Finite Group Databasebeta

← C7×#13387 #13190603

Subgroups — C7×#13387 #13190603

Order 1792.

428 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3058C211yesmaximal, characteristic
2C7×#2159C211yesmaximal, characteristic, complemented
3C2×C7×#361C211yesmaximal, characteristic, complemented
4C2×C7×#357C211yesmaximal, characteristic, complemented
5C2×C7×#350C211yesmaximal, characteristic, complemented
6C7×#1816C211yesmaximal, characteristic, complemented
7C7×#1382C211yesmaximal, characteristic, complemented
8C22×C7×D8C2211yescharacteristic
9C2×C7×#131C2211yescharacteristic
10C7×#535—21no—
11C2×C7×M32C2211yescharacteristic
12C22×C7×C16C2211yescharacteristic
13C7×#469—22no—
14C7×#466—21no—
15C7×#466—21no—
16C2×C7×#108C2211yescharacteristic
17C7×#427C2211yescharacteristic
18C2×C7×#107C2211yescharacteristic
19C7×#418—21no—
20C7×#417—21no—
21C7×#367—21no—
22C7×#366—21no—
23C7×#361—22no—
24C7×#357—22no—
25C7×#350—22no—
26C7×#343—21no—
27#13387C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41D411yescharacteristic
29C22×C7×D4—21no—
30C2×C7×D8—21no—
31C2×C7×D8—21no—
32C2×C7×D8—21no—
33C2×C7×D8—21no—
34C2×C7×D8—21no—
35C2×C7×D8D412yes—
36C7×#131—21no—
37C7×#131—21no—
38C7×#131—21no—
39C7×#131—21no—
40C2×C7×M16—21no—
41C2×C7×M16—21no—
42C2×C7×M16D411yescharacteristic
43C22×C7×C8—21no—
44C22×C7×C8C2311yescharacteristic
45C7×#127—41no—
46C7×#124—41no—
47C7×#121—22no—
48C7×#121—22no—
49C7×#120—42no—
50C7×#117—21no—
51C2×C7×C4⋊C4—21no—
52C7×M32—21no—
53C7×M32D412yescomplemented
54C2×C7×C16—21no—
55C2×C7×C16—21no—
56C2×C7×C16—21no—
57C2×C7×C16—21no—
58C7×#108—21no—
59C7×#108—21no—
60C7×#107—21no—
61C7×#107—21no—
62C7×#107D411yescharacteristic
63C7×#107D411yescharacteristic
64C7×#106—21no—
65C7×#103—41no—
66C7×#102—22no—
67C7×#102—22no—
68C7×#102—41no—
69#3058C2×C711yescharacteristic
70#2159C2×C711yescharacteristic, complemented
71C2×#361C2×C711yescharacteristic, complemented
72C2×#357C2×C711yescharacteristic, complemented
73C2×#350C2×C711yescharacteristic, complemented
74#1816C2×C711yescharacteristic, complemented
75#1382C2×C711yescharacteristic, complemented
76C24×C7—41no—
77C7×#41—21no—
78C7×#41—21no—
79C7×#41—21no—
80C7×#41—21no—
81C2×C7×Q8D811yescharacteristic
82C2×C7×D4—21no—
83C2×C7×D4—22no—
84C2×C7×D4—42no—
85C2×C7×D4—42no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4D811yescharacteristic
92C22×C4×C7—21no—
93C22×C4×C7—41no—
94C22×C4×C7C2×D411yescharacteristic
95C7×D8—22no—
96C7×D8—22no—
97C7×D8—41no—
98C7×D8—41no—
99C7×M16—21no—
100C7×M16—21no—
101C7×M16—41no—
102C7×M16—41no—
103C7×M16—41no—
104C7×M16—41no—
105C2×C7×C8—21no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8C2×D411yescharacteristic
110C2×C7×C8C2×D411yescharacteristic
111C2×C7×C8C2×D411yescharacteristic
112C2×C7×C8#4111yescharacteristic
113C2×C7×C8#4111yescharacteristic
114C2×C7×C8#4111yescharacteristic
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—41no—
122C42×C7—41no—
123C7×C16—22no—
124C7×C16—41no—
125C22×D8C22×C711yescharacteristic
126C2×#131C22×C711yescharacteristic
127#535—21no—
128C2×M32C22×C711yescharacteristic
129C22×C16C22×C711yescharacteristic
130#469—22no—
131#466—21no—
132#466—21no—
133C2×#108C22×C711yescharacteristic
134#427C22×C711yescharacteristic
135C2×#107C22×C711yescharacteristic
136#418—21no—
137#417—21no—
138#367—21no—
139#366—21no—
140#361—22no—
141#357—22no—
142#350—22no—
143#343—21no—
144C23×C7—21no—
145C23×C7—42no—
146C23×C7—81no—
147C23×C7—81no—
148C23×C7—161no—
149C23×C7—161no—
150C23×C7AGL(1,ℤ/8)11yescharacteristic
151C7×Q8—21no—
152C7×Q8—21no—
153C7×D4—21no—
154C7×D4—21no—
155C7×D4—41no—
156C7×D4—41no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4—81no—
160C7×D4—81no—
161C7×D4—162no—
162C7×D4—162no—
163C2×C4×C7—21no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—81no—
169C2×C4×C7—161no—
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12311yescharacteristic
173C2×C4×C7C2×D811yescharacteristic
174C2×C4×C7#13511yescharacteristic
175C2×C4×C7#13711yescharacteristic
176C7×C8—41no—
