Finite Group Databasebeta

← C7×#21911 #13199127

Subgroups — C7×#21911 #13199127

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2297C211yesmaximal, characteristic
2C7×#1924C214yesmaximal, complemented
3C7×#1923C214yesmaximal, complemented
4C7×#1797C212yesmaximal, complemented
5C7×#1780C212yesmaximal
6C7×#1750C212yesmaximal, complemented
7C22×C7×#41C2211yescharacteristic
8C2×C7×#117C2211yescharacteristic
9C22×C42×C7C2211yescharacteristic
10C2×C7×C4⋊C8C2214yes—
11C2×C7×C22⋊C8C2214yes—
12C2×C7×C22⋊C8C2212yes—
13C2×C7×C22⋊C8C2214yes—
14C2×C7×C22⋊C8C2212yes—
15C2×C7×#97C2212yes—
16C2×C4×C7×C8C2212yes—
17C7×#391C412yes—
18C7×#390C412yes—
19C7×#386C412yes—
20C7×#386C412yes—
21C4×C7×C22⋊C4C2214yes—
22C7×#336C2214yes—
23C7×#336C2212yes—
24C7×#336C2212yes—
25#21911C711yesmaximal, characteristic, direct product factor, Sylow
26C2×C7×#41—24no—
27C2×C7×#41—42no—
28C2×C7×#41—42no—
29C22×C7×Q8C2×C411yescharacteristic
30C22×C7×D4C2×C412yes—
31C22×C7×D4C2×C411yescharacteristic
32C23×C4×C7C2×C411yescharacteristic
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7C2312yes—
35C23×C4×C7C2311yescharacteristic
36C22×C7×C8C2314yes—
37C22×C7×C8C2314yes—
38C7×#117—42no—
39C2×C7×C4⋊C4C2×C412yes—
40C2×C7×C22⋊C4C2×C414yes—
41C2×C7×C22⋊C4C2×C414yes—
42C2×C7×C22⋊C4C2×C412yes—
43C2×C42×C7—42no—
44C2×C42×C7C2×C412yes—
45C2×C42×C7C2312yes—
46C2×C42×C7C2312yes—
47C7×C4⋊C8—44no—
48C7×C22⋊C8—28no—
49C7×C22⋊C8—44no—
50C7×C22⋊C8—42no—
51C7×C22⋊C8—42no—
52C7×#97—42no—
53C4×C7×C8—42no—
54C7×#95C2×C414yes—
55C7×#95C2×C414yes—
56#2297C2×C711yescharacteristic
57#1924C2×C714yescomplemented
58#1923C2×C714yescomplemented
59#1797C2×C712yescomplemented
60#1780C2×C712yes—
61#1750C2×C712yescomplemented
62C24×C7C22×C412yes—
63C24×C7C22×C411yescharacteristic
64C7×#41—416no—
65C2×C7×Q8—22no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×D4—24no—
69C2×C7×D4—22no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—24no—
77C22×C4×C7—24no—
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—24no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—42no—
84C22×C4×C7C22×C412yes—
85C22×C4×C7C22×C412yes—
86C22×C4×C7C22×C414yes—
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C22×C412yes—
89C22×C4×C7#4114yes—
90C22×C4×C7C2411yescharacteristic
91C2×C7×C8—24no—
92C2×C7×C8—24no—
93C2×C7×C8—24no—
94C2×C7×C8—24no—
95C2×C7×C8—24no—
96C2×C7×C8—24no—
97C7×C4⋊C4—42no—
98C7×C22⋊C4—28no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—42no—
101C42×C7—22no—
102C42×C7—22no—
103C42×C7—24no—
104C42×C7—42no—
105C22×#41C22×C711yescharacteristic
106C2×#117C22×C711yescharacteristic
107C22×C42C22×C711yescharacteristic
108C2×C4⋊C8C22×C714yes—
109C2×C22⋊C8C22×C714yes—
110C2×C22⋊C8C22×C712yes—
111C2×C22⋊C8C22×C714yes—
112C2×C22⋊C8C22×C712yes—
113C2×#97C22×C712yes—
114C2×C4×C8C22×C712yes—
115#391C4×C712yes—
116#390C4×C712yes—
117#386C4×C712yes—
118#386C4×C712yes—
119C4×C22⋊C4C22×C714yes—
120#336C22×C712yes—
121#336C22×C712yes—
122#336C22×C714yes—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—22no—
126C23×C7—22no—
127C23×C7—22no—
128C23×C7—22no—
129C23×C7—22no—
130C23×C7—24no—
131C23×C7—24no—
132C23×C7#11712yes—
133C23×C7C23×C411yescharacteristic
134C7×Q8—44no—
135C7×D4—44no—
136C7×D4—48no—
137C2×C4×C7—22no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—22no—
141C2×C4×C7—28no—
142C2×C4×C7—22no—
143C2×C4×C7—22no—
144C2×C4×C7—24no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—24no—
148C2×C4×C7—22no—
149C2×C4×C7—24no—
