Finite Group Databasebeta

← C7×#13391 #13190607

Subgroups — C7×#13391 #13190607

Order 1792.

390 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3067C211yesmaximal, characteristic
2C7×#2161C211yesmaximal, characteristic, complemented
3C7×#2148C211yesmaximal, characteristic, complemented
4C2×C7×#360C211yesmaximal, characteristic, complemented
5C7×#2115C211yesmaximal, characteristic, complemented
6C7×#1815C211yesmaximal, characteristic, complemented
7C7×#1383C211yesmaximal, characteristic, complemented
8C7×#575—21no—
9C2×C7×#135C2211yescharacteristic
10C2×C7×#131C2211yescharacteristic
11C2×C7×M32C2211yescharacteristic
12C2×C7×M32C2211yescharacteristic
13C7×#473—21no—
14C7×#472—21no—
15C7×#471—22no—
16C2×C7×#108C2211yescharacteristic
17C2×C7×#108C2211yescharacteristic
18C2×C7×#106C2211yescharacteristic
19C7×#417—21no—
20C7×#413—21no—
21C7×#368—22no—
22C7×#362—21no—
23C7×#361—21no—
24C7×#360—22no—
25C7×#349—22no—
26C7×#343—21no—
27#13391C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41—21no—
29C2×C7×#41D411yescharacteristic
30C7×#137—22no—
31C7×#137—41no—
32C7×AGL(1,ℤ/8)—22no—
33C7×AGL(1,ℤ/8)—41no—
34C7×#135—21no—
35C7×#135—21no—
36C7×#135—21no—
37C7×#135—21no—
38C7×#135—42no—
39C2×C7×Q16D411yescharacteristic
40C2×C7×SD16—21no—
41C2×C7×D8D411yescharacteristic
42C7×#131—21no—
43C7×#131—21no—
44C7×#131—21no—
45C7×#131—21no—
46C2×C7×M16—21no—
47C2×C7×M16—21no—
48C2×C7×M16—21no—
49C2×C7×M16D411yescharacteristic
50C22×C7×C8—21no—
51C22×C7×C8C2311yescharacteristic
52C2×C7×C4⋊C4—21no—
53C7×M32—21no—
54C7×M32—21no—
55C7×M32—21no—
56C7×M32D412yes—
57C2×C7×C16—21no—
58C2×C7×C16—21no—
59C7×#108—21no—
60C7×#108—21no—
61C7×#108—21no—
62C7×#108D412yes—
63C7×#106—21no—
64C7×#106—21no—
65C7×#103—41no—
66C7×#102—41no—
67C7×#101—22no—
68C7×#100—22no—
69#3067C2×C711yescharacteristic
70#2161C2×C711yescharacteristic, complemented
71#2148C2×C711yescharacteristic, complemented
72C2×#360C2×C711yescharacteristic, complemented
73#2115C2×C711yescharacteristic, complemented
74#1815C2×C711yescharacteristic, complemented
75#1383C2×C711yescharacteristic, complemented
76C7×#41—21no—
77C7×#41—21no—
78C7×#41—21no—
79C7×#41—21no—
80C7×#41—41no—
81C7×#41—41no—
82C7×#41—41no—
83C7×#41—41no—
84C2×C7×Q8—21no—
85C2×C7×Q8D811yescharacteristic
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—41no—
89C2×C7×D4D811yescharacteristic
90C22×C4×C7—21no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7C2×D411yescharacteristic
94C7×Q16—21no—
95C7×Q16—21no—
96C7×Q16—42no—
97C7×SD16—42no—
98C7×SD16—42no—
99C7×SD16—41no—
100C7×SD16—41no—
101C7×D8—21no—
102C7×D8—21no—
103C7×D8—42no—
104C7×M16—21no—
105C7×M16—22no—
106C7×M16—21no—
107C7×M16—41no—
108C7×M16—41no—
109C7×M16—41no—
110C7×M16—41no—
111C7×M16—41no—
112C2×C7×C8—21no—
113C2×C7×C8—41no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C2×C7×C8C2×D411yescharacteristic
118C2×C7×C8C2×D411yescharacteristic
119C2×C7×C8C2×D411yescharacteristic
120C2×C7×C8#4111yescharacteristic
121C2×C7×C8#4111yescharacteristic
122C2×C7×C8#4111yescharacteristic
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C16—22no—
126C7×C16—41no—
127#575—21no—
128C2×#135C22×C711yescharacteristic
129C2×#131C22×C711yescharacteristic
130C2×M32C22×C711yescharacteristic
131C2×M32C22×C711yescharacteristic
132#473—21no—
133#472—21no—
134#471—22no—
135C2×#108C22×C711yescharacteristic
136C2×#108C22×C711yescharacteristic
137C2×#106C22×C711yescharacteristic
138#417—21no—
139#413—21no—
140#368—22no—
141#362—21no—
142#361—21no—
143#360—22no—
144#349—22no—
145#343—21no—
146C23×C7—21no—
147C23×C7—41no—
148C23×C7#13511yescharacteristic
149C7×Q8—21no—
150C7×Q8—21no—
151C7×Q8—41no—
152C7×Q8—41no—
153C7×D4—21no—
154C7×D4—21no—
155C7×D4—41no—
156C7×D4—41no—
157C7×D4—41no—
158C7×D4—41no—
159C7×D4—81no—
160C7×D4—81no—
161C2×C4×C7—21no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—81no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7—161no—
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12311yescharacteristic
172C2×C4×C7C2×D811yescharacteristic
