Finite Group Databasebeta

← C7×#10288 #13187504

Subgroups — C7×#10288 #13187504

Order 1792.

554 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3226C211yesmaximal, characteristic
2C7×#3206C211yesmaximal, characteristic
3C7×#2862C211yesmaximal, characteristic, complemented
4C7×#1962C211yesmaximal, characteristic
5C7×#1548C211yesmaximal, characteristic, complemented
6C7×#1546C211yesmaximal, characteristic, complemented
7C7×#1276C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C7×#500—21no—
12C7×#492—21no—
13C7×#487—21no—
14C7×#486—21no—
15C7×#480—22no—
16C7×#474—22no—
17C7×#461—22no—
18C7×#448—22no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#423C2211yescharacteristic
22C2×C7×#104—22no—
23C7×#391—22no—
24C7×#391—22no—
25C7×#382—22no—
26C43×C7C2211yescharacteristic
27C7×#333—22no—
28C7×#332—22no—
29#10288C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8—21no—
32C22×C7×Q8D412yes—
33C2×C7×SD16—24no—
34C2×C7×M16—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8D411yescharacteristic
44C7×#127D411yescharacteristic
45C7×#124—21no—
46C7×#121—21no—
47C7×#121—21no—
48C4×C7×D4—21no—
49C4×C7×D4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4D412yes—
54C2×C42×C7—21no—
55C2×C42×C7—21no—
56C2×C42×C7D412yes—
57C2×C42×C7C2311yescharacteristic
58C7×#107—24no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8D412yes—
62C7×C4⋊C8D412yes—
63C7×#104—44no—
64C7×#103—24no—
65C7×#102—24no—
66C7×C22⋊C8—21no—
67C7×C22⋊C8—21no—
68C4×C7×C8—21no—
69C4×C7×C8—21no—
70C7×#95—28no—
71#3226C2×C711yescharacteristic
72#3206C2×C711yescharacteristic
73#2862C2×C711yescharacteristic, complemented
74#1962C2×C711yescharacteristic
75#1548C2×C711yescharacteristic, complemented
76#1546C2×C711yescharacteristic, complemented
77#1276C2×C711yescharacteristic, complemented
78C7×#41—42no—
79C7×#41—42no—
80C2×C7×Q8—22no—
81C2×C7×Q8—21no—
82C2×C7×Q8—22no—
83C2×C7×Q8—44no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—41no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C7×SD16—84no—
100C7×M16—42no—
101C2×C7×C8—22no—
102C2×C7×C8—22no—
103C2×C7×C8—41no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C42×C7—24no—
117C42×C7—22no—
118C42×C7—22no—
119C42×C7—42no—
120C42×C7C2×D411yescharacteristic
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7#4111yescharacteristic
124#532C22×C711yescharacteristic
125C2×C4⋊Q8C22×C711yescharacteristic
126C2×C4⋊Q8C22×C711yescharacteristic
127#500—21no—
128#492—21no—
129#487—21no—
130#486—21no—
131#480—22no—
132#474—22no—
133#461—22no—
134#448—22no—
135#431C22×C711yescharacteristic
136#431C22×C711yescharacteristic
137#423C22×C711yescharacteristic
138C2×#104—22no—
139#391—22no—
140#391—22no—
141#382—22no—
142C43C22×C711yescharacteristic
143#333—22no—
144#332—22no—
145C23×C7—41no—
146C23×C7#12011yescharacteristic
147C7×Q8—44no—
148C7×Q8—42no—
149C7×Q8—82no—
150C7×Q8—82no—
151C7×Q8—81no—
152C7×Q8—81no—
153C7×D4—42no—
154C7×D4—42no—
155C7×D4—42no—
156C7×D4—82no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—24no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—44no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—44no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12711yescharacteristic
175C2×C4×C7#13512yes—
176C2×C4×C7AGL(1,ℤ/8)12yes—
177C7×C8—42no—
178C7×C8—42no—
179C7×C8—81no—
180C2×#41—21no—
181C22×Q8—21no—
182C22×Q8C7×D412yes—
183C2×SD16—24no—
184C2×M16—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C7×D411yescharacteristic
194#127C7×D411yescharacteristic
195#124—21no—
196#121—21no—
197#121—21no—
198C4×D4—21no—
199C4×D4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C7×D412yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C7×D412yes—
207C2×C42C23×C711yescharacteristic
208#107—24no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8C7×D412yes—
212C4⋊C8C7×D412yes—
213#104—44no—
214#103—24no—
215#102—24no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95—28no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7#39211yescharacteristic
227C22×C7#44912yes—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—44no—
231C4×C7—82no—
232C4×C7—82no—
233C4×C7—81no—
234C4×C7#45512yes—
235C4×C7#45612yes—
236C4×C7#46312yes—
237C4×C7#49511yescharacteristic
238C4×C7#49611yescharacteristic
239#41—42no—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4C2×C7×D411yescharacteristic
258C22×C4C2×C7×D411yescharacteristic
259C22×C4C2×C7×D411yescharacteristic
260SD16—84no—
261M16—42no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—24no—
278C42—22no—
279C42—22no—
280C42—42no—
281C42C2×C7×D411yescharacteristic
282C42C2×C7×D411yescharacteristic
283C42C2×C7×D411yescharacteristic
284C42C7×#4111yescharacteristic
285C2×C7—22no—
286C2×C7—161no—
287C2×C7#200411yescharacteristic
288C2×C7#200511yescharacteristic
289C2×C7#201211yescharacteristic
290C23—41no—
291C23C7×#12011yescharacteristic
292Q8—44no—
293Q8—42no—
294Q8—82no—
295Q8—81no—
296Q8—81no—
297Q8—82no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4C7×#12011yescharacteristic
315C2×C4C7×#12011yescharacteristic
316C2×C4C7×#12111yescharacteristic
317C2×C4C7×#12111yescharacteristic
318C2×C4C7×#12111yescharacteristic
319C2×C4C7×#12711yescharacteristic
320C2×C4C7×#13512yes—
321C2×C4C7×AGL(1,ℤ/8)12yes—
322C8—42no—
323C8—42no—
324C8—81no—
325C7#1028811yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22C7×#39211yescharacteristic
332C22C7×#44912yes—
333C4—21no—
334C4—21no—
335C4—44no—
336C4—82no—
337C4—82no—
338C4—81no—
339C4C7×#45512yes—
340C4C7×#45612yes—
341C4C7×#46312yes—
342C4C7×#49511yescharacteristic
343C4C7×#49611yescharacteristic
344C2—22no—
345C2—161no—
346C2C7×#200411yescharacteristic
347C2C7×#200511yescharacteristic
348C2C7×#201211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ 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×#2862C2semidirect
5. C7×#1548C2semidirect
6. C7×#1546C2semidirect
7. C7×#1276C2semidirect
29. #10288C7direct (N × H)
73. #2862C2×C7semidirect
75. #1548C2×C7semidirect
76. #1546C2×C7semidirect
77. #1276C2×C7semidirect
325. C7#10288direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32262
C7×#32062
C7×#28622
C7×#19622
C7×#15482
C7×#15462
C7×#12762
#102887