Finite Group Databasebeta

← C7×#10552 #13187768

Subgroups — C7×#10552 #13187768

Order 1792.

386 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3141C211yesmaximal, characteristic
2C7×#1950C211yesmaximal, characteristic, complemented
3C7×#1548C211yesmaximal, characteristic, complemented
4C7×#1547C211yesmaximal, characteristic
5C2×C7×#332C211yesmaximal, characteristic, complemented
6C7×#1480C211yesmaximal, characteristic
7C7×#1283C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C7×#489—22no—
11C7×#487—21no—
12C7×#486—21no—
13C7×#431C2211yescharacteristic
14C2×C7×#108—21no—
15C7×#423C2211yescharacteristic
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C7×#417—21no—
19C7×#417—21no—
20C4×C7×M16C2211yescharacteristic
21C7×#341—21no—
22C7×#333—22no—
23C7×#332—21no—
24C7×#332—22no—
25C7×#331—21no—
26C7×D4⋊C8—22no—
27#10552C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41—21no—
29C22×C7×Q8—21no—
30C2×C7×M16—21no—
31C2×C7×M16D411yescharacteristic
32C2×C7×M16D411yescharacteristic
33C22×C7×C8—21no—
34C22×C7×C8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8D411yescharacteristic
39C7×C4⋊Q8D412yes—
40C7×#127D411yescharacteristic
41C7×#124—21no—
42C7×#121—21no—
43C7×#121—21no—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C42×C7C2311yescharacteristic
49C7×#108—42no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C4⋊C8D412yes—
57C7×#103—22no—
58C7×#103—22no—
59C7×#102—22no—
60C7×#102—22no—
61C7×C22⋊C8—21no—
62C7×#97—21no—
63C4×C7×C8—21no—
64C4×C7×C8—21no—
65#3141C2×C711yescharacteristic
66#1950C2×C711yescharacteristic, complemented
67#1548C2×C711yescharacteristic, complemented
68#1547C2×C711yescharacteristic
69C2×#332C2×C711yescharacteristic, complemented
70#1480C2×C711yescharacteristic
71#1283C2×C711yescharacteristic, complemented
72C7×#41—41no—
73C7×#41—41no—
74C7×#41—41no—
75C7×#41—41no—
76C2×C7×Q8—21no—
77C2×C7×Q8—22no—
78C2×C7×Q8—41no—
79C2×C7×Q8—81no—
80C2×C7×Q8—81no—
81C2×C7×D4—21no—
82C2×C7×D4—21no—
83C2×C7×D4—21no—
84C2×C7×D4—41no—
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C7×M16—21no—
91C7×M16—21no—
92C7×M16—21no—
93C7×M16—21no—
94C7×M16—41no—
95C7×M16—41no—
96C2×C7×C8—21no—
97C2×C7×C8—21no—
98C2×C7×C8—22no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C2×C7×C8—81no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C42×C7C2×D411yescharacteristic
114C42×C7C2×D411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7#4111yescharacteristic
117#532C22×C711yescharacteristic
118C2×C4⋊Q8C22×C711yescharacteristic
119#489—22no—
120#487—21no—
121#486—21no—
122#431C22×C711yescharacteristic
123C2×#108—21no—
124#423C22×C711yescharacteristic
125C2×C4⋊C8C22×C711yescharacteristic
126C2×C4⋊C8C22×C711yescharacteristic
127#417—21no—
128#417—21no—
129C4×M16C22×C711yescharacteristic
130#341—21no—
131#333—22no—
132#332—21no—
133#332—22no—
134#331—21no—
135D4⋊C8—22no—
136C23×C7—41no—
137C23×C7#12411yescharacteristic
138C7×Q8—41no—
139C7×Q8—41no—
140C7×Q8—82no—
141C7×Q8—81no—
142C7×Q8—81no—
143C7×D4—41no—
144C7×D4—41no—
145C7×D4—41no—
146C7×D4—41no—
147C7×D4—81no—
148C7×D4—81no—
149C7×D4—81no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—42no—
155C2×C4×C7—82no—
156C2×C4×C7—81no—
157C2×C4×C7—81no—
158C2×C4×C7#12011yescharacteristic
159C2×C4×C7#12111yescharacteristic
160C2×C4×C7#12111yescharacteristic
161C2×C4×C7#12111yescharacteristic
162C2×C4×C7#12311yescharacteristic
163C2×C4×C7#12311yescharacteristic
164C2×C4×C7#13511yescharacteristic
165C2×C4×C7#13511yescharacteristic
166C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C7#13711yescharacteristic
