Finite Group Databasebeta

← C7×#8739 #13185955

Subgroups — C7×#8739 #13185955

Order 1792.

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

normal subgroups only

This group has 656 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×#2297C211yesmaximal, characteristic
2C7×#1785C211yesmaximal, characteristic
3C7×#1782C211yesmaximal, characteristic
4C7×#1284C214yesmaximal, complemented
5C22×C7×#41C2211yescharacteristic
6C2×C7×#117C411yescharacteristic, complemented
7C22×C42×C7C411yescharacteristic, complemented
8C2×C7×#101—21no—
9C2×C7×#100—21no—
10C2×C7×#99—21no—
11C2×C7×#99—21no—
12C7×#407C2212yes—
13C7×#391—21no—
14C7×#390—21no—
15C7×#386—21no—
16C7×#386—21no—
17C7×#381C2212yes—
18C4×C7×C22⋊C4C412yescomplemented
19C4×C7×C22⋊C4C2212yes—
20#8739C711yesmaximal, characteristic, direct product factor, Sylow
21C2×C7×#41—24no—
22C2×C7×#41—42no—
23C2×C7×#41—42no—
24C22×C7×Q8D411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C22×C7×D4D411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7C2×C411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C2×C7×M16—22no—
33C7×#117—42no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C42×C7—21no—
40C2×C42×C7—42no—
41C2×C42×C7C2×C412yes—
42C2×C42×C7C2×C412yes—
43C7×#101—42no—
44C7×#100—42no—
45C7×#99—42no—
46C7×#99—42no—
47C7×C22⋊C8—24no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—24no—
51#2297C2×C711yescharacteristic
52#1785C2×C711yescharacteristic
53#1782C2×C711yescharacteristic
54#1284C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C22⋊C411yescharacteristic
57C24×C7C2×D411yescharacteristic
58C7×#41—416no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8—41no—
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—41no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7—44no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7#4114yes—
89C7×M16—42no—
90C7×M16—42no—
91C2×C7×C8—42no—
92C7×C4⋊C4—42no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C42×C7—42no—
99C42×C7—42no—
100C42×C7—42no—
101C22×#41C22×C711yescharacteristic
102C2×#117C4×C711yescharacteristic, complemented
103C22×C42C4×C711yescharacteristic, complemented
104C2×#101—21no—
105C2×#100—21no—
106C2×#99—21no—
107C2×#99—21no—
108#407C22×C712yes—
109#391—21no—
110#390—21no—
111#386—21no—
112#386—21no—
113#381C22×C712yes—
114C4×C22⋊C4C4×C712yescomplemented
115C4×C22⋊C4C22×C712yes—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—42no—
126C23×C7#9912yes—
127C23×C7C2×C22⋊C411yescharacteristic
128C23×C7#12312yes—
129C7×Q8—44no—
130C7×D4—44no—
131C7×D4—44no—
132C7×D4—44no—
133C2×C4×C7—28no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—24no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—44no—
145C2×C4×C7—44no—
146C2×C4×C7—42no—
147C2×C4×C7—48no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—82no—
153C2×C4×C7—84no—
154C2×C4×C7#9912yes—
155C2×C4×C7#11712yes—
156C2×C4×C7#12312yes—
157C7×C8—82no—
158C2×#41—24no—
159C2×#41—42no—
160C2×#41—42no—
161C22×Q8C7×D411yescharacteristic
162C22×D4C7×D411yescharacteristic
163C22×D4C7×D411yescharacteristic
164C22×D4C7×D411yescharacteristic
165C23×C4—21no—
166C23×C4C2×C4×C711yescharacteristic
167C23×C4C2×C4×C711yescharacteristic
168C23×C4C23×C711yescharacteristic
169C2×M16—22no—
170#117—42no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42C2×C4×C712yes—
179C2×C42C2×C4×C712yes—
180#101—42no—
181#100—42no—
182#99—42no—
183#99—42no—
184C22⋊C8—24no—
185#95—22no—
186#95—22no—
187#95—24no—
188C22×C7—22no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7—24no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—42no—
195C22×C7—41no—
196C22×C7—42no—
197C22×C7—81no—
198C22×C7—81no—
199C22×C7—81no—
200C22×C7—81no—
201C22×C7#38111yescharacteristic
202C22×C7C2×#9911yescharacteristic
203C22×C7C2×#9911yescharacteristic
204C4×C7—22no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—44no—
208C4×C7—41no—
209C4×C7—82no—
210C4×C7—162no—
211C24C7×C22⋊C411yescharacteristic
212C24C7×C22⋊C411yescharacteristic
213C24C2×C7×D411yescharacteristic
214#41—416no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C7×C22⋊C411yescharacteristic
241C22×C4C7×C22⋊C411yescharacteristic
242C22×C4C22×C4×C711yescharacteristic
243C22×C4C2×C7×D411yescharacteristic
244C22×C4C7×#4114yes—
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—42no—
255C42—42no—
256C42—42no—
257C2×C7—21no—
258C2×C7—21no—
259C2×C7—42no—
260C2×C7—42no—
261C2×C7—81no—
262C2×C7#181911yescharacteristic
263C2×C7#182712yes—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23C7×#9912yes—
275C23C2×C7×C22⋊C411yescharacteristic
276C23C7×#12312yes—
277Q8—44no—
278D4—44no—
279D4—44no—
280D4—44no—
281C2×C4—28no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4C7×#9912yes—
303C2×C4C7×#11712yes—
304C2×C4C7×#12312yes—
305C8—82no—
306C7#873911yescharacteristic, direct product factor, Sylow
307C22—22no—
308C22—22no—
309C22—22no—
310C22—24no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22C7×#38111yescharacteristic
321C22C2×C7×#9911yescharacteristic
322C22C2×C7×#9911yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—41no—
328C4—82no—
329C4—162no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2—81no—
335C2C7×#181911yescharacteristic
336C2C7×#182712yes—
subgroupG/H
center, socleC22×C7#381
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1284C2semidirect
6. C2×C7×#117C4semidirect
7. C22×C42×C7C4semidirect
18. C4×C7×C22⋊C4C4semidirect
20. #8739C7direct (N × H)
54. #1284C2×C7semidirect
102. C2×#117C4×C7semidirect
103. C22×C42C4×C7semidirect
114. C4×C22⋊C4C4×C7semidirect
306. C7#8739direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#22972
C7×#17852
C7×#17822
C7×#12842
#87397