Finite Group Databasebeta

← C7×#9283 #13186499

Subgroups — C7×#9283 #13186499

Order 1792.

448 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3207C211yesmaximal, characteristic
2C7×#2981C211yesmaximal, characteristic, complemented
3C7×#1530C211yesmaximal, characteristic, complemented
4C7×#1530C211yesmaximal, characteristic, complemented
5C7×#1496C211yesmaximal, characteristic, complemented
6C7×#1287C211yesmaximal, characteristic, complemented
7C7×#1275C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C7×#532C2211yescharacteristic
10C7×#493C411yescharacteristic
11C7×#492C411yescharacteristic
12C7×#487C411yescharacteristic
13C7×#486C411yescharacteristic
14C7×#463—22no—
15C7×#449—22no—
16C7×#436—21no—
17C7×#435—22no—
18C7×C8×D4—21no—
19C7×#432C2211yescharacteristic
20C7×#432C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#408—22no—
24C2×C7×#97C2211yescharacteristic
25C7×#329—22no—
26C7×#329—22no—
27C7×D4⋊C8—21no—
28C7×D4⋊C8—21no—
29#9283C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C2×C7×#41D412yes—
32C2×C7×SD16—22no—
33C2×C7×D8—22no—
34C2×C7×M16—22no—
35C22×C7×C8—21no—
36C22×C7×C8—22no—
37C7×C4⋊Q8C2×C411yescharacteristic
38C7×C4⋊Q8C2×C411yescharacteristic
39C7×#127C2×C411yescharacteristic
40C7×#127C2×C411yescharacteristic
41C7×#124—21no—
42C7×#124D412yes—
43C7×#121—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C2×C42×C7C2311yescharacteristic
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×#103—22no—
57C7×#102—22no—
58C7×C22⋊C8—22no—
59C7×C22⋊C8—22no—
60C7×C22⋊C8—21no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×C22⋊C8—21no—
64C7×#97—21no—
65C7×#97—21no—
66C7×#97—21no—
67C7×#97D412yes—
68C4×C7×C8C2×C411yescharacteristic
69C4×C7×C8C2×C411yescharacteristic
70C4×C7×C8D411yescharacteristic
71C4×C7×C8D411yescharacteristic
72#3207C2×C711yescharacteristic
73#2981C2×C711yescharacteristic, complemented
74#1530C2×C711yescharacteristic, complemented
75#1530C2×C711yescharacteristic, complemented
76#1496C2×C711yescharacteristic, complemented
77#1287C2×C711yescharacteristic, complemented
78#1275C2×C711yescharacteristic, complemented
79C7×#41—82no—
80C7×#41—82no—
81C2×C7×Q8—21no—
82C2×C7×Q8C22⋊C412yes—
83C2×C7×D4—22no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—41no—
90C2×C7×D4C22⋊C412yes—
91C22×C4×C7—22no—
92C22×C4×C7—41no—
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C22×C4×C7#4111yescharacteristic
96C7×SD16—82no—
97C7×D8—82no—
98C7×M16—82no—
99C2×C7×C8—22no—
100C2×C7×C8—21no—
101C2×C7×C8—21no—
102C2×C7×C8—21no—
103C2×C7×C8—21no—
104C2×C7×C8—22no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—82no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C42×C7C22×C411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7C2×D411yescharacteristic
120C42×C7C2×D411yescharacteristic
121#532C22×C711yescharacteristic
122#532C22×C711yescharacteristic
123#493C4×C711yescharacteristic
124#492C4×C711yescharacteristic
125#487C4×C711yescharacteristic
126#486C4×C711yescharacteristic
127#463—22no—
128#449—22no—
129#436—21no—
130#435—22no—
131C8×D4—21no—
132#432C22×C711yescharacteristic
133#432C22×C711yescharacteristic
134#431C22×C711yescharacteristic
135#431C22×C711yescharacteristic
136#408—22no—
137C2×#97C22×C711yescharacteristic
138#329—22no—
139#329—22no—
140D4⋊C8—21no—
141D4⋊C8—21no—
142C23×C7—22no—
143C23×C7—41no—
144C23×C7#12311yescharacteristic
145C7×Q8—42no—
146C7×Q8—81no—
147C7×D4—42no—
148C7×D4—42no—
149C7×D4—41no—
150C7×D4—41no—
151C7×D4—41no—
152C7×D4—41no—
153C7×D4—81no—
154C7×D4—81no—
155C7×D4—82no—
156C7×D4—82no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—41no—
