Finite Group Databasebeta

← C7×#18999 #13196215

Subgroups — C7×#18999 #13196215

Order 1792.

850 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#563C211yesmaximal, characteristic
2C7×#3236C212yesmaximal, complemented
3C7×#3140C211yesmaximal, characteristic
4C7×#1664C218yesmaximal, complemented
5C7×#1533C211yesmaximal, characteristic
6C7×#1528C212yesmaximal, complemented
7C7×#563—24no—
8C7×#563—22no—
9C7×#563C2214yes—
10C7×#533C2211yescharacteristic
11C2×C7×C4⋊Q8—21no—
12C2×C7×#126—22no—
13C2×C7×#125C2212yes—
14C2×C7×#124C2211yescharacteristic
15C2×C7×#123—21no—
16C2×C7×C22⋊Q8—22no—
17C7×#501C2214yescomplemented
18C7×#491C2214yescomplemented
19C7×#489C2214yes—
20C7×#488—22no—
21C7×#484—22no—
22C7×#483—22no—
23C7×#479—24no—
24C7×#432C2212yes—
25C7×#419—21no—
26C7×#418—21no—
27C4×C7×M16C2211yescharacteristic
28C7×#330C2218yescomplemented
29C7×#329C2214yes—
30#18999C711yesmaximal, characteristic, direct product factor, Sylow
31C22×C7×Q8—21no—
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C2×C7×M16—21no—
36C7×C4⋊Q8—24no—
37C7×C4⋊Q8—24no—
38C7×C4⋊Q8C2312yes—
39C7×#126—28no—
40C7×#126—28no—
41C7×#125—22no—
42C7×#125—22no—
43C7×#125—21no—
44C7×#125D414yes—
45C7×#125C2314yes—
46C7×#124—22no—
47C7×#124D412yes—
48C7×#124C2312yes—
49C7×#123—24no—
50C7×#123—24no—
51C7×C22⋊Q8—28no—
52C7×C22⋊Q8—28no—
53C7×C22⋊Q8—22no—
54C4×C7×Q8—22no—
55C7×#117—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4D412yes—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—22no—
64C2×C42×C7—21no—
65C2×C42×C7C2311yescharacteristic
66C7×#107—24no—
67C7×#106—24no—
68C7×C4⋊C8—22no—
69C7×#103—24no—
70C7×#103—24no—
71C7×#103—24no—
72C7×#102—24no—
73C7×C22⋊C8—22no—
74C7×#97C2312yes—
75C7×#97C2314yes—
76C4×C7×C8—21no—
77C2×#563C2×C711yescharacteristic
78#3236C2×C712yescomplemented
79#3140C2×C711yescharacteristic
80#1664C2×C718yescomplemented
81#1533C2×C711yescharacteristic
82#1528C2×C712yescomplemented
83C24×C7—21no—
84C2×C7×Q8—24no—
85C2×C7×Q8—21no—
86C2×C7×Q8—22no—
87C2×C7×Q8—42no—
88C2×C7×Q8—82no—
89C2×C7×Q8C2×D412yes—
90C2×C7×D4—21no—
91C2×C7×D4—42no—
92C2×C7×D4C2×D412yes—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—22no—
96C22×C4×C7—24no—
97C22×C4×C7—82no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D412yes—
100C7×M16—42no—
101C2×C7×C8—22no—
102C2×C7×C8—24no—
103C7×C4⋊C4—28no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—24no—
109C7×C4⋊C4—24no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—42no—
113C7×C4⋊C4C2×D414yes—
114C7×C4⋊C4C2×D414yes—
115C7×C22⋊C4—28no—
116C7×C22⋊C4—24no—
117C7×C22⋊C4—24no—
118C7×C22⋊C4—24no—
119C7×C22⋊C4—41no—
120C42×C7—24no—
121C42×C7—41no—
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D412yes—
124C42×C7C2411yescharacteristic
125#563—24no—
126#563—22no—
127#563C22×C714yes—
128#533C22×C711yescharacteristic
129C2×C4⋊Q8—21no—
130C2×#126—22no—
131C2×#125C22×C712yes—
132C2×#124C22×C711yescharacteristic
133C2×#123—21no—
134C2×C22⋊Q8—22no—
135#501C22×C714yescomplemented
136#491C22×C714yescomplemented
137#489C22×C714yes—
138#488—22no—
139#484—22no—
140#483—22no—
141#479—24no—
142#432C22×C712yes—
143#419—21no—
144#418—21no—
145C4×M16C22×C711yescharacteristic
146#330C22×C718yescomplemented
147#329C22×C714yes—
148C23×C7—22no—
149C23×C7—44no—
150C23×C7—42no—
151C23×C7#12011yescharacteristic
152C7×Q8—42no—
