Finite Group Databasebeta

← C7×#12217 #13189433

Subgroups — C7×#12217 #13189433

Order 1792.

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2840C211yesmaximal, characteristic
2C7×#2029C211yesmaximal, characteristic
3C7×#1602C212yesmaximal, complemented
4C7×#1351C212yesmaximal, complemented
5C7×#1316C211yesmaximal, characteristic
6C7×#543C2211yescharacteristic
7C7×#451—22no—
8C7×#393—22no—
9C7×#393—21no—
10C7×#393—22no—
11C7×#393C2212yes—
12C7×#379—21no—
13C7×#379—21no—
14C7×#379C2211yescharacteristic
15C7×#342—21no—
16C7×#342C2211yescharacteristic
17C7×#328—22no—
18C7×#323C2212yes—
19#12217C711yesmaximal, characteristic, direct product factor, Sylow
20C25×C7D411yescharacteristic
21C22×C7×Q8—21no—
22C22×C7×Q8D412yes—
23C2×C7×Q16—42no—
24C7×C22⋊Q8—22no—
25C7×C22⋊Q8—42no—
26C2×C7×C22⋊C4—21no—
27C2×C7×C22⋊C4—21no—
28C2×C7×C22⋊C4—21no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—21no—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—42no—
34C2×C7×C22⋊C4D411yescharacteristic
35C2×C7×C22⋊C4C2311yescharacteristic
36C7×#103—42no—
37C7×C22⋊C8—22no—
38C7×#95—21no—
39C7×#95—22no—
40C7×#95—22no—
41C7×#95D412yes—
42#2840C2×C711yescharacteristic
43#2029C2×C711yescharacteristic
44#1602C2×C712yescomplemented
45#1351C2×C712yescomplemented
46#1316C2×C711yescharacteristic
47C24×C7—21no—
48C24×C7—21no—
49C24×C7—41no—
50C24×C7—41no—
51C24×C7—82no—
52C24×C7Q1612yescomplemented
53C24×C7C2×D411yescharacteristic
54C2×C7×Q8—22no—
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C2×C7×Q8—82no—
58C2×C7×Q8—81no—
59C22×C4×C7—22no—
60C22×C4×C7—21no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7C2×D412yes—
65C7×Q16—84no—
66C2×C7×C8—42no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—42no—
69C7×C22⋊C4—41no—
70C7×C22⋊C4—41no—
71C7×C22⋊C4—41no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4—41no—
74C7×C22⋊C4—41no—
75C7×C22⋊C4—82no—
76C7×C22⋊C4—82no—
77C7×C22⋊C4—82no—
78C7×C22⋊C4—81no—
79C7×C22⋊C4—81no—
80C7×C22⋊C4—84no—
81#543C22×C711yescharacteristic
82#451—22no—
83#393—22no—
84#393—21no—
85#393—22no—
86#393C22×C712yes—
87#379—21no—
88#379—21no—
89#379C22×C711yescharacteristic
90#342—21no—
91#342C22×C711yescharacteristic
92#328—22no—
93#323C22×C712yes—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7—22no—
97C23×C7—42no—
98C23×C7—42no—
99C23×C7—42no—
100C23×C7—42no—
101C23×C7—44no—
102C23×C7—41no—
103C23×C7—41no—
104C23×C7—81no—
105C23×C7—82no—
106C23×C7—82no—
107C23×C7—82no—
108C23×C7—84no—
109C23×C7#12011yescharacteristic
110C23×C7C2×Q1611yescharacteristic
111C23×C7#13711yescharacteristic
112C7×Q8—44no—
113C7×Q8—82no—
114C7×Q8—82no—
115C7×Q8—84no—
116C2×C4×C7—22no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—41no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—82no—
128C2×C4×C7—81no—
129C2×C4×C7—82no—
130C7×C8—82no—
131C25C7×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8C7×D412yes—
134C2×Q16—42no—
135C22⋊Q8—22no—
136C22⋊Q8—42no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4C7×D411yescharacteristic
146C2×C22⋊C4C23×C711yescharacteristic
147#103—42no—
148C22⋊C8—22no—
149#95—21no—
150#95—22no—
151#95—22no—
152#95C7×D412yes—
153C22×C7—22no—
154C22×C7—21no—
155C22×C7—21no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7—42no—
161C22×C7—44no—
162C22×C7—42no—
163C22×C7—81no—
164C22×C7—82no—
165C22×C7—82no—
166C22×C7—84no—
167C22×C7—82no—
168C22×C7#45111yescharacteristic
169C22×C7#45511yescharacteristic
170C22×C7B(4,𝔽2)11yescharacteristic
171C4×C7—42no—
172C4×C7—81no—
173C4×C7—82no—
174C4×C7—82no—
175C4×C7—82no—
176C4×C7—161no—
177C24—21no—
178C24—21no—
179C24—41no—
180C24—41no—
181C24—82no—
182C24C7×Q1612yescomplemented
183C24C2×C7×D411yescharacteristic
184C2×Q8—22no—
185C2×Q8—42no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—81no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C2×C7×D412yes—
195Q16—84no—
196C2×C8—42no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—84no—
211C2×C7—21no—
212C2×C7—21no—
213C2×C7—41no—
214C2×C7—41no—
215C2×C7—82no—
216C2×C7#160211yescharacteristic
217C2×C7#219912yes—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—44no—
226C23—41no—
227C23—41no—
228C23—81no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23—84no—
233C23C7×#12011yescharacteristic
234C23C2×C7×Q1611yescharacteristic
235C23C7×#13711yescharacteristic
236Q8—44no—
237Q8—82no—
238Q8—82no—
239Q8—84no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—82no—
254C8—82no—
255C7#1221711yescharacteristic, direct product factor, Sylow
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22C7×#45111yescharacteristic
272C22C7×#45511yescharacteristic
273C22C7×B(4,𝔽2)11yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—82no—
278C4—82no—
279C4—161no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2C7×#160211yescharacteristic
286C2C7×#219912yes—
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC25×C7D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1602C2semidirect
4. C7×#1351C2semidirect
19. #12217C7direct (N × H)
44. #1602C2×C7semidirect
45. #1351C2×C7semidirect
52. C24×C7Q16semidirect
182. C24C7×Q16semidirect
255. C7#12217direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28402
C7×#20292
C7×#16022
C7×#13512
C7×#13162
#122177