Finite Group Databasebeta

← C71×#7767 #31828639

Subgroups — C71×#7767 #31828639

Order 18176.

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

normal subgroups only

This group has 392 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×#2805C211yesmaximal, characteristic
2C71×#2058C211yesmaximal, characteristic
3C71×#2043C211yesmaximal, characteristic
4C71×#2024C211yesmaximal, characteristic
5C71×#1272C211yesmaximal, characteristic
6C71×#1271C211yesmaximal, characteristic
7C71×#1271C211yesmaximal, characteristic
8C2×C4⋊Q8×C71C2211yescharacteristic
9C2×#106×C71—21no—
10C2×#103×C71—21no—
11C2×#103×C71—21no—
12C2×#103×C71—21no—
13#400×C71—21no—
14#400×C71—21no—
15#395×C71—21no—
16#391×C71C2211yescharacteristic
17#389×C71—21no—
18#389×C71C2211yescharacteristic
19#389×C71C2211yescharacteristic
20#387×C71—21no—
21#387×C71—21no—
22#340×C71—21no—
23#340×C71—21no—
24#336×C71C2211yescharacteristic
25#336×C71C2211yescharacteristic
26#336×C71C2211yescharacteristic
27C22×Q8×C71—21no—
28C22×Q8×C71D411yescharacteristic
29C22×C8×C71—21no—
30C22×C8×C71—21no—
31C22×C8×C71—21no—
32C4⋊Q8×C71—42no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C42×C71—21no—
45C2×C42×C71C2311yescharacteristic
46#106×C71—42no—
47#103×C71—42no—
48#103×C71—42no—
49#103×C71—42no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59C2×Q8×C71—22no—
60C2×Q8×C71—21no—
61C2×Q8×C71—21no—
62C2×Q8×C71—41no—
63C2×Q8×C71—41no—
64C2×Q8×C71—41no—
65C2×Q8×C71—41no—
66C2×Q8×C71—41no—
67C2×Q8×C71—41no—
68C2×Q8×C71—41no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—41no—
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C2×C8×C71—22no—
80C2×C8×C71—41no—
81C2×C8×C71—41no—
82C2×C8×C71—41no—
83C2×C8×C71—41no—
84C2×C8×C71—41no—
85C2×C8×C71—41no—
86C2×C8×C71—41no—
87C2×C8×C71—41no—
88C4⋊C4×C71—21no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—22no—
91C4⋊C4×C71—21no—
92C4⋊C4×C71—21no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—41no—
102C4⋊C4×C71—81no—
103C4⋊C4×C71—81no—
104C4⋊C4×C71SD1612yes—
105C42×C71—41no—
106C42×C71—81no—
107C23×C71#12011yescharacteristic
108Q8×C71—42no—
109Q8×C71—42no—
110Q8×C71—42no—
111Q8×C71—81no—
112Q8×C71—81no—
113Q8×C71—81no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—22no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—42no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—81no—
140C2×C4×C71—81no—
141C2×C4×C71C2×SD1611yescharacteristic
142C2×C4×C71#13511yescharacteristic
143C2×C4×C71#13511yescharacteristic
144C2×C4×C71#13711yescharacteristic
145C2×C4×C71#13711yescharacteristic
146C2×C4×C71#13711yescharacteristic
147C8×C71—42no—
148C8×C71—81no—
149C8×C71—81no—
150C22×C71#44811yescharacteristic
151C22×C71#45211yescharacteristic
152C22×C71#45211yescharacteristic
153C22×C71#45411yescharacteristic
154C22×C71#45511yescharacteristic
155C22×C71#45611yescharacteristic
156C22×C71B(4,𝔽2)11yescharacteristic
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—42no—
164C4×C71—82no—
165C4×C71—81no—
166C4×C71—81no—
167C4×C71—81no—
168C4×C71—81no—
169C4×C71—81no—
170#7767C7111yesmaximal, characteristic, direct product factor, Sylow
171C2×C71#159911yescharacteristic
172C2×C71#160311yescharacteristic
173C2×C71#160611yescharacteristic
174C2×C71#163811yescharacteristic
175C2×C71#164311yescharacteristic
176C2×C71#165311yescharacteristic
177C2×C71#166411yescharacteristic
178#2805C2×C7111yescharacteristic
179#2058C2×C7111yescharacteristic
180#2043C2×C7111yescharacteristic
181#2024C2×C7111yescharacteristic
182#1272C2×C7111yescharacteristic
183#1271C2×C7111yescharacteristic
184#1271C2×C7111yescharacteristic
185C71#776711yescharacteristic, direct product factor, Sylow
186C2×C4⋊Q8C22×C7111yescharacteristic
187C2×#106—21no—
188C2×#103—21no—
189C2×#103—21no—
190C2×#103—21no—
191#400—21no—
192#400—21no—
193#395—21no—
194#391C22×C7111yescharacteristic
195#389—21no—
196#389C22×C7111yescharacteristic
197#389C22×C7111yescharacteristic
198#387—21no—
199#387—21no—
200#340—21no—
201#340—21no—
202#336C22×C7111yescharacteristic
203#336C22×C7111yescharacteristic
204#336C22×C7111yescharacteristic
205C22×Q8—21no—
206C22×Q8D4×C7111yescharacteristic
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C4⋊Q8—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4D4×C7111yescharacteristic
218C2×C4⋊C4D4×C7111yescharacteristic
219C2×C4⋊C4D4×C7111yescharacteristic
220C2×C4⋊C4D4×C7111yescharacteristic
221C2×C4⋊C4D4×C7111yescharacteristic
222C2×C42—21no—
223C2×C42C23×C7111yescharacteristic
224#106—42no—
225#103—42no—
226#103—42no—
227#103—42no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4C2×D4×C7111yescharacteristic
255C22×C4C2×D4×C7111yescharacteristic
256C22×C4C2×D4×C7111yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C4⋊C4SD16×C7112yes—
283C42—41no—
284C42—81no—
285C23#120×C7111yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—42no—
289Q8—81no—
290Q8—81no—
291Q8—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4C2×SD16×C7111yescharacteristic
320C2×C4#135×C7111yescharacteristic
321C2×C4#135×C7111yescharacteristic
322C2×C4#137×C7111yescharacteristic
323C2×C4#137×C7111yescharacteristic
324C2×C4#137×C7111yescharacteristic
325C8—42no—
326C8—81no—
327C8—81no—
328C22#448×C7111yescharacteristic
329C22#452×C7111yescharacteristic
330C22#452×C7111yescharacteristic
331C22#454×C7111yescharacteristic
332C22#455×C7111yescharacteristic
333C22#456×C7111yescharacteristic
334C22B(4,𝔽2)×C7111yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C2C71×#159911yescharacteristic
349C2C71×#160311yescharacteristic
350C2C71×#160611yescharacteristic
351C2C71×#163811yescharacteristic
352C2C71×#164311yescharacteristic
353C2C71×#165311yescharacteristic
354C2C71×#166411yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C71#120
commutator, Frattini, cosocleC2×C42C23×C71
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
170. #7767C71direct (N × H)
185. C71#7767direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#28052
C71×#20582
C71×#20432
C71×#20242
C71×#12722
C71×#12712
#776771