153C7×Q8—44no—
154C7×Q8—81no—
155C7×Q8—84no—
156C7×D4—42no—
157C7×D4—81no—
158C2×C4×C7—22no—
159C2×C4×C7—24no—
160C2×C4×C7—22no—
161C2×C4×C7—24no—
162C2×C4×C7—42no—
163C2×C4×C7—44no—
164C2×C4×C7—44no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—84no—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12012yes—
170C2×C4×C7C22×D412yes—
171C2×C4×C7C22×D411yescharacteristic
172C7×C8—44no—
173C7×C8—42no—
174C22×Q8—21no—
175C22×Q8C7×D411yescharacteristic
176C22×D4C7×D411yescharacteristic
177C23×C4—21no—
178C2×M16—21no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8C23×C712yes—
182#126—28no—
183#126—28no—
184#125—22no—
185#125—22no—
186#125—21no—
187#125C7×D414yes—
188#125C23×C714yes—
189#124—22no—
190#124C7×D412yes—
191#124C23×C712yes—
192#123—24no—
193#123—24no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—22no—
197C4×Q8—22no—
198#117—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C7×D412yes—
203C2×C4⋊C4C7×D412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C42—21no—
208C2×C42C23×C711yescharacteristic
209#107—24no—
210#106—24no—
211C4⋊C8—22no—
212#103—24no—
213#103—24no—
214#103—24no—
215#102—24no—
216C22⋊C8—22no—
217#97C23×C714yes—
218#97C23×C712yes—
219C4×C8—21no—
220C22×C7—22no—
221C22×C7—42no—
222C22×C7—44no—
223C22×C7—84no—
224C22×C7#45512yes—
225C22×C7C2×#12011yescharacteristic
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—44no—
231C4×C7—42no—
232C4×C7—44no—
233C4×C7—82no—
234C24—21no—
235C2×Q8—24no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×Q8C2×C7×D412yes—
241C2×D4—21no—
242C2×D4—42no—
243C2×D4C2×C7×D412yes—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—82no—
249C22×C4C2×C7×D411yescharacteristic
250C22×C4C2×C7×D412yes—
251M16—42no—
252C2×C8—22no—
253C2×C8—24no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4C2×C7×D414yes—
265C4⋊C4C2×C7×D414yes—
266C22⋊C4—28no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—41no—
271C42—24no—
272C42—41no—
273C42C2×C7×D411yescharacteristic
274C42C2×C7×D412yes—
275C42C24×C711yescharacteristic
276C2×C7—22no—
277C2×C7—82no—
278C2×C7C2×#45511yescharacteristic
279C2×C7#302012yes—
280C23—22no—
281C23—44no—
282C23—42no—
283C23C7×#12011yescharacteristic
284Q8—42no—
285Q8—44no—
286Q8—81no—
287Q8—84no—
288D4—42no—
289D4—81no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4C7×#12011yescharacteristic
301C2×C4C7×#12012yes—
302C2×C4C22×C7×D412yes—
303C2×C4C22×C7×D411yescharacteristic
304C8—44no—
305C8—42no—
306C7#1899911yescharacteristic, direct product factor, Sylow
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22C7×#45512yes—
312C22C2×C7×#12011yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—82no—
321C2—22no—
322C2—82no—
323C2C2×C7×#45511yescharacteristic
324C2C7×#302012yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC22×C7×D4D4
LCSG ▷ 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×#3236C2semidirect
4. C7×#1664C2semidirect
6. C7×#1528C2semidirect
17. C7×#501C22semidirect
18. C7×#491C22semidirect
28. C7×#330C22semidirect
30. #18999C7direct (N × H)
78. #3236C2×C7semidirect
80. #1664C2×C7semidirect
82. #1528C2×C7semidirect
135. #501C22×C7semidirect
136. #491C22×C7semidirect
146. #330C22×C7semidirect
306. C7#18999direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5632
C7×#32362
C7×#31402
C7×#16642
C7×#15332
C7×#15282
#189997