Finite Group Databasebeta

← C7×#9266 #13186482

Subgroups — C7×#9266 #13186482

Order 1792.

400 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3232C211yesmaximal, characteristic
2C7×#2968C211yesmaximal, characteristic
3C7×#1532C211yesmaximal, characteristic
4C2×C7×#330C211yesmaximal, characteristic, complemented
5C7×#1470C211yesmaximal, characteristic, complemented
6C7×#1302C211yesmaximal, characteristic, complemented
7C7×#1281C211yesmaximal, characteristic, complemented
8C7×#529C2211yescharacteristic
9C2×C7×#125C2211yescharacteristic
10C7×#499—21no—
11C7×#488—21no—
12C7×#484C411yescharacteristic
13C7×#483C411yescharacteristic
14C7×#481C411yescharacteristic
15C7×#480C411yescharacteristic
16C7×#479—21no—
17C7×#478—21no—
18C7×#446—21no—
19C7×#435—21no—
20C7×#435—21no—
21C7×C8×D4—21no—
22C7×#433—21no—
23C7×#432C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C7×#431C2211yescharacteristic
26C7×#405—21no—
27C2×C7×#97C2211yescharacteristic
28C2×C7×#97C2211yescharacteristic
29C7×#330—21no—
30C7×#330—21no—
31C7×#330—21no—
32C7×#329—21no—
33C7×#324—21no—
34C7×#324—21no—
35#9266C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×M16—21no—
37C22×C7×C8—21no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C7×#126—21no—
41C7×#125—21no—
42C7×#125—21no—
43C7×#125—21no—
44C7×#125—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⋊C4—21no—
56C2×C7×C4⋊C4C2×C411yescharacteristic
57C2×C7×C4⋊C4C2×C411yescharacteristic
58C2×C42×C7C2311yescharacteristic
59C7×#107—21no—
60C7×#107—21no—
61C7×#106—21no—
62C7×#106—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×#103—21no—
69C7×#103—21no—
70C7×#102—21no—
71C7×#102—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8C2×C411yescharacteristic
76C7×C22⋊C8C2×C411yescharacteristic
77C7×C22⋊C8D411yescharacteristic
78C7×C22⋊C8D411yescharacteristic
79C7×#97—21no—
80C7×#97—21no—
81C7×#97—21no—
82C7×#97—21no—
83C7×#97—21no—
84C7×#97D411yescharacteristic
85C7×#97D411yescharacteristic
86C4×C7×C8—21no—
87C4×C7×C8—21no—
88C4×C7×C8—21no—
89#3232C2×C711yescharacteristic
90#2968C2×C711yescharacteristic
91#1532C2×C711yescharacteristic
92C2×#330C2×C711yescharacteristic, complemented
93#1470C2×C711yescharacteristic, complemented
94#1302C2×C711yescharacteristic, complemented
95#1281C2×C711yescharacteristic, complemented
96C2×C7×Q8C22⋊C411yescharacteristic
97C2×C7×D4—21no—
98C2×C7×D4C22⋊C411yescharacteristic
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7C22×C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7#4111yescharacteristic
105C7×M16—42no—
106C2×C7×C8—21no—
107C2×C7×C8—22no—
108C2×C7×C8—21no—
109C2×C7×C8—21no—
110C2×C7×C8—21no—
111C2×C7×C8—21no—
112C2×C7×C8—21no—
113C2×C7×C8—42no—
114C2×C7×C8—42no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C2×C7×C8—41no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4C22⋊C411yescharacteristic
131C7×C4⋊C4C22⋊C411yescharacteristic
132C7×C22⋊C4—21no—
133C7×C22⋊C4—21no—
134C7×C22⋊C4—21no—
135C7×C22⋊C4—21no—
136C7×C22⋊C4—21no—
137C42×C7—21no—
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7C2×D411yescharacteristic
141C42×C7#4111yescharacteristic
142#529C22×C711yescharacteristic
143C2×#125C22×C711yescharacteristic
144#499—21no—
145#488—21no—
146#484C4×C711yescharacteristic
147#483C4×C711yescharacteristic
148#481C4×C711yescharacteristic
149#480C4×C711yescharacteristic
150#479—21no—
151#478—21no—
152#446—21no—
153#435—21no—
154#435—21no—
155C8×D4—21no—
156#433—21no—
157#432C22×C711yescharacteristic
158#431C22×C711yescharacteristic
159#431C22×C711yescharacteristic
160#405—21no—
161C2×#97C22×C711yescharacteristic
