Finite Group Databasebeta

← C7×#11798 #13189014

Subgroups — C7×#11798 #13189014

Order 1792.

486 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2392C211yesmaximal, characteristic
2C7×#1820C211yesmaximal, characteristic, complemented
3C7×#1464C211yesmaximal, characteristic, complemented
4C7×#1344C212yesmaximal, complemented
5C7×#1343C212yesmaximal
6C2×C4×C7×D4—21no—
7C2×C4×C7×D4C411yescharacteristic, complemented
8C2×C7×#117C2211yescharacteristic
9C7×#432—21no—
10C7×#398C411yescharacteristic, complemented
11C7×#397C2211yescharacteristic
12C7×#396—21no—
13C7×#394—21no—
14C7×#394C411yescharacteristic, complemented
15C7×#392C2211yescharacteristic
16C7×#381—21no—
17C4×C7×C4⋊C4C411yescharacteristic, complemented
18C4×C7×C22⋊C4—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11798C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7—21no—
27C23×C4×C7D411yescharacteristic
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C4×C7×D4—44no—
31C7×#117—41no—
32C7×#117—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C7×C22⋊C4C2311yescharacteristic
46C2×C42×C7—21no—
47C2×C42×C7C2×C411yescharacteristic
48C2×C42×C7D411yescharacteristic
49C7×C4⋊C8—41no—
50C7×C22⋊C8—22no—
51C7×#97—41no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—41no—
55C7×#95—41no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2392C2×C711yescharacteristic
59#1820C2×C711yescharacteristic, complemented
60#1464C2×C711yescharacteristic, complemented
61#1344C2×C712yescomplemented
62#1343C2×C712yes—
63C24×C7—21no—
64C24×C7C2×D411yescharacteristic
65C2×C7×D4—42no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—42no—
70C22×C4×C7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—41no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7—42no—
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C2×C7×C8—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—21no—
107C42×C7—42no—
108C2×C4×D4—21no—
109C2×C4×D4C4×C711yescharacteristic, complemented
110C2×#117C22×C711yescharacteristic
111#432—21no—
112#398C4×C711yescharacteristic, complemented
113#397C22×C711yescharacteristic
114#396—21no—
115#394—21no—
116#394C4×C711yescharacteristic, complemented
117#392C22×C711yescharacteristic
118#381—21no—
119C4×C4⋊C4C4×C711yescharacteristic, complemented
120C4×C22⋊C4—21no—
121#342—21no—
122#342C22×C712yes—
123#324—21no—
124#323C22×C712yes—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—22no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7C2×C22⋊C411yescharacteristic
137C23×C7AGL(1,ℤ/8)12yes—
138C7×D4—42no—
139C7×D4—48no—
140C7×D4—42no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—22no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
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—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—82no—
166C2×C4×C7—82no—
167C2×C4×C7—82no—
168C2×C4×C7—82no—
169C2×C4×C7—82no—
170C2×C4×C7#9911yescharacteristic
171C2×C4×C7#9911yescharacteristic
172C7×C8—82no—
173C22×D4—21no—
174C22×D4C2×C4×C711yescharacteristic
175C23×C4—21no—
176C23×C4C7×D411yescharacteristic
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—44no—
180#117—41no—
181#117—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×C4×C711yescharacteristic
185C2×C4⋊C4C2×C4×C711yescharacteristic
186C2×C4⋊C4C7×D411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C7×D411yescharacteristic
194C2×C22⋊C4C23×C711yescharacteristic
195C2×C42—21no—
196C2×C42C2×C4×C711yescharacteristic
197C2×C42C7×D411yescharacteristic
198C4⋊C8—41no—
199C22⋊C8—22no—
200#97—41no—
201#95—21no—
202#95—21no—
203#95—41no—
204#95—41no—
205#95C2×C4×C711yescharacteristic
206#95C2×C4×C711yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—22no—
210C22×C7—42no—
211C22×C7—42no—
212C22×C7—42no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—42no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7C2×#9911yescharacteristic
222C22×C7#41811yescharacteristic
223C22×C7#42111yescharacteristic
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—42no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7—162no—
235C24—21no—
236C24C2×C7×D411yescharacteristic
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4C7×C22⋊C411yescharacteristic
255C22×C4C7×C22⋊C411yescharacteristic
256C22×C4C7×C22⋊C411yescharacteristic
257C22×C4C7×C22⋊C411yescharacteristic
258C22×C4C22×C4×C711yescharacteristic
259C22×C4C2×C7×D411yescharacteristic
260C2×C8—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—42no—
280C2×C7—21no—
281C2×C7—21no—
282C2×C7—42no—
283C2×C7—82no—
284C2×C7#151011yescharacteristic
285C2×C7#212011yescharacteristic
286C2×C7#212611yescharacteristic
287C23—21no—
288C23—21no—
289C23—22no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23C2×C7×C22⋊C411yescharacteristic
299C23C7×AGL(1,ℤ/8)12yes—
300D4—42no—
301D4—48no—
302D4—42no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4C7×#9911yescharacteristic
333C2×C4C7×#9911yescharacteristic
334C8—82no—
335C7#1179811yescharacteristic, direct product factor, Sylow
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×C7×#9911yescharacteristic
351C22C7×#41811yescharacteristic
352C22C7×#42111yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—162no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—82no—
368C2C7×#151011yescharacteristic
369C2C7×#212011yescharacteristic
370C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1820C2semidirect
3. C7×#1464C2semidirect
4. C7×#1344C2semidirect
7. C2×C4×C7×D4C4semidirect
10. C7×#398C4semidirect
14. C7×#394C4semidirect
17. C4×C7×C4⋊C4C4semidirect
23. #11798C7direct (N × H)
59. #1820C2×C7semidirect
60. #1464C2×C7semidirect
61. #1344C2×C7semidirect
109. C2×C4×D4C4×C7semidirect
112. #398C4×C7semidirect
116. #394C4×C7semidirect
119. C4×C4⋊C4C4×C7semidirect
335. C7#11798direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23922
C7×#18202
C7×#14642
C7×#13442
C7×#13432
#117987