Finite Group Databasebeta

← C7×#9197 #13186413

Subgroups — C7×#9197 #13186413

Order 1792.

624 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2905C211yesmaximal, characteristic, complemented
2C7×#2619C211yesmaximal, characteristic
3C7×#1545C212yesmaximal, complemented
4C7×#1458C211yesmaximal, characteristic, complemented
5C7×#1286C212yesmaximal, complemented
6C22×C7×M16—21no—
7C2×C7×#127C2211yescharacteristic
8C2×C7×#121—21no—
9C2×C7×#117C2211yescharacteristic
10C7×#431C2212yes—
11C7×#424—24no—
12C7×#423—21no—
13C7×#423C2212yes—
14C7×#394—21no—
15C7×#394C412yes—
16C7×#392—21no—
17C7×#392C412yes—
18C7×#390—21no—
19C7×#382C2211yescharacteristic
20C7×#331—24no—
21C7×#324—21no—
22C7×#323—21no—
23#9197C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—22no—
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4C2×C412yes—
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—22no—
30C2×C7×M16—22no—
31C2×C7×M16—42no—
32C2×C7×M16—42no—
33C22×C7×C8—22no—
34C7×#127—22no—
35C7×#127—22no—
36C7×#121—42no—
37C7×#121—42no—
38C7×#117—22no—
39C7×#117—22no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C42×C7D411yescharacteristic
49C2×C42×C7C2311yescharacteristic
50C7×C4⋊C8—24no—
51C7×C4⋊C8—22no—
52C7×C4⋊C8—22no—
53C7×C4⋊C8D414yescomplemented
54C7×C22⋊C8—22no—
55C4×C7×C8—22no—
56C7×#95—21no—
57C7×#95C2×C412yes—
58#2905C2×C711yescharacteristic, complemented
59#2619C2×C711yescharacteristic
60#1545C2×C712yescomplemented
61#1458C2×C711yescharacteristic, complemented
62#1286C2×C712yescomplemented
63C24×C7—22no—
64C24×C7C22⋊C411yescharacteristic
65C2×C7×D4—24no—
66C2×C7×D4—44no—
67C2×C7×D4—41no—
68C2×C7×D4—42no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C2×C7×D4—82no—
75C2×C7×D4—82no—
76C2×C7×D4—82no—
77C2×C7×D4—82no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7D812yes—
87C22×C4×C7SD1612yes—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C7×M16—44no—
92C7×M16—42no—
93C7×M16—42no—
94C2×C7×C8—24no—
95C2×C7×C8—44no—
96C2×C7×C8—42no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—44no—
104C7×C22⋊C4—82no—
105C7×C22⋊C4—82no—
106C42×C7—21no—
107C42×C7—21no—
108C42×C7C2×D412yes—
109C42×C7#4112yes—
110C22×M16—21no—
111C2×#127C22×C711yescharacteristic
112C2×#121—21no—
113C2×#117C22×C711yescharacteristic
114#431C22×C712yes—
115#424—24no—
116#423—21no—
117#423C22×C712yes—
118#394—21no—
119#394C4×C712yes—
120#392—21no—
121#392C4×C712yes—
122#390—21no—
123#382C22×C711yescharacteristic
124#331—24no—
125#324—21no—
126#323—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—42no—
132C23×C7—42no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7#10212yes—
139C23×C7C2×C22⋊C411yescharacteristic
140C7×D4—84no—
141C7×D4—84no—
142C7×D4—88no—
143C7×D4—81no—
144C7×D4—81no—
145C7×D4—81no—
146C7×D4—81no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—22no—
152C2×C4×C7—22no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7#10212yes—
163C2×C4×C7C4×D412yes—
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12112yes—
166C2×C4×C7#12311yescharacteristic
167C2×C4×C7C2×D811yescharacteristic
168C2×C4×C7C2×SD1611yescharacteristic
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C7#13711yescharacteristic
171C7×C8—44no—
172C7×C8—82no—
173C22×D4—22no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×C4×C712yes—
177C23×C4C7×D411yescharacteristic
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—22no—
183#127—22no—
184#127—22no—
185#121—42no—
