Finite Group Databasebeta

← C7×#11901 #13189117

Subgroups — C7×#11901 #13189117

Order 1792.

508 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 118 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
3C7×#1459C211yesmaximal, characteristic
4C7×#1352C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C7×C22⋊Q8—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#99—21no—
10C2×C7×C22⋊C8—21no—
11C7×#399C412yescomplemented
12C7×#399C2212yes—
13C7×#390—21no—
14C7×#384—21no—
15C7×#380—21no—
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11901C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C22×C7×C8—41no—
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C7×C22⋊Q8—44no—
29C7×C22⋊Q8—44no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—41no—
37C2×C7×C4⋊C4C2×C412yes—
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7—21no—
45C2×C42×C7C2×C411yescharacteristic
46C7×#99—42no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—41no—
49C7×#95—41no—
50C7×#95C2×C412yes—
51#2542C2×C711yescharacteristic
52#1799C2×C711yescharacteristic
53#1459C2×C711yescharacteristic
54#1352C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—42no—
58C2×C7×Q8—42no—
59C2×C7×Q8—41no—
60C2×C7×Q8—41no—
61C2×C7×D4—41no—
62C2×C7×D4—41no—
63C2×C7×D4SD1612yes—
64C2×C7×D4Q1612yes—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—22no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C2×C7×C8—42no—
80C2×C7×C8—82no—
81C7×C4⋊C4—24no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—44no—
85C7×C4⋊C4—44no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—82no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—44no—
91C7×C22⋊C4—44no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7—42no—
97C2×C4⋊Q8C4×C711yescharacteristic, complemented
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C4×D4C22×C711yescharacteristic
101C2×#99—21no—
102C2×C22⋊C8—21no—
103#399C4×C712yescomplemented
104#399C22×C712yes—
105#390—21no—
106#384—21no—
107#380—21no—
108C4×C4⋊C4C4×C711yescharacteristic, complemented
109#342C22×C712yes—
110#324—21no—
111#323C22×C712yes—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7#10312yes—
121C23×C7C2×C22⋊C411yescharacteristic
122C23×C7C2×SD1611yescharacteristic
123C23×C7C2×Q1611yescharacteristic
124C7×Q8—82no—
125C7×Q8—82no—
126C7×D4—42no—
127C7×D4—42no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—22no—
131C2×C4×C7—24no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—81no—
146C2×C4×C7—81no—
147C2×C4×C7—82no—
148C2×C4×C7—82no—
149C2×C4×C7—82no—
150C2×C4×C7#9911yescharacteristic
151C2×C4×C7#9911yescharacteristic
152C2×C4×C7#10312yes—
153C7×C8—82no—
154C22×Q8—21no—
155C22×D4C7×D411yescharacteristic
156C23×C4C7×D411yescharacteristic
157C23×C4C7×D411yescharacteristic
158C22×C8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C22⋊Q8—44no—
162C22⋊Q8—44no—
163C4×D4—41no—
164C4×D4—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×C4×C712yes—
171C2×C4⋊C4C7×D411yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C2×C4×C711yescharacteristic
176C2×C22⋊C4C23×C711yescharacteristic
177C2×C42—21no—
178C2×C42C2×C4×C711yescharacteristic
179#99—42no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182#95—41no—
183#95C2×C4×C712yes—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—22no—
188C22×C7—21no—
189C22×C7—42no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—81no—
197C22×C7—81no—
198C22×C7C2×#9911yescharacteristic
199C22×C7C2×#10311yescharacteristic
200C22×C7#42111yescharacteristic
201C4×C7—21no—
202C4×C7—22no—
203C4×C7—44no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—82no—
208C4×C7—162no—
209C4×C7#35312yes—
210C24C7×C22⋊C411yescharacteristic
211C24C2×C7×D411yescharacteristic
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4C7×SD1612yes—
219C2×D4C7×Q1612yes—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C7×C22⋊C411yescharacteristic
230C22×C4C7×C22⋊C411yescharacteristic
231C22×C4C7×C22⋊C411yescharacteristic
232C22×C4C22×C4×C711yescharacteristic
233C22×C4C2×C7×D411yescharacteristic
234C2×C8—42no—
235C2×C8—82no—
236C4⋊C4—24no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—82no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C42—21no—
250C42—21no—
251C42—42no—
252C2×C7—21no—
253C2×C7—21no—
254C2×C7—41no—
255C2×C7—42no—
256C2×C7—41no—
257C2×C7#152011yescharacteristic
258C2×C7#212311yescharacteristic
259C2×C7C2×#35311yescharacteristic
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23C7×#10312yes—
269C23C2×C7×C22⋊C411yescharacteristic
270C23C2×C7×SD1611yescharacteristic
271C23C2×C7×Q1611yescharacteristic
272Q8—82no—
273Q8—82no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4C7×#9911yescharacteristic
299C2×C4C7×#9911yescharacteristic
300C2×C4C7×#10312yes—
301C8—82no—
302C7#1190111yescharacteristic, direct product factor, Sylow
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×C7×#9911yescharacteristic
318C22C2×C7×#10311yescharacteristic
319C22C7×#42111yescharacteristic
320C4—22no—
321C4—21no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—162no—
328C4C7×#35312yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2C7×#152011yescharacteristic
335C2C7×#212311yescharacteristic
336C2C2×C7×#35311yescharacteristic
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. C2×C7×C4⋊Q8C4semidirect
11. C7×#399C4semidirect
16. C4×C7×C4⋊C4C4semidirect
20. #11901C7direct (N × H)
54. #1352C2×C7semidirect
97. C2×C4⋊Q8C4×C7semidirect
103. #399C4×C7semidirect
108. C4×C4⋊C4C4×C7semidirect
302. C7#11901direct (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
C7×#14592
C7×#13522
#119017