Finite Group Databasebeta

← C7×#9273 #13186489

Subgroups — C7×#9273 #13186489

Order 1792.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3205C211yesmaximal, characteristic
2C7×#2935C211yesmaximal, characteristic, complemented
3C7×#1965C211yesmaximal, characteristic, complemented
4C7×#1547C211yesmaximal, characteristic, complemented
5C2×C7×#331C211yesmaximal, characteristic, complemented
6C7×#1301C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C2×C7×#131—21no—
9C7×#532C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C7×#493—21no—
12C7×#486—21no—
13C7×#459—22no—
14C7×#447C414yes—
15C7×#436—21no—
16C7×#435—22no—
17C7×C8×D4—21no—
18C7×#431C2211yescharacteristic
19C7×#423C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#100—21no—
23C2×C7×#100—21no—
24C7×#408—21no—
25C4×C7×M16C2211yescharacteristic
26C7×#332—21no—
27C7×#331—21no—
28C7×#331—21no—
29C7×#331—21no—
30#9273C711yesmaximal, characteristic, direct product factor, Sylow
31C2×C7×#41—21no—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4C2×C412yes—
35C2×C7×D8—24no—
36C7×#131—42no—
37C7×#131—42no—
38C2×C7×M16—22no—
39C2×C7×M16—21no—
40C2×C7×M16D411yescharacteristic
41C2×C7×M16D411yescharacteristic
42C22×C7×C8—22no—
43C22×C7×C8—21no—
44C22×C7×C8—21no—
45C7×C4⋊Q8D411yescharacteristic
46C7×#127—21no—
47C7×#127—21no—
48C7×#127—21no—
49C7×#127—21no—
50C7×#127D411yescharacteristic
51C7×#124—21no—
52C7×#121—21no—
53C7×#121C2×C412yes—
54C4×C7×D4—21no—
55C4×C7×D4D412yes—
56C2×C42×C7C2311yescharacteristic
57C7×C4⋊C8—21no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8D411yescharacteristic
64C7×C4⋊C8D411yescharacteristic
65C7×#102—24no—
66C7×#100—42no—
67C7×#100—42no—
68C7×C22⋊C8—22no—
69C7×C22⋊C8—22no—
70C7×C22⋊C8C2×C412yes—
71C7×#97—21no—
72C4×C7×C8—21no—
73C4×C7×C8—21no—
74#3205C2×C711yescharacteristic
75#2935C2×C711yescharacteristic, complemented
76#1965C2×C711yescharacteristic, complemented
77#1547C2×C711yescharacteristic, complemented
78C2×#331C2×C711yescharacteristic, complemented
79#1301C2×C711yescharacteristic, complemented
80#1283C2×C711yescharacteristic, complemented
81C24×C7—22no—
82C7×#41—41no—
83C7×#41—41no—
84C7×#41—41no—
85C7×#41—41no—
86C2×C7×Q8—21no—
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—81no—
101C2×C7×D4—81no—
102C2×C7×D4—82no—
103C2×C7×D4C22⋊C412yes—
104C22×C4×C7—22no—
105C22×C4×C7C22×C411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7#4111yescharacteristic
108C7×D8—84no—
109C7×M16—21no—
110C7×M16—21no—
111C7×M16—21no—
112C7×M16—44no—
113C7×M16—41no—
114C7×M16—41no—
115C7×M16D812yescomplemented
116C2×C7×C8—21no—
117C2×C7×C8—22no—
118C2×C7×C8—21no—
119C2×C7×C8—21no—
120C2×C7×C8—21no—
121C2×C7×C8—44no—
122C2×C7×C8—41no—
123C2×C7×C8—41no—
124C2×C7×C8—41no—
125C2×C7×C8—41no—
126C2×C7×C8—81no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4C22⋊C412yes—
129C7×C22⋊C4—22no—
130C7×C22⋊C4—22no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7#4111yescharacteristic
135C2×#131—21no—
136#532C22×C711yescharacteristic
137C2×#127C22×C711yescharacteristic
138#493—21no—
139#486—21no—
140#459—22no—
141#447C4×C714yes—
142#436—21no—
143#435—22no—
144C8×D4—21no—
145#431C22×C711yescharacteristic
146#423C22×C711yescharacteristic
147C2×C4⋊C8C22×C711yescharacteristic
148C2×C4⋊C8C22×C711yescharacteristic
149C2×#100—21no—
150C2×#100—21no—
151#408—21no—
152C4×M16C22×C711yescharacteristic
153#332—21no—
154#331—21no—
155#331—21no—
156#331—21no—
157C23×C7—22no—
158C23×C7—42no—
159C23×C7—42no—
160C23×C7—42no—
161C23×C7—82no—
162C23×C7—82no—
163C23×C7C4×D411yescharacteristic
164C7×Q8—41no—
165C7×Q8—41no—
166C7×D4—42no—
167C7×D4—44no—
168C7×D4—42no—
169C7×D4—42no—
170C7×D4—41no—
171C7×D4—41no—
172C7×D4—41no—
173C7×D4—41no—
174C7×D4—82no—
175C7×D4—82no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—81no—
179C7×D4—81no—
180C7×D4—81no—
181C7×D4—81no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—22no—
185C2×C4×C7—41no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7C2×C22⋊C411yescharacteristic
189C2×C4×C7C4×D411yescharacteristic
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12311yescharacteristic
194C2×C4×C7C2×D811yescharacteristic
195C2×C4×C7#13511yescharacteristic