162C2×#97C22×C711yescharacteristic
163#330—21no—
164#330—21no—
165#330—21no—
166#329—21no—
167#324—21no—
168#324—21no—
169C23×C7—21no—
170C23×C7C4×D411yescharacteristic
171C7×Q8—41no—
172C7×D4—41no—
173C7×D4—41no—
174C7×D4—41no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—81no—
189C2×C4×C7C2×C22⋊C411yescharacteristic
190C2×C4×C7C4×D411yescharacteristic
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12311yescharacteristic
195C7×C8—22no—
196C7×C8—41no—
197C7×C8—41no—
198C7×C8—42no—
199C7×C8—41no—
200C7×C8—41no—
201C7×C8—41no—
202C7×C8—81no—
203C2×M16—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207#126—21no—
208#125—21no—
209#125—21no—
210#125—21no—
211#125—21no—
212#125C7×D411yescharacteristic
213#124C7×D411yescharacteristic
214#123—21no—
215C22⋊Q8C2×C4×C711yescharacteristic
216#121C2×C4×C711yescharacteristic
217C4×Q8C7×D411yescharacteristic
218C4×D4—21no—
219C4×D4C7×D411yescharacteristic
220#117—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×C4×C711yescharacteristic
224C2×C4⋊C4C2×C4×C711yescharacteristic
225C2×C42C23×C711yescharacteristic
226#107—21no—
227#107—21no—
228#106—21no—
229#106—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8C2×C4×C711yescharacteristic
243C22⋊C8C2×C4×C711yescharacteristic
244C22⋊C8C7×D411yescharacteristic
245C22⋊C8C7×D411yescharacteristic
246#97—21no—
247#97—21no—
248#97—21no—
249#97—21no—
250#97—21no—
251#97C7×D411yescharacteristic
252#97C7×D411yescharacteristic
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7#38611yescharacteristic
262C22×C7#44311yescharacteristic
263C22×C7#44411yescharacteristic
264C4×C7—21no—
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—41no—
271C4×C7—41no—
272C4×C7—41no—
273C4×C7—81no—
274C4×C7—81no—
275C2×Q8C7×C22⋊C411yescharacteristic
276C2×D4—21no—
277C2×D4C7×C22⋊C411yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4C22×C4×C711yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283C22×C4C7×#4111yescharacteristic
284M16—42no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4C7×C22⋊C411yescharacteristic
310C4⋊C4C7×C22⋊C411yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C42—21no—
317C42C2×C7×D411yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C2×C7×D411yescharacteristic
320C42C7×#4111yescharacteristic
321C2×C7—21no—
322C2×C7—21no—
323C2×C7—81no—
324C2×C7#187611yescharacteristic
325C2×C7#190611yescharacteristic
326C2×C7#191511yescharacteristic
327C23—21no—
328C23C4×C7×D411yescharacteristic
329Q8—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4C2×C7×C22⋊C411yescharacteristic
348C2×C4C4×C7×D411yescharacteristic
349C2×C4C7×#12011yescharacteristic
350C2×C4C7×#12111yescharacteristic
351C2×C4C7×#12111yescharacteristic
352C2×C4C7×#12311yescharacteristic
353C8—22no—
354C8—41no—
355C8—41no—
356C8—42no—
357C8—41no—
358C8—41no—
359C8—41no—
360C8—81no—
361C7#926611yescharacteristic, direct product factor, Sylow
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22C7×#38611yescharacteristic
368C22C7×#44311yescharacteristic
369C22C7×#44411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—81no—
384C2C7×#187611yescharacteristic
385C2C7×#190611yescharacteristic
386C2C7×#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
4. C2×C7×#330C2semidirect
5. C7×#1470C2semidirect
6. C7×#1302C2semidirect
7. C7×#1281C2semidirect
35. #9266C7direct (N × H)
92. C2×#330C2×C7semidirect
93. #1470C2×C7semidirect
94. #1302C2×C7semidirect
95. #1281C2×C7semidirect
361. C7#9266direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32322
C7×#29682
C7×#15322
C2×C7×#3302
C7×#14702
C7×#13022
C7×#12812
#92667