Finite Group Databasebeta

← C2×C7×#2102 #13200966

Subgroups — C2×C7×#2102 #13200966

Order 1792.

710 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#430C211yesmaximal, characteristic
2C2×C7×#423C211yesmaximal, characteristic
3C7×#2102C218yesmaximal, direct product factor
4C2×C7×#344C212yesmaximal
5C2×C7×#339C212yesmaximal
6C2×C7×H(3,ℤ/4)C211yesmaximal, characteristic
7C22×C7×M16C2211yescharacteristic
8C22×C7×M16C2212yes—
9C22×C7×M16C2211yescharacteristic
10C2×C7×#117C2212yes—
11C22×C42×C7C2211yescharacteristic
12C7×#430—22no—
13C7×#430C2214yes—
14C2×C7×#108—22no—
15C7×#423—22no—
16C7×#423C2214yes—
17C2×C7×C4⋊C8—22no—
18C7×#344C2218yes—
19C7×#339C2218yes—
20C7×H(3,ℤ/4)C2214yes—
21C2×#2102C711yesmaximal, characteristic, direct product factor, Sylow
22C23×C4×C7—21no—
23C23×C4×C7C2311yescharacteristic
24C2×C7×M16—22no—
25C2×C7×M16—22no—
26C2×C7×M16—21no—
27C2×C7×M16—22no—
28C2×C7×M16—22no—
29C2×C7×M16—22no—
30C2×C7×M16—22no—
31C2×C7×M16—22no—
32C2×C7×M16—22no—
33C2×C7×M16—22no—
34C2×C7×M16D412yes—
35C2×C7×M16C2314yes—
36C2×C7×M16C2312yes—
37C2×C7×M16C2312yes—
38C22×C7×C8—21no—
39C22×C7×C8—22no—
40C22×C7×C8D412yes—
41C7×#117—22no—
42C7×#117—22no—
43C7×#117—22no—
44C7×#117C2314yes—
45C2×C7×C4⋊C4—22no—
46C2×C7×C22⋊C4—22no—
47C2×C42×C7—21no—
48C2×C42×C7—21no—
49C2×C42×C7—21no—
50C2×C42×C7—22no—
51C2×C42×C7Q811yescharacteristic
52C2×C42×C7Q812yes—
53C2×C42×C7Q811yescharacteristic
54C2×C42×C7C2312yes—
55C7×#108—28no—
56C7×C4⋊C8—28no—
57C2×#430C2×C711yescharacteristic
58C2×#423C2×C711yescharacteristic
59#2102C2×C718yesdirect product factor
60C2×#344C2×C712yes—
61C2×#339C2×C712yes—
62C2×H(3,ℤ/4)C2×C711yescharacteristic
63C24×C7#4111yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—44no—
69C22×C4×C7—42no—
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7C2×Q811yescharacteristic
73C22×C4×C7C2×Q811yescharacteristic
74C22×C4×C7#4111yescharacteristic
75C22×C4×C7#4112yes—
76C22×C4×C7#4112yes—
77C22×C4×C7#4112yes—
78C22×C4×C7#4112yes—
79C22×C4×C7C2411yescharacteristic
80C7×M16—24no—
81C7×M16—24no—
82C7×M16—24no—
83C7×M16—24no—
84C7×M16—24no—
85C7×M16—42no—
86C7×M16—42no—
87C7×M16—42no—
88C2×C7×C8—22no—
89C2×C7×C8—22no—
90C2×C7×C8—22no—
91C2×C7×C8—24no—
92C2×C7×C8—41no—
93C2×C7×C8—41no—
94C2×C7×C8—42no—
95C2×C7×C8—42no—
96C2×C7×C8C2×D414yes—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—42no—
99C7×C22⋊C4—24no—
100C7×C22⋊C4—42no—
101C42×C7—21no—
102C42×C7—22no—
103C42×C7—21no—
104C42×C7—22no—
105C42×C7—24no—
106C42×C7—42no—
107C42×C7C2×Q812yes—
108C42×C7C2×Q812yes—
109C22×M16C22×C711yescharacteristic
110C22×M16C22×C712yes—
111C22×M16C22×C711yescharacteristic
112C2×#117C22×C712yes—
113C22×C42C22×C711yescharacteristic
114#430—22no—
115#430C22×C714yes—
116C2×#108—22no—
117#423—22no—
118#423C22×C714yes—
119C2×C4⋊C8—22no—
120#344C22×C718yes—
121#339C22×C718yes—
122H(3,ℤ/4)C22×C714yes—
123C23×C7—42no—
124C23×C7C22⋊Q811yescharacteristic
125C23×C7C22⋊Q811yescharacteristic
126C23×C7#12612yes—
127C23×C7#12612yes—
128C23×C7C2×#4111yescharacteristic
129C2×C4×C7—24no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—44no—
133C2×C4×C7—44no—
134C2×C4×C7—42no—
135C2×C4×C7—44no—
136C2×C4×C7—84no—
137C2×C4×C7C22⋊Q811yescharacteristic
138C2×C4×C7C22⋊Q811yescharacteristic
139C2×C4×C7#12311yescharacteristic
140C2×C4×C7#12311yescharacteristic
141C2×C4×C7#12312yes—
142C2×C4×C7#12412yes—
143C2×C4×C7#12411yescharacteristic
144C2×C4×C7#12411yescharacteristic
145C2×C4×C7#12512yes—
146C2×C4×C7#12512yes—
147C2×C4×C7#12512yes—
148C2×C4×C7#12512yes—
149C2×C4×C7#12612yes—
150C2×C4×C7#12612yes—
151C2×C4×C7C22×D411yescharacteristic
152C2×C4×C7C22×Q811yescharacteristic
153C2×C4×C7C2×#4112yes—
154C2×C4×C7C2×#4112yes—
155C7×C8—24no—
156C7×C8—42no—
