Finite Group Databasebeta

← C7×#10553 #13187769

Subgroups — C7×#10553 #13187769

Order 1792.

416 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×#3138C211yesmaximal, characteristic
2C7×#1950C211yesmaximal, characteristic, complemented
3C7×#1550C211yesmaximal, characteristic, complemented
4C7×#1545C211yesmaximal, characteristic
5C2×C7×#331C211yesmaximal, characteristic
6C7×#1481C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×#127C2211yescharacteristic
10C7×#490—22no—
11C7×#486—21no—
12C7×#486—21no—
13C7×#431C2211yescharacteristic
14C2×C7×#108—21no—
15C7×#423C2211yescharacteristic
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C7×#418—21no—
19C7×#418—21no—
20C4×C7×M16C2211yescharacteristic
21C7×#341—21no—
22C7×#333—21no—
23C7×#332—21no—
24C7×#331—22no—
25C7×#331—22no—
26C7×Q8⋊C8—22no—
27#10553C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C2×C7×M16—21no—
32C2×C7×M16D411yescharacteristic
33C2×C7×M16D411yescharacteristic
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C7×C4⋊Q8D411yescharacteristic
37C7×C4⋊Q8D411yescharacteristic
38C7×#127—21no—
39C7×#127—21no—
40C7×#127—21no—
41C7×#127D412yes—
42C7×#125—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C4×C7×Q8—21no—
46C4×C7×Q8—21no—
47C7×#117—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×#102—22no—
58C7×#102—22no—
59C7×#102—22no—
60C7×#102—22no—
61C7×C22⋊C8—21no—
62C7×#97—21no—
63C4×C7×C8—21no—
64C4×C7×C8—21no—
65#3138C2×C711yescharacteristic
66#1950C2×C711yescharacteristic, complemented
67#1550C2×C711yescharacteristic, complemented
68#1545C2×C711yescharacteristic
69C2×#331C2×C711yescharacteristic
70#1481C2×C711yescharacteristic, complemented
71#1283C2×C711yescharacteristic, complemented
72C24×C7—41no—
73C2×C7×Q8—21no—
74C2×C7×Q8—21no—
75C2×C7×D4—22no—
76C2×C7×D4—22no—
77C2×C7×D4—22no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—81no—
86C2×C7×D4—81no—
87C2×C7×D4—82no—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C22×C4×C7#4111yescharacteristic
91C7×M16—21no—
92C7×M16—21no—
93C7×M16—21no—
94C7×M16—21no—
95C7×M16—41no—
96C7×M16—41no—
97C2×C7×C8—21no—
98C2×C7×C8—21no—
99C2×C7×C8—22no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C2×C7×C8—81no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C22⋊C4—41no—
113C42×C7—41no—
114C42×C7C2×D411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7C2×D411yescharacteristic
117C42×C7#4111yescharacteristic
118#533C22×C711yescharacteristic
119C2×#127C22×C711yescharacteristic
120#490—22no—
121#486—21no—
122#486—21no—
123#431C22×C711yescharacteristic
124C2×#108—21no—
125#423C22×C711yescharacteristic
126C2×C4⋊C8C22×C711yescharacteristic
127C2×C4⋊C8C22×C711yescharacteristic
128#418—21no—
129#418—21no—
130C4×M16C22×C711yescharacteristic
131#341—21no—
132#333—21no—
133#332—21no—
134#331—22no—
135#331—22no—
136Q8⋊C8—22no—
137C23×C7—42no—
138C23×C7—82no—
139C23×C7—82no—
140C23×C7—81no—
141C23×C7—81no—
142C23×C7#12411yescharacteristic
143C7×Q8—41no—
144C7×Q8—41no—
145C7×Q8—81no—
146C7×D4—42no—
147C7×D4—42no—
148C7×D4—82no—
149C7×D4—82no—
150C7×D4—82no—
151C7×D4—81no—
152C7×D4—81no—
153C7×D4—81no—
154C7×D4—81no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7#12011yescharacteristic
161C2×C4×C7#12111yescharacteristic
162C2×C4×C7#12111yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12311yescharacteristic
165C2×C4×C7#12311yescharacteristic
166C2×C4×C7#13511yescharacteristic
167C2×C4×C7#13511yescharacteristic
168C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C7#13711yescharacteristic
170C7×C8—41no—
171C7×C8—41no—
