Finite Group Databasebeta

← C7×#9455 #13186671

Subgroups — C7×#9455 #13186671

Order 1792.

554 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C7×#2973C211yesmaximal, characteristic
2C7×#2542C211yesmaximal, characteristic
3C7×#1556C212yesmaximal, complemented
4C7×#1466C211yesmaximal, characteristic
5C7×#1285C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×C22⋊Q8—21no—
8C2×C7×C22⋊Q8—21no—
9C2×C4×C7×D4C2211yescharacteristic
10C7×#435—24no—
11C7×#432C2212yes—
12C7×#432C2212yes—
13C7×#407—21no—
14C2×C7×C22⋊C8—21no—
15C7×#399—21no—
16C7×#399—21no—
17C7×#390C412yes—
18C7×#384C412yes—
19C4×C7×C4⋊C4C2211yescharacteristic
20C7×#332—24no—
21C7×#323—21no—
22C7×#323—21no—
23#9455C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—22no—
29C22×C7×C8—22no—
30C7×C4⋊Q8—22no—
31C7×C4⋊Q8—22no—
32C7×C22⋊Q8—44no—
33C7×C22⋊Q8—44no—
34C4×C7×D4—22no—
35C4×C7×D4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4C2×C412yes—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C42×C7C2×C412yes—
47C2×C42×C7C2311yescharacteristic
48C7×C4⋊C8—22no—
49C7×C4⋊C8D414yes—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×C22⋊C8—22no—
53C7×C22⋊C8—22no—
54C7×#97—22no—
55C7×#97—22no—
56C7×#95—21no—
57#2973C2×C711yescharacteristic
58#2542C2×C711yescharacteristic
59#1556C2×C712yescomplemented
60#1466C2×C711yescharacteristic
61#1285C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C24×C7C22⋊C411yescharacteristic
64C2×C7×Q8—42no—
65C2×C7×Q8—42no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—41no—
71C22×C4×C7—22no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C7×M16—82no—
85C2×C7×C8—24no—
86C2×C7×C8—42no—
87C2×C7×C8—82no—
88C7×C4⋊C4—24no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—24no—
91C7×C4⋊C4—44no—
92C7×C4⋊C4—44no—
93C7×C4⋊C4—42no—
94C7×C22⋊C4—22no—
95C7×C22⋊C4—22no—
96C7×C22⋊C4—44no—
97C7×C22⋊C4—44no—
98C42×C7—42no—
99C42×C7C2×D412yes—
100C42×C7#4112yes—
101C2×C4⋊Q8C22×C711yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×D4C22×C711yescharacteristic
105#435—24no—
106#432C22×C712yes—
107#432C22×C712yes—
108#407—21no—
109C2×C22⋊C8—21no—
110#399—21no—
111#399—21no—
112#390C4×C712yes—
113#384C4×C712yes—
114C4×C4⋊C4C22×C711yescharacteristic
115#332—24no—
116#323—21no—
117#323—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7#10112yes—
128C23×C7C2×C22⋊C411yescharacteristic
129C7×Q8—82no—
130C7×Q8—82no—
131C7×D4—44no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—24no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—81no—
149C2×C4×C7—81no—
150C2×C4×C7C4×D412yes—
151C2×C4×C7#12011yescharacteristic
152C2×C4×C7#12112yes—
153C2×C4×C7#12311yescharacteristic
154C7×C8—44no—
155C7×C8—82no—
156C22×Q8—21no—
157C22×D4C2×C4×C711yescharacteristic
158C23×C4C7×D411yescharacteristic
159C23×C4C7×D411yescharacteristic
160C2×M16—22no—
161C22×C8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C22⋊Q8—44no—
165C22⋊Q8—44no—
166C4×D4—22no—
167C4×D4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×C4×C712yes—
173C2×C4⋊C4C2×C4×C711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C7×D411yescharacteristic
177C2×C22⋊C4C7×D411yescharacteristic
178C2×C42C2×C4×C712yes—
179C2×C42C23×C711yescharacteristic
180C4⋊C8—22no—
181C4⋊C8C7×D414yes—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186#97—22no—
187#97—22no—
188#95—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—22no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—81no—
200C22×C7—81no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10111yescharacteristic
203C22×C7#41311yescharacteristic
204C4×C7—21no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—44no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—82no—
211C4×C7#45312yes—
212C24C7×C22⋊C411yescharacteristic
213C24C7×C22⋊C411yescharacteristic
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C7×C22⋊C411yescharacteristic
230C22×C4C7×C22⋊C411yescharacteristic
231C22×C4C22×C4×C711yescharacteristic
232C22×C4C2×C7×D411yescharacteristic
233C22×C4C2×C7×D411yescharacteristic
234M16—82no—
235C2×C8—24no—
236C2×C8—42no—
237C2×C8—82no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C42—42no—
249C42C2×C7×D412yes—
250C42C7×#4112yes—
251C2×C7—21no—
252C2×C7—21no—
253C2×C7—42no—
254C2×C7—81no—
255C2×C7#188711yescharacteristic
256C2×C7#190111yescharacteristic
257C2×C7#190911yescharacteristic
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23C7×#10112yes—
268C23C2×C7×C22⋊C411yescharacteristic
269Q8—82no—
270Q8—82no—
271D4—44no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4C4×C7×D412yes—
291C2×C4C7×#12011yescharacteristic
292C2×C4C7×#12112yes—
293C2×C4C7×#12311yescharacteristic
294C8—44no—
295C8—82no—
296C7#945511yescharacteristic, direct product factor, Sylow
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22C7×#38611yescharacteristic
310C22C2×C7×#10111yescharacteristic
311C22C7×#41311yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4C7×#45312yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2C7×#188711yescharacteristic
325C2C7×#190111yescharacteristic
326C2C7×#190911yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1556C2semidirect
5. C7×#1285C2semidirect
23. #9455C7direct (N × H)
59. #1556C2×C7semidirect
61. #1285C2×C7semidirect
296. C7#9455direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29732
C7×#25422
C7×#15562
C7×#14662
C7×#12852
#94557