Finite Group Databasebeta

← C7×#13298 #13190514

Subgroups — C7×#13298 #13190514

Order 1792.

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

normal subgroups only

This group has 556 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×#2542C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C4×C7×#99C211yesmaximal, characteristic
4C7×#1393C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C7×C22⋊Q8—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#99C2212yes—
10C2×C7×#99C2212yes—
11C7×#399—21no—
12C7×#399—21no—
13C7×#390C2212yes—
14C7×#384C412yescomplemented
15C7×#380—21no—
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C4×C7×C22⋊C4—21no—
18C7×#342—21no—
19C7×#342—21no—
20#13298C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4C2311yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—41no—
27C7×C22⋊Q8—44no—
28C7×C22⋊Q8—44no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—41no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7—41no—
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7C2×C412yes—
47C7×#99—22no—
48C7×#99—24no—
49C7×#99Q814yescomplemented
50C7×#95—21no—
51C7×#95—41no—
52C7×#95—41no—
53#2542C2×C711yescharacteristic
54#1799C2×C711yescharacteristic
55C4×#99C2×C711yescharacteristic
56#1393C2×C714yes—
57C24×C7C2×D411yescharacteristic
58C24×C7C2×D411yescharacteristic
59C2×C7×Q8—42no—
60C2×C7×Q8—42no—
61C2×C7×Q8—41no—
62C2×C7×Q8—41no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C2×C7×D4C2×Q812yes—
66C2×C7×D4#4112yes—
67C22×C4×C7—22no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—22no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C7×C4⋊C4—24no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4—44no—
86C7×C4⋊C4—44no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—82no—
89C7×C22⋊C4—24no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—44no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C42×C7—21no—
98C42×C7—21no—
99C42×C7—42no—
100C42×C7—82no—
101C2×C4⋊Q8C4×C711yescharacteristic, complemented
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×D4C22×C711yescharacteristic
105C2×#99C22×C712yes—
106C2×#99C22×C712yes—
107#399—21no—
108#399—21no—
109#390C22×C712yes—
110#384C4×C712yescomplemented
111#380—21no—
112C4×C4⋊C4C4×C711yescharacteristic, complemented
113C4×C22⋊C4—21no—
114#342—21no—
115#342—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7C2×C22⋊C411yescharacteristic
125C23×C7C22⋊Q812yes—
126C23×C7#12411yescharacteristic
127C23×C7C4⋊Q811yescharacteristic
128C7×Q8—82no—
129C7×Q8—82no—
130C7×D4—42no—
131C7×D4—42no—
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—21no—
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—41no—
146C2×C4×C7—44no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—81no—
151C2×C4×C7—81no—
152C2×C4×C7—82no—
153C2×C4×C7—82no—
154C2×C4×C7—82no—
155C2×C4×C7—82no—
156C2×C4×C7—82no—
157C2×C4×C7#9911yescharacteristic
158C2×C4×C7#9911yescharacteristic
159C2×C4×C7C4×Q812yes—
160C22×Q8—21no—
161C22×D4C23×C711yescharacteristic
162C23×C4C7×D411yescharacteristic
163C23×C4C7×D411yescharacteristic
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C2×C4×C712yes—
176C2×C4⋊C4C2×C4×C711yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C7×D411yescharacteristic
182C2×C22⋊C4C7×D411yescharacteristic
183C2×C42—41no—
184C2×C42C2×C4×C711yescharacteristic
185C2×C42C2×C4×C712yes—
186#99—22no—
187#99—24no—
188#99C7×Q814yescomplemented
189#95—21no—
190#95—41no—
191#95—41no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—42no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7#39111yescharacteristic
207C22×C7C2×#9911yescharacteristic
208C22×C7#41011yescharacteristic
209C4×C7—21no—
210C4×C7—22no—
211C4×C7—44no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—84no—
216C4×C7—82no—
217C4×C7—162no—
218C4×C7#35312yes—
219C24C2×C7×D411yescharacteristic
220C24C2×C7×D411yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4C2×C7×Q812yes—
228C2×D4C7×#4112yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C7×C22⋊C411yescharacteristic
240C22×C4C7×C22⋊C411yescharacteristic
241C22×C4C7×C22⋊C411yescharacteristic
242C22×C4C7×C22⋊C411yescharacteristic
243C22×C4C22×C4×C711yescharacteristic
244C4⋊C4—24no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—82no—
251C22⋊C4—24no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—21no—
260C42—21no—
261C42—42no—
262C42—82no—
263C2×C7—21no—
264C2×C7—21no—
265C2×C7—42no—
266C2×C7—41no—
267C2×C7—41no—
268C2×C7#199611yescharacteristic
269C2×C7C2×#35311yescharacteristic
270C2×C7#212911yescharacteristic
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C7×C22⋊C411yescharacteristic
280C23C7×C22⋊Q812yes—
281C23C7×#12411yescharacteristic
282C23C7×C4⋊Q811yescharacteristic
283Q8—82no—
284Q8—82no—
285D4—42no—
286D4—42no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4C7×#9911yescharacteristic
313C2×C4C7×#9911yescharacteristic
314C2×C4C4×C7×Q812yes—
315C7#1329811yescharacteristic, direct product factor, Sylow
316C22—21no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22C7×#39111yescharacteristic
331C22C2×C7×#9911yescharacteristic
332C22C7×#41011yescharacteristic
333C4—22no—
334C4—21no—
335C4—44no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—84no—
340C4—82no—
341C4—162no—
342C4C7×#35312yes—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—41no—
347C2—41no—
348C2C7×#199611yescharacteristic
349C2C2×C7×#35311yescharacteristic
350C2C7×#212911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ 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
5. C2×C7×C4⋊Q8C4semidirect
14. C7×#384C4semidirect
16. C4×C7×C4⋊C4C4semidirect
20. #13298C7direct (N × H)
49. C7×#99Q8semidirect
101. C2×C4⋊Q8C4×C7semidirect
110. #384C4×C7semidirect
112. C4×C4⋊C4C4×C7semidirect
188. #99C7×Q8semidirect
315. C7#13298direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25422
C7×#17992
C4×C7×#992
C7×#13932
#132987