172C7×C8—42no—
173C7×C8—81no—
174C7×C8—81no—
175C7×C8—81no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C2×M16—21no—
180C2×M16C7×D411yescharacteristic
181C2×M16C7×D411yescharacteristic
182C22×C8—21no—
183C22×C8—21no—
184C4⋊Q8C7×D411yescharacteristic
185C4⋊Q8C7×D411yescharacteristic
186#127—21no—
187#127—21no—
188#127—21no—
189#127C7×D412yes—
190#125—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195#117—21no—
196C2×C42C23×C711yescharacteristic
197#108—42no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8C7×D412yes—
205#102—22no—
206#102—22no—
207#102—22no—
208#102—22no—
209C22⋊C8—21no—
210#97—21no—
211C4×C8—21no—
212C4×C8—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—82no—
216C22×C7—82no—
217C22×C7—82no—
218C22×C7—162no—
219C22×C7—162no—
220C22×C7#39411yescharacteristic
221C22×C7#48811yescharacteristic
222C22×C7#48911yescharacteristic
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7#44911yescharacteristic
229C4×C7#45211yescharacteristic
230C4×C7#46311yescharacteristic
231C4×C7#46411yescharacteristic
232C4×C7#47111yescharacteristic
233C4×C7#47211yescharacteristic
234C24—41no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—81no—
248C2×D4—81no—
249C2×D4—82no—
250C22×C4C2×C7×D411yescharacteristic
251C22×C4C7×#4111yescharacteristic
252C22×C4C7×#4111yescharacteristic
253M16—21no—
254M16—21no—
255M16—21no—
256M16—21no—
257M16—41no—
258M16—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—81no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C22⋊C4—41no—
275C42—41no—
276C42C2×C7×D411yescharacteristic
277C42C2×C7×D411yescharacteristic
278C42C2×C7×D411yescharacteristic
279C42C7×#4111yescharacteristic
280C2×C7—21no—
281C2×C7—21no—
282C2×C7—162no—
283C2×C7#204211yescharacteristic
284C2×C7#204811yescharacteristic
285C2×C7#205111yescharacteristic
286C23—42no—
287C23—82no—
288C23—82no—
289C23—81no—
290C23—81no—
291C23C7×#12411yescharacteristic
292Q8—41no—
293Q8—41no—
294Q8—81no—
295D4—42no—
296D4—42no—
297D4—82no—
298D4—82no—
299D4—82no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—81no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4C7×#12011yescharacteristic
310C2×C4C7×#12111yescharacteristic
311C2×C4C7×#12111yescharacteristic
312C2×C4C7×#12111yescharacteristic
313C2×C4C7×#12311yescharacteristic
314C2×C4C7×#12311yescharacteristic
315C2×C4C7×#13511yescharacteristic
316C2×C4C7×#13511yescharacteristic
317C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
318C2×C4C7×#13711yescharacteristic
319C8—41no—
320C8—41no—
321C8—42no—
322C8—81no—
323C8—81no—
324C8—81no—
325C7#1055311yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C7×#39411yescharacteristic
334C22C7×#48811yescharacteristic
335C22C7×#48911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4C7×#44911yescharacteristic
342C4C7×#45211yescharacteristic
343C4C7×#46311yescharacteristic
344C4C7×#46411yescharacteristic
345C4C7×#47111yescharacteristic
346C4C7×#47211yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2C7×#204211yescharacteristic
351C2C7×#204811yescharacteristic
352C2C7×#205111yescharacteristic
subgroupG/H
center, socleC22×C7#394
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
2. C7×#1950C2semidirect
3. C7×#1550C2semidirect
6. C7×#1481C2semidirect
7. C7×#1283C2semidirect
27. #10553C7direct (N × H)
66. #1950C2×C7semidirect
67. #1550C2×C7semidirect
70. #1481C2×C7semidirect
71. #1283C2×C7semidirect
325. C7#10553direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31382
C7×#19502
C7×#15502
C7×#15452
C2×C7×#3312
C7×#14812
C7×#12832
#105537