Finite Group Databasebeta

← C7×#23669 #13200885

Subgroups — C7×#23669 #13200885

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#427C211yesmaximal, characteristic
2C2×C7×#424C211yesmaximal, characteristic
3C7×#2316C212yesmaximal
4C7×#2088C214yesmaximal, complemented
5C7×#2087C212yesmaximal, complemented
6C7×#2086C212yesmaximal, complemented
7C7×#1736C212yesmaximal
8C2×C7×#340C211yesmaximal, characteristic
9C22×C7×M16C2211yescharacteristic
10C23×C7×C8C2211yescharacteristic
11C2×C7×#117C2212yes—
12C2×C7×#117C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C7×#427—24no—
15C2×C7×#107C2211yescharacteristic
16C2×C7×#107C2211yescharacteristic
17C2×C7×#106C2212yes—
18C7×#424—24no—
19C2×C7×C4⋊C8C2212yes—
20C2×C7×C4⋊C8C2212yes—
21C7×#389—22no—
22C7×#389—22no—
23C7×#389C2214yes—
24C7×#389C2214yes—
25C7×#380—24no—
26C7×#340C2212yes—
27C7×#340C2212yes—
28C7×#340C2214yes—
29C7×#340C2214yes—
30#23669C711yesmaximal, characteristic, direct product factor, Sylow
31C23×C4×C7—22no—
32C23×C4×C7C2311yescharacteristic
33C2×C7×M16—22no—
34C2×C7×M16—24no—
35C22×C7×C8—24no—
36C22×C7×C8D414yes—
37C22×C7×C8C2312yes—
38C22×C7×C8C2311yescharacteristic
39C22×C7×C8C2311yescharacteristic
40C7×#117—22no—
41C7×#117—22no—
42C7×#117—24no—
43C7×#117—24no—
44C2×C7×C4⋊C4—22no—
45C2×C7×C4⋊C4—24no—
46C2×C7×C4⋊C4—24no—
47C2×C7×C4⋊C4—24no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C4⋊C4Q818yes—
51C2×C7×C4⋊C4C2312yes—
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C4⋊C4C2314yes—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4D412yes—
56C2×C42×C7—22no—
57C2×C42×C7C2312yes—
58C7×#107—22no—
59C7×#107—22no—
60C7×#106—24no—
61C7×C4⋊C8—24no—
62C7×C4⋊C8—24no—
63C7×#95—24no—
64C7×#95—24no—
65C2×#427C2×C711yescharacteristic
66C2×#424C2×C711yescharacteristic
67#2316C2×C712yes—
68#2088C2×C714yescomplemented
69#2087C2×C712yescomplemented
70#2086C2×C712yescomplemented
71#1736C2×C712yes—
72C2×#340C2×C711yescharacteristic
73C24×C7C2×D411yescharacteristic
74C22×C4×C7—24no—
75C22×C4×C7—28no—
76C22×C4×C7—22no—
77C22×C4×C7—22no—
78C22×C4×C7—82no—
79C22×C4×C7—82no—
80C22×C4×C7C2×D412yes—
81C22×C4×C7C2×D412yes—
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D412yes—
84C22×C4×C7C2×Q814yes—
85C22×C4×C7#4112yes—
86C22×C4×C7#4114yes—
87C22×C4×C7C2411yescharacteristic
88C7×M16—44no—
89C2×C7×C8—28no—
90C2×C7×C8—22no—
91C2×C7×C8—22no—
92C2×C7×C8—22no—
93C2×C7×C8—24no—
94C2×C7×C8C2×D412yes—
95C2×C7×C8C2×D412yes—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—28no—
99C7×C4⋊C4—24no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—22no—
104C7×C4⋊C4—84no—
105C7×C4⋊C4—84no—
106C7×C4⋊C4C2×Q818yes—
107C7×C22⋊C4—24no—
108C7×C22⋊C4—44no—
109C42×C7—22no—
110C42×C7—22no—
111C42×C7—44no—
112C22×M16C22×C711yescharacteristic
113C23×C8C22×C711yescharacteristic
114C2×#117C22×C712yes—
115C2×#117C22×C711yescharacteristic
116C22×C4⋊C4C22×C712yes—
117#427—24no—
118C2×#107C22×C711yescharacteristic
119C2×#107C22×C711yescharacteristic
120C2×#106C22×C712yes—
121#424—24no—
122C2×C4⋊C8C22×C712yes—
123C2×C4⋊C8C22×C712yes—
124#389—22no—
125#389—22no—
126#389C22×C714yes—
127#389C22×C714yes—
128#380—24no—
129#340C22×C712yes—
130#340C22×C714yes—
131#340C22×C714yes—
132#340C22×C712yes—
133C23×C7—21no—
134C23×C7—22no—
135C23×C7—21no—
136C23×C7#12112yes—
137C23×C7C22⋊Q814yes—
138C23×C7C22×D411yescharacteristic
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—28no—
142C2×C4×C7—22no—
143C2×C4×C7—24no—
144C2×C4×C7—22no—
145C2×C4×C7—44no—
146C2×C4×C7—44no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—88no—
154C2×C4×C7#12112yes—
