Finite Group Databasebeta

← C7×#87520 #72230288

Subgroups — C7×#87520 #72230288

Order 2688.

534 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 32 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
1C7×#5950C211yesmaximal, characteristic, complemented
2C7×#5935C211yesmaximal, characteristic, complemented
3C7×#5933C211yesmaximal, characteristic, complemented
4C7×#3018—31nomaximal, Hall
5C7×#997C2211yescharacteristic
6C7×#985—42nomaximal
7C4×C7×S4—41nomaximal
8C7×#585D311yescharacteristic, complemented
9C7×#575—32no—
10C7×#532—31no—
11C7×#456—32no—
12C7×#455—31no—
13C7×#455—31no—
14C7×#454—32no—
15C7×#453—31no—
16C7×#453—31no—
17C2×C7×#104—32no—
18C7×#413—31no—
19#87520C711yesmaximal, characteristic, direct product factor, Hall
20C2×C7×S4—41no—
21C7×#41⋊C3—42no—
22C4×C7×A4—41no—
23C4×C7×A4—81no—
24C7×SL(2,ℤ/4)—41no—
25C7×GL(2,𝔽3)—42no—
26C7×#226—42no—
27C7×2−1+4—32no—
28C7×H(5,𝔽2)—31no—
29C7×H(5,𝔽2)—61no—
30C7×H(5,𝔽2)C2×D311yescharacteristic
31C2×C7×#41—31no—
32C2×C7×#41—32no—
33C2×C7×#41—31no—
34C2×C7×#41—61no—
35C7×#137—32no—
36C7×#137—32no—
37C7×#137—62no—
38C7×AGL(1,ℤ/8)—32no—
39C7×AGL(1,ℤ/8)—32no—
40C7×AGL(1,ℤ/8)—62no—
41C7×#135—62no—
42C7×#135—62no—
43C2×C7×M16—32no—
44C7×C4⋊Q8—31no—
45C7×#127—31no—
46C7×#124—31no—
47C7×#124—31no—
48C7×#121—61no—
49C7×#121—61no—
50C4×C7×D4—61no—
51C4×C7×D4—61no—
52C2×C42×C7—31no—
53C7×#104—32no—
54C7×#104—32no—
55C7×#104—32no—
56C7×#104—32no—
57C7×#101—32no—
58C7×#100—31no—
59C7×#100—31no—
60#5950C2×C711yescharacteristic, complemented
61#5935C2×C711yescharacteristic, complemented
62#5933C2×C711yescharacteristic, complemented
63C2×C7×A4—41no—
64C2×C7×A4—81no—
65C7×S4—42no—
66C4×D3×C7—161no—
67C7×SL(2,𝔽3)—42no—
68#3018—31noSylow
69C7×#41—32no—
70C7×#41—32no—
71C7×#41—32no—
72C7×#41—62no—
73C7×#41—62no—
74C7×#41—62no—
75C7×#41—62no—
76C7×#41—62no—
77C7×#41—62no—
78C7×#41—62no—
79C7×#41S412yescomplemented
80C2×C7×Q8—31no—
81C2×C7×Q8—32no—
82C2×C7×Q8—31no—
83C2×C7×Q8—61no—
84C2×C7×D4—32no—
85C2×C7×D4—31no—
86C2×C7×D4—31no—
87C2×C7×D4—31no—
88C2×C7×D4—31no—
89C2×C7×D4—61no—
90C2×C7×D4—62no—
91C2×C7×D4—61no—
92C2×C7×D4—61no—
93C2×C7×D4—61no—
94C2×C7×D4—61no—
95C2×C7×D4—61no—
96C22×C4×C7—21no—
97C22×C4×C7—31no—
98C22×C4×C7—31no—
99C22×C4×C7—61no—
100C22×C4×C7—61no—
101C22×C4×C7—61no—
102C22×C4×C7S411yescharacteristic
103C7×Q16—62no—
104C7×Q16—62no—
105C7×SD16—62no—
106C7×SD16—62no—
107C7×SD16—62no—
108C7×SD16—62no—
109C7×D8—62no—
110C7×D8—62no—
111C7×M16—32no—
112C7×M16—32no—
113C7×M16—62no—
114C2×C7×C8—62no—
115C7×C4⋊C4—61no—
116C7×C4⋊C4—61no—
117C7×C22⋊C4—61no—
118C7×C22⋊C4—61no—
119C7×C22⋊C4—61no—
120C7×C22⋊C4—61no—
121C42×C7—31no—
122C42×C7—31no—
123C42×C7—61no—
124#997C22×C711yescharacteristic
125#985—42no—
126C4×S4—41no—
127C2×D3×C7—161no—
128C7×A4—41no—
129C7×A4—81no—
130C3×C4×C7—161no—
131C7×Dic3—161no—
132#585D3×C711yescharacteristic, complemented
133#575—32no—
134#532—31no—
135#456—32no—
136#455—31no—
137#455—31no—
138#454—32no—
139#453—31no—
140#453—31no—
141C2×#104—32no—
142#413—31no—
143C23×C7—21no—
144C23×C7—31no—
145C23×C7—31no—
146C23×C7—61no—
147C23×C7—61no—
148C23×C7C2×S411yescharacteristic
149C7×Q8—32no—
150C7×Q8—62no—
151C7×Q8—62no—
152C7×Q8C2×S412yes—
153C7×D4—32no—
154C7×D4—32no—
155C7×D4—62no—
156C7×D4—62no—
157C7×D4—62no—
158C7×D4—62no—
159C7×D4—62no—
160C7×D4—122no—
161C7×D4—122no—
162C7×D4—122no—
163C2×C4×C7—31no—
164C2×C4×C7—32no—
165C2×C4×C7—31no—
166C2×C4×C7—32no—
167C2×C4×C7—31no—
168C2×C4×C7—31no—
169C2×C4×C7—61no—
170C2×C4×C7—62no—
171C2×C4×C7—61no—
172C2×C4×C7—61no—
173C2×C4×C7—61no—
174C2×C4×C7—61no—
175C2×C4×C7—61no—
176C2×C4×C7—61no—
177C2×C4×C7—121no—
