Finite Group Databasebeta

← C7×#10338 #13187554

Subgroups — C7×#10338 #13187554

Order 1792.

998 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2483C211yesmaximal, characteristic, complemented
2C7×#2405C212yesmaximal
3C7×#1819C212yesmaximal, complemented
4C7×#1781C211yesmaximal, characteristic
5C7×#1304C211yesmaximal, characteristic, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#123—22no—
8C22×C7×C22⋊C4C2212yes—
9C2×C7×#99—22no—
10C7×#397—22no—
11C7×#394—24no—
12C7×#386—22no—
13C7×#385—22no—
14C7×#385—22no—
15C7×#381—22no—
16C7×#379—22no—
17C7×#379C2211yescharacteristic
18C7×#379C2212yes—
19C7×#379C2211yescharacteristic
20C7×#342—24no—
21#10338C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C25×C7C2311yescharacteristic
24C22×C7×D4—22no—
25C22×C7×D4—21no—
26C22×C7×D4—22no—
27C22×C7×D4—21no—
28C22×C7×D4—44no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D412yes—
31C23×C4×C7D412yes—
32C7×#123—48no—
33C2×C7×C4⋊C4—24no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C22⋊C4—24no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—24no—
48C2×C7×C22⋊C4D414yes—
49C2×C7×C22⋊C4D412yes—
50C7×#99—44no—
51C7×#95—22no—
52C7×#95—24no—
53C7×#95—24no—
54#2483C2×C711yescharacteristic, complemented
55#2405C2×C712yes—
56#1819C2×C712yescomplemented
57#1781C2×C711yescharacteristic
58#1304C2×C711yescharacteristic, complemented
59C24×C7—22no—
60C24×C7—21no—
61C24×C7—42no—
62C24×C7—42no—
63C24×C7—44no—
64C24×C7—44no—
65C24×C7C2×D411yescharacteristic
66C24×C7C2×D412yes—
67C24×C7C2×D411yescharacteristic
68C24×C7C2×D412yes—
69C24×C7#4111yescharacteristic
70C2×C7×D4—24no—
71C2×C7×D4—28no—
72C2×C7×D4—44no—
73C2×C7×D4—42no—
74C2×C7×D4—44no—
75C2×C7×D4—42no—
76C2×C7×D4—44no—
77C2×C7×D4—42no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C2×C7×D4—41no—
81C22×C4×C7—22no—
82C22×C4×C7—21no—
83C22×C4×C7—22no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—24no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—42no—
92C22×C4×C7—44no—
93C22×C4×C7—44no—
94C22×C4×C7—41no—
95C7×C4⋊C4—48no—
96C7×C4⋊C4—82no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—44no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—48no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—44no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—44no—
109C7×C22⋊C4—84no—
110C23×D4C22×C711yescharacteristic
111C2×#123—22no—
112C22×C22⋊C4C22×C712yes—
113C2×#99—22no—
114#397—22no—
115#394—24no—
116#386—22no—
117#385—22no—
118#385—22no—
119#381—22no—
120#379—22no—
121#379C22×C711yescharacteristic
122#379C22×C712yes—
123#379C22×C711yescharacteristic
124#342—24no—
125C23×C7—24no—
126C23×C7—24no—
127C23×C7—24no—
128C23×C7—22no—
129C23×C7—24no—
130C23×C7—24no—
131C23×C7—22no—
132C23×C7—41no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—41no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7—42no—
139C23×C7—42no—
140C23×C7—41no—
141C23×C7—44no—
142C23×C7—41no—
143C23×C7—44no—
144C23×C7—82no—
145C23×C7—88no—
146C23×C7—88no—
147C23×C7#12011yescharacteristic
148C23×C7#12012yes—
149C23×C7#12111yescharacteristic
150C23×C7#12112yes—
151C23×C7#12711yescharacteristic
152C7×D4—48no—
153C7×D4—44no—
154C7×D4—44no—
155C2×C4×C7—24no—
