Finite Group Databasebeta

← C7×#19058 #13196274

Subgroups — C7×#19058 #13196274

Order 1792.

840 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 222 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×#3550C211yesmaximal, characteristic
2C7×#3237C212yesmaximal, complemented
3C7×#3135C211yesmaximal, characteristic
4C7×#1664C218yesmaximal, complemented
5C7×#1536C211yesmaximal, characteristic
6C7×#1531C212yesmaximal, complemented
7C7×#563C2214yes—
8C7×#559—21no—
9C7×Q82—21no—
10C7×#556—22no—
11C7×D4×Q8—22no—
12C7×#542—24no—
13C7×#536—22no—
14C7×#533C2211yescharacteristic
15C7×#533C2212yes—
16C7×#532C2211yescharacteristic
17C7×#501C2214yes—
18C7×#491—21no—
19C7×#491C2214yes—
20C7×#490—21no—
21C7×#489C2214yes—
22C7×#485—24no—
23C7×#477—22no—
24C7×#475—22no—
25C7×#432C2212yes—
26C7×#416—22no—
27C2×C7×#97C2211yescharacteristic
28C7×#330C2218yes—
29C7×#329C2214yes—
30#19058C711yesmaximal, characteristic, direct product factor, Sylow
31C2×C7×#41D412yes—
32C22×C7×Q8—22no—
33C22×C7×C8—21no—
34C7×C4⋊Q8—24no—
35C7×C4⋊Q8—24no—
36C7×C4⋊Q8D411yescharacteristic
37C7×C4⋊Q8D412yes—
38C7×C4⋊Q8D412yes—
39C7×C4⋊Q8C2312yes—
40C7×#127D411yescharacteristic
41C7×#126—28no—
42C7×#125—21no—
43C7×#125C2314yes—
44C7×#124—24no—
45C7×#124—24no—
46C7×#124C2312yes—
47C7×#123—24no—
48C7×#123—24no—
49C7×C22⋊Q8—24no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—28no—
53C7×C22⋊Q8—22no—
54C7×#121—22no—
55C4×C7×Q8—24no—
56C4×C7×Q8—22no—
57C4×C7×Q8—22no—
58C4×C7×Q8—22no—
59C4×C7×D4—24no—
60C4×C7×D4—22no—
61C7×#117—22no—
62C7×#117—21no—
63C7×#117D414yes—
64C2×C42×C7C2311yescharacteristic
65C7×#107—24no—
66C7×#106—24no—
67C7×C4⋊C8—22no—
68C7×#103—24no—
69C7×#103—24no—
70C7×#103—24no—
71C7×#102—24no—
72C7×C22⋊C8—22no—
73C7×#97—21no—
74C7×#97C2312yes—
75C7×#97C2314yes—
76#3550C2×C711yescharacteristic
77#3237C2×C712yescomplemented
78#3135C2×C711yescharacteristic
79#1664C2×C718yescomplemented
80#1536C2×C711yescharacteristic
81#1531C2×C712yescomplemented
82C7×#41—44no—
83C2×C7×Q8—24no—
84C2×C7×Q8—22no—
85C2×C7×Q8—22no—
86C2×C7×Q8—22no—
87C2×C7×Q8—84no—
88C2×C7×Q8C2×D412yes—
89C2×C7×D4—22no—
90C2×C7×D4—22no—
91C2×C7×D4C2×D412yes—
92C22×C4×C7—24no—
93C22×C4×C7—22no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D412yes—
96C2×C7×C8—22no—
97C2×C7×C8—24no—
98C2×C7×C8—42no—
99C7×C4⋊C4—28no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—24no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4C2×D414yes—
109C7×C4⋊C4C2×D414yes—
110C7×C22⋊C4—24no—
111C7×C22⋊C4—28no—
112C7×C22⋊C4—24no—
113C7×C22⋊C4—24no—
114C7×C22⋊C4—41no—
115C42×C7—24no—
116C42×C7—24no—
117C42×C7—41no—
118C42×C7C2×D411yescharacteristic
119C42×C7C2×D412yes—
120C42×C7C2411yescharacteristic
121#563C22×C714yes—
122#559—21no—
123Q82—21no—
124#556—22no—
125D4×Q8—22no—
126#542—24no—
127#536—22no—
128#533C22×C711yescharacteristic
129#533C22×C712yes—
130#532C22×C711yescharacteristic
131#501C22×C714yes—
132#491—21no—
133#491C22×C714yes—
134#490—21no—
135#489C22×C714yes—
136#485—24no—
137#477—22no—
138#475—22no—
139#432C22×C712yes—
140#416—22no—
141C2×#97C22×C711yescharacteristic
142#330C22×C718yes—
143#329C22×C714yes—
144C23×C7—22no—
145C23×C7#12011yescharacteristic