168C7×C8—41no—
169C7×C8—41no—
170C7×C8—42no—
171C7×C8—81no—
172C7×C8—81no—
173C7×C8—81no—
174C2×#41—21no—
175C22×Q8—21no—
176C2×M16—21no—
177C2×M16C7×D411yescharacteristic
178C2×M16C7×D411yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C7×D412yes—
185C4⋊Q8C7×D411yescharacteristic
186#127C7×D411yescharacteristic
187#124—21no—
188#121—21no—
189#121—21no—
190C4×D4—21no—
191C4×D4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C42C23×C711yescharacteristic
195#108—42no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8C7×D412yes—
203#103—22no—
204#103—22no—
205#102—22no—
206#102—22no—
207C22⋊C8—21no—
208#97—21no—
209C4×C8—21no—
210C4×C8—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7#39411yescharacteristic
217C22×C7#48811yescharacteristic
218C22×C7#48911yescharacteristic
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—82no—
223C4×C7—81no—
224C4×C7#44911yescharacteristic
225C4×C7#45211yescharacteristic
226C4×C7#46311yescharacteristic
227C4×C7#46411yescharacteristic
228C4×C7#47111yescharacteristic
229C4×C7#47211yescharacteristic
230#41—41no—
231#41—41no—
232#41—41no—
233#41—41no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—81no—
238C2×Q8—81no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4C2×C7×D411yescharacteristic
246C22×C4C7×#4111yescharacteristic
247C22×C4C7×#4111yescharacteristic
248M16—21no—
249M16—21no—
250M16—21no—
251M16—21no—
252M16—41no—
253M16—41no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—81no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C42C2×C7×D411yescharacteristic
272C42C2×C7×D411yescharacteristic
273C42C2×C7×D411yescharacteristic
274C42C7×#4111yescharacteristic
275C2×C7—21no—
276C2×C7—21no—
277C2×C7—161no—
278C2×C7#204211yescharacteristic
279C2×C7#204911yescharacteristic
280C2×C7#205111yescharacteristic
281C23—41no—
282C23C7×#12411yescharacteristic
283Q8—41no—
284Q8—41no—
285Q8—82no—
286Q8—81no—
287Q8—81no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—81no—
293D4—81no—
294D4—81no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4C7×#12011yescharacteristic
304C2×C4C7×#12111yescharacteristic
305C2×C4C7×#12111yescharacteristic
306C2×C4C7×#12111yescharacteristic
307C2×C4C7×#12311yescharacteristic
308C2×C4C7×#12311yescharacteristic
309C2×C4C7×#13511yescharacteristic
310C2×C4C7×#13511yescharacteristic
311C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
312C2×C4C7×#13711yescharacteristic
313C8—41no—
314C8—41no—
315C8—42no—
316C8—81no—
317C8—81no—
318C8—81no—
319C7#1055211yescharacteristic, direct product factor, Sylow
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C7×#39411yescharacteristic
326C22C7×#48811yescharacteristic
327C22C7×#48911yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—82no—
333C4C7×#44911yescharacteristic
334C4C7×#45211yescharacteristic
335C4C7×#46311yescharacteristic
336C4C7×#46411yescharacteristic
337C4C7×#47111yescharacteristic
338C4C7×#47211yescharacteristic
339C2—21no—
340C2—21no—
341C2—161no—
342C2C7×#204211yescharacteristic
343C2C7×#204911yescharacteristic
344C2C7×#205111yescharacteristic
subgroupG/H
center, socleC22×C7#394
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
2. C7×#1950C2semidirect
3. C7×#1548C2semidirect
5. C2×C7×#332C2semidirect
7. C7×#1283C2semidirect
27. #10552C7direct (N × H)
66. #1950C2×C7semidirect
67. #1548C2×C7semidirect
69. C2×#332C2×C7semidirect
71. #1283C2×C7semidirect
319. C7#10552direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31412
C7×#19502
C7×#15482
C7×#15472
C2×C7×#3322
C7×#14802
C7×#12832
#105527