Finite Group Databasebeta

← C7×#15176 #13192392

Subgroups — C7×#15176 #13192392

Order 1792.

1842 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×Q8C211yesmaximal, characteristic, complemented
2C2×C7×#395C211yesmaximal, characteristic, complemented
3C2×C7×#393C211yesmaximal, characteristic
4C2×C7×#388C213yesmaximal
5C7×#1441C213yesmaximal, complemented
6C7×#1439C213yesmaximal
7C7×#1434C213yesmaximal, complemented
8C23×C7×Q8C2211yescharacteristic
9C2×C4×C7×Q8—212no—
10C22×C7×C4⋊C4C2213yes—
11C22×C7×C4⋊C4C2213yes—
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2213yes—
14C22×C7×C22⋊C4C2213yes—
15C22×C42×C7C2213yes—
16C7×#395—24no—
17C7×#393—24no—
18C7×#388—212no—
19C2×C7×#95C2213yes—
20C2×C7×#95C2213yes—
21C2×C7×#95C2213yes—
22C2×C7×#95C2216yes—
23C2×C7×#95C2213yes—
24#15176C711yesmaximal, characteristic, direct product factor, Sylow
25C25×C7C2311yescharacteristic
26C22×C7×Q8—26no—
27C22×C7×Q8—26no—
28C22×C7×Q8D414yes—
29C23×C4×C7C2313yes—
30C23×C4×C7C2311yescharacteristic
31C23×C4×C7C2313yes—
32C23×C4×C7C2313yes—
33C23×C4×C7C2311yescharacteristic
34C23×C4×C7C2313yes—
35C4×C7×Q8—232no—
36C2×C7×C4⋊C4—212no—
37C2×C7×C4⋊C4—26no—
38C2×C7×C4⋊C4—26no—
39C2×C7×C4⋊C4—26no—
40C2×C7×C4⋊C4—26no—
41C2×C7×C22⋊C4—26no—
42C2×C7×C22⋊C4—26no—
43C2×C7×C22⋊C4—26no—
44C2×C7×C22⋊C4—26no—
45C2×C7×C22⋊C4—26no—
46C2×C7×C22⋊C4—26no—
47C2×C7×C22⋊C4—26no—
48C2×C42×C7—212no—
49C2×C42×C7D4112yes—
50C7×#95—26no—
51C7×#95—26no—
52C7×#95—26no—
53C7×#95—26no—
54C7×#95—212no—
55C22×C4×Q8C2×C711yescharacteristic, complemented
56C2×#395C2×C711yescharacteristic, complemented
57C2×#393C2×C711yescharacteristic
58C2×#388C2×C713yes—
59#1441C2×C713yescomplemented
60#1439C2×C713yes—
61#1434C2×C713yescomplemented
62C24×C7—23no—
63C24×C7—21no—
64C24×C7—23no—
65C24×C7—23no—
66C24×C7—21no—
67C24×C7—23no—
68C24×C7#4112yes—
69C24×C7C2411yescharacteristic
70C2×C7×Q8—28no—
71C2×C7×Q8—224no—
72C2×C7×Q8—224no—
73C22×C4×C7—26no—
74C22×C4×C7—26no—
75C22×C4×C7—21no—
76C22×C4×C7—23no—
77C22×C4×C7—23no—
78C22×C4×C7—23no—
79C22×C4×C7—26no—
80C22×C4×C7—26no—
81C22×C4×C7—23no—
82C22×C4×C7—23no—
83C22×C4×C7—23no—
84C22×C4×C7—23no—
85C22×C4×C7—23no—
86C22×C4×C7—23no—
87C22×C4×C7—23no—
88C22×C4×C7—23no—
89C22×C4×C7—23no—
90C22×C4×C7—21no—
91C22×C4×C7—26no—
92C22×C4×C7—23no—
93C22×C4×C7—23no—
94C22×C4×C7—26no—
95C22×C4×C7C2×D416yes—
96C22×C4×C7C2×D416yes—
97C22×C4×C7C2×D416yes—
98C22×C4×C7C2×D416yes—
99C22×C4×C7#4116yes—
100C22×C4×C7#4116yes—
101C7×C4⋊C4—224no—
102C7×C4⋊C4—412no—
103C7×C22⋊C4—224no—
104C7×C22⋊C4—44no—
105C7×C22⋊C4—412no—
