Finite Group Databasebeta

← C7×#9339 #13186555

Subgroups — C7×#9339 #13186555

Order 1792.

416 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
1C7×#3239C211yesmaximal, characteristic
2C7×#2995C211yesmaximal, characteristic
3C7×#1966C211yesmaximal, characteristic, complemented
4C7×#1556C211yesmaximal, characteristic, complemented
5C7×#1529C211yesmaximal, characteristic
6C7×#1307C211yesmaximal, characteristic, complemented
7C7×#1291C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C7×#529C2211yescharacteristic
10C7×#501—21no—
11C7×#489—21no—
12C7×#484C412yes—
13C7×#481C412yes—
14C7×#475—21no—
15C7×#474—21no—
16C7×#446—21no—
17C7×#436—21no—
18C7×#435—21no—
19C7×#435—21no—
20C7×#433—21no—
21C7×#432C2211yescharacteristic
22C7×#432C2211yescharacteristic
23C7×#432C2211yescharacteristic
24C7×#424—21no—
25C7×#423C2211yescharacteristic
26C2×C7×#101—21no—
27C2×C7×#101—21no—
28C4×C7×M16C2211yescharacteristic
29C7×#332—21no—
30C7×#332—21no—
31C7×#330—21no—
32C7×#329—21no—
33#9339C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8—21no—
35C2×C7×M16—21no—
36C2×C7×M16—21no—
37C2×C7×M16D411yescharacteristic
38C2×C7×M16D411yescharacteristic
39C22×C7×C8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×#126—21no—
45C7×#125D411yescharacteristic
46C7×#124D411yescharacteristic
47C7×#123—21no—
48C7×C22⋊Q8C2×C411yescharacteristic
49C7×#121C2×C411yescharacteristic
50C4×C7×Q8D411yescharacteristic
51C4×C7×D4—21no—
52C4×C7×D4D411yescharacteristic
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2×C412yes—
56C2×C42×C7C2311yescharacteristic
57C7×#107—22no—
58C7×#106—22no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8D411yescharacteristic
66C7×C4⋊C8D411yescharacteristic
67C7×#103—22no—
68C7×#102—22no—
69C7×#101—42no—
70C7×#101—42no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8C2×C412yes—
76C7×#97—21no—
77C7×#97—21no—
78C7×#97—21no—
79C7×#97—21no—
80C7×#97—21no—
81C4×C7×C8—21no—
82C4×C7×C8—21no—
83#3239C2×C711yescharacteristic
84#2995C2×C711yescharacteristic
85#1966C2×C711yescharacteristic, complemented
86#1556C2×C711yescharacteristic, complemented
87#1529C2×C711yescharacteristic
88#1307C2×C711yescharacteristic, complemented
89#1291C2×C711yescharacteristic, complemented
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C2×C7×Q8—41no—
95C2×C7×Q8—41no—
96C2×C7×Q8C22⋊C411yescharacteristic
97C2×C7×D4—21no—
98C2×C7×D4C22⋊C411yescharacteristic
99C22×C4×C7—21no—
100C22×C4×C7—22no—
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7#4111yescharacteristic
104C7×M16—22no—
105C7×M16—41no—
106C7×M16—41no—
107C7×M16—41no—
108C7×M16—81no—
109C2×C7×C8—22no—
110C2×C7×C8—21no—
111C2×C7×C8—21no—
112C2×C7×C8—21no—
113C2×C7×C8—21no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—81no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4C22⋊C411yescharacteristic
126C7×C4⋊C4C22⋊C411yescharacteristic
127C7×C22⋊C4—21no—
128C7×C22⋊C4—21no—
129C7×C22⋊C4—21no—
130C7×C22⋊C4—21no—
131C7×C22⋊C4—21no—
132C42×C7—21no—
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7#4111yescharacteristic
137C2×C4⋊Q8C22×C711yescharacteristic
138#529C22×C711yescharacteristic
139#501—21no—
140#489—21no—
141#484C4×C712yes—
142#481C4×C712yes—
143#475—21no—
144#474—21no—
145#446—21no—
146#436—21no—
147#435—21no—
148#435—21no—
149#433—21no—
150#432C22×C711yescharacteristic
151#432C22×C711yescharacteristic
152#432C22×C711yescharacteristic
153#424—21no—
154#423C22×C711yescharacteristic
155C2×#101—21no—
156C2×#101—21no—
157C4×M16C22×C711yescharacteristic
158#332—21no—
159#332—21no—
160#330—21no—
161#329—21no—
162C23×C7—21no—
