Finite Group Databasebeta

← C2×C7×#1698 #13196367

Subgroups — C2×C7×#1698 #13196367

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#486C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C2×C7×#460C211yesmaximal, characteristic
4C2×C7×#446C211yesmaximal, characteristic, complemented
5C7×#1698C218yesmaximal, direct product factor
6C2×C7×#335C211yesmaximal, characteristic, complemented
7C2×C7×#333C211yesmaximal, characteristic, complemented
8C2×C7×#331C211yesmaximal, characteristic
9C22×C7×SD16—21no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×#127C2211yescharacteristic
12C2×C4×C7×Q8C2211yescharacteristic
13C7×#486—22no—
14C7×#486C2214yes—
15C7×#477—22no—
16C7×#477C2214yescomplemented
17C7×#460—22no—
18C7×#460C2214yes—
19C7×#446—22no—
20C7×#446C2214yescomplemented
21C2×C7×#107—21no—
22C2×C7×C4⋊C8—21no—
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#103—21no—
27C2×C7×#102—21no—
28C2×C7×#102—21no—
29C2×C7×#102—21no—
30C2×C7×#97—21no—
31C2×C4×C7×C8C2211yescharacteristic
32C7×#335C2214yescomplemented
33C7×#333C2214yescomplemented
34C7×#331C2214yes—
35C2×#1698C711yesmaximal, characteristic, direct product factor, Sylow
36C22×C7×Q8—21no—
37C22×C7×Q8D411yescharacteristic
38C22×C7×D4—21no—
39C22×C7×D4—21no—
40C22×C7×D4—21no—
41C2×C7×SD16—24no—
42C2×C7×SD16—44no—
43C22×C7×C8—21no—
44C22×C7×C8—21no—
45C22×C7×C8—21no—
46C22×C7×C8D411yescharacteristic
47C22×C7×C8D411yescharacteristic
48C7×C4⋊Q8—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8C2312yes—
52C7×#127—21no—
53C7×#127—21no—
54C7×#127—21no—
55C7×#127C2312yes—
56C4×C7×Q8—21no—
57C4×C7×Q8—21no—
58C4×C7×Q8D412yes—
59C4×C7×Q8C2312yes—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C42×C7—21no—
65C2×C42×C7C2311yescharacteristic
66C7×#107—24no—
67C7×C4⋊C8—24no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8C2312yes—
72C7×C4⋊C8C2312yes—
73C7×C4⋊C8C2312yes—
74C7×#103—24no—
75C7×#102—24no—
76C7×#102—24no—
77C7×#102—24no—
78C7×#97—24no—
79C4×C7×C8D412yes—
80C4×C7×C8C2312yes—
81C2×#486C2×C711yescharacteristic
82C2×#477C2×C711yescharacteristic, complemented
83C2×#460C2×C711yescharacteristic
84C2×#446C2×C711yescharacteristic, complemented
85#1698C2×C718yesdirect product factor
86C2×#335C2×C711yescharacteristic, complemented
87C2×#333C2×C711yescharacteristic, complemented
88C2×#331C2×C711yescharacteristic
89C24×C7—41no—
90C2×C7×Q8—21no—
91C2×C7×Q8—22no—
92C2×C7×Q8—22no—
93C2×C7×Q8—22no—
94C2×C7×Q8—41no—
95C2×C7×Q8—82no—
96C2×C7×Q8C2×D412yes—
97C2×C7×D4—22no—
98C2×C7×D4—22no—
99C2×C7×D4—22no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—42no—
104C2×C7×D4—42no—
105C2×C7×D4—42no—
106C2×C7×D4—42no—
107C2×C7×D4—82no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—41no—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7#4111yescharacteristic
114C7×SD16—48no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8—42no—
121C2×C7×C8—42no—
122C2×C7×C8—42no—
123C2×C7×C8C2×D412yes—
124C2×C7×C8C2×D412yes—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C7×C4⋊C4C2×D412yes—
133C42×C7—22no—
134C42×C7—22no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7#4111yescharacteristic
138C42×C7C2411yescharacteristic
139C22×SD16—21no—
140C2×C4⋊Q8C22×C711yescharacteristic
141C2×#127C22×C711yescharacteristic
142C2×C4×Q8C22×C711yescharacteristic
143#486—22no—
144#486C22×C714yes—
145#477—22no—
146#477C22×C714yescomplemented
147#460—22no—
148#460C22×C714yes—
149#446—22no—
150#446C22×C714yescomplemented
151C2×#107—21no—
152C2×C4⋊C8—21no—
153C2×C4⋊C8C22×C711yescharacteristic
154C2×C4⋊C8C22×C711yescharacteristic
155C2×C4⋊C8C22×C711yescharacteristic
156C2×#103—21no—
157C2×#102—21no—
158C2×#102—21no—
159C2×#102—21no—
160C2×#97—21no—
161C2×C4×C8C22×C711yescharacteristic
162#335C22×C714yescomplemented
163#333C22×C714yescomplemented
164#331C22×C714yes—
165C23×C7—42no—
166C23×C7—82no—
167C23×C7—82no—
168C23×C7—82no—
169C23×C7#12111yescharacteristic
170C7×Q8—22no—
171C7×Q8—22no—
172C7×Q8—42no—
173C7×Q8—82no—
174C7×Q8—82no—
175C7×D4—42no—
176C7×D4—42no—
177C7×D4—42no—
178C7×D4—42no—
179C7×D4—82no—
180C7×D4—82no—
181C7×D4—82no—
182C7×D4—82no—
183C2×C4×C7—22no—
184C2×C4×C7—22no—
185C2×C4×C7—22no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—82no—
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#13512yes—
194C2×C4×C7AGL(1,ℤ/8)12yes—
