Finite Group Databasebeta

← C7×#13303 #13190519

Subgroups — C7×#13303 #13190519

Order 1792.

396 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2565C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic
3C4×C7×#101C211yesmaximal, characteristic
4C7×#1394C212yesmaximal
5C7×#1394C212yesmaximal
6C2×C7×#125C411yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#423—21no—
9C2×C7×#101C2212yes—
10C2×C7×#101C2212yes—
11C4×C7×M16—21no—
12C7×#398—21no—
13C7×#398—21no—
14C7×#391C2211yescharacteristic
15C7×#391C2211yescharacteristic
16C7×#387—21no—
17C7×#387—21no—
18C7×#384C411yescharacteristic
19C7×#382C411yescharacteristic
20C4×C7×C4⋊C4C411yescharacteristic
21C7×#343—21no—
22C7×#343—21no—
23#13303C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2311yescharacteristic
25C2×C7×M16—22no—
26C2×C7×M16—22no—
27C7×#125—41no—
28C7×#125—41no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7C2×C411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×C4⋊C8—41no—
48C7×#101—22no—
49C7×#101—24no—
50C7×#101Q814yes—
51C7×#97—41no—
52C4×C7×C8—41no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60#2565C2×C711yescharacteristic
61#1801C2×C711yescharacteristic
62C4×#101C2×C711yescharacteristic
63#1394C2×C712yes—
64#1394C2×C712yes—
65C2×C7×Q8—41no—
66C2×C7×Q8—41no—
67C2×C7×Q8C2×Q812yes—
68C2×C7×Q8#4112yes—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×M16—24no—
81C7×M16—42no—
82C7×M16—42no—
83C7×M16—42no—
84C2×C7×C8—42no—
85C2×C7×C8—42no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C42×C7—21no—
98C42×C7—21no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—41no—
104C42×C7—41no—
105C2×#125C4×C711yescharacteristic
106C2×C4×Q8C22×C711yescharacteristic
107#423—21no—
108C2×#101C22×C712yes—
109C2×#101C22×C712yes—
110C4×M16—21no—
111#398—21no—
112#398—21no—
113#391C22×C711yescharacteristic
114#391C22×C711yescharacteristic
115#387—21no—
116#387—21no—
117#384C4×C711yescharacteristic
118#382C4×C711yescharacteristic
119C4×C4⋊C4C4×C711yescharacteristic
120#343—21no—
121#343—21no—
122C23×C7C2×C22⋊C411yescharacteristic
123C7×Q8—42no—
124C7×Q8—42no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7#9911yescharacteristic
150C2×C4×C7#9911yescharacteristic
151C2×C4×C7C4×Q812yes—
152C2×C4×C7C22⋊Q812yes—
153C2×C4×C7#12411yescharacteristic
154C2×C4×C7C4⋊Q811yescharacteristic
155C7×C8—44no—
156C7×C8—82no—
157C22×Q8C23×C711yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160#125—41no—
161#125—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×C4×C711yescharacteristic
170C2×C4⋊C4C2×C4×C711yescharacteristic
171C2×C4⋊C4C2×C4×C711yescharacteristic
172C2×C4⋊C4C7×D411yescharacteristic
173C2×C4⋊C4C7×D411yescharacteristic
174C2×C42C2×C4×C711yescharacteristic
175C2×C42C2×C4×C711yescharacteristic
176C2×C42C2×C4×C711yescharacteristic
177C2×C42C7×D411yescharacteristic
178C2×C42C7×D411yescharacteristic
179C4⋊C8—41no—
180C4⋊C8—41no—
181#101—22no—
182#101—24no—
183#101C7×Q814yes—
184#97—41no—
185C4×C8—41no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7#39111yescharacteristic
196C22×C7C2×#9911yescharacteristic
197C22×C7#41011yescharacteristic
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—22no—
201C4×C7—21no—
202C4×C7—22no—
203C4×C7—21no—
204C4×C7—42no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—81no—
209C4×C7—81no—
210C4×C7—81no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8C2×C7×Q812yes—
214C2×Q8C7×#4112yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4C7×C22⋊C411yescharacteristic
220C22×C4C7×C22⋊C411yescharacteristic
221C22×C4C7×C22⋊C411yescharacteristic
222C22×C4C7×C22⋊C411yescharacteristic
223C22×C4C22×C4×C711yescharacteristic
224C22×C4C2×C7×D411yescharacteristic
225C22×C4C2×C7×D411yescharacteristic
226M16—24no—
227M16—42no—
228M16—42no—
229M16—42no—
230C2×C8—42no—
231C2×C8—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—41no—
250C42—41no—
251C2×C7—21no—
252C2×C7—21no—
253C2×C7#199611yescharacteristic
254C2×C7#213211yescharacteristic
255C2×C7#213311yescharacteristic
256C23C2×C7×C22⋊C411yescharacteristic
257Q8—42no—
258Q8—42no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4C7×#9911yescharacteristic
284C2×C4C7×#9911yescharacteristic
285C2×C4C4×C7×Q812yes—
286C2×C4C7×C22⋊Q812yes—
287C2×C4C7×#12411yescharacteristic
288C2×C4C7×C4⋊Q811yescharacteristic
289C8—44no—
290C8—82no—
291C7#1330311yescharacteristic, direct product factor, Sylow
292C22—21no—
293C22—21no—
294C22C7×#39111yescharacteristic
295C22C2×C7×#9911yescharacteristic
296C22C7×#41011yescharacteristic
297C4—22no—
298C4—21no—
299C4—22no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2C7×#199611yescharacteristic
313C2C7×#213211yescharacteristic
314C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×Q8C23×C7
prime radicalC23×C7C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
23. #13303C7direct (N × H)
291. C7#13303direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25652
C7×#18012
C4×C7×#1012
C7×#13942
#133037