163C23×C7C4×D411yescharacteristic
164C7×Q8—41no—
165C7×Q8—81no—
166C7×Q8—81no—
167C7×Q8—81no—
168C7×Q8—81no—
169C7×D4—41no—
170C7×D4—81no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7C2×C22⋊C411yescharacteristic
184C2×C4×C7C4×D411yescharacteristic
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12311yescharacteristic
189C7×C8—42no—
190C7×C8—42no—
191C7×C8—41no—
192C7×C8—41no—
193C7×C8—41no—
194C7×C8—41no—
195C7×C8—81no—
196C22×Q8—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16C7×D411yescharacteristic
200C2×M16C7×D411yescharacteristic
201C22×C8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206#126—21no—
207#125C7×D411yescharacteristic
208#124C7×D411yescharacteristic
209#123—21no—
210C22⋊Q8C2×C4×C711yescharacteristic
211#121C2×C4×C711yescharacteristic
212C4×Q8C7×D411yescharacteristic
213C4×D4—21no—
214C4×D4C7×D411yescharacteristic
215#117—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×C4×C712yes—
218C2×C42C23×C711yescharacteristic
219#107—22no—
220#106—22no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8C7×D411yescharacteristic
228C4⋊C8C7×D411yescharacteristic
229#103—22no—
230#102—22no—
231#101—42no—
232#101—42no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8C2×C4×C712yes—
238#97—21no—
239#97—21no—
240#97—21no—
241#97—21no—
242#97—21no—
243C4×C8—21no—
244C4×C8—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7#38611yescharacteristic
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7—41no—
260C4×C7—82no—
261C4×C7#45311yescharacteristic
262C4×C7#45411yescharacteristic
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8C7×C22⋊C411yescharacteristic
270C2×D4—21no—
271C2×D4C7×C22⋊C411yescharacteristic
272C22×C4—21no—
273C22×C4—22no—
274C22×C4C22×C4×C711yescharacteristic
275C22×C4C2×C7×D411yescharacteristic
276C22×C4C7×#4111yescharacteristic
277M16—22no—
278M16—41no—
279M16—41no—
280M16—41no—
281M16—81no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—81no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4C7×C22⋊C411yescharacteristic
299C4⋊C4C7×C22⋊C411yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C42—21no—
306C42C2×C7×D411yescharacteristic
307C42C2×C7×D411yescharacteristic
308C42C2×C7×D411yescharacteristic
309C42C7×#4111yescharacteristic
310C2×C7—41no—
311C2×C7—81no—
312C2×C7#188311yescharacteristic
313C2×C7#190211yescharacteristic
314C2×C7#191511yescharacteristic
315C23—21no—
316C23C4×C7×D411yescharacteristic
317Q8—41no—
318Q8—81no—
319Q8—81no—
320Q8—81no—
321Q8—81no—
322D4—41no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4C2×C7×C22⋊C411yescharacteristic
337C2×C4C4×C7×D411yescharacteristic
338C2×C4C7×#12011yescharacteristic
339C2×C4C7×#12111yescharacteristic
340C2×C4C7×#12111yescharacteristic
341C2×C4C7×#12311yescharacteristic
342C8—42no—
343C8—42no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C8—81no—
349C7#933911yescharacteristic, direct product factor, Sylow
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22C7×#38611yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4C7×#45311yescharacteristic
367C4C7×#45411yescharacteristic
368C2—41no—
369C2—81no—
370C2C7×#188311yescharacteristic
371C2C7×#190211yescharacteristic
372C2C7×#191511yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1966C2semidirect
4. C7×#1556C2semidirect
6. C7×#1307C2semidirect
7. C7×#1291C2semidirect
33. #9339C7direct (N × H)
85. #1966C2×C7semidirect
86. #1556C2×C7semidirect
88. #1307C2×C7semidirect
89. #1291C2×C7semidirect
349. C7#9339direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32392
C7×#29952
C7×#19662
C7×#15562
C7×#15292
C7×#13072
C7×#12912
#93397