Finite Group Databasebeta

← C7×#13037 #13190253

Subgroups — C7×#13037 #13190253

Order 1792.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#509C211yesmaximal, characteristic
2C2×C7×#442C211yesmaximal, characteristic, complemented
3C2×C7×#361C211yesmaximal, characteristic, complemented
4C2×C7×#360C211yesmaximal, characteristic, complemented
5C7×#1975C211yesmaximal, characteristic, complemented
6C7×#1383C211yesmaximal, characteristic, complemented
7C7×#1375C211yesmaximal, characteristic, complemented
8C2×C7×#135C2211yescharacteristic
9C22×C7×D8C2211yescharacteristic
10C2×C4×C7×D4—21no—
11C7×#509—22no—
12C7×#509—22no—
13C2×C7×SD32C412yescomplemented
14C2×C7×D16C412yescomplemented
15C2×C7×M32C2211yescharacteristic
16C2×C7×M32C2211yescharacteristic
17C7×#442—22no—
18C7×#442—22no—
19C2×C7×#108C2211yescharacteristic
20C2×C7×#106C2211yescharacteristic
21C2×C7×#104—21no—
22C2×C7×#102—21no—
23C7×#408—21no—
24C2×C7×#97C2211yescharacteristic
25C7×#361—22no—
26C7×#360—22no—
27#13037C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41—21no—
29C22×C7×D4—21no—
30C23×C4×C7—41no—
31C7×#135—21no—
32C7×#135—21no—
33C7×#135—21no—
34C7×#135—21no—
35C2×C7×Q16C2×C411yescharacteristic, complemented
36C2×C7×SD16—21no—
37C2×C7×D8—21no—
38C2×C7×D8—21no—
39C2×C7×D8—21no—
40C2×C7×D8C2×C412yes—
41C2×C7×D8C2×C411yescharacteristic, complemented
42C2×C7×D8D412yes—
43C2×C7×D8D412yes—
44C2×C7×M16—21no—
45C22×C7×C8D411yescharacteristic
46C22×C7×C8C2311yescharacteristic
47C4×C7×D4—24no—
48C4×C7×D4—42no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C22⋊C4—41no—
51C2×C42×C7D411yescharacteristic
52C7×SD32—24no—
53C7×D16—24no—
54C7×M32—21no—
55C7×M32—21no—
56C7×M32—21no—
57C7×M32—21no—
58C2×C7×C16—21no—
59C2×C7×C16C2×C412yescomplemented
60C7×#108—21no—
61C7×#108—21no—
62C7×#106—21no—
63C7×#106D412yes—
64C7×#104—42no—
65C7×#102—24no—
66C7×C22⋊C8—42no—
67C7×#97—21no—
68C7×#97—21no—
69C2×#509C2×C711yescharacteristic
70C2×#442C2×C711yescharacteristic, complemented
71C2×#361C2×C711yescharacteristic, complemented
72C2×#360C2×C711yescharacteristic, complemented
73#1975C2×C711yescharacteristic, complemented
74#1383C2×C711yescharacteristic, complemented
75#1375C2×C711yescharacteristic, complemented
76C24×C7—41no—
77C7×#41—41no—
78C7×#41—41no—
79C7×#41—41no—
80C7×#41—41no—
81C2×C7×Q8—21no—
82C2×C7×D4—22no—
83C2×C7×D4—21no—
84C2×C7×D4—22no—
85C2×C7×D4—22no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C22×C4×C7—21no—
91C22×C4×C7—44no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—84no—
95C22×C4×C7C2×D411yescharacteristic
96C7×Q16—21no—
97C7×Q16—21no—
98C7×SD16—41no—
99C7×SD16—41no—
100C7×D8—21no—
101C7×D8—22no—
102C7×D8—21no—
103C7×D8—22no—
104C7×D8—22no—
105C7×D8C22⋊C414yes—
106C7×M16—41no—
107C7×M16—41no—
108C2×C7×C8—21no—
109C2×C7×C8—21no—
110C2×C7×C8—21no—
111C2×C7×C8—41no—
112C2×C7×C8C22×C411yescharacteristic
113C2×C7×C8C2×D411yescharacteristic