157C7×C8—44no—
158C23×C4—21no—
159C23×C4C23×C711yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16C7×D412yes—
171C2×M16C23×C714yes—
172C2×M16C23×C712yes—
173C2×M16C23×C712yes—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8C7×D412yes—
177#117—22no—
178#117—22no—
179#117—22no—
180#117C23×C714yes—
181C2×C4⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42C7×Q811yescharacteristic
188C2×C42C7×Q812yes—
189C2×C42C7×Q811yescharacteristic
190C2×C42C23×C712yes—
191#108—28no—
192C4⋊C8—28no—
193C22×C7—22no—
194C22×C7—24no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—42no—
198C22×C7#40011yescharacteristic
199C22×C7#40012yes—
200C22×C7#40011yescharacteristic
201C22×C7C2×C22⋊Q811yescharacteristic
202C22×C7C2×#12612yes—
203C4×C7—24no—
204C4×C7—42no—
205C4×C7—84no—
206C4×C7#40011yescharacteristic
207C4×C7#40012yes—
208C4×C7#40011yescharacteristic
209C4×C7C2×#12311yescharacteristic
210C4×C7C2×#12411yescharacteristic
211C4×C7C2×#12512yes—
212C24C7×#4111yescharacteristic
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4C2×C7×D411yescharacteristic
220C22×C4C2×C7×D411yescharacteristic
221C22×C4C2×C7×Q811yescharacteristic
222C22×C4C2×C7×Q811yescharacteristic
223C22×C4C7×#4111yescharacteristic
224C22×C4C7×#4112yes—
225C22×C4C7×#4112yes—
226C22×C4C7×#4112yes—
227C22×C4C7×#4112yes—
228C22×C4C24×C711yescharacteristic
229M16—24no—
230M16—24no—
231M16—24no—
232M16—24no—
233M16—24no—
234M16—42no—
235M16—42no—
236M16—42no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8C2×C7×D414yes—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C22⋊C4—24no—
249C22⋊C4—42no—
250C42—21no—
251C42—22no—
252C42—21no—
253C42—22no—
254C42—24no—
255C42—42no—
256C42C2×C7×Q812yes—
257C42C2×C7×Q812yes—
258C2×C7—21no—
259C2×C7—22no—
260C2×C7—21no—
261C2×C7—22no—
262C2×C7#210212yesdirect product factor
263C2×C7C2×#40011yescharacteristic
264C23—42no—
265C23C7×C22⋊Q811yescharacteristic
266C23C7×C22⋊Q811yescharacteristic
267C23C7×#12612yes—
268C23C7×#12612yes—
269C23C2×C7×#4111yescharacteristic
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—84no—
278C2×C4C7×C22⋊Q811yescharacteristic
279C2×C4C7×C22⋊Q811yescharacteristic
280C2×C4C7×#12311yescharacteristic
281C2×C4C7×#12311yescharacteristic
282C2×C4C7×#12312yes—
283C2×C4C7×#12411yescharacteristic
284C2×C4C7×#12412yes—
285C2×C4C7×#12411yescharacteristic
286C2×C4C7×#12512yes—
287C2×C4C7×#12512yes—
288C2×C4C7×#12512yes—
289C2×C4C7×#12512yes—
290C2×C4C7×#12612yes—
291C2×C4C7×#12612yes—
292C2×C4C22×C7×D411yescharacteristic
293C2×C4C22×C7×Q811yescharacteristic
294C2×C4C2×C7×#4112yes—
295C2×C4C2×C7×#4112yes—
296C8—24no—
297C8—42no—
298C8—44no—
299C7C2×#210211yescharacteristic, direct product factor, Sylow
300C22—22no—
301C22—24no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22C7×#40011yescharacteristic
306C22C7×#40012yes—
307C22C7×#40011yescharacteristic
308C22C2×C7×C22⋊Q811yescharacteristic
309C22C2×C7×#12612yes—
310C4—24no—
311C4—42no—
312C4—84no—
313C4C7×#40011yescharacteristic
314C4C7×#40011yescharacteristic
315C4C7×#40012yes—
316C4C2×C7×#12311yescharacteristic
317C4C2×C7×#12411yescharacteristic
318C4C2×C7×#12512yes—
319C2—21no—
320C2—22no—
321C2—21no—
322C2—22no—
323C2C7×#210212yesdirect product factor
324C2C2×C7×#40011yescharacteristic
subgroupG/H
centerC2×C4×C7#124
commutator, Frattini, cosocleC22×C4C24×C7
socleC22×C7#400
prime radicalC24×C7#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#2102C2direct (N × H)
21. C2×#2102C7direct (N × H)
59. #2102C2×C7direct (N × H)
262. C2×C7#2102direct (N × H)
299. C7C2×#2102direct (N × H)
323. C2C7×#2102direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4302
C2×C7×#4232
C7×#21022
C2×C7×#3442
C2×C7×#3392
C2×C7×H(3,ℤ/4)2
C2×#21027