Finite Group Databasebeta

← C7×#10292 #13187508

Subgroups — C7×#10292 #13187508

Order 1792.

640 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3230C211yesmaximal, characteristic, complemented
2C7×#3208C211yesmaximal, characteristic
3C7×#2858C211yesmaximal, characteristic
4C7×#1994C211yesmaximal, characteristic, complemented
5C7×#1549C211yesmaximal, characteristic, complemented
6C7×#1545C211yesmaximal, characteristic
7C7×#1276C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C2×C7×#124—22no—
12C7×#500—21no—
13C7×#492—21no—
14C7×#487—21no—
15C7×#486—21no—
16C7×#484—22no—
17C7×#477—22no—
18C7×#460—22no—
19C7×#450—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#390—22no—
24C7×#390—22no—
25C43×C7C2211yescharacteristic
26C7×#339—22no—
27C7×#332—22no—
28C7×#331—22no—
29#10292C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4D412yes—
34C2×C7×SD16—24no—
35C2×C7×M16—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×#127—21no—
43C7×#127—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#125—21no—
47C7×#124—48no—
48C7×C22⋊Q8—21no—
49C7×C22⋊Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C7×#117—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4D412yes—
55C2×C7×C22⋊C4—28no—
56C2×C42×C7—21no—
57C2×C42×C7—21no—
58C2×C42×C7D412yes—
59C2×C42×C7C2311yescharacteristic
60C7×#107—24no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8D412yes—
64C7×C4⋊C8D412yes—
65C7×#103—24no—
66C7×#102—24no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8—21no—
69C4×C7×C8—21no—
70C4×C7×C8—21no—
71#3230C2×C711yescharacteristic, complemented
72#3208C2×C711yescharacteristic
73#2858C2×C711yescharacteristic
74#1994C2×C711yescharacteristic, complemented
75#1549C2×C711yescharacteristic, complemented
76#1545C2×C711yescharacteristic
77#1276C2×C711yescharacteristic, complemented
78C24×C7—22no—
79C2×C7×Q8—21no—
80C2×C7×Q8—21no—
81C2×C7×Q8—44no—
82C2×C7×Q8—41no—
83C2×C7×Q8—41no—
84C2×C7×D4—22no—
85C2×C7×D4—22no—
86C2×C7×D4—44no—
87C2×C7×D4—44no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—41no—
92C2×C7×D4—42no—
93C2×C7×D4—42no—
94C22×C4×C7—22no—
95C22×C4×C7—22no—
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—21no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C22⋊C4—416no—
115C7×C22⋊C4—41no—
116C42×C7—24no—
117C42×C7—22no—
118C42×C7—22no—
119C42×C7—42no—
120C42×C7—41no—
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124C42×C7#4111yescharacteristic
125#533C22×C711yescharacteristic
126C2×C4⋊Q8C22×C711yescharacteristic
127C2×#127C22×C711yescharacteristic
128C2×#124—22no—
129#500—21no—
130#492—21no—
131#487—21no—
132#486—21no—
133#484—22no—
134#477—22no—
135#460—22no—
136#450—22no—
137#431C22×C711yescharacteristic
138#431C22×C711yescharacteristic
139#423C22×C711yescharacteristic
140#390—22no—
141#390—22no—
142C43C22×C711yescharacteristic
143#339—22no—
144#332—22no—
145#331—22no—
146C23×C7—44no—
147C23×C7—44no—
148C23×C7—42no—
149C23×C7—44no—
150C23×C7#12011yescharacteristic
151C7×Q8—42no—
152C7×Q8—42no—
153C7×Q8—82no—
154C7×Q8—82no—
155C7×D4—44no—
156C7×D4—82no—
157C7×D4—82no—
158C7×D4—82no—
159C7×D4—82no—
160C7×D4—81no—
161C7×D4—81no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—24no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—44no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7#12011yescharacteristic
172C2×C4×C7#12011yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12111yescharacteristic