114C2×C7×C8C2×D411yescharacteristic
115C2×C7×C8C2×D411yescharacteristic
116C2×C7×C8#4111yescharacteristic
117C2×C7×C8#4111yescharacteristic
118C7×C4⋊C4—22no—
119C7×C4⋊C4—41no—
120C7×C22⋊C4—44no—
121C42×C7—21no—
122C42×C7D812yes—
123C7×C16—22no—
124C7×C16—41no—
125C2×#135C22×C711yescharacteristic
126C22×D8C22×C711yescharacteristic
127C2×C4×D4—21no—
128#509—22no—
129#509—22no—
130C2×SD32C4×C712yescomplemented
131C2×D16C4×C712yescomplemented
132C2×M32C22×C711yescharacteristic
133C2×M32C22×C711yescharacteristic
134#442—22no—
135#442—22no—
136C2×#108C22×C711yescharacteristic
137C2×#106C22×C711yescharacteristic
138C2×#104—21no—
139C2×#102—21no—
140#408—21no—
141C2×#97C22×C711yescharacteristic
142#361—22no—
143#360—22no—
144C23×C7—42no—
145C23×C7—42no—
146C23×C7—42no—
147C23×C7—41no—
148C23×C7—82no—
149C23×C7—82no—
150C23×C7AGL(1,ℤ/8)11yescharacteristic
151C7×Q8—41no—
152C7×Q8—41no—
153C7×D4—24no—
154C7×D4—41no—
155C7×D4—42no—
156C7×D4—41no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4—81no—
160C7×D4—81no—
161C2×C4×C7—22no—
162C2×C4×C7—42no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—88no—
166C2×C4×C7—82no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7C4×D411yescharacteristic
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7C2×D811yescharacteristic
173C2×C4×C7#13511yescharacteristic
174C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
175C7×C8—41no—
176C7×C8—81no—
177C7×C8C2×C22⋊C411yescharacteristic
178C7×C8C4×D411yescharacteristic
179C7×C8#12111yescharacteristic
180C7×C8#12311yescharacteristic
181C2×#41—21no—
182C22×D4—21no—
183C23×C4—41no—
184#135—21no—
185#135—21no—
186#135—21no—
187#135—21no—
188C2×Q16C2×C4×C711yescharacteristic, complemented
189C2×SD16—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8—21no—
193C2×D8C2×C4×C712yes—
194C2×D8C2×C4×C711yescharacteristic, complemented
195C2×D8C7×D412yes—
196C2×D8C7×D412yes—
197C2×M16—21no—
198C22×C8C7×D411yescharacteristic
199C22×C8C23×C711yescharacteristic
200C4×D4—24no—
201C4×D4—42no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—41no—
204C2×C42C7×D411yescharacteristic
205SD32—24no—
206D16—24no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211C2×C16—21no—
212C2×C16C2×C4×C712yescomplemented
213#108—21no—
214#108—21no—
215#106—21no—
216#106C7×D412yes—
217#104—42no—
218#102—24no—
219C22⋊C8—42no—
220#97—21no—
221#97—21no—
222C22×C7—22no—
223C22×C7—44no—
224C22×C7—82no—
225C22×C7—82no—
226C22×C7—84no—
227C22×C7—84no—
228C22×C7—81no—
229C22×C7—82no—
230C22×C7#44211yescharacteristic
231C22×C7#44711yescharacteristic
232C22×C7#46311yescharacteristic
233C4×C7—42no—
234C4×C7—82no—
235C4×C7—81no—
236C4×C7#38611yescharacteristic
237C4×C7C4×D811yescharacteristic
238C4×C7#44911yescharacteristic
239C4×C7#45911yescharacteristic
240C24—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—21no—
255C22×C4—44no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—84no—