196C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
198C7×C8—22no—
199C7×C8—42no—
200C7×C8—41no—
201C7×C8—41no—
202C7×C8—41no—
203C7×C8—81no—
204C7×C8—81no—
205C2×#41—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C2×C4×C712yes—
209C2×D8—24no—
210#131—42no—
211#131—42no—
212C2×M16—22no—
213C2×M16—21no—
214C2×M16C7×D411yescharacteristic
215C2×M16C7×D411yescharacteristic
216C22×C8—22no—
217C22×C8—21no—
218C22×C8—21no—
219C4⋊Q8C7×D411yescharacteristic
220#127—21no—
221#127—21no—
222#127—21no—
223#127—21no—
224#127C7×D411yescharacteristic
225#124—21no—
226#121—21no—
227#121C2×C4×C712yes—
228C4×D4—21no—
229C4×D4C7×D412yes—
230C2×C42C23×C711yescharacteristic
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C7×D411yescharacteristic
238C4⋊C8C7×D411yescharacteristic
239#102—24no—
240#100—42no—
241#100—42no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8C2×C4×C712yes—
245#97—21no—
246C4×C8—21no—
247C4×C8—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—42no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—82no—
254C22×C7—82no—
255C22×C7—82no—
256C22×C7—162no—
257C22×C7—162no—
258C22×C7#38611yescharacteristic
259C22×C7C4×D811yescharacteristic
260C22×C7#44211yescharacteristic
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—42no—
265C4×C7#44711yescharacteristic
266C4×C7#44911yescharacteristic
267C4×C7#45911yescharacteristic
268C4×C7#46311yescharacteristic
269C4×C7#47111yescharacteristic
270C4×C7#47211yescharacteristic
271C24—22no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4—82no—
293C2×D4C7×C22⋊C412yes—
294C22×C4—22no—
295C22×C4C22×C4×C711yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297C22×C4C7×#4111yescharacteristic
298D8—84no—
299M16—21no—
300M16—21no—
301M16—21no—
302M16—44no—
303M16—41no—
304M16—41no—
305M16C7×D812yescomplemented
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—44no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—81no—
317C4⋊C4—21no—
318C4⋊C4C7×C22⋊C412yes—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C42C2×C7×D411yescharacteristic
322C42C2×C7×D411yescharacteristic
323C42C2×C7×D411yescharacteristic
324C42C7×#4111yescharacteristic
325C2×C7—21no—
326C2×C7—21no—
327C2×C7—82no—
328C2×C7—162no—
329C2×C7#187911yescharacteristic
330C2×C7#190711yescharacteristic
331C2×C7#191411yescharacteristic
332C23—22no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—82no—
337C23—82no—
338C23C4×C7×D411yescharacteristic
339Q8—41no—
340Q8—41no—
341D4—42no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—82no—
350D4—82no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4C2×C7×C22⋊C411yescharacteristic
364C2×C4C4×C7×D411yescharacteristic
365C2×C4C7×#12011yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12111yescharacteristic
368C2×C4C7×#12311yescharacteristic
369C2×C4C2×C7×D811yescharacteristic
370C2×C4C7×#13511yescharacteristic
371C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
373C8—22no—
374C8—42no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—81no—
379C8—81no—
380C7#927311yescharacteristic, direct product factor, Sylow
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22C7×#38611yescharacteristic
392C22C4×C7×D811yescharacteristic
393C22C7×#44211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4C7×#44711yescharacteristic
399C4C7×#44911yescharacteristic
400C4C7×#45911yescharacteristic
401C4C7×#46311yescharacteristic
402C4C7×#47111yescharacteristic
403C4C7×#47211yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—162no—
408C2C7×#187911yescharacteristic
409C2C7×#190711yescharacteristic
410C2C7×#191411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3205C2
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
2. C7×#2935C2semidirect
3. C7×#1965C2semidirect
4. C7×#1547C2semidirect
5. C2×C7×#331C2semidirect
6. C7×#1301C2semidirect
7. C7×#1283C2semidirect
30. #9273C7direct (N × H)
75. #2935C2×C7semidirect
76. #1965C2×C7semidirect
77. #1547C2×C7semidirect
78. C2×#331C2×C7semidirect
79. #1301C2×C7semidirect
80. #1283C2×C7semidirect
115. C7×M16D8semidirect
305. M16C7×D8semidirect
380. C7#9273direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32052
C7×#29352
C7×#19652
C7×#15472
C2×C7×#3312
C7×#13012
C7×#12832
#92737