150C2×C4×C7—22no—
151C2×C4×C7—44no—
152C2×C4×C7—44no—
153C2×C4×C7—44no—
154C2×C4×C7—44no—
155C2×C4×C7#11712yes—
156C2×C4×C7#13114yes—
157C2×C4×C7C2×#4112yes—
158C2×C4×C7H(5,𝔽2)12yes—
159C2×C4×C7H(5,𝔽2)12yes—
160C7×C8—44no—
161C7×C8—44no—
162C2×#41—24no—
163C2×#41—42no—
164C2×#41—42no—
165C22×Q8C2×C4×C711yescharacteristic
166C22×D4C2×C4×C712yes—
167C22×D4C2×C4×C711yescharacteristic
168C23×C4C2×C4×C711yescharacteristic
169C23×C4C2×C4×C711yescharacteristic
170C23×C4C23×C712yes—
171C23×C4C23×C711yescharacteristic
172C22×C8C23×C714yes—
173C22×C8C23×C714yes—
174#117—42no—
175C2×C4⋊C4C2×C4×C712yes—
176C2×C22⋊C4C2×C4×C714yes—
177C2×C22⋊C4C2×C4×C712yes—
178C2×C22⋊C4C2×C4×C714yes—
179C2×C42—42no—
180C2×C42C2×C4×C712yes—
181C2×C42C23×C712yes—
182C2×C42C23×C712yes—
183C4⋊C8—44no—
184C22⋊C8—28no—
185C22⋊C8—44no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188#97—42no—
189C4×C8—42no—
190#95C2×C4×C714yes—
191#95C2×C4×C714yes—
192C22×C7—22no—
193C22×C7—24no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—42no—
197C22×C7—42no—
198C22×C7—44no—
199C22×C7—42no—
200C22×C7—42no—
201C22×C7—44no—
202C22×C7C2×#11711yescharacteristic
203C22×C7#51811yescharacteristic
204C22×C7#51811yescharacteristic
205C22×C7C2×#13112yes—
206C22×C7#56812yes—
207C4×C7—22no—
208C4×C7—24no—
209C4×C7—42no—
210C4×C7—42no—
211C4×C7—44no—
212C4×C7#53814yes—
213C24C22×C4×C712yes—
214C24C22×C4×C711yescharacteristic
215#41—416no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4C22×C4×C712yes—
236C22×C4C22×C4×C712yes—
237C22×C4C22×C4×C714yes—
238C22×C4C22×C4×C711yescharacteristic
239C22×C4C22×C4×C712yes—
240C22×C4C7×#4114yes—
241C22×C4C24×C711yescharacteristic
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C4⋊C4—42no—
249C22⋊C4—28no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C42—22no—
253C42—22no—
254C42—24no—
255C42—42no—
256C2×C7—42no—
257C2×C7—82no—
258C2×C7#231211yescharacteristic
259C2×C7#292512yes—
260C2×C7#297711yescharacteristic
261C2×C7#297811yescharacteristic
262C2×C7#298112yes—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23C7×#11712yes—
273C23C23×C4×C711yescharacteristic
274Q8—44no—
275D4—44no—
276D4—48no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4C7×#11712yes—
296C2×C4C7×#13114yes—
297C2×C4C2×C7×#4112yes—
298C2×C4C7×H(5,𝔽2)12yes—
299C2×C4C7×H(5,𝔽2)12yes—
300C8—44no—
301C8—44no—
302C7#2191111yescharacteristic, direct product factor, Sylow
303C22—24no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—42no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—44no—
312C22—42no—
313C22C2×C7×#11711yescharacteristic
314C22C7×#51811yescharacteristic
315C22C7×#51811yescharacteristic
316C22C2×C7×#13112yes—
317C22C7×#56812yes—
318C4—24no—
319C4—22no—
320C4—42no—
321C4—44no—
322C4—42no—
323C4C7×#53814yes—
324C2—42no—
325C2—82no—
326C2C7×#231211yescharacteristic
327C2C7×#292512yes—
328C2C7×#297711yescharacteristic
329C2C7×#297811yescharacteristic
330C2C7×#298112yes—
subgroupG/H
centerC22×C4×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC22×C4C24×C7
socleC23×C7C23×C4
prime radicalC22×C7×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1924C2semidirect
3. C7×#1923C2semidirect
4. C7×#1797C2semidirect
6. C7×#1750C2semidirect
25. #21911C7direct (N × H)
57. #1924C2×C7semidirect
58. #1923C2×C7semidirect
59. #1797C2×C7semidirect
61. #1750C2×C7semidirect
302. C7#21911direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#22972
C7×#19242
C7×#19232
C7×#17972
C7×#17802
C7×#17502
#219117