Finite Group Databasebeta

← C7×#13573 #13190789

Subgroups — C7×#13573 #13190789

Order 1792.

470 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×C7×#456C211yesmaximal, characteristic
2C2×C7×#356C211yesmaximal, characteristic, complemented
3C7×#1558C211yesmaximal, characteristic, complemented
4C7×#1405C214yesmaximal, complemented
5C2×C7×2−1+4C411yescharacteristic
6C2×C7×#137—21no—
7C2×C7×C4⋊Q8C2211yescharacteristic
8C7×#456—21no—
9C7×#456—21no—
10C7×#456C412yes—
11C7×#456C2212yes—
12C7×#432—21no—
13C2×C7×#104—21no—
14C2×C7×#101—21no—
15C2×C7×#101C411yescharacteristic
16C7×#356—21no—
17C7×#356C2212yescomplemented
18C7×#333—21no—
19C7×#333C2212yescomplemented
20#13573C711yesmaximal, characteristic, direct product factor, Sylow
21C7×2−1+4—21no—
22C7×2−1+4—41no—
23C7×2−1+4—42no—
24C7×2−1+4C2×C412yes—
25C2×C7×#41—21no—
26C2×C7×#41—41no—
27C2×C7×#41—41no—
28C22×C7×Q8—41no—
29C22×C7×Q8C2×C411yescharacteristic
30C22×C7×Q8D411yescharacteristic
31C7×#137—22no—
32C7×#137—22no—
33C7×#137—42no—
34C2×C7×Q16—41no—
35C2×C7×SD16—41no—
36C2×C7×M16—21no—
37C2×C7×M16—41no—
38C7×C4⋊Q8—41no—
39C7×C4⋊Q8C2×C411yescharacteristic
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8C2311yescharacteristic
43C2×C7×C4⋊C4—41no—
44C2×C42×C7D411yescharacteristic
45C7×C4⋊C8—22no—
46C7×#104—22no—
47C7×#104—22no—
48C7×#101—21no—
49C7×#101—22no—
50C7×#101—41no—
51C7×#101C2×C412yes—
52C7×C22⋊C8—41no—
53C7×#97—41no—
54C2×#456C2×C711yescharacteristic
55C2×#356C2×C711yescharacteristic, complemented
56#1558C2×C711yescharacteristic, complemented
57#1405C2×C714yescomplemented
58C7×#41—22no—
59C7×#41—22no—
60C7×#41—41no—
61C7×#41—42no—
62C7×#41—42no—
63C7×#41—42no—
64C7×#41—42no—
65C7×#41—44no—
66C7×#41—42no—
67C7×#41—42no—
68C7×#41—41no—
69C2×C7×Q8—21no—
70C2×C7×Q8—21no—
71C2×C7×Q8—21no—
72C2×C7×Q8—41no—
73C2×C7×Q8—41no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×Q8—42no—
77C2×C7×Q8—42no—
78C2×C7×Q8—44no—
79C2×C7×Q8—41no—
80C2×C7×Q8C22⋊C411yescharacteristic
81C2×C7×Q8C22⋊C411yescharacteristic
82C2×C7×Q8C22×C411yescharacteristic
83C2×C7×Q8C2×D411yescharacteristic
84C2×C7×D4—21no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—41no—
91C22×C4×C7—42no—
92C22×C4×C7C22⋊C411yescharacteristic
93C7×Q16—42no—
94C7×Q16—42no—
95C7×SD16—42no—
96C7×SD16—42no—
97C7×M16—22no—
98C7×M16—41no—
99C7×M16—42no—
100C7×M16—81no—
101C2×C7×C8—41no—
102C2×C7×C8—42no—
103C2×C7×C8—81no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C42×C7—21no—
109C42×C7C22⋊C411yescharacteristic
110C42×C7C2×D411yescharacteristic
111C2×2−1+4C4×C711yescharacteristic
112C2×#137—21no—
113C2×C4⋊Q8C22×C711yescharacteristic
114#456—21no—
115#456—21no—
116#456C4×C712yes—
117#456C22×C712yes—
118#432—21no—
119C2×#104—21no—
120C2×#101—21no—
121C2×#101C4×C711yescharacteristic
122#356—21no—
123#356C22×C712yescomplemented
124#333—21no—
125#333C22×C712yescomplemented
126C23×C7—41no—
127C23×C7#9911yescharacteristic
128C7×Q8—22no—
129C7×Q8—41no—
130C7×Q8—41no—
131C7×Q8—41no—
132C7×Q8—41no—
133C7×Q8—42no—
134C7×Q8—41no—
135C7×Q8—42no—
136C7×Q8—42no—
137C7×Q8—81no—
138C7×D4—22no—
139C7×D4—41no—
140C7×D4—42no—
141C7×D4—42no—
142C7×D4—42no—
143C7×D4—42no—
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—21no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—44no—
