Finite Group Databasebeta

← C7×#15171 #13192387

Subgroups — C7×#15171 #13192387

Order 1792.

1302 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#401C211yesmaximal, characteristic, complemented
2C2×C7×#396C211yesmaximal, characteristic
3C7×#1442C213yesmaximal, complemented
4C7×#1440C213yesmaximal
5C7×#1437C213yesmaximal
6C7×#1436C213yesmaximal, complemented
7C4×C7×#95C211yesmaximal, characteristic, complemented
8C22×C7×C4⋊C4C2213yes—
9C22×C7×C22⋊C4C2211yescharacteristic
10C22×C7×C22⋊C4C2213yes—
11C22×C7×C22⋊C4C2213yes—
12C22×C42×C7C2213yes—
13C7×#401—24no—
14C7×#396—24no—
15C2×C7×#95C2213yes—
16C2×C7×#95C2213yes—
17C2×C7×#95C2213yes—
18C2×C7×#95C2213yes—
19C2×C7×#95C2213yes—
20C2×C7×#95C2213yes—
21C2×C7×#95C2213yes—
22C2×C7×#95C2211yescharacteristic
23#15171C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7C2311yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C23×C4×C7C2313yes—
27C23×C4×C7C2313yes—
28C23×C4×C7C2313yes—
29C23×C4×C7C2313yes—
30C23×C4×C7C2311yescharacteristic
31C2×C7×C4⋊C4—26no—
32C2×C7×C4⋊C4—26no—
33C2×C7×C4⋊C4—43no—
34C2×C7×C22⋊C4—26no—
35C2×C7×C22⋊C4—26no—
36C2×C7×C22⋊C4—26no—
37C2×C7×C22⋊C4—26no—
38C2×C7×C22⋊C4—26no—
39C2×C7×C22⋊C4—43no—
40C2×C7×C22⋊C4—43no—
41C2×C42×C7—26no—
42C2×C42×C7—43no—
43C2×C42×C7—43no—
44C7×#95—26no—
45C7×#95—26no—
46C7×#95—26no—
47C7×#95—26no—
48C7×#95—26no—
49C7×#95—26no—
50C7×#95—43no—
51C7×#95D414yes—
52C2×#401C2×C711yescharacteristic, complemented
53C2×#396C2×C711yescharacteristic
54#1442C2×C713yescomplemented
55#1440C2×C713yes—
56#1437C2×C713yes—
57#1436C2×C713yescomplemented
58C4×#95C2×C711yescharacteristic, complemented
59C24×C7—21no—
60C24×C7—23no—
61C24×C7—23no—
62C24×C7—23no—
63C24×C7—23no—
64C24×C7—21no—
65C24×C7#4112yes—
66C24×C7C2411yescharacteristic
67C22×C4×C7—23no—
68C22×C4×C7—21no—
69C22×C4×C7—23no—
70C22×C4×C7—23no—
71C22×C4×C7—23no—
72C22×C4×C7—23no—
73C22×C4×C7—23no—
74C22×C4×C7—23no—
75C22×C4×C7—23no—
76C22×C4×C7—23no—
77C22×C4×C7—23no—
78C22×C4×C7—23no—
79C22×C4×C7—23no—
80C22×C4×C7—23no—
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—21no—
88C22×C4×C7—23no—
89C22×C4×C7—23no—
90C22×C4×C7—23no—
91C22×C4×C7—23no—
92C22×C4×C7—23no—
93C22×C4×C7—23no—
94C22×C4×C7—23no—
95C22×C4×C7—23no—
96C22×C4×C7—23no—
97C22×C4×C7C2×D416yes—
98C22×C4×C7#4116yes—
99C22×C4×C7#4116yes—
100C22×C4×C7#4116yes—
101C7×C4⋊C4—412no—
102C7×C22⋊C4—412no—
103C7×C22⋊C4—412no—
104C7×C22⋊C4—44no—
105C42×C7—412no—
106C22×C4⋊C4C22×C713yes—
107C22×C22⋊C4C22×C711yescharacteristic
108C22×C22⋊C4C22×C713yes—
109C22×C22⋊C4C22×C713yes—
110C22×C42C22×C713yes—
111#401—24no—
112#396—24no—
113C2×#95C22×C713yes—
114C2×#95C22×C713yes—
115C2×#95C22×C713yes—
116C2×#95C22×C713yes—
117C2×#95C22×C713yes—
118C2×#95C22×C713yes—
119C2×#95C22×C713yes—
120C2×#95C22×C711yescharacteristic
121C23×C7—26no—
122C23×C7—22no—
123C23×C7—26no—
124C23×C7—41no—
125C23×C7—43no—
126C23×C7—43no—
127C23×C7—43no—
128C23×C7—43no—
129C23×C7—43no—
130C23×C7—43no—
131C23×C7—43no—
132C23×C7—43no—
133C23×C7—43no—
134C23×C7C22×D411yescharacteristic
135C23×C7C2×#4113yes—
136C23×C7C2×#4113yes—
137C23×C7C2×#4111yescharacteristic
138C23×C7C2×#4113yes—
139C23×C7H(5,𝔽2)13yes—
140C23×C72−1+411yescharacteristic
141C2×C4×C7—26no—
142C2×C4×C7—26no—
143C2×C4×C7—26no—
144C2×C4×C7—26no—
145C2×C4×C7—26no—
146C2×C4×C7—26no—
147C2×C4×C7—26no—
148C2×C4×C7—26no—
149C2×C4×C7—26no—
150C2×C4×C7—26no—
151C2×C4×C7—26no—
152C2×C4×C7—43no—
153C2×C4×C7—43no—
154C2×C4×C7—43no—
155C2×C4×C7—43no—
156C2×C4×C7—43no—
157C2×C4×C7—43no—
158C2×C4×C7—43no—
159C2×C4×C7—43no—
160C2×C4×C7—43no—
161C2×C4×C7—43no—
162C2×C4×C7—43no—
163C2×C4×C7—43no—
164C2×C4×C7—41no—
165C2×C4×C7—43no—
166C2×C4×C7—43no—
167C2×C4×C7—43no—
168C2×C4×C7—43no—
169C2×C4×C7—43no—
170C2×C4×C7—43no—