259C22×C4C2×C7×D411yescharacteristic
260Q16—21no—
261Q16—21no—
262SD16—41no—
263SD16—41no—
264D8—21no—
265D8—22no—
266D8—21no—
267D8—22no—
268D8—22no—
269D8C7×C22⋊C414yes—
270M16—41no—
271M16—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—41no—
276C2×C8C22×C4×C711yescharacteristic
277C2×C8C2×C7×D411yescharacteristic
278C2×C8C2×C7×D411yescharacteristic
279C2×C8C2×C7×D411yescharacteristic
280C2×C8C7×#4111yescharacteristic
281C2×C8C7×#4111yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—41no—
284C22⋊C4—44no—
285C42—21no—
286C42C7×D812yes—
287C16—22no—
288C16—41no—
289C2×C7—21no—
290C2×C7—21no—
291C2×C7—84no—
292C2×C7—161no—
293C2×C7#187911yescharacteristic
294C2×C7AGL(1,ℤ/16)12yes—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—41no—
299C23—82no—
300C23—82no—
301C23C7×AGL(1,ℤ/8)11yescharacteristic
302Q8—41no—
303Q8—41no—
304D4—24no—
305D4—41no—
306D4—42no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—81no—
311D4—81no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—88no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4C4×C7×D411yescharacteristic
321C2×C4C7×#12011yescharacteristic
322C2×C4C7×#12111yescharacteristic
323C2×C4C2×C7×D811yescharacteristic
324C2×C4C7×#13511yescharacteristic
325C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
326C8—41no—
327C8—81no—
328C8C2×C7×C22⋊C411yescharacteristic
329C8C4×C7×D411yescharacteristic
330C8C7×#12111yescharacteristic
331C8C7×#12311yescharacteristic
332C7#1303711yescharacteristic, direct product factor, Sylow
333C22—22no—
334C22—44no—
335C22—82no—
336C22—82no—
337C22—84no—
338C22—84no—
339C22—81no—
340C22—82no—
341C22C7×#44211yescharacteristic
342C22C7×#44711yescharacteristic
343C22C7×#46311yescharacteristic
344C4—42no—
345C4—82no—
346C4—81no—
347C4C7×#38611yescharacteristic
348C4C4×C7×D811yescharacteristic
349C4C7×#44911yescharacteristic
350C4C7×#45911yescharacteristic
351C2—21no—
352C2—21no—
353C2—84no—
354C2—161no—
355C2C7×#187911yescharacteristic
356C2C7×AGL(1,ℤ/16)12yes—
subgroupG/H
center, socleC22×C7#442
commutatorC2×C8C22×C4×C7
Frattini, cosocleC22×C8C23×C7
prime radicalC2×C7×#509C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C7×#442C2semidirect
3. C2×C7×#361C2semidirect
4. C2×C7×#360C2semidirect
5. C7×#1975C2semidirect
6. C7×#1383C2semidirect
7. C7×#1375C2semidirect
13. C2×C7×SD32C4semidirect
14. C2×C7×D16C4semidirect
27. #13037C7direct (N × H)
35. C2×C7×Q16C2×C4semidirect
41. C2×C7×D8C2×C4semidirect
59. C2×C7×C16C2×C4semidirect
70. C2×#442C2×C7semidirect
71. C2×#361C2×C7semidirect
72. C2×#360C2×C7semidirect
73. #1975C2×C7semidirect
74. #1383C2×C7semidirect
75. #1375C2×C7semidirect
130. C2×SD32C4×C7semidirect
131. C2×D16C4×C7semidirect
188. C2×Q16C2×C4×C7semidirect
194. C2×D8C2×C4×C7semidirect
212. C2×C16C2×C4×C7semidirect
332. C7#13037direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5092
C2×C7×#4422
C2×C7×#3612
C2×C7×#3602
C7×#19752
C7×#13832
C7×#13752
#130377