Finite Group Databasebeta

← C7×#11902 #13189118

Subgroups — C7×#11902 #13189118

Order 1792.

452 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2564C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1460C211yesmaximal, characteristic
4C7×#1352C212yesmaximal, complemented
5C7×#1351C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#431—21no—
9C2×C7×#99—21no—
10C7×#400C411yescharacteristic, complemented
11C7×#399C2211yescharacteristic
12C7×#395C411yescharacteristic, complemented
13C7×#393C2211yescharacteristic
14C7×#387—21no—
15C7×#386—21no—
16C7×#385—21no—
17C7×#380—21no—
18C7×#380—21no—
19C2×C7×#95C411yescharacteristic, complemented
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11902C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C23×C4×C7—21no—
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C7×C22⋊Q8—41no—
30C7×C22⋊Q8—41no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—41no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C7×C22⋊C4C2311yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×#99—42no—
48C7×C22⋊C8—22no—
49C4×C7×C8—41no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—41no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2564C2×C711yescharacteristic
59#1799C2×C711yescharacteristic
60#1460C2×C711yescharacteristic
61#1352C2×C712yescomplemented
62#1351C2×C712yescomplemented
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—41no—
66C2×C7×Q8—81no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4SD1612yes—
70C2×C7×D4Q1612yes—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—42no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7—81no—
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C2×C7×C8—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—81no—
97C7×C4⋊C4—82no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—81no—
105C42×C7—21no—
106C42×C7—21no—
107C2×C22⋊Q8C4×C711yescharacteristic, complemented
108C2×C4×D4C22×C711yescharacteristic
109#431—21no—
110C2×#99—21no—
111#400C4×C711yescharacteristic, complemented
112#399C22×C711yescharacteristic
113#395C4×C711yescharacteristic, complemented
114#393C22×C711yescharacteristic
115#387—21no—
116#386—21no—
117#385—21no—
118#380—21no—
119#380—21no—
120C2×#95C4×C711yescharacteristic, complemented
121#342C22×C712yes—
122#323—21no—
123#323C22×C712yes—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7#10312yes—
133C23×C7C2×C22⋊C411yescharacteristic
134C23×C7C2×SD1611yescharacteristic
135C23×C7C2×Q1611yescharacteristic
136C7×Q8—82no—
137C7×D4—42no—
138C7×D4—42no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—22no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—41no—
146C2×C4×C7—42no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—81no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7—84no—
165C2×C4×C7—82no—
166C2×C4×C7—82no—
167C2×C4×C7—82no—
168C2×C4×C7#9911yescharacteristic
169C2×C4×C7#9911yescharacteristic
170C2×C4×C7#10312yes—
171C7×C8—82no—
172C22×Q8C2×C4×C711yescharacteristic
173C22×D4C7×D411yescharacteristic
174C23×C4—21no—
175C23×C4C2×C4×C711yescharacteristic
176C23×C4C7×D411yescharacteristic
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C2×C4×C711yescharacteristic
186C2×C4⋊C4C2×C4×C711yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4C7×D411yescharacteristic
192C2×C22⋊C4C23×C711yescharacteristic
193C2×C42C7×D411yescharacteristic
194C4⋊C8—41no—
195#99—42no—
196C22⋊C8—22no—
197C4×C8—41no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—41no—
204#95C2×C4×C711yescharacteristic
205#95C2×C4×C711yescharacteristic
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—22no—
210C22×C7—21no—
211C22×C7—42no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—42no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—81no—
220C22×C7C2×#9911yescharacteristic
221C22×C7C2×#10311yescharacteristic
222C22×C7#42111yescharacteristic
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—22no—
226C4×C7—42no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—81no—
231C4×C7—81no—
232C4×C7—81no—
233C4×C7—162no—
234C24C7×C22⋊C411yescharacteristic
235C24C2×C7×D411yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C7×SD1612yes—
241C2×D4C7×Q1612yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4C7×C22⋊C411yescharacteristic
257C22×C4C7×C22⋊C411yescharacteristic
258C22×C4C7×C22⋊C411yescharacteristic
259C22×C4C22×C4×C711yescharacteristic
260C22×C4C2×C7×D411yescharacteristic
261C2×C8—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—82no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C2×C7—21no—
279C2×C7—21no—
280C2×C7—41no—
281C2×C7—42no—
282C2×C7—41no—
283C2×C7#152011yescharacteristic
284C2×C7#212111yescharacteristic
285C2×C7#212611yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23C7×#10312yes—
295C23C2×C7×C22⋊C411yescharacteristic
296C23C2×C7×SD1611yescharacteristic
297C23C2×C7×Q1611yescharacteristic
298Q8—82no—
299D4—42no—
300D4—42no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C7×#9911yescharacteristic
331C2×C4C7×#9911yescharacteristic
332C2×C4C7×#10312yes—
333C8—82no—
334C7#1190211yescharacteristic, direct product factor, Sylow
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22C2×C7×#9911yescharacteristic
350C22C2×C7×#10311yescharacteristic
351C22C7×#42111yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—162no—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—42no—
367C2—41no—
368C2C7×#152011yescharacteristic
369C2C7×#212111yescharacteristic
370C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1352C2semidirect
5. C7×#1351C2semidirect
6. C2×C7×C22⋊Q8C4semidirect
10. C7×#400C4semidirect
12. C7×#395C4semidirect
19. C2×C7×#95C4semidirect
23. #11902C7direct (N × H)
61. #1352C2×C7semidirect
62. #1351C2×C7semidirect
107. C2×C22⋊Q8C4×C7semidirect
111. #400C4×C7semidirect
113. #395C4×C7semidirect
120. C2×#95C4×C7semidirect
334. C7#11902direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25642
C7×#17992
C7×#14602
C7×#13522
C7×#13512
#119027