Finite Group Databasebeta

← C7×#14637 #13191853

Subgroups — C7×#14637 #13191853

Order 1792.

1378 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#396C211yesmaximal, characteristic
2C7×#1445C213yesmaximal, complemented
3C7×#1444C211yesmaximal, characteristic, complemented
4C7×#1443C211yesmaximal, characteristic, complemented
5C7×#1441C213yesmaximal, complemented
6C7×#1439C213yesmaximal
7C7×C22⋊#95C213yesmaximal
8C24×C4×C7C2211yescharacteristic
9C22×C7×C4⋊C4C2213yes—
10C22×C7×C4⋊C4C2213yes—
11C22×C7×C22⋊C4C2213yes—
12C22×C7×C22⋊C4C2213yes—
13C7×#396C418yescomplemented
14C2×C7×#95C2213yes—
15C2×C7×#95C2213yes—
16C2×C7×#95C2213yes—
17C2×C7×#95C2213yes—
18C2×C7×#95C2213yes—
19C2×C7×#95C2213yes—
20C2×C7×#95C2213yes—
21C2×C7×#95C2211yescharacteristic
22#14637C711yesmaximal, characteristic, direct product factor, Sylow
23C25×C7C2311yescharacteristic
24C23×C4×C7—43no—
25C23×C4×C7—43no—
26C23×C4×C7Q814yes—
27C23×C4×C7C2313yes—
28C23×C4×C7C2313yes—
29C23×C4×C7C2313yes—
30C23×C4×C7C2313yes—
31C23×C4×C7C2311yescharacteristic
32C23×C4×C7C2311yescharacteristic
33C2×C7×C4⋊C4—43no—
34C2×C7×C4⋊C4—43no—
35C2×C7×C4⋊C4—43no—
36C2×C7×C4⋊C4—43no—
37C2×C7×C4⋊C4—43no—
38C2×C7×C4⋊C4C2×C4112yescomplemented
39C2×C7×C22⋊C4—26no—
40C2×C7×C22⋊C4—43no—
41C2×C7×C22⋊C4—43no—
42C2×C7×C22⋊C4—43no—
43C2×C7×C22⋊C4—43no—
44C2×C7×C22⋊C4C2×C4112yes—
45C7×#95—43no—
46C7×#95—43no—
47C7×#95—43no—
48C7×#95—43no—
49C7×#95—43no—
50C7×#95—43no—
51C7×#95—43no—
52C7×#95C2×C414yescomplemented
53C2×#396C2×C711yescharacteristic
54#1445C2×C713yescomplemented
55#1444C2×C711yescharacteristic, complemented
56#1443C2×C711yescharacteristic, complemented
57#1441C2×C713yescomplemented
58#1439C2×C713yes—
59C22⋊#95C2×C713yes—
60C24×C7—23no—
61C24×C7—23no—
62C24×C7—23no—
63C24×C7—23no—
64C24×C7C22×C412yes—
65C24×C7C2×Q812yes—
66C24×C7#4112yes—
67C24×C7C2411yescharacteristic
68C22×C4×C7—23no—
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—23no—
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×C7—23no—
98C22×C4×C7—412no—
99C22×C4×C7—44no—
100C22×C4×C7—412no—
101C22×C4×C7C22×C416yes—
102C22×C4×C7C22×C416yes—
103C22×C4×C7C2×Q812yes—
104C22×C4×C7C2×Q812yes—
105C7×C4⋊C4—412no—
106C7×C4⋊C4—412no—
107C7×C22⋊C4—412no—
108C7×C22⋊C4—412no—
109C24×C4C22×C711yescharacteristic
110C22×C4⋊C4C22×C713yes—
111C22×C4⋊C4C22×C713yes—
112C22×C22⋊C4C22×C713yes—
113C22×C22⋊C4C22×C713yes—
114#396C4×C718yescomplemented
115C2×#95C22×C713yes—
116C2×#95C22×C713yes—
117C2×#95C22×C713yes—
118C2×#95C22×C713yes—
119C2×#95C22×C713yes—
120C2×#95C22×C713yes—
121C2×#95C22×C713yes—
122C2×#95C22×C711yescharacteristic
123C23×C7—26no—
124C23×C7—26no—
125C23×C7—26no—
126C23×C7—26no—
127C23×C7—26no—
128C23×C7—26no—
129C23×C7—41no—
130C23×C7—43no—
131C23×C7—43no—
132C23×C7—43no—
133C23×C7—43no—
134C23×C7—43no—
135C23×C7C4×Q814yes—
136C23×C7C23×C411yescharacteristic
137C23×C7C22×Q811yescharacteristic
138C23×C7C2×#4111yescharacteristic
139C23×C7H(5,𝔽2)13yes—
140C23×C7H(5,𝔽2)13yes—
141C23×C7H(5,𝔽2)13yes—
142C23×C72−1+413yes—
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—416no—
152C2×C4×C7—43no—
153C2×C4×C7—43no—
154C2×C4×C7—43no—
155C2×C4×C7—43no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
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—43no—
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—43no—
175C2×C4×C7—43no—
176C2×C4×C7—43no—
177C2×C4×C7—43no—
178C25C23×C711yescharacteristic
179C23×C4—43no—
180C23×C4—43no—
181C23×C4C7×Q814yes—
182C23×C4C23×C713yes—
183C23×C4C23×C713yes—
184C23×C4C23×C713yes—
185C23×C4C23×C713yes—
186C23×C4C23×C711yescharacteristic
187C23×C4C23×C711yescharacteristic
188C2×C4⋊C4—43no—
189C2×C4⋊C4—43no—
