Finite Group Databasebeta

← C7×#10321 #13187537

Subgroups — C7×#10321 #13187537

Order 1792.

726 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3186C211yesmaximal, characteristic
2C7×#2854C212yesmaximal, complemented
3C7×#1817C211yesmaximal, characteristic, complemented
4C7×#1519C212yesmaximal, complemented
5C7×#1289C211yesmaximal, characteristic, complemented
6C2×C7×#123—22no—
7C2×C7×#121C2212yes—
8C22×C42×C7C2211yescharacteristic
9C7×#480—22no—
10C7×#466—22no—
11C7×#447—22no—
12C2×C7×C22⋊C8C2211yescharacteristic
13C2×C7×C22⋊C8C2211yescharacteristic
14C7×#390—22no—
15C7×#389—22no—
16C7×#388—24no—
17C7×#386—22no—
18C7×#380C2212yes—
19C7×#336—22no—
20C7×#327—24no—
21#10321C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×D4—22no—
23C22×C7×D4D412yes—
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C2×C7×D8—22no—
28C22×C7×C8—21no—
29C22×C7×C8D412yes—
30C7×#123—44no—
31C7×#123—44no—
32C7×#121—22no—
33C7×#121—22no—
34C7×#121—22no—
35C7×#121—22no—
36C2×C7×C4⋊C4—24no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4D412yes—
39C2×C7×C22⋊C4—24no—
40C2×C7×C22⋊C4—24no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C42×C7—21no—
44C2×C42×C7—22no—
45C2×C42×C7—22no—
46C2×C42×C7—21no—
47C7×#107—22no—
48C7×#102—24no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×#95—24no—
52C7×#95D414yes—
53#3186C2×C711yescharacteristic
54#2854C2×C712yescomplemented
55#1817C2×C711yescharacteristic, complemented
56#1519C2×C712yescomplemented
57#1289C2×C711yescharacteristic, complemented
58C24×C7—22no—
59C24×C7C2×D411yescharacteristic
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—44no—
63C2×C7×D4—42no—
64C2×C7×D4—42no—
65C2×C7×D4—42no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—24no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7D812yescomplemented
80C22×C4×C7C2×D412yes—
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D412yes—
83C22×C4×C7#4111yescharacteristic
84C7×D8—82no—
85C2×C7×C8—22no—
86C2×C7×C8—42no—
87C2×C7×C8—42no—
88C2×C7×C8—41no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—44no—
92C7×C4⋊C4—48no—
93C7×C22⋊C4—48no—
94C7×C22⋊C4—48no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—42no—
99C42×C7—41no—
100C42×C7—42no—
101C42×C7—41no—
102C2×#123—22no—
103C2×#121C22×C712yes—
104C22×C42C22×C711yescharacteristic
105#480—22no—
106#466—22no—
107#447—22no—
108C2×C22⋊C8C22×C711yescharacteristic
109C2×C22⋊C8C22×C711yescharacteristic
110#390—22no—
111#389—22no—
112#388—24no—
113#386—22no—
114#380C22×C712yes—
115#336—22no—
116#327—24no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—44no—
120C23×C7—44no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7#12012yes—
125C23×C7#12111yescharacteristic
126C23×C7C2×D811yescharacteristic
127C23×C7#13511yescharacteristic
128C23×C7AGL(1,ℤ/8)12yes—
129C7×D4—44no—
130C7×D4—82no—
131C7×D4—82no—
132C7×D4—82no—
133C7×D4—82no—
134C7×D4—82no—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—24no—
138C2×C4×C7—22no—
139C2×C4×C7—24no—
140C2×C4×C7—22no—
141C2×C4×C7—42no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—44no—
150C2×C4×C7—44no—
151C2×C4×C7—42no—
152C2×C4×C7—44no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7#12011yescharacteristic
156C2×C4×C7#12112yes—
157C2×C4×C7#12711yescharacteristic
158C7×C8—42no—
159C7×C8—81no—
160C22×D4—22no—
161C22×D4C7×D412yes—
162C23×C4C7×D411yescharacteristic
163C23×C4C7×D411yescharacteristic
164C23×C4C23×C711yescharacteristic
165C2×D8—22no—
166C22×C8—21no—
167C22×C8C7×D412yes—
168#123—44no—
169#123—44no—
170#121—22no—
171#121—22no—
172#121—22no—
173#121—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C7×D412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185#107—22no—
186#102—24no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189#95—24no—
190#95C7×D414yes—
191C22×C7—22no—
192C22×C7—21no—
193C22×C7—22no—
194C22×C7—21no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—42no—
198C22×C7—84no—
199C22×C7—84no—
200C22×C7—82no—
201C22×C7—82no—
202C22×C7—82no—
203C22×C7#39211yescharacteristic
204C22×C7#44712yes—
205C22×C7#44912yes—
206C22×C7#46611yescharacteristic
207C22×C7#46911yescharacteristic
208C4×C7—21no—
209C4×C7—22no—
210C4×C7—42no—
211C4×C7—42no—
212C4×C7—84no—
213C4×C7—82no—
214C4×C7#45812yes—
215C24—22no—
216C24C2×C7×D411yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C7×D812yescomplemented
237C22×C4C2×C7×D412yes—
238C22×C4C2×C7×D411yescharacteristic
239C22×C4C2×C7×D412yes—
240C22×C4C7×#4111yescharacteristic
241D8—82no—
242C2×C8—22no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—41no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—48no—
249C4⋊C4—44no—
250C22⋊C4—48no—
251C22⋊C4—48no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C42—41no—
257C42—42no—
258C42—41no—
259C2×C7—22no—
260C2×C7—22no—
261C2×C7—21no—
262C2×C7—21no—
263C2×C7—162no—
264C2×C7#199911yescharacteristic
265C2×C7#201011yescharacteristic
266C2×C7#202211yescharacteristic
267C23—41no—
268C23—41no—
269C23—44no—
270C23—44no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23C7×#12012yes—
275C23C7×#12111yescharacteristic
276C23C2×C7×D811yescharacteristic
277C23C7×#13511yescharacteristic
278C23C7×AGL(1,ℤ/8)12yes—
279D4—44no—
280D4—82no—
281D4—82no—
282D4—82no—
283D4—82no—
284D4—82no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4C7×#12011yescharacteristic
306C2×C4C7×#12112yes—
307C2×C4C7×#12711yescharacteristic
308C8—42no—
309C8—81no—
310C7#1032111yescharacteristic, direct product factor, Sylow
311C22—22no—
312C22—21no—
313C22—22no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22C7×#39211yescharacteristic
324C22C7×#44712yes—
325C22C7×#44912yes—
326C22C7×#46611yescharacteristic
327C22C7×#46911yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—82no—
334C4C7×#45812yes—
335C2—22no—
336C2—22no—
337C2—21no—
338C2—21no—
339C2—162no—
340C2C7×#199911yescharacteristic
341C2C7×#201011yescharacteristic
342C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC7×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2854C2semidirect
3. C7×#1817C2semidirect
4. C7×#1519C2semidirect
5. C7×#1289C2semidirect
21. #10321C7direct (N × H)
54. #2854C2×C7semidirect
55. #1817C2×C7semidirect
56. #1519C2×C7semidirect
57. #1289C2×C7semidirect
79. C22×C4×C7D8semidirect
236. C22×C4C7×D8semidirect
310. C7#10321direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31862
C7×#28542
C7×#18172
C7×#15192
C7×#12892
#103217