178C2×C4×C7—122no—
179C2×C4×C7—121no—
180C7×C8—62no—
181C7×C8—62no—
182C2×S4—41no—
183#41⋊C3—42no—
184C4×A4—41no—
185C4×A4—81no—
186SL(2,ℤ/4)—41no—
187GL(2,𝔽3)—42no—
188#226—42no—
189C2×C3×C7—161no—
190D3×C7—162no—
1912−1+4—32no—
192H(5,𝔽2)—31no—
193H(5,𝔽2)—61no—
194H(5,𝔽2)C2×D3×C711yescharacteristic
195C2×#41—31no—
196C2×#41—32no—
197C2×#41—31no—
198C2×#41—61no—
199#137—32no—
200#137—32no—
201#137—62no—
202AGL(1,ℤ/8)—32no—
203AGL(1,ℤ/8)—32no—
204AGL(1,ℤ/8)—62no—
205#135—62no—
206#135—62no—
207C2×M16—32no—
208C4⋊Q8—31no—
209#127—31no—
210#124—31no—
211#124—31no—
212#121—61no—
213#121—61no—
214C4×D4—61no—
215C4×D4—61no—
216C2×C42—31no—
217#104—32no—
218#104—32no—
219#104—32no—
220#104—32no—
221#101—32no—
222#100—31no—
223#100—31no—
224C22×C7—31no—
225C22×C7—32no—
226C22×C7—41no—
227C22×C7—61no—
228C22×C7—81no—
229C22×C7—121no—
230C22×C7—121no—
231C22×C7—121no—
232C22×C7—121no—
233C22×C7—121no—
234C22×C7—241no—
235C4×C7—32no—
236C4×C7—31no—
237C4×C7—62no—
238C4×C7—61no—
239C4×C7—121no—
240C4×C7—121no—
241C4×C7#102011yescharacteristic
242C2×A4—41no—
243C2×A4—81no—
244S4—42no—
245C4×D3—161no—
246SL(2,𝔽3)—42no—
247C3×C7—161noHall
248#41—32no—
249#41—32no—
250#41—32no—
251#41—62no—
252#41—62no—
253#41—62no—
254#41—62no—
255#41—62no—
256#41—62no—
257#41—62no—
258#41C7×S412yescomplemented
259C2×Q8—31no—
260C2×Q8—32no—
261C2×Q8—31no—
262C2×Q8—61no—
263C2×D4—32no—
264C2×D4—31no—
265C2×D4—31no—
266C2×D4—31no—
267C2×D4—31no—
268C2×D4—61no—
269C2×D4—62no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C22×C4—21no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4C7×S411yescharacteristic
282Q16—62no—
283Q16—62no—
284SD16—62no—
285SD16—62no—
286SD16—62no—
287SD16—62no—
288D8—62no—
289D8—62no—
290M16—32no—
291M16—32no—
292M16—62no—
293C2×C8—62no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C42—31no—
301C42—31no—
302C42—61no—
303C2×C7—61no—
304C2×C7—62no—
305C2×C7—121no—
306C2×C7—241no—
307C2×C7C2×#102011yescharacteristic
308C2×D3—161no—
309A4—41no—
310A4—81no—
311C3×C4—161no—
312Dic3—161no—
313C23—21no—
314C23—31no—
315C23—31no—
316C23—61no—
317C23—61no—
318C23C2×C7×S411yescharacteristic
319Q8—32no—
320Q8—62no—
321Q8—62no—
322Q8C2×C7×S412yes—
323D4—32no—
324D4—32no—
325D4—62no—
326D4—62no—
327D4—62no—
328D4—62no—
329D4—62no—
330D4—122no—
331D4—122no—
332D4—122no—
333C2×C4—31no—
334C2×C4—32no—
335C2×C4—31no—
336C2×C4—32no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—61no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—121no—
350C8—62no—
351C8—62no—
352C7#8752011yescharacteristic, direct product factor, Sylow
353C2×C3—161no—
354D3—162no—
355C22—31no—
356C22—32no—
357C22—41no—
358C22—61no—
359C22—81no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—241no—
366C4—32no—
367C4—31no—
368C4—62no—
369C4—61no—
370C4—121no—
371C4—121no—
372C4C7×#102011yescharacteristic
373C3—161noSylow
374C2—61no—
375C2—62no—
376C2—121no—
377C2—241no—
378C2C2×C7×#102011yescharacteristic
subgroupG/H
center, hypercenterC4×C7#1020
commutator, cosocle, hypocenter#997C22×C7
FrattiniC2C2×C7×#1020
FittingC7×#585D3
socleC2×C7C2×#1020
LCSG ▷ #997
UCSC1 ◁ C4×C7

Complements

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

NH ≅ G/Ntype
1. C7×#5950C2semidirect
2. C7×#5935C2semidirect
3. C7×#5933C2semidirect
8. C7×#585D3semidirect
19. #87520C7direct (N × H)
60. #5950C2×C7semidirect
61. #5935C2×C7semidirect
62. #5933C2×C7semidirect
79. C7×#41S4semidirect
132. #585D3×C7semidirect
258. #41C7×S4semidirect
352. C7#87520direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#59502
C7×#59352
C7×#59332
C7×#30183
C7×#9854
C4×C7×S44
#875207