Finite Group Databasebeta

← C7×#10337 #13187553

Subgroups — C7×#10337 #13187553

Order 1792.

1274 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1274 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×#2405C212yesmaximal, complemented
2C7×#2403C211yesmaximal, characteristic
3C7×#1819C211yesmaximal, characteristic, complemented
4C7×#1781C212yesmaximal, complemented
5C7×#1304C211yesmaximal, characteristic, complemented
6C23×C7×D4C2212yes—
7C2×C7×#123—22no—
8C2×C7×#120—22no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C2×C7×#99—24no—
11C7×#394—24no—
12C7×#392—22no—
13C7×#386—22no—
14C7×#386—22no—
15C7×#385—22no—
16C7×#379—22no—
17C7×#379C2211yescharacteristic
18C7×#379C2211yescharacteristic
19C7×#379C2212yes—
20C7×#342—22no—
21#10337C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D412yes—
23C25×C7C2311yescharacteristic
24C22×C7×D4—22no—
25C22×C7×D4—22no—
26C22×C7×D4—22no—
27C22×C7×D4—22no—
28C22×C7×D4—22no—
29C22×C7×D4—22no—
30C22×C7×D4—22no—
31C22×C7×D4—48no—
32C23×C4×C7D412yes—
33C23×C4×C7D412yescomplemented
34C7×#123—48no—
35C7×#120—48no—
36C2×C7×C4⋊C4—24no—
37C2×C7×C22⋊C4—24no—
38C2×C7×C22⋊C4—24no—
39C2×C7×C22⋊C4—21no—
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—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—22no—
48C2×C7×C22⋊C4—24no—
49C2×C7×C22⋊C4D412yes—
50C2×C7×C22⋊C4D414yescomplemented
51C7×#99—48no—
52C7×#95—24no—
53C7×#95—22no—
54#2405C2×C712yescomplemented
55#2403C2×C711yescharacteristic
56#1819C2×C711yescharacteristic, complemented
57#1781C2×C712yescomplemented
58#1304C2×C711yescharacteristic, complemented
59C24×C7—22no—
60C24×C7—22no—
61C24×C7—22no—
62C24×C7—44no—
63C24×C7—44no—
64C24×C7—42no—
65C24×C7—44no—
66C24×C7—44no—
67C24×C7C2×D412yes—
68C24×C7C2×D411yescharacteristic
69C24×C7C2×D411yescharacteristic
70C24×C7C2×D412yes—
71C24×C7#4111yescharacteristic
72C2×C7×D4—28no—
73C2×C7×D4—28no—
74C2×C7×D4—28no—
75C2×C7×D4—44no—
76C2×C7×D4—44no—
77C2×C7×D4—44no—
78C2×C7×D4—44no—
79C2×C7×D4—44no—
80C2×C7×D4—44no—
81C2×C7×D4—44no—
82C2×C7×D4—44no—
83C2×C7×D4—44no—
84C2×C7×D4—42no—
85C2×C7×D4—42no—
86C2×C7×D4—42no—
87C2×C7×D4—42no—
88C2×C7×D4—48no—
89C2×C7×D4—82no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—24no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—44no—
99C22×C4×C7—44no—
100C22×C4×C7—41no—
101C7×C4⋊C4—48no—
102C7×C22⋊C4—48no—
103C7×C22⋊C4—44no—
104C7×C22⋊C4—44no—
105C7×C22⋊C4—44no—
106C7×C22⋊C4—48no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—44no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—44no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—84no—
115C23×D4C22×C712yes—
116C2×#123—22no—
117C2×#120—22no—
118C22×C22⋊C4C22×C711yescharacteristic
119C2×#99—24no—
120#394—24no—
121#392—22no—
122#386—22no—
123#386—22no—
124#385—22no—
125#379—22no—
126#379C22×C711yescharacteristic
127#379C22×C711yescharacteristic
128#379C22×C712yes—
129#342—22no—
130C23×C7—24no—
131C23×C7—24no—
132C23×C7—24no—
133C23×C7—24no—
134C23×C7—22no—
135C23×C7—24no—
136C23×C7—24no—
137C23×C7—24no—
138C23×C7—42no—
139C23×C7—41no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—42no—
144C23×C7—41no—
145C23×C7—44no—
146C23×C7—42no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—44no—
150C23×C7—42no—
151C23×C7—44no—
152C23×C7—44no—
153C23×C7—82no—
154C23×C7—82no—
155C23×C7—88no—
156C23×C7—816no—
157C23×C7#12012yes—
158C23×C7#12011yescharacteristic
159C23×C7#12112yes—
160C23×C7#12111yescharacteristic
161C23×C7#12711yescharacteristic
162C7×D4—48no—
163C7×D4—48no—
164C7×D4—48no—
165C7×D4—48no—