146C7×Q8—42no—
147C7×Q8—44no—
148C7×Q8—44no—
149C7×Q8—44no—
150C7×Q8—84no—
151C7×D4—42no—
152C7×D4—44no—
153C7×D4—82no—
154C2×C4×C7—22no—
155C2×C4×C7—24no—
156C2×C4×C7—22no—
157C2×C4×C7—24no—
158C2×C4×C7—42no—
159C2×C4×C7—44no—
160C2×C4×C7—42no—
161C2×C4×C7—84no—
162C2×C4×C7#12012yes—
163C2×C4×C7#12011yescharacteristic
164C2×C4×C7C22×D412yes—
165C2×C4×C7C22×D411yescharacteristic
166C7×C8—44no—
167C7×C8—42no—
168C2×#41C7×D412yes—
169C22×Q8—22no—
170C22×C8—21no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8C7×D411yescharacteristic
174C4⋊Q8C7×D412yes—
175C4⋊Q8C7×D412yes—
176C4⋊Q8C23×C712yes—
177#127C7×D411yescharacteristic
178#126—28no—
179#125—21no—
180#125C23×C714yes—
181#124—24no—
182#124—24no—
183#124C23×C712yes—
184#123—24no—
185#123—24no—
186C22⋊Q8—24no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—28no—
190C22⋊Q8—22no—
191#121—22no—
192C4×Q8—24no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×D4—24no—
197C4×D4—22no—
198#117—22no—
199#117—21no—
200#117C7×D414yes—
201C2×C42C23×C711yescharacteristic
202#107—24no—
203#106—24no—
204C4⋊C8—22no—
205#103—24no—
206#103—24no—
207#103—24no—
208#102—24no—
209C22⋊C8—22no—
210#97—21no—
211#97C23×C714yes—
212#97C23×C712yes—
213C22×C7—21no—
214C22×C7—22no—
215C22×C7—42no—
216C22×C7—44no—
217C22×C7C2×#12011yescharacteristic
218C4×C7—21no—
219C4×C7—22no—
220C4×C7—22no—
221C4×C7—42no—
222C4×C7—44no—
223C4×C7—44no—
224C4×C7—82no—
225C4×C7#45512yes—
226#41—44no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—84no—
232C2×Q8C2×C7×D412yes—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4C2×C7×D412yes—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4C2×C7×D411yescharacteristic
239C22×C4C2×C7×D412yes—
240C2×C8—22no—
241C2×C8—24no—
242C2×C8—42no—
243C4⋊C4—28no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4C2×C7×D414yes—
253C4⋊C4C2×C7×D414yes—
254C22⋊C4—24no—
255C22⋊C4—28no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C42—24no—
260C42—24no—
261C42—41no—
262C42C2×C7×D411yescharacteristic
263C42C2×C7×D412yes—
264C42C24×C711yescharacteristic
265C2×C7—41no—
266C2×C7—82no—
267C2×C7C2×#45511yescharacteristic
268C2×C7#302012yes—
269C23—22no—
270C23C7×#12011yescharacteristic
271Q8—42no—
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—84no—
276D4—42no—
277D4—44no—
278D4—82no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—84no—
287C2×C4C7×#12012yes—
288C2×C4C7×#12011yescharacteristic
289C2×C4C22×C7×D412yes—
290C2×C4C22×C7×D411yescharacteristic
291C8—44no—
292C8—42no—
293C7#1905811yescharacteristic, direct product factor, Sylow
294C22—21no—
295C22—22no—
296C22—42no—
297C22—44no—
298C22C2×C7×#12011yescharacteristic
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—44no—
304C4—44no—
305C4—82no—
306C4C7×#45512yes—
307C2—41no—
308C2—82no—
309C2C2×C7×#45511yescharacteristic
310C2C7×#302012yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3237C2semidirect
4. C7×#1664C2semidirect
6. C7×#1531C2semidirect
30. #19058C7direct (N × H)
77. #3237C2×C7semidirect
79. #1664C2×C7semidirect
81. #1531C2×C7semidirect
293. C7#19058direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35502
C7×#32372
C7×#31352
C7×#16642
C7×#15362
C7×#15312
#190587