171C2×C4×C7—43no—
172C2×C4×C7—43no—
173C2×C4×C7—43no—
174C2×C4×C7#12614yes—
175C25C23×C711yescharacteristic
176C23×C4C23×C711yescharacteristic
177C23×C4C23×C713yes—
178C23×C4C23×C713yes—
179C23×C4C23×C713yes—
180C23×C4C23×C713yes—
181C23×C4C23×C711yescharacteristic
182C2×C4⋊C4—26no—
183C2×C4⋊C4—26no—
184C2×C4⋊C4—43no—
185C2×C22⋊C4—26no—
186C2×C22⋊C4—26no—
187C2×C22⋊C4—26no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—43no—
191C2×C22⋊C4—43no—
192C2×C42—26no—
193C2×C42—43no—
194C2×C42—43no—
195#95—26no—
196#95—26no—
197#95—26no—
198#95—26no—
199#95—26no—
200#95—26no—
201#95—43no—
202#95C7×D414yes—
203C22×C7—46no—
204C22×C7—46no—
205C22×C7—42no—
206C22×C7—83no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7—83no—
210C22×C7C2×#12611yescharacteristic
211C22×C7#52913yes—
212C22×C7#52913yes—
213C22×C7#52913yes—
214C22×C7#53213yes—
215C22×C7#53813yes—
216C22×C7#53913yes—
217C22×C7#54613yes—
218C22×C7#54811yescharacteristic
219C22×C7#55113yes—
220C22×C7#55313yes—
221C22×C7#55313yes—
222C22×C7#55613yes—
223C4×C7—24no—
224C4×C7—46no—
225C4×C7—46no—
226C4×C7—46no—
227C4×C7—83no—
228C4×C7—81no—
229C24—21no—
230C24—23no—
231C24—23no—
232C24—23no—
233C24—23no—
234C24—21no—
235C24C7×#4112yes—
236C24C24×C711yescharacteristic
237C22×C4—23no—
238C22×C4—21no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—23no—
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—23no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—21no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4C2×C7×D416yes—
268C22×C4C7×#4116yes—
269C22×C4C7×#4116yes—
270C22×C4C7×#4116yes—
271C4⋊C4—412no—
272C22⋊C4—412no—
273C22⋊C4—412no—
274C22⋊C4—44no—
275C42—412no—
276C2×C7—82no—
277C2×C7#240111yescharacteristic
278C2×C7#255913yes—
279C2×C7#256013yes—
280C2×C7#257613yes—
281C2×C7#259413yes—
282C2×C7#264311yescharacteristic
283C2×C7#286011yescharacteristic
284C23—26no—
285C23—22no—
286C23—26no—
287C23—41no—
288C23—43no—
289C23—43no—
290C23—43no—
291C23—43no—
292C23—43no—
293C23—43no—
294C23—43no—
295C23—43no—
296C23—43no—
297C23C22×C7×D411yescharacteristic
298C23C2×C7×#4113yes—
299C23C2×C7×#4113yes—
300C23C2×C7×#4111yescharacteristic
301C23C2×C7×#4113yes—
302C23C7×H(5,𝔽2)13yes—
303C23C7×2−1+411yescharacteristic
304C2×C4—26no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—43no—
316C2×C4—43no—
317C2×C4—43no—
318C2×C4—43no—
319C2×C4—43no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—43no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—41no—
328C2×C4—43no—
329C2×C4—43no—
330C2×C4—43no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—43no—
337C2×C4C7×#12614yes—
338C7#1517111yescharacteristic, direct product factor, Sylow
339C22—42no—
340C22—46no—
341C22—46no—
342C22—81no—
343C22—83no—
344C22—83no—
345C22—81no—
346C22C2×C7×#12611yescharacteristic
347C22C7×#52913yes—
348C22C7×#52913yes—
349C22C7×#52913yes—
350C22C7×#53213yes—
351C22C7×#53813yes—
352C22C7×#53913yes—
353C22C7×#54613yes—
354C22C7×#54811yescharacteristic
355C22C7×#55113yes—
356C22C7×#55313yes—
357C22C7×#55313yes—
358C22C7×#55613yes—
359C4—24no—
360C4—46no—
361C4—46no—
362C4—46no—
363C4—83no—
364C4—81no—
365C2—82no—
366C2C7×#240111yescharacteristic
367C2C7×#255913yes—
368C2C7×#256013yes—
369C2C7×#257613yes—
370C2C7×#259413yes—
371C2C7×#264311yescharacteristic
372C2C7×#286011yescharacteristic
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. C2×C7×#401C2semidirect
3. C7×#1442C2semidirect
6. C7×#1436C2semidirect
7. C4×C7×#95C2semidirect
23. #15171C7direct (N × H)
52. C2×#401C2×C7semidirect
54. #1442C2×C7semidirect
57. #1436C2×C7semidirect
58. C4×#95C2×C7semidirect
338. C7#15171direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4012
C2×C7×#3962
C7×#14422
C7×#14402
C7×#14372
C7×#14362
C4×C7×#952
#151717