Finite Group Databasebeta

← C7×#10551 #13187767

Subgroups — C7×#10551 #13187767

Order 1792.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3141C211yesmaximal, characteristic
2C7×#1950C211yesmaximal, characteristic, complemented
3C7×#1549C211yesmaximal, characteristic, complemented
4C7×#1547C211yesmaximal, characteristic
5C2×C7×#333C211yesmaximal, 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#10551C711yesmaximal, 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#1549C2×C711yescharacteristic, complemented
68#1547C2×C711yescharacteristic
69C2×#333C2×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—82no—
80C2×C7×D4—21no—
81C2×C7×D4—21no—
82C2×C7×D4—21no—
83C2×C7×D4—41no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C7×M16—21no—
90C7×M16—21no—
91C7×M16—21no—
92C7×M16—21no—
93C7×M16—41no—
94C7×M16—41no—
95C2×C7×C8—21no—
96C2×C7×C8—21no—
97C2×C7×C8—22no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—81no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C42×C7C2×D411yescharacteristic
113C42×C7C2×D411yescharacteristic
114C42×C7C2×D411yescharacteristic
115C42×C7#4111yescharacteristic
116#532C22×C711yescharacteristic
117C2×C4⋊Q8C22×C711yescharacteristic
118#489—22no—
119#487—21no—
120#486—21no—
121#431C22×C711yescharacteristic
122C2×#108—21no—
123#423C22×C711yescharacteristic
124C2×C4⋊C8C22×C711yescharacteristic
125C2×C4⋊C8C22×C711yescharacteristic
126#417—21no—
127#417—21no—
128C4×M16C22×C711yescharacteristic
129#341—21no—
130#333—22no—
131#332—21no—
132#332—22no—
133#331—21no—
134D4⋊C8—22no—
135C23×C7—41no—
136C23×C7#12411yescharacteristic
137C7×Q8—41no—
138C7×Q8—41no—
139C7×Q8—82no—
140C7×Q8—82no—
141C7×D4—41no—
142C7×D4—41no—
143C7×D4—41no—
144C7×D4—41no—
145C7×D4—81no—
146C7×D4—81no—
147C7×D4—81no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—82no—
154C2×C4×C7—81no—
155C2×C4×C7—81no—
156C2×C4×C7#12011yescharacteristic
157C2×C4×C7#12111yescharacteristic
158C2×C4×C7#12111yescharacteristic
159C2×C4×C7#12111yescharacteristic
160C2×C4×C7#12311yescharacteristic
161C2×C4×C7#12311yescharacteristic
162C2×C4×C7#13511yescharacteristic
163C2×C4×C7#13511yescharacteristic
164C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C7#13711yescharacteristic
166C7×C8—41no—
167C7×C8—41no—
168C7×C8—42no—
169C7×C8—81no—
170C7×C8—81no—
171C7×C8—81no—
172C2×#41—21no—
173C22×Q8—21no—
174C2×M16—21no—
175C2×M16C7×D411yescharacteristic
176C2×M16C7×D411yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C7×D412yes—
183C4⋊Q8C7×D411yescharacteristic
184#127C7×D411yescharacteristic
185#124—21no—
186#121—21no—
187#121—21no—
188C4×D4—21no—
189C4×D4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C42C23×C711yescharacteristic
193#108—42no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8C7×D412yes—
201#103—22no—
202#103—22no—
203#102—22no—
204#102—22no—
205C22⋊C8—21no—
206#97—21no—
207C4×C8—21no—
208C4×C8—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—81no—
212C22×C7—81no—
213C22×C7—81no—
214C22×C7#39411yescharacteristic
215C22×C7#48811yescharacteristic
216C22×C7#48911yescharacteristic
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—82no—
221C4×C7—81no—
222C4×C7#44911yescharacteristic
223C4×C7#45211yescharacteristic
224C4×C7#46311yescharacteristic
225C4×C7#46411yescharacteristic
226C4×C7#47111yescharacteristic
227C4×C7#47311yescharacteristic
228#41—41no—
229#41—41no—
230#41—41no—
231#41—41no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—82no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4C2×C7×D411yescharacteristic
243C22×C4C7×#4111yescharacteristic
244C22×C4C7×#4111yescharacteristic
245M16—21no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—41no—
250M16—41no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—81no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42C2×C7×D411yescharacteristic
269C42C2×C7×D411yescharacteristic
270C42C2×C7×D411yescharacteristic
271C42C7×#4111yescharacteristic
272C2×C7—21no—
273C2×C7—21no—
274C2×C7—161no—
275C2×C7#204211yescharacteristic
276C2×C7#204811yescharacteristic
277C2×C7#205211yescharacteristic
278C23—41no—
279C23C7×#12411yescharacteristic
280Q8—41no—
281Q8—41no—
282Q8—82no—
283Q8—82no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4C7×#12011yescharacteristic
300C2×C4C7×#12111yescharacteristic
301C2×C4C7×#12111yescharacteristic
302C2×C4C7×#12111yescharacteristic
303C2×C4C7×#12311yescharacteristic
304C2×C4C7×#12311yescharacteristic
305C2×C4C7×#13511yescharacteristic
306C2×C4C7×#13511yescharacteristic
307C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
308C2×C4C7×#13711yescharacteristic
309C8—41no—
310C8—41no—
311C8—42no—
312C8—81no—
313C8—81no—
314C8—81no—
315C7#1055111yescharacteristic, direct product factor, Sylow
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C7×#39411yescharacteristic
322C22C7×#48811yescharacteristic
323C22C7×#48911yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—81no—
328C4—82no—
329C4C7×#44911yescharacteristic
330C4C7×#45211yescharacteristic
331C4C7×#46311yescharacteristic
332C4C7×#46411yescharacteristic
333C4C7×#47111yescharacteristic
334C4C7×#47311yescharacteristic
335C2—21no—
336C2—21no—
337C2—161no—
338C2C7×#204211yescharacteristic
339C2C7×#204811yescharacteristic
340C2C7×#205211yescharacteristic
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×#1549C2semidirect
5. C2×C7×#333C2semidirect
7. C7×#1283C2semidirect
27. #10551C7direct (N × H)
66. #1950C2×C7semidirect
67. #1549C2×C7semidirect
69. C2×#333C2×C7semidirect
71. #1283C2×C7semidirect
315. C7#10551direct (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×#15492
C7×#15472
C2×C7×#3332
C7×#14802
C7×#12832
#105517