156C2×C4×C7—44no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7#9911yescharacteristic
163C2×C4×C7C2×C22⋊C411yescharacteristic
164C7×C8—42no—
165C7×C8—82no—
166C7×C8—82no—
1672−1+4—21no—
1682−1+4—41no—
1692−1+4—42no—
1702−1+4C2×C4×C712yes—
171C2×#41—21no—
172C2×#41—41no—
173C2×#41—41no—
174C22×Q8—41no—
175C22×Q8C2×C4×C711yescharacteristic
176C22×Q8C7×D411yescharacteristic
177#137—22no—
178#137—22no—
179#137—42no—
180C2×Q16—41no—
181C2×SD16—41no—
182C2×M16—21no—
183C2×M16—41no—
184C4⋊Q8—41no—
185C4⋊Q8C2×C4×C711yescharacteristic
186C4⋊Q8C7×D411yescharacteristic
187C4⋊Q8C7×D411yescharacteristic
188C4⋊Q8C23×C711yescharacteristic
189C2×C4⋊C4—41no—
190C2×C42C7×D411yescharacteristic
191C4⋊C8—22no—
192#104—22no—
193#104—22no—
194#101—21no—
195#101—22no—
196#101—41no—
197#101C2×C4×C712yes—
198C22⋊C8—41no—
199#97—41no—
200C22×C7—41no—
201C22×C7—42no—
202C22×C7—82no—
203C22×C7AΣL(1,𝔽16)11yescharacteristic
204C22×C7AΣL(1,𝔽16)11yescharacteristic
205C22×C7C2×#9911yescharacteristic
206C4×C7—22no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—42no—
215#41—22no—
216#41—22no—
217#41—41no—
218#41—42no—
219#41—42no—
220#41—42no—
221#41—42no—
222#41—44no—
223#41—42no—
224#41—42no—
225#41—41no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8C7×C22⋊C411yescharacteristic
238C2×Q8C7×C22⋊C411yescharacteristic
239C2×Q8C22×C4×C711yescharacteristic
240C2×Q8C2×C7×D411yescharacteristic
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4C7×C22⋊C411yescharacteristic
250Q16—42no—
251Q16—42no—
252SD16—42no—
253SD16—42no—
254M16—22no—
255M16—41no—
256M16—42no—
257M16—81no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—81no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C42—21no—
266C42C7×C22⋊C411yescharacteristic
267C42C2×C7×D411yescharacteristic
268C2×C7—21no—
269C2×C7—21no—
270C2×C7—21no—
271C2×C7—82no—
272C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
273C23—41no—
274C23C7×#9911yescharacteristic
275Q8—22no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—42no—
281Q8—41no—
282Q8—42no—
283Q8—42no—
284Q8—81no—
285D4—22no—
286D4—41no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—42no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4C7×#9911yescharacteristic
310C2×C4C2×C7×C22⋊C411yescharacteristic
311C8—42no—
312C8—82no—
313C8—82no—
314C7#1357311yescharacteristic, direct product factor, Sylow
315C22—41no—
316C22—42no—
317C22—82no—
318C22C7×AΣL(1,𝔽16)11yescharacteristic
319C22C7×AΣL(1,𝔽16)11yescharacteristic
320C22C2×C7×#9911yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—42no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—82no—
334C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC2×C7C2×AΣL(1,𝔽16)
commutatorC2×Q8C22×C4×C7
Frattini, cosocleC4⋊Q8C23×C7
prime radicalC2×C7×2−1+4C4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×C4⋊Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#356C2semidirect
3. C7×#1558C2semidirect
4. C7×#1405C2semidirect
17. C7×#356C22semidirect
19. C7×#333C22semidirect
20. #13573C7direct (N × H)
55. C2×#356C2×C7semidirect
56. #1558C2×C7semidirect
57. #1405C2×C7semidirect
123. #356C22×C7semidirect
125. #333C22×C7semidirect
314. C7#13573direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4562
C2×C7×#3562
C7×#15582
C7×#14052
#135737