106C42×C7—224no—
107C23×Q8C22×C711yescharacteristic
108C2×C4×Q8—212no—
109C22×C4⋊C4C22×C713yes—
110C22×C4⋊C4C22×C713yes—
111C22×C22⋊C4C22×C713yes—
112C22×C22⋊C4C22×C713yes—
113C22×C22⋊C4C22×C711yescharacteristic
114C22×C42C22×C713yes—
115#395—24no—
116#393—24no—
117#388—212no—
118C2×#95C22×C713yes—
119C2×#95C22×C713yes—
120C2×#95C22×C713yes—
121C2×#95C22×C716yes—
122C2×#95C22×C713yes—
123C23×C7—26no—
124C23×C7—22no—
125C23×C7—26no—
126C23×C7—41no—
127C23×C7—43no—
128C23×C7—43no—
129C23×C7—43no—
130C23×C7—43no—
131C23×C7—43no—
132C23×C7—43no—
133C23×C7—43no—
134C23×C7—46no—
135C23×C7C22×D413yes—
136C23×C7C22×D411yescharacteristic
137C23×C7C2×#4113yes—
138C23×C7C2×#4111yescharacteristic
139C23×C7C2×#4113yes—
140C23×C72−1+411yescharacteristic
141C23×C72−1+413yes—
142C7×Q8—232no—
143C2×C4×C7—26no—
144C2×C4×C7—212no—
145C2×C4×C7—26no—
146C2×C4×C7—212no—
147C2×C4×C7—26no—
148C2×C4×C7—26no—
149C2×C4×C7—26no—
150C2×C4×C7—26no—
151C2×C4×C7—26no—
152C2×C4×C7—26no—
153C2×C4×C7—26no—
154C2×C4×C7—26no—
155C2×C4×C7—43no—
156C2×C4×C7—46no—
157C2×C4×C7—46no—
158C2×C4×C7—43no—
159C2×C4×C7—43no—
160C2×C4×C7—43no—
161C2×C4×C7—41no—
162C2×C4×C7—43no—
163C2×C4×C7—43no—
164C2×C4×C7—43no—
165C2×C4×C7—43no—
166C2×C4×C7—43no—
167C2×C4×C7—46no—
168C2×C4×C7—46no—
169C2×C4×C7#12014yes—
170C2×C4×C7#121112yes—
171C25C23×C711yescharacteristic
172C22×Q8—26no—
173C22×Q8—26no—
174C22×Q8C7×D414yes—
175C23×C4C23×C711yescharacteristic
176C23×C4C23×C713yes—
177C23×C4C23×C713yes—
178C23×C4C23×C713yes—
179C23×C4C23×C711yescharacteristic
180C23×C4C23×C713yes—
181C4×Q8—232no—
182C2×C4⋊C4—212no—
183C2×C4⋊C4—26no—
184C2×C4⋊C4—26no—
185C2×C4⋊C4—26no—
186C2×C4⋊C4—26no—
187C2×C22⋊C4—26no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—26no—
191C2×C22⋊C4—26no—
192C2×C22⋊C4—26no—
193C2×C22⋊C4—26no—
194C2×C42—212no—
195C2×C42C7×D4112yes—
196#95—26no—
197#95—26no—
198#95—26no—
199#95—26no—
200#95—212no—
201C22×C7—46no—
202C22×C7—42no—
203C22×C7—46no—
204C22×C7—81no—
205C22×C7—83no—
206C22×C7—81no—
207C22×C7—83no—
208C22×C7C2×#12011yescharacteristic
209C22×C7C2×#12113yes—
210C22×C7#52513yes—
211C22×C7#52513yes—
212C22×C7#53213yes—
213C22×C7#53613yes—
214C22×C7#53613yes—
215C22×C7#54811yescharacteristic
216C22×C7#54813yes—
217C22×C7#55213yes—
218C22×C7#55213yes—
219C22×C7#55616yes—
220C4×C7—212no—
221C4×C7—24no—
222C4×C7—46no—
223C4×C7—46no—
224C4×C7—81no—
225C4×C7—83no—
226C24—23no—
227C24—21no—
228C24—23no—
229C24—23no—
230C24—21no—
231C24—23no—
232C24C7×#4112yes—
233C24C24×C711yescharacteristic
234C2×Q8—28no—
235C2×Q8—224no—