177C7×C8—81no—
178C7×C8#12111yescharacteristic
179C7×C8#12111yescharacteristic
180C7×C8#12311yescharacteristic
181C7×C8#12411yescharacteristic
182C2×#41C7×D411yescharacteristic
183C22×D4—21no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8C7×D412yes—
190#131—21no—
191#131—21no—
192#131—21no—
193#131—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16C7×D411yescharacteristic
197C22×C8—21no—
198C22×C8C23×C711yescharacteristic
199#127—41no—
200#124—41no—
201#121—22no—
202#121—22no—
203#120—42no—
204#117—21no—
205C2×C4⋊C4—21no—
206M32—21no—
207M32C7×D412yescomplemented
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212#108—21no—
213#108—21no—
214#107—21no—
215#107—21no—
216#107C7×D411yescharacteristic
217#107C7×D411yescharacteristic
218#106—21no—
219#103—41no—
220#102—22no—
221#102—22no—
222#102—41no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—82no—
229C22×C7—82no—
230C22×C7—82no—
231C22×C7—162no—
232C22×C7—162no—
233C22×C7#44711yescharacteristic
234C22×C7#46811yescharacteristic
235C22×C7#48211yescharacteristic
236C4×C7—41no—
237C4×C7—81no—
238C4×C7—81no—
239C4×C7—161no—
240C4×C7#39411yescharacteristic
241C4×C7#45211yescharacteristic
242C4×C7#46611yescharacteristic
243C4×C7#48011yescharacteristic
244C24—41no—
245#41—21no—
246#41—21no—
247#41—21no—
248#41—21no—
249C2×Q8C7×D811yescharacteristic
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4C7×D811yescharacteristic
260C22×C4—21no—
261C22×C4—41no—
262C22×C4C2×C7×D411yescharacteristic
263D8—22no—
264D8—22no—
265D8—41no—
266D8—41no—
267M16—21no—
268M16—21no—
269M16—41no—
270M16—41no—
271M16—41no—
272M16—41no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C2×C7×D411yescharacteristic
278C2×C8C2×C7×D411yescharacteristic
279C2×C8C2×C7×D411yescharacteristic
280C2×C8C7×#4111yescharacteristic
281C2×C8C7×#4111yescharacteristic
282C2×C8C7×#4111yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C42—41no—
291C16—22no—
292C16—41no—
293C2×C7—21no—
294C2×C7—21no—
295C2×C7—81no—
296C2×C7—162no—
297C2×C7#203311yescharacteristic
298C2×C7#219211yescharacteristic
299C2×C7#219411yescharacteristic
300C23—21no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—161no—
305C23—161no—
306C23C7×AGL(1,ℤ/8)11yescharacteristic
307Q8—21no—
308Q8—21no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—81no—
316D4—81no—
317D4—162no—
318D4—162no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—161no—
326C2×C4C7×#12011yescharacteristic
327C2×C4C7×#12111yescharacteristic
328C2×C4C7×#12311yescharacteristic
329C2×C4C2×C7×D811yescharacteristic
330C2×C4C7×#13511yescharacteristic
331C2×C4C7×#13711yescharacteristic
332C8—41no—
333C8—81no—
334C8C7×#12111yescharacteristic
335C8C7×#12111yescharacteristic
336C8C7×#12311yescharacteristic
337C8C7×#12411yescharacteristic
338C7#1338711yescharacteristic, direct product factor, Sylow
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—162no—
348C22—162no—
349C22C7×#44711yescharacteristic
350C22C7×#46811yescharacteristic
351C22C7×#48211yescharacteristic
352C4—41no—
353C4—81no—
354C4—81no—
355C4—161no—
356C4C7×#39411yescharacteristic
357C4C7×#45211yescharacteristic
358C4C7×#46611yescharacteristic
359C4C7×#48011yescharacteristic
360C2—21no—
361C2—21no—
362C2—81no—
363C2—162no—
364C2C7×#203311yescharacteristic
365C2C7×#219211yescharacteristic
366C2C7×#219411yescharacteristic
subgroupG/H
center, socleC22×C7#447
commutator, Frattini, cosocleC22×C8C23×C7
prime radicalC7×#3058C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2159C2semidirect
3. C2×C7×#361C2semidirect
4. C2×C7×#357C2semidirect
5. C2×C7×#350C2semidirect
6. C7×#1816C2semidirect
7. C7×#1382C2semidirect
27. #13387C7direct (N × H)
53. C7×M32D4semidirect
70. #2159C2×C7semidirect
71. C2×#361C2×C7semidirect
72. C2×#357C2×C7semidirect
73. C2×#350C2×C7semidirect
74. #1816C2×C7semidirect
75. #1382C2×C7semidirect
207. M32C7×D4semidirect
338. C7#13387direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30582
C7×#21592
C2×C7×#3612
C2×C7×#3572
C2×C7×#3502
C7×#18162
C7×#13822
#133877