155C2×C4×C7C22⋊Q814yes—
156C2×C4×C7#12314yes—
157C2×C4×C7#12518yes—
158C2×C4×C7C4⋊Q814yes—
159C2×C4×C7C22×D411yescharacteristic
160C2×C4×C7C22×Q812yes—
161C2×C4×C7C2×#4112yes—
162C2×C4×C7C2×#4111yescharacteristic
163C7×C8—22no—
164C7×C8—22no—
165C7×C8—42no—
166C23×C4—22no—
167C23×C4C23×C711yescharacteristic
168C2×M16—22no—
169C2×M16—24no—
170C22×C8—24no—
171C22×C8C7×D414yes—
172C22×C8C23×C712yes—
173C22×C8C23×C711yescharacteristic
174C22×C8C23×C711yescharacteristic
175#117—22no—
176#117—22no—
177#117—24no—
178#117—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4C7×D411yescharacteristic
184C2×C4⋊C4C7×D411yescharacteristic
185C2×C4⋊C4C7×Q818yes—
186C2×C4⋊C4C23×C712yes—
187C2×C4⋊C4C23×C712yes—
188C2×C4⋊C4C23×C714yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C7×D412yes—
191C2×C42—22no—
192C2×C42C23×C712yes—
193#107—22no—
194#107—22no—
195#106—24no—
196C4⋊C8—24no—
197C4⋊C8—24no—
198#95—24no—
199#95—24no—
200C22×C7—24no—
201C22×C7—24no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7#39814yes—
205C22×C7C2×#12111yescharacteristic
206C22×C7C2×C22⋊Q812yes—
207C22×C7#57611yescharacteristic
208C22×C7#57712yes—
209C22×C7#57811yescharacteristic
210C4×C7—48no—
211C4×C7—42no—
212C4×C7—42no—
213C4×C7—82no—
214C4×C7—82no—
215C4×C7#39814yes—
216C4×C7C2×#12311yescharacteristic
217C4×C7C2×#12512yes—
218C4×C7C2×C4⋊Q811yescharacteristic
219C24C2×C7×D411yescharacteristic
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4C2×C7×D412yes—
227C22×C4C2×C7×D412yes—
228C22×C4C2×C7×D411yescharacteristic
229C22×C4C2×C7×D412yes—
230C22×C4C2×C7×Q814yes—
231C22×C4C7×#4112yes—
232C22×C4C7×#4114yes—
233C22×C4C24×C711yescharacteristic
234M16—44no—
235C2×C8—28no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—24no—
240C2×C8C2×C7×D412yes—
241C2×C8C2×C7×D412yes—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4C2×C7×Q818yes—
253C22⋊C4—24no—
254C22⋊C4—44no—
255C42—22no—
256C42—22no—
257C42—44no—
258C2×C7—24no—
259C2×C7C2×#39811yescharacteristic
260C2×C7#306011yescharacteristic
261C2×C7#306111yescharacteristic
262C2×C7#308112yes—
263C2×C7#308212yes—
264C23—21no—
265C23—22no—
266C23—21no—
267C23C7×#12112yes—
268C23C7×C22⋊Q814yes—
269C23C22×C7×D411yescharacteristic
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—28no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—88no—
285C2×C4C7×#12112yes—
286C2×C4C7×C22⋊Q814yes—
287C2×C4C7×#12314yes—
288C2×C4C7×#12518yes—
289C2×C4C7×C4⋊Q814yes—
290C2×C4C22×C7×D411yescharacteristic
291C2×C4C22×C7×Q812yes—
292C2×C4C2×C7×#4112yes—
293C2×C4C2×C7×#4111yescharacteristic
294C8—22no—
295C8—22no—
296C8—42no—
297C7#2366911yescharacteristic, direct product factor, Sylow
298C22—22no—
299C22—22no—
300C22—24no—
301C22—24no—
302C22C7×#39814yes—
303C22C2×C7×#12111yescharacteristic
304C22C2×C7×C22⋊Q812yes—
305C22C7×#57611yescharacteristic
306C22C7×#57712yes—
307C22C7×#57811yescharacteristic
308C4—42no—
309C4—42no—
310C4—48no—
311C4—82no—
312C4—82no—
313C4C7×#39814yes—
314C4C2×C7×#12311yescharacteristic
315C4C2×C7×#12512yes—
316C4C2×C7×C4⋊Q811yescharacteristic
317C2—24no—
318C2C2×C7×#39811yescharacteristic
319C2C7×#306011yescharacteristic
320C2C7×#306111yescharacteristic
321C2C7×#308112yes—
322C2C7×#308212yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#2088C2semidirect
5. C7×#2087C2semidirect
6. C7×#2086C2semidirect
30. #23669C7direct (N × H)
68. #2088C2×C7semidirect
69. #2087C2×C7semidirect
70. #2086C2×C7semidirect
297. C7#23669direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4272
C2×C7×#4242
C7×#23162
C7×#20882
C7×#20872
C7×#20862
C7×#17362
C2×C7×#3402
#236697