195C2×C4×C7#13712yes—
196C2×C4×C7#13712yes—
197C2×C4×C7C22×D411yescharacteristic
198C2×C4×C7C22×D411yescharacteristic
199C2×C4×C7C2×#4111yescharacteristic
200C7×C8—22no—
201C7×C8—22no—
202C7×C8—42no—
203C7×C8—42no—
204C7×C8—42no—
205C22×Q8—21no—
206C22×Q8C7×D411yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C2×SD16—24no—
211C2×SD16—44no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8C7×D411yescharacteristic
216C22×C8C7×D411yescharacteristic
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8C23×C712yes—
221#127—21no—
222#127—21no—
223#127—21no—
224#127C23×C712yes—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8C7×D412yes—
228C4×Q8C23×C712yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C7×D411yescharacteristic
233C2×C42—21no—
234C2×C42C23×C711yescharacteristic
235#107—24no—
236C4⋊C8—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8C23×C712yes—
241C4⋊C8C23×C712yes—
242C4⋊C8C23×C712yes—
243#103—24no—
244#102—24no—
245#102—24no—
246#102—24no—
247#97—24no—
248C4×C8C7×D412yes—
249C4×C8C23×C712yes—
250C22×C7—82no—
251C22×C7—82no—
252C22×C7—82no—
253C22×C7—164no—
254C22×C7#46412yes—
255C22×C7#46812yes—
256C22×C7#47212yes—
257C22×C7C2×#12111yescharacteristic
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—42no—
261C4×C7—42no—
262C4×C7—82no—
263C4×C7#46412yes—
264C4×C7#46812yes—
265C4×C7C2×#13511yescharacteristic
266C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
267C4×C7C2×#13711yescharacteristic
268C4×C7C2×#13711yescharacteristic
269C24—41no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×Q8C2×C7×D412yes—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—82no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4C2×C7×D411yescharacteristic
292C22×C4C2×C7×D411yescharacteristic
293C22×C4C7×#4111yescharacteristic
294SD16—48no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8C2×C7×D412yes—
304C2×C8C2×C7×D412yes—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4C2×C7×D412yes—
313C42—22no—
314C42—22no—
315C42C2×C7×D411yescharacteristic
316C42C2×C7×D411yescharacteristic
317C42C7×#4111yescharacteristic
318C42C24×C711yescharacteristic
319C2×C7—162no—
320C2×C7#169814yesdirect product factor
321C2×C7C2×#46411yescharacteristic
322C2×C7C2×#46811yescharacteristic
323C2×C7C2×#47211yescharacteristic
324C23—42no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23C7×#12111yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—82no—
339D4—82no—
340D4—82no—
341D4—82no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4C7×#12111yescharacteristic
350C2×C4C7×#12111yescharacteristic
351C2×C4C7×#12111yescharacteristic
352C2×C4C7×#13512yes—
353C2×C4C7×AGL(1,ℤ/8)12yes—
354C2×C4C7×#13712yes—
355C2×C4C7×#13712yes—
356C2×C4C22×C7×D411yescharacteristic
357C2×C4C22×C7×D411yescharacteristic
358C2×C4C2×C7×#4111yescharacteristic
359C8—22no—
360C8—22no—
361C8—42no—
362C8—42no—
363C8—42no—
364C7C2×#169811yescharacteristic, direct product factor, Sylow
365C22—82no—
366C22—82no—
367C22—82no—
368C22—164no—
369C22C7×#46412yes—
370C22C7×#46812yes—
371C22C7×#47212yes—
372C22C2×C7×#12111yescharacteristic
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4C7×#46412yes—
379C4C7×#46812yes—
380C4C2×C7×#13511yescharacteristic
381C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
382C4C2×C7×#13711yescharacteristic
383C4C2×C7×#13711yescharacteristic
384C2—162no—
385C2C7×#169814yesdirect product factor
386C2C2×C7×#46411yescharacteristic
387C2C2×C7×#46811yescharacteristic
388C2C2×C7×#47211yescharacteristic
subgroupG/H
center, socleC23×C7#121
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
4. C2×C7×#446C2semidirect
5. C7×#1698C2direct (N × H)
6. C2×C7×#335C2semidirect
7. C2×C7×#333C2semidirect
16. C7×#477C22semidirect
20. C7×#446C22semidirect
32. C7×#335C22semidirect
33. C7×#333C22semidirect
35. C2×#1698C7direct (N × H)
82. C2×#477C2×C7semidirect
84. C2×#446C2×C7semidirect
85. #1698C2×C7direct (N × H)
86. C2×#335C2×C7semidirect
87. C2×#333C2×C7semidirect
146. #477C22×C7semidirect
150. #446C22×C7semidirect
162. #335C22×C7semidirect
163. #333C22×C7semidirect
320. C2×C7#1698direct (N × H)
364. C7C2×#1698direct (N × H)
385. C2C7×#1698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4862
C2×C7×#4772
C2×C7×#4602
C2×C7×#4462
C7×#16982
C2×C7×#3352
C2×C7×#3332
C2×C7×#3312
C2×#16987