190C2×C4⋊C4—43no—
191C2×C4⋊C4—43no—
192C2×C4⋊C4—43no—
193C2×C4⋊C4C2×C4×C7112yescomplemented
194C2×C22⋊C4—26no—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—43no—
197C2×C22⋊C4—43no—
198C2×C22⋊C4—43no—
199C2×C22⋊C4C2×C4×C7112yes—
200#95—43no—
201#95—43no—
202#95—43no—
203#95—43no—
204#95—43no—
205#95—43no—
206#95—43no—
207#95C2×C4×C714yescomplemented
208C22×C7—212no—
209C22×C7—42no—
210C22×C7—46no—
211C22×C7—46no—
212C22×C7—42no—
213C22×C7—46no—
214C22×C7—42no—
215C22×C7C2×C4×Q811yescharacteristic
216C22×C7#51813yes—
217C22×C7#52013yes—
218C22×C7#53813yes—
219C22×C7#53913yes—
220C22×C7#54313yes—
221C22×C7#54413yes—
222C22×C7#56013yes—
223C22×C7#56011yescharacteristic
224C22×C7#56013yes—
225C22×C7#56213yes—
226C22×C7#56213yes—
227C22×C7#56313yes—
228C4×C7—42no—
229C4×C7—42no—
230C4×C7—46no—
231C4×C7—46no—
232C4×C7—83no—
233C4×C7—83no—
234C24—23no—
235C24—23no—
236C24—23no—
237C24—23no—
238C24C22×C4×C712yes—
239C24C2×C7×Q812yes—
240C24C7×#4112yes—
241C24C24×C711yescharacteristic
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—23no—
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×C4—23no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—23no—
271C22×C4—23no—
272C22×C4—412no—
273C22×C4—44no—
274C22×C4—412no—
275C22×C4C22×C4×C716yes—
276C22×C4C22×C4×C716yes—
277C22×C4C2×C7×Q812yes—
278C22×C4C2×C7×Q812yes—
279C4⋊C4—412no—
280C4⋊C4—412no—
281C22⋊C4—412no—
282C22⋊C4—412no—
283C2×C7—44no—
284C2×C7#232913yes—
285C2×C7#237713yes—
286C2×C7#238311yescharacteristic
287C2×C7#282511yescharacteristic
288C2×C7#282713yes—
289C2×C7#284113yes—
290C2×C7#284211yescharacteristic
291C23—26no—
292C23—26no—
293C23—26no—
294C23—26no—
295C23—26no—
296C23—26no—
297C23—41no—
298C23—43no—
299C23—43no—
300C23—43no—
301C23—43no—
302C23—43no—
303C23C4×C7×Q814yes—
304C23C23×C4×C711yescharacteristic
305C23C22×C7×Q811yescharacteristic
306C23C2×C7×#4111yescharacteristic
307C23C7×H(5,𝔽2)13yes—
308C23C7×H(5,𝔽2)13yes—
309C23C7×H(5,𝔽2)13yes—
310C23C7×2−1+413yes—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—26no—
319C2×C4—416no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—43no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—43no—
327C2×C4—43no—
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×C4—43no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—43no—
343C2×C4—43no—
344C2×C4—43no—
345C2×C4—43no—
346C7#1463711yescharacteristic, direct product factor, Sylow
347C22—212no—
348C22—42no—
349C22—46no—
350C22—46no—
351C22—42no—
352C22—46no—
353C22—42no—
354C22C2×C4×C7×Q811yescharacteristic
355C22C7×#51813yes—
356C22C7×#52013yes—
357C22C7×#53813yes—
358C22C7×#53913yes—
359C22C7×#54313yes—
360C22C7×#54413yes—
361C22C7×#56013yes—
362C22C7×#56011yescharacteristic
363C22C7×#56013yes—
364C22C7×#56213yes—
365C22C7×#56213yes—
366C22C7×#56313yes—
367C4—42no—
368C4—42no—
369C4—46no—
370C4—46no—
371C4—83no—
372C4—83no—
373C2—44no—
374C2C7×#232913yes—
375C2C7×#237713yes—
376C2C7×#238311yescharacteristic
377C2C7×#282511yescharacteristic
378C2C7×#282713yes—
379C2C7×#284113yes—
380C2C7×#284211yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1445C2semidirect
3. C7×#1444C2semidirect
4. C7×#1443C2semidirect
5. C7×#1441C2semidirect
13. C7×#396C4semidirect
22. #14637C7direct (N × H)
38. C2×C7×C4⋊C4C2×C4semidirect
52. C7×#95C2×C4semidirect
54. #1445C2×C7semidirect
55. #1444C2×C7semidirect
56. #1443C2×C7semidirect
57. #1441C2×C7semidirect
114. #396C4×C7semidirect
193. C2×C4⋊C4C2×C4×C7semidirect
207. #95C2×C4×C7semidirect
346. C7#14637direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3962
C7×#14452
C7×#14442
C7×#14432
C7×#14412
C7×#14392
C7×C22⋊#952
#146377