175C2×C4×C7#12111yescharacteristic
176C2×C4×C7#12711yescharacteristic
177C2×C4×C7#13512yes—
178C2×C4×C7#13712yes—
179C7×C8—42no—
180C7×C8—42no—
181C7×C8—81no—
182C22×Q8—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C7×D412yes—
186C2×SD16—24no—
187C2×M16—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C7×D411yescharacteristic
193C4⋊Q8C7×D411yescharacteristic
194#127—21no—
195#127—21no—
196#127—21no—
197#127—21no—
198#125—21no—
199#124—48no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C7×D412yes—
207C2×C22⋊C4—28no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42C7×D412yes—
211C2×C42C23×C711yescharacteristic
212#107—24no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8C7×D412yes—
216C4⋊C8C7×D412yes—
217#103—24no—
218#102—24no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—88no—
226C22×C7—82no—
227C22×C7—82no—
228C22×C7—82no—
229C22×C7#39211yescharacteristic
230C22×C7#45212yes—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—44no—
234C4×C7—82no—
235C4×C7—81no—
236C4×C7—81no—
237C4×C7#45312yes—
238C4×C7#45412yes—
239C4×C7#46412yes—
240C4×C7#49511yescharacteristic
241C4×C7#49711yescharacteristic
242C24—22no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—44no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4C2×C7×D411yescharacteristic
261C22×C4C2×C7×D411yescharacteristic
262C22×C4C2×C7×D411yescharacteristic
263SD16—84no—
264M16—42no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C22⋊C4—416no—
279C22⋊C4—41no—
280C42—24no—
281C42—22no—
282C42—22no—
283C42—42no—
284C42—41no—
285C42C2×C7×D411yescharacteristic
286C42C2×C7×D411yescharacteristic
287C42C2×C7×D411yescharacteristic
288C42C7×#4111yescharacteristic
289C2×C7—22no—
290C2×C7—162no—
291C2×C7#200211yescharacteristic
292C2×C7#200311yescharacteristic
293C2×C7#201511yescharacteristic
294C23—44no—
295C23—44no—
296C23—42no—
297C23—44no—
298C23C7×#12011yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—82no—
302Q8—82no—
303D4—44no—
304D4—82no—
305D4—82no—
306D4—81no—
307D4—81no—
308D4—82no—
309D4—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4C7×#12011yescharacteristic
320C2×C4C7×#12011yescharacteristic
321C2×C4C7×#12111yescharacteristic
322C2×C4C7×#12111yescharacteristic
323C2×C4C7×#12111yescharacteristic
324C2×C4C7×#12711yescharacteristic
325C2×C4C7×#13512yes—
326C2×C4C7×#13712yes—
327C8—42no—
328C8—42no—
329C8—81no—
330C7#1029211yescharacteristic, direct product factor, Sylow
331C22—21no—
332C22—21no—
333C22—88no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22C7×#39211yescharacteristic
338C22C7×#45212yes—
339C4—21no—
340C4—21no—
341C4—44no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4C7×#45312yes—
346C4C7×#45412yes—
347C4C7×#46412yes—
348C4C7×#49511yescharacteristic
349C4C7×#49711yescharacteristic
350C2—22no—
351C2—162no—
352C2C7×#200211yescharacteristic
353C2C7×#200311yescharacteristic
354C2C7×#201511yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
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
1. C7×#3230C2semidirect
4. C7×#1994C2semidirect
5. C7×#1549C2semidirect
7. C7×#1276C2semidirect
29. #10292C7direct (N × H)
71. #3230C2×C7semidirect
74. #1994C2×C7semidirect
75. #1549C2×C7semidirect
77. #1276C2×C7semidirect
330. C7#10292direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32302
C7×#32082
C7×#28582
C7×#19942
C7×#15492
C7×#15452
C7×#12762
#102927