166C7×D4—88no—
167C2×C4×C7—24no—
168C2×C4×C7—24no—
169C2×C4×C7—24no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—44no—
175C2×C4×C7—44no—
176C2×C4×C7—44no—
177C2×C4×C7—44no—
178C2×C4×C7—42no—
179C2×C4×C7—44no—
180C2×C4×C7—44no—
181C2×C4×C7—42no—
182C2×C4×C7—82no—
183C2×C4×C7—82no—
184C2×C4×C7—81no—
185C25C7×D412yes—
186C25C23×C711yescharacteristic
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—48no—
195C23×C4C7×D412yes—
196C23×C4C7×D412yescomplemented
197#123—48no—
198#120—48no—
199C2×C4⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C7×D412yes—
213C2×C22⋊C4C7×D414yescomplemented
214#99—48no—
215#95—24no—
216#95—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—21no—
225C22×C7—22no—
226C22×C7—21no—
227C22×C7—44no—
228C22×C7—48no—
229C22×C7—48no—
230C22×C7—48no—
231C22×C7—44no—
232C22×C7—82no—
233C22×C7—84no—
234C22×C7—84no—
235C22×C7—84no—
236C22×C7—82no—
237C22×C7—84no—
238C22×C7—84no—
239C22×C7—82no—
240C22×C7—82no—
241C22×C7#39211yescharacteristic
242C22×C7B(4,𝔽2)14yes—
243C22×C7#45812yes—
244C4×C7—44no—
245C4×C7—84no—
246C4×C7—82no—
247C4×C7—82no—
248C4×C7—161no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—44no—
253C24—44no—
254C24—42no—
255C24—44no—
256C24—44no—
257C24C2×C7×D412yes—
258C24C2×C7×D411yescharacteristic
259C24C2×C7×D411yescharacteristic
260C24C2×C7×D412yes—
261C24C7×#4111yescharacteristic
262C2×D4—28no—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—48no—
279C2×D4—82no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—41no—
291C4⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—48no—
296C22⋊C4—44no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—44no—
301C22⋊C4—42no—
302C22⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—84no—
305C2×C7—22no—
306C2×C7—24no—
307C2×C7—41no—
308C2×C7—41no—
309C2×C7—42no—
310C2×C7—84no—
311C2×C7#202112yes—
312C2×C7#202211yescharacteristic
313C23—24no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—42no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23—44no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—44no—
333C23—42no—
334C23—44no—
335C23—44no—
336C23—82no—
337C23—82no—
338C23—88no—
339C23—816no—
340C23C7×#12012yes—
341C23C7×#12011yescharacteristic
342C23C7×#12112yes—
343C23C7×#12111yescharacteristic
344C23C7×#12711yescharacteristic
345D4—48no—
346D4—48no—
347D4—48no—
348D4—48no—
349D4—88no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C7#1033711yescharacteristic, direct product factor, Sylow
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—48no—
380C22—44no—
381C22—48no—
382C22—48no—
383C22—44no—
384C22—82no—
385C22—84no—
386C22—84no—
387C22—84no—
388C22—82no—
389C22—84no—
390C22—84no—
391C22—82no—
392C22—82no—
393C22C7×#39211yescharacteristic
394C22C7×B(4,𝔽2)14yes—
395C22C7×#45812yes—
396C4—44no—
397C4—82no—
398C4—84no—
399C4—82no—
400C4—161no—
401C2—22no—
402C2—24no—
403C2—41no—
404C2—42no—
405C2—41no—
406C2—84no—
407C2C7×#202112yes—
408C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC25C23×C7
prime radicalC7×#2403C2
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×#2405C2semidirect
3. C7×#1819C2semidirect
4. C7×#1781C2semidirect
5. C7×#1304C2semidirect
21. #10337C7direct (N × H)
33. C23×C4×C7D4semidirect
50. C2×C7×C22⋊C4D4semidirect
54. #2405C2×C7semidirect
56. #1819C2×C7semidirect
57. #1781C2×C7semidirect
58. #1304C2×C7semidirect
196. C23×C4C7×D4semidirect
213. C2×C22⋊C4C7×D4semidirect
368. C7#10337direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24052
C7×#24032
C7×#18192
C7×#17812
C7×#13042
#103377