Finite Group Databasebeta

← C7×#9338 #13186554

Subgroups — C7×#9338 #13186554

Order 1792.

400 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3238C211yesmaximal, characteristic
2C7×#2995C211yesmaximal, characteristic
3C7×#1997C211yesmaximal, 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×#483C412yes—
13C7×#480C412yes—
14C7×#477—21no—
15C7×#476—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
26C4×C7×M16C2211yescharacteristic
27C7×#343—21no—
28C7×#343—21no—
29C7×#332—21no—
30C7×#332—21no—
31C7×#330—21no—
32C7×#329—21no—
33#9338C711yesmaximal, 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×C22⋊C8—21no—
70C7×C22⋊C8—21no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8C2×C412yes—
74C7×#97—21no—
75C7×#97—21no—
76C7×#97—21no—
77C7×#97—21no—
78C7×#97—21no—
79C4×C7×C8—21no—
80C4×C7×C8—21no—
81#3238C2×C711yescharacteristic
82#2995C2×C711yescharacteristic
83#1997C2×C711yescharacteristic, complemented
84#1556C2×C711yescharacteristic, complemented
85#1529C2×C711yescharacteristic
86#1307C2×C711yescharacteristic, complemented
87#1291C2×C711yescharacteristic, complemented
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8—82no—
91C2×C7×Q8C22⋊C411yescharacteristic
92C2×C7×D4—21no—
93C2×C7×D4C22⋊C411yescharacteristic
94C22×C4×C7—21no—
95C22×C4×C7—22no—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7#4111yescharacteristic
99C7×M16—22no—
100C7×M16—41no—
101C7×M16—41no—
102C7×M16—41no—
103C7×M16—81no—
104C2×C7×C8—22no—
105C2×C7×C8—21no—
106C2×C7×C8—21no—
107C2×C7×C8—21no—
108C2×C7×C8—21no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C2×C7×C8—81no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4C22⋊C411yescharacteristic
121C7×C4⋊C4C22⋊C411yescharacteristic
122C7×C22⋊C4—21no—
123C7×C22⋊C4—21no—
124C7×C22⋊C4—21no—
125C7×C22⋊C4—21no—
126C7×C22⋊C4—21no—
127C42×C7—21no—
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7C2×D411yescharacteristic
131C42×C7#4111yescharacteristic
132C2×C4⋊Q8C22×C711yescharacteristic
133#529C22×C711yescharacteristic
134#501—21no—
135#489—21no—
136#483C4×C712yes—
137#480C4×C712yes—
138#477—21no—
139#476—21no—
140#446—21no—
141#436—21no—
142#435—21no—
143#435—21no—
144#433—21no—
145#432C22×C711yescharacteristic
146#432C22×C711yescharacteristic
147#432C22×C711yescharacteristic
148#424—21no—
149#423C22×C711yescharacteristic
150C4×M16C22×C711yescharacteristic
151#343—21no—
152#343—21no—
153#332—21no—
154#332—21no—
155#330—21no—
156#329—21no—
157C23×C7—21no—
158C23×C7C4×D411yescharacteristic
159C7×Q8—41no—
160C7×Q8—82no—
161C7×Q8—82no—
162C7×D4—41no—
163C7×D4—81no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—82no—
175C2×C4×C7C2×C22⋊C411yescharacteristic
176C2×C4×C7C4×D411yescharacteristic
177C2×C4×C7#12011yescharacteristic
178C2×C4×C7#12111yescharacteristic
179C2×C4×C7#12111yescharacteristic
180C2×C4×C7#12311yescharacteristic
181C7×C8—42no—
182C7×C8—42no—
183C7×C8—41no—
184C7×C8—41no—
185C7×C8—41no—
186C7×C8—41no—
187C7×C8—81no—
188C22×Q8—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16C7×D411yescharacteristic
192C2×M16C7×D411yescharacteristic
193C22×C8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198#126—21no—
199#125C7×D411yescharacteristic
200#124C7×D411yescharacteristic
201#123—21no—
202C22⋊Q8C2×C4×C711yescharacteristic
203#121C2×C4×C711yescharacteristic
204C4×Q8C7×D411yescharacteristic
205C4×D4—21no—
206C4×D4C7×D411yescharacteristic
207#117—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×C4×C712yes—
210C2×C42C23×C711yescharacteristic
211#107—22no—
212#106—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8C7×D411yescharacteristic
220C4⋊C8C7×D411yescharacteristic
221#103—22no—
222#102—22no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8C2×C4×C712yes—
228#97—21no—
229#97—21no—
230#97—21no—
231#97—21no—
232#97—21no—
233C4×C8—21no—
234C4×C8—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7#38611yescharacteristic
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7—82no—
251C4×C7#45511yescharacteristic
252C4×C7#45611yescharacteristic
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—82no—
256C2×Q8C7×C22⋊C411yescharacteristic
257C2×D4—21no—
258C2×D4C7×C22⋊C411yescharacteristic
259C22×C4—21no—
260C22×C4—22no—
261C22×C4C22×C4×C711yescharacteristic
262C22×C4C2×C7×D411yescharacteristic
263C22×C4C7×#4111yescharacteristic
264M16—22no—
265M16—41no—
266M16—41no—
267M16—41no—
268M16—81no—
269C2×C8—22no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—81no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4C7×C22⋊C411yescharacteristic
286C4⋊C4C7×C22⋊C411yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C42—21no—
293C42C2×C7×D411yescharacteristic
294C42C2×C7×D411yescharacteristic
295C42C2×C7×D411yescharacteristic
296C42C7×#4111yescharacteristic
297C2×C7—41no—
298C2×C7—81no—
299C2×C7#188211yescharacteristic
300C2×C7#190311yescharacteristic
301C2×C7#191511yescharacteristic
302C23—21no—
303C23C4×C7×D411yescharacteristic
304Q8—41no—
305Q8—82no—
306Q8—82no—
307D4—41no—
308D4—81no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4C2×C7×C22⋊C411yescharacteristic
321C2×C4C4×C7×D411yescharacteristic
322C2×C4C7×#12011yescharacteristic
323C2×C4C7×#12111yescharacteristic
324C2×C4C7×#12111yescharacteristic
325C2×C4C7×#12311yescharacteristic
326C8—42no—
327C8—42no—
328C8—41no—
329C8—41no—
330C8—41no—
331C8—41no—
332C8—81no—
333C7#933811yescharacteristic, direct product factor, Sylow
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22C7×#38611yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—82no—
350C4C7×#45511yescharacteristic
351C4C7×#45611yescharacteristic
352C2—41no—
353C2—81no—
354C2C7×#188211yescharacteristic
355C2C7×#190311yescharacteristic
356C2C7×#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×#1997C2semidirect
4. C7×#1556C2semidirect
6. C7×#1307C2semidirect
7. C7×#1291C2semidirect
33. #9338C7direct (N × H)
83. #1997C2×C7semidirect
84. #1556C2×C7semidirect
86. #1307C2×C7semidirect
87. #1291C2×C7semidirect
333. C7#9338direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32382
C7×#29952
C7×#19972
C7×#15562
C7×#15292
C7×#13072
C7×#12912
#93387