173C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C7#13711yescharacteristic
175C7×C8—41no—
176C7×C8—42no—
177C7×C8—81no—
178C7×C8#12111yescharacteristic
179C7×C8#12111yescharacteristic
180C7×C8#12311yescharacteristic
181C7×C8#12411yescharacteristic
182C2×#41—21no—
183C2×#41C7×D411yescharacteristic
184#137—22no—
185#137—41no—
186AGL(1,ℤ/8)—22no—
187AGL(1,ℤ/8)—41no—
188#135—21no—
189#135—21no—
190#135—21no—
191#135—21no—
192#135—42no—
193C2×Q16C7×D411yescharacteristic
194C2×SD16—21no—
195C2×D8C7×D411yescharacteristic
196#131—21no—
197#131—21no—
198#131—21no—
199#131—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16C7×D411yescharacteristic
204C22×C8—21no—
205C22×C8C23×C711yescharacteristic
206C2×C4⋊C4—21no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32C7×D412yes—
211C2×C16—21no—
212C2×C16—21no—
213#108—21no—
214#108—21no—
215#108—21no—
216#108C7×D412yes—
217#106—21no—
218#106—21no—
219#103—41no—
220#102—41no—
221#101—22no—
222#100—22no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7—161no—
228C22×C7#45211yescharacteristic
229C22×C7#46611yescharacteristic
230C22×C7#48211yescharacteristic
231C4×C7—41no—
232C4×C7—81no—
233C4×C7—161no—
234C4×C7#39411yescharacteristic
235C4×C7#44711yescharacteristic
236C4×C7#46811yescharacteristic
237C4×C7#48011yescharacteristic
238#41—21no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8C7×D811yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4C7×D811yescharacteristic
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×C7×D411yescharacteristic
256Q16—21no—
257Q16—21no—
258Q16—42no—
259SD16—42no—
260SD16—42no—
261SD16—41no—
262SD16—41no—
263D8—21no—
264D8—21no—
265D8—42no—
266M16—21no—
267M16—22no—
268M16—21no—
269M16—41no—
270M16—41no—
271M16—41no—
272M16—41no—
273M16—41no—
274C2×C8—21no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8C2×C7×D411yescharacteristic
280C2×C8C2×C7×D411yescharacteristic
281C2×C8C2×C7×D411yescharacteristic
282C2×C8C7×#4111yescharacteristic
283C2×C8C7×#4111yescharacteristic
284C2×C8C7×#4111yescharacteristic
285C4⋊C4—41no—
286C4⋊C4—41no—
287C16—22no—
288C16—41no—
289C2×C7—21no—
290C2×C7—21no—
291C2×C7—21no—
292C2×C7—81no—
293C2×C7—161no—
294C2×C7#203311yescharacteristic
295C23—21no—
296C23—41no—
297C23C7×#13511yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—41no—
301Q8—41no—
302D4—21no—
303D4—21no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—81no—
309D4—81no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—161no—
318C2×C4C7×#12011yescharacteristic
319C2×C4C7×#12111yescharacteristic
320C2×C4C7×#12311yescharacteristic
321C2×C4C2×C7×D811yescharacteristic
322C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
323C2×C4C7×#13711yescharacteristic
324C8—41no—
325C8—42no—
326C8—81no—
327C8C7×#12111yescharacteristic
328C8C7×#12111yescharacteristic
329C8C7×#12311yescharacteristic
330C8C7×#12411yescharacteristic
331C7#1339111yescharacteristic, direct product factor, Sylow
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22—161no—
337C22C7×#45211yescharacteristic
338C22C7×#46611yescharacteristic
339C22C7×#48211yescharacteristic
340C4—41no—
341C4—81no—
342C4—161no—
343C4C7×#39411yescharacteristic
344C4C7×#44711yescharacteristic
345C4C7×#46811yescharacteristic
346C4C7×#48011yescharacteristic
347C2—21no—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—161no—
352C2C7×#203311yescharacteristic
subgroupG/H
center, socleC2×C7#2033
commutator, Frattini, cosocleC22×C8C23×C7
prime radicalC7×#3067C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2161C2semidirect
3. C7×#2148C2semidirect
4. C2×C7×#360C2semidirect
5. C7×#2115C2semidirect
6. C7×#1815C2semidirect
7. C7×#1383C2semidirect
27. #13391C7direct (N × H)
70. #2161C2×C7semidirect
71. #2148C2×C7semidirect
72. C2×#360C2×C7semidirect
73. #2115C2×C7semidirect
74. #1815C2×C7semidirect
75. #1383C2×C7semidirect
331. C7#13391direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30672
C7×#21612
C7×#21482
C2×C7×#3602
C7×#21152
C7×#18152
C7×#13832
#133917