156C2×C4×C7—22no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—44no—
161C2×C4×C7—44no—
162C2×C4×C7—44no—
163C2×C4×C7—44no—
164C2×C4×C7—44no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—42no—
168C2×C4×C7—44no—
169C2×C4×C7—44no—
170C2×C4×C7—44no—
171C2×C4×C7—82no—
172C2×C4×C7—82no—
173C2×C4×C7—81no—
174C2×C4×C7—82no—
175C25C7×D411yescharacteristic
176C25C23×C711yescharacteristic
177C22×D4—22no—
178C22×D4—21no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—44no—
182C23×C4C7×D412yes—
183C23×C4C7×D412yes—
184C23×C4C7×D411yescharacteristic
185#123—48no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4C7×D414yes—
202C2×C22⋊C4C7×D412yes—
203#99—44no—
204#95—22no—
205#95—24no—
206#95—24no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—21no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—22no—
217C22×C7—48no—
218C22×C7—44no—
219C22×C7—48no—
220C22×C7—42no—
221C22×C7—84no—
222C22×C7—82no—
223C22×C7—84no—
224C22×C7—82no—
225C22×C7—82no—
226C22×C7—82no—
227C22×C7—82no—
228C22×C7#39211yescharacteristic
229C22×C7B(4,𝔽2)12yes—
230C22×C7#45814yes—
231C4×C7—42no—
232C4×C7—84no—
233C4×C7—82no—
234C4×C7—82no—
235C4×C7—82no—
236C4×C7—161no—
237C24—22no—
238C24—21no—
239C24—42no—
240C24—42no—
241C24—44no—
242C24—44no—
243C24C2×C7×D411yescharacteristic
244C24C2×C7×D412yes—
245C24C2×C7×D411yescharacteristic
246C24C2×C7×D412yes—
247C24C7×#4111yescharacteristic
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—44no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—41no—
273C4⋊C4—48no—
274C4⋊C4—82no—
275C22⋊C4—44no—
276C22⋊C4—48no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—84no—
288C2×C7—24no—
289C2×C7—22no—
290C2×C7—41no—
291C2×C7—42no—
292C2×C7—41no—
293C2×C7—82no—
294C2×C7#202111yescharacteristic
295C2×C7#202212yes—
296C23—24no—
297C23—24no—
298C23—24no—
299C23—22no—
300C23—24no—
301C23—24no—
302C23—22no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—44no—
313C23—41no—
314C23—44no—
315C23—82no—
316C23—88no—
317C23—88no—
318C23C7×#12011yescharacteristic
319C23C7×#12012yes—
320C23C7×#12111yescharacteristic
321C23C7×#12112yes—
322C23C7×#12711yescharacteristic
323D4—48no—
324D4—44no—
325D4—44no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—82no—
346C7#1033811yescharacteristic, direct product factor, Sylow
347C22—22no—
348C22—22no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—44no—
358C22—48no—
359C22—48no—
360C22—42no—
361C22—84no—
362C22—82no—
363C22—84no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22C7×#39211yescharacteristic
369C22C7×B(4,𝔽2)12yes—
370C22C7×#45814yes—
371C4—42no—
372C4—84no—
373C4—82no—
374C4—82no—
375C4—82no—
376C4—161no—
377C2—24no—
378C2—22no—
379C2—41no—
380C2—41no—
381C2—42no—
382C2—82no—
383C2C7×#202111yescharacteristic
384C2C7×#202212yes—
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC25C23×C7
prime radicalC23×C7×D4C22
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2483C2semidirect
3. C7×#1819C2semidirect
5. C7×#1304C2semidirect
21. #10338C7direct (N × H)
54. #2483C2×C7semidirect
56. #1819C2×C7semidirect
58. #1304C2×C7semidirect
346. C7#10338direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24832
C7×#24052
C7×#18192
C7×#17812
C7×#13042
#103387