186#121—42no—
187#117—22no—
188#117—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×C4×C711yescharacteristic
191C2×C4⋊C4C7×D411yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C2×C4×C711yescharacteristic
196C2×C22⋊C4C7×D411yescharacteristic
197C2×C42C7×D411yescharacteristic
198C2×C42C23×C711yescharacteristic
199C4⋊C8—24no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8C7×D414yescomplemented
203C22⋊C8—22no—
204C4×C8—22no—
205#95—21no—
206#95C2×C4×C712yes—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—22no—
210C22×C7—42no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—81no—
214C22×C7—82no—
215C22×C7—82no—
216C22×C7—82no—
217C22×C7—82no—
218C22×C7—82no—
219C22×C7—82no—
220C22×C7—82no—
221C22×C7#38611yescharacteristic
222C22×C7C2×#10211yescharacteristic
223C22×C7#41711yescharacteristic
224C4×C7—22no—
225C4×C7—22no—
226C4×C7—22no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—82no—
230C4×C7#44711yescharacteristic
231C4×C7#44811yescharacteristic
232C4×C7#48011yescharacteristic
233C4×C7#48111yescharacteristic
234C24—22no—
235C24C7×C22⋊C411yescharacteristic
236C2×D4—24no—
237C2×D4—44no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4C7×C22⋊C411yescharacteristic
255C22×C4C7×C22⋊C411yescharacteristic
256C22×C4C7×C22⋊C411yescharacteristic
257C22×C4C7×D812yes—
258C22×C4C7×SD1612yes—
259C22×C4C22×C4×C711yescharacteristic
260C22×C4C2×C7×D411yescharacteristic
261C22×C4C2×C7×D411yescharacteristic
262M16—44no—
263M16—42no—
264M16—42no—
265C2×C8—24no—
266C2×C8—44no—
267C2×C8—42no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—44no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C42—21no—
278C42—21no—
279C42C2×C7×D412yes—
280C42C7×#4112yes—
281C2×C7—21no—
282C2×C7—21no—
283C2×C7—42no—
284C2×C7—162no—
285C2×C7#187411yescharacteristic
286C2×C7#191011yescharacteristic
287C2×C7#191411yescharacteristic
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23C7×#10212yes—
300C23C2×C7×C22⋊C411yescharacteristic
301D4—84no—
302D4—84no—
303D4—88no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4C7×#10212yes—
324C2×C4C4×C7×D412yes—
325C2×C4C7×#12011yescharacteristic
326C2×C4C7×#12112yes—
327C2×C4C7×#12311yescharacteristic
328C2×C4C2×C7×D811yescharacteristic
329C2×C4C2×C7×SD1611yescharacteristic
330C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
331C2×C4C7×#13711yescharacteristic
332C8—44no—
333C8—82no—
334C7#919711yescharacteristic, direct product factor, Sylow
335C22—21no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22C7×#38611yescharacteristic
350C22C2×C7×#10211yescharacteristic
351C22C7×#41711yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—82no—
358C4C7×#44711yescharacteristic
359C4C7×#44811yescharacteristic
360C4C7×#48011yescharacteristic
361C4C7×#48111yescharacteristic
362C2—21no—
363C2—21no—
364C2—42no—
365C2—162no—
366C2C7×#187411yescharacteristic
367C2C7×#191011yescharacteristic
368C2C7×#191411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#2619C2
LCSG ▷ C22×C4 ▷ 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×#2905C2semidirect
3. C7×#1545C2semidirect
4. C7×#1458C2semidirect
5. C7×#1286C2semidirect
23. #9197C7direct (N × H)
53. C7×C4⋊C8D4semidirect
58. #2905C2×C7semidirect
60. #1545C2×C7semidirect
61. #1458C2×C7semidirect
62. #1286C2×C7semidirect
202. C4⋊C8C7×D4semidirect
334. C7#9197direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29052
C7×#26192
C7×#15452
C7×#14582
C7×#12862
#91977