236C2×Q8—224no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—21no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—23no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—21no—
255C22×C4—26no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—26no—
259C22×C4C2×C7×D416yes—
260C22×C4C2×C7×D416yes—
261C22×C4C2×C7×D416yes—
262C22×C4C2×C7×D416yes—
263C22×C4C7×#4116yes—
264C22×C4C7×#4116yes—
265C4⋊C4—224no—
266C4⋊C4—412no—
267C22⋊C4—224no—
268C22⋊C4—44no—
269C22⋊C4—412no—
270C42—224no—
271C2×C7—82no—
272C2×C7#238811yescharacteristic
273C2×C7#253613yes—
274C2×C7#254513yes—
275C2×C7#259913yes—
276C2×C7#260413yes—
277C2×C7#261411yescharacteristic
278C2×C7#284711yescharacteristic
279C23—26no—
280C23—22no—
281C23—26no—
282C23—41no—
283C23—43no—
284C23—43no—
285C23—43no—
286C23—43no—
287C23—43no—
288C23—43no—
289C23—43no—
290C23—46no—
291C23C22×C7×D413yes—
292C23C22×C7×D411yescharacteristic
293C23C2×C7×#4113yes—
294C23C2×C7×#4111yescharacteristic
295C23C2×C7×#4113yes—
296C23C7×2−1+411yescharacteristic
297C23C7×2−1+413yes—
298Q8—232no—
299C2×C4—26no—
300C2×C4—212no—
301C2×C4—26no—
302C2×C4—212no—
303C2×C4—26no—
304C2×C4—26no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—43no—
312C2×C4—46no—
313C2×C4—46no—
314C2×C4—43no—
315C2×C4—43no—
316C2×C4—43no—
317C2×C4—41no—
318C2×C4—43no—
319C2×C4—43no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—46no—
324C2×C4—46no—
325C2×C4C7×#12014yes—
326C2×C4C7×#121112yes—
327C7#1517611yescharacteristic, direct product factor, Sylow
328C22—42no—
329C22—46no—
330C22—46no—
331C22—83no—
332C22—81no—
333C22—83no—
334C22—81no—
335C22C2×C7×#12011yescharacteristic
336C22C2×C7×#12113yes—
337C22C7×#52513yes—
338C22C7×#52513yes—
339C22C7×#53213yes—
340C22C7×#53613yes—
341C22C7×#53613yes—
342C22C7×#54811yescharacteristic
343C22C7×#54813yes—
344C22C7×#55213yes—
345C22C7×#55213yes—
346C22C7×#55616yes—
347C4—212no—
348C4—24no—
349C4—46no—
350C4—46no—
351C4—83no—
352C4—81no—
353C2—82no—
354C2C7×#238811yescharacteristic
355C2C7×#253613yes—
356C2C7×#254513yes—
357C2C7×#259913yes—
358C2C7×#260413yes—
359C2C7×#261411yescharacteristic
360C2C7×#284711yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C4×C7×Q8C2semidirect
2. C2×C7×#395C2semidirect
5. C7×#1441C2semidirect
7. C7×#1434C2semidirect
24. #15176C7direct (N × H)
55. C22×C4×Q8C2×C7semidirect
56. C2×#395C2×C7semidirect
59. #1441C2×C7semidirect
61. #1434C2×C7semidirect
327. C7#15176direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C7×Q82
C2×C7×#3952
C2×C7×#3932
C2×C7×#3882
C7×#14412
C7×#14392
C7×#14342
#151767