165C2×C4×C7—82no—
166C2×C4×C7C2×C22⋊C411yescharacteristic
167C2×C4×C7C4×D411yescharacteristic
168C2×C4×C7C4×D411yescharacteristic
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C7×C8—42no—
173C7×C8—41no—
174C7×C8—42no—
175C7×C8—41no—
176C7×C8—41no—
177C7×C8—41no—
178C7×C8—81no—
179C2×#41—21no—
180C2×#41C7×D412yes—
181C2×SD16—22no—
182C2×D8—22no—
183C2×M16—22no—
184C22×C8—21no—
185C22×C8—22no—
186C4⋊Q8C2×C4×C711yescharacteristic
187C4⋊Q8C2×C4×C711yescharacteristic
188#127C2×C4×C711yescharacteristic
189#127C2×C4×C711yescharacteristic
190#124—21no—
191#124C7×D412yes—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C2×C42C23×C711yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205#103—22no—
206#102—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213#97—21no—
214#97—21no—
215#97—21no—
216#97C7×D412yes—
217C4×C8C2×C4×C711yescharacteristic
218C4×C8C2×C4×C711yescharacteristic
219C4×C8C7×D411yescharacteristic
220C4×C8C7×D411yescharacteristic
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—42no—
225C22×C7—42no—
226C22×C7—42no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7#38611yescharacteristic
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—42no—
237C4×C7—81no—
238C4×C7#44311yescharacteristic
239C4×C7#44411yescharacteristic
240#41—82no—
241#41—82no—
242C2×Q8—21no—
243C2×Q8C7×C22⋊C412yes—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4C7×C22⋊C412yes—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4C2×C7×D411yescharacteristic
255C22×C4C7×#4111yescharacteristic
256C22×C4C7×#4111yescharacteristic
257SD16—82no—
258D8—82no—
259M16—82no—
260C2×C8—22no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—22no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—82no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42C22×C4×C711yescharacteristic
279C42C2×C7×D411yescharacteristic
280C42C2×C7×D411yescharacteristic
281C42C2×C7×D411yescharacteristic
282C2×C7—41no—
283C2×C7—82no—
284C2×C7—161no—
285C2×C7#187611yescharacteristic
286C2×C7#190311yescharacteristic
287C2×C7#191111yescharacteristic
288C23—22no—
289C23—41no—
290C23C7×#12311yescharacteristic
291Q8—42no—
292Q8—81no—
293D4—42no—
294D4—42no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—81no—
300D4—81no—
301D4—82no—
302D4—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4C2×C7×C22⋊C411yescharacteristic
313C2×C4C4×C7×D411yescharacteristic
314C2×C4C4×C7×D411yescharacteristic
315C2×C4C7×#12011yescharacteristic
316C2×C4C7×#12111yescharacteristic
317C2×C4C7×#12111yescharacteristic
318C8—42no—
319C8—41no—
320C8—42no—
321C8—41no—
322C8—41no—
323C8—41no—
324C8—81no—
325C7#928311yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—21no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C7×#38611yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—81no—
343C4C7×#44311yescharacteristic
344C4C7×#44411yescharacteristic
345C2—41no—
346C2—82no—
347C2—161no—
348C2C7×#187611yescharacteristic
349C2C7×#190311yescharacteristic
350C2C7×#191111yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3207C2
LCSG ▷ C42 ▷ 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×#2981C2semidirect
3. C7×#1530C2semidirect
4. C7×#1530C2semidirect
5. C7×#1496C2semidirect
6. C7×#1287C2semidirect
7. C7×#1275C2semidirect
29. #9283C7direct (N × H)
73. #2981C2×C7semidirect
74. #1530C2×C7semidirect
75. #1530C2×C7semidirect
76. #1496C2×C7semidirect
77. #1287C2×C7semidirect
78. #1275C2×C7semidirect
325. C7#9283direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32072
C7×#29812
C7×#15302
C7×#14962
C7×#12872
C7×#12752
#92837