Finite Group Databasebeta

← C7×#12854 #13190070

Subgroups — C7×#12854 #13190070

Order 1792.

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

normal subgroups only

This group has 784 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×#2308C211yesmaximal, characteristic
2C7×#1783C211yesmaximal, characteristic
3C7×#1781C211yesmaximal, characteristic
4C7×#1390C214yesmaximal, complemented
5C23×C7×D4C2211yescharacteristic
6C22×C7×C22⋊C4C411yescharacteristic, complemented
7C22×C7×C22⋊C4C411yescharacteristic, complemented
8C2×C7×#100—21no—
9C2×C7×#100C2212yes—
10C2×C7×#99—21no—
11C2×C7×#99C2212yes—
12C7×#407—21no—
13C7×#386—21no—
14C7×#386—21no—
15C7×#386C412yescomplemented
16C7×#386C2212yes—
17C7×#381—21no—
18C7×#379—21no—
19C7×#379—21no—
20#12854C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7C2×C411yescharacteristic
22C25×C7D411yescharacteristic
23C22×C7×D4—24no—
24C22×C7×D4—42no—
25C22×C7×D4—42no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C22×C7×D4D411yescharacteristic
29C22×C7×D4C2311yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7C2×C412yes—
32C23×C4×C7D411yescharacteristic
33C2×C7×M16—22no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—42no—
41C2×C7×C22⋊C4—42no—
42C2×C7×C22⋊C4—42no—
43C2×C7×C22⋊C4C2×C412yes—
44C7×#100—24no—
45C7×#100—42no—
46C7×#99—24no—
47C7×#99—42no—
48C7×C22⋊C8—42no—
49C7×#95—22no—
50C7×#95—22no—
51#2308C2×C711yescharacteristic
52#1783C2×C711yescharacteristic
53#1781C2×C711yescharacteristic
54#1390C2×C714yescomplemented
55C24×C7—22no—
56C24×C7—22no—
57C24×C7—22no—
58C24×C7—22no—
59C24×C7—22no—
60C24×C7—41no—
61C24×C7—82no—
62C24×C7—82no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C22⋊C411yescharacteristic
65C24×C7C22⋊C411yescharacteristic
66C24×C7C22×C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—22no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—416no—
80C2×C7×D4#4114yes—
81C22×C4×C7—22no—
82C22×C4×C7—22no—
83C22×C4×C7—22no—
84C22×C4×C7—22no—
85C22×C4×C7—41no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7—42no—
89C22×C4×C7—82no—
90C22×C4×C7—82no—
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C7×M16—42no—
94C7×M16—42no—
95C2×C7×C8—42no—
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—82no—
103C7×C22⋊C4—82no—
104C7×C22⋊C4—84no—
105C23×D4C22×C711yescharacteristic
106C22×C22⋊C4C4×C711yescharacteristic, complemented
107C22×C22⋊C4C4×C711yescharacteristic, complemented
108C2×#100—21no—
109C2×#100C22×C712yes—
110C2×#99—21no—
111C2×#99C22×C712yes—
112#407—21no—
113#386—21no—
114#386—21no—
115#386C4×C712yescomplemented
116#386C22×C712yes—
117#381—21no—
118#379—21no—
119#379—21no—
120C23×C7—28no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—42no—
133C23×C7—44no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7—44no—
139C23×C7—82no—
140C23×C7—82no—
141C23×C7—82no—
142C23×C7—84no—
143C23×C7—84no—
144C23×C7—84no—
145C23×C7—84no—
146C23×C7#9912yes—
147C23×C7#9912yes—
148C23×C7C2×C22⋊C411yescharacteristic
149C23×C7#11712yes—
150C23×C7#12312yes—
151C7×D4—44no—
152C7×D4—44no—
153C7×D4—44no—
154C7×D4—44no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—44no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—84no—
168C2×C4×C7—84no—
169C2×C4×C7—82no—
170C2×C4×C7—84no—
171C2×C4×C7#12312yes—
172C7×C8—82no—
173C25C2×C4×C711yescharacteristic
174C25C7×D411yescharacteristic
175C22×D4—24no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4C2×C4×C711yescharacteristic
179C22×D4C7×D411yescharacteristic
180C22×D4C7×D411yescharacteristic
181C22×D4C23×C711yescharacteristic
182C23×C4—21no—
183C23×C4C2×C4×C712yes—
184C23×C4C7×D411yescharacteristic
185C2×M16—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4C2×C4×C712yes—
196#100—24no—
197#100—42no—
198#99—24no—
199#99—42no—
200C22⋊C8—42no—
201#95—22no—
202#95—22no—
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—42no—
211C22×C7—42no—
212C22×C7—42no—
213C22×C7—42no—
214C22×C7—42no—
215C22×C7—42no—
216C22×C7—42no—
217C22×C7—42no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—42no—
221C22×C7—41no—
222C22×C7—42no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—88no—
226C22×C7—88no—
227C22×C7#38111yescharacteristic
228C22×C7C2×#9911yescharacteristic
229C22×C7C2×#9911yescharacteristic
230C4×C7—22no—
231C4×C7—41no—
232C4×C7—82no—
233C4×C7—82no—
234C4×C7—162no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—41no—
241C24—82no—
242C24—82no—
243C24C7×C22⋊C411yescharacteristic
244C24C7×C22⋊C411yescharacteristic
245C24C7×C22⋊C411yescharacteristic
246C24C22×C4×C711yescharacteristic
247C24C2×C7×D411yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—416no—
260C2×D4C7×#4114yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C7×C22⋊C411yescharacteristic
272C22×C4C2×C7×D411yescharacteristic
273M16—42no—
274M16—42no—
275C2×C8—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—84no—
285C2×C7—21no—
286C2×C7—21no—
287C2×C7—42no—
288C2×C7—42no—
289C2×C7—42no—
290C2×C7—42no—
291C2×C7—81no—
292C2×C7#181911yescharacteristic
293C2×C7#212112yes—
294C23—28no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—44no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—44no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—84no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23C7×#9912yes—
321C23C7×#9912yes—
322C23C2×C7×C22⋊C411yescharacteristic
323C23C7×#11712yes—
324C23C7×#12312yes—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—44no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4C7×#12312yes—
346C8—82no—
347C7#1285411yescharacteristic, direct product factor, Sylow
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—88no—
371C22—88no—
372C22C7×#38111yescharacteristic
373C22C2×C7×#9911yescharacteristic
374C22C2×C7×#9911yescharacteristic
375C4—22no—
376C4—41no—
377C4—82no—
378C4—82no—
379C4—162no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—42no—
385C2—42no—
386C2—81no—
387C2C7×#181911yescharacteristic
388C2C7×#212112yes—
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC23×C7×D4C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C23×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1390C2semidirect
6. C22×C7×C22⋊C4C4semidirect
7. C22×C7×C22⋊C4C4semidirect
15. C7×#386C4semidirect
20. #12854C7direct (N × H)
54. #1390C2×C7semidirect
106. C22×C22⋊C4C4×C7semidirect
107. C22×C22⋊C4C4×C7semidirect
115. #386C4×C7semidirect
347. C7#12854direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23082
C7×#17832
C7×#17812
C7×#13902
#128547