteristic
183C2×C71#202011yescharacteristic
184C2×C71#202211yescharacteristic
185#3195C2×C7111yescharacteristic
186C2×#484C2×C7111yescharacteristic
187C2×#483C2×C7111yescharacteristic
188#1821C2×C7111yescharacteristic
189#1472C2×C7111yescharacteristic
190#1472C2×C7111yescharacteristic
191#1298C2×C7111yescharacteristic
192C71#999511yescharacteristic, direct product factor, Sylow
193C2×C22⋊Q8C22×C7111yescharacteristic
194C2×C22⋊Q8C22×C7111yescharacteristic
195C22×C4⋊C4C22×C7111yescharacteristic
196#484—22no—
197#484—22no—
198#483—22no—
199#483—21no—
200#483—21no—
201#483—22no—
202#470—22no—
203#451—21no—
204#451—21no—
205C2×#107—21no—
206C2×#106—21no—
207C2×#103—21no—
208C2×#103—21no—
209#407C22×C7111yescharacteristic
210#407C22×C7111yescharacteristic
211C2×C22⋊C8C22×C7111yescharacteristic
212C2×C22⋊C8C22×C7111yescharacteristic
213#343—22no—
214#324—22no—
215#324—22no—
216C22×Q8D4×C7111yescharacteristic
217C22×Q8D4×C7111yescharacteristic
218C23×C4—21no—
219C23×C4C23×C7111yescharacteristic
220C2×Q16—22no—
221C2×M16—21no—
222C2×M16D4×C7112yes—
223C22×C8—21no—
224C22×C8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4D4×C7112yes—
241C2×C4⋊C4D4×C7111yescharacteristic
242C2×C4⋊C4D4×C7111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245#107—24no—
246#106—24no—
247#106—22no—
248#103—24no—
249#103—24no—
250#103—22no—
251#103—22no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8D4×C7112yes—
259C22⋊C8D4×C7112yes—
260C24#41×C7111yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—42no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—82no—
276C22×C4C2×D4×C7111yescharacteristic
277C22×C4C2×D4×C7111yescharacteristic
278C22×C4C2×D4×C7111yescharacteristic
279C22×C4C2×D4×C7111yescharacteristic
280C22×C4C2×D4×C7111yescharacteristic
281C22×C4C2×D4×C7111yescharacteristic
282Q16—82no—
283M16—42no—
284M16—42no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23#121×C7111yescharacteristic
311C23#121×C7111yescharacteristic
312C23#121×C7111yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—42no—
316Q8—42no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4#120×C7111yescharacteristic
336C2×C4#120×C7111yescharacteristic
337C2×C4#120×C7111yescharacteristic
338C2×C4#127×C7111yescharacteristic
339C8—42no—
340C8—42no—
341C8—42no—
342C8—81no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22#392×C7111yescharacteristic
352C22#471×C7112yes—
353C22#473×C7112yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4#458×C7112yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2C71×#201911yescharacteristic
367C2C71×#202011yescharacteristic
368C2C71×#202211yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC23×C4C23×C71
prime radicalC24×C71#41
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
178. #9995C71direct (N × H)
192. C71#9995direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#31952
C2×#484×C712
C2×#483×C712
C71×#18212
C71×#14722
C71×#12982
#999571