Finite Group Databasebeta

← C7×#10325 #13187541

Subgroups — C7×#10325 #13187541

Order 1792.

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

normal subgroups only

This group has 602 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×#3189C211yesmaximal, characteristic
2C7×#2855C212yesmaximal
3C7×#1817C211yesmaximal, characteristic
4C7×#1520C212yesmaximal
5C7×#1289C211yesmaximal, characteristic
6C2×C7×C22⋊Q8C2212yes—
7C22×C42×C7C2211yescharacteristic
8C7×#484—22no—
9C7×#467—22no—
10C7×#451—22no—
11C2×C7×C22⋊C8C2211yescharacteristic
12C2×C7×C22⋊C8C2211yescharacteristic
13C7×#391—22no—
14C7×#389—22no—
15C7×#387—24no—
16C7×#385—22no—
17C7×#381—22no—
18C7×#380C2212yes—
19C7×#336—22no—
20C7×#328—24no—
21#10325C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D412yes—
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C2×C7×Q16—22no—
27C22×C7×C8—21no—
28C22×C7×C8D412yes—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C7×C22⋊Q8—22no—
32C7×C22⋊Q8—22no—
33C2×C7×C4⋊C4—24no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4D412yes—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C42×C7—21no—
40C2×C42×C7—22no—
41C2×C42×C7—22no—
42C2×C42×C7—21no—
43C7×#107—22no—
44C7×#103—24no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7×#95—24no—
48C7×#95—24no—
49C7×#95—24no—
50C7×#95D414yes—
51#3189C2×C711yescharacteristic
52#2855C2×C712yes—
53#1817C2×C711yescharacteristic
54#1520C2×C712yes—
55#1289C2×C711yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—44no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—24no—
68C22×C4×C7—22no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7Q1612yescomplemented
72C22×C4×C7C2×D412yes—
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7#4111yescharacteristic
76C7×Q16—82no—
77C2×C7×C8—22no—
78C2×C7×C8—42no—
79C2×C7×C8—42no—
80C2×C7×C8—41no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—44no—
84C7×C4⋊C4—48no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—42no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C42×C7—41no—
92C42×C7—42no—
93C42×C7—41no—
94C2×C22⋊Q8C22×C712yes—
95C22×C42C22×C711yescharacteristic
96#484—22no—
97#467—22no—
98#451—22no—
99C2×C22⋊C8C22×C711yescharacteristic
100C2×C22⋊C8C22×C711yescharacteristic
101#391—22no—
102#389—22no—
103#387—24no—
104#385—22no—
105#381—22no—
106#380C22×C712yes—
107#336—22no—
108#328—24no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7#12012yes—
112C23×C7#12111yescharacteristic
113C23×C7C2×Q1611yescharacteristic
114C23×C7#13511yescharacteristic
115C23×C7#13712yes—
116C7×Q8—44no—
117C7×Q8—82no—
118C2×C4×C7—24no—
119C2×C4×C7—22no—
120C2×C4×C7—24no—
121C2×C4×C7—22no—
122C2×C4×C7—24no—
123C2×C4×C7—22no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—41no—
132C2×C4×C7—44no—
133C2×C4×C7—44no—
134C2×C4×C7—44no—
135C2×C4×C7—44no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—44no—
139C2×C4×C7#12011yescharacteristic
140C2×C4×C7#12112yes—
141C2×C4×C7#12711yescharacteristic
142C7×C8—42no—
143C7×C8—81no—
144C22×Q8C7×D412yes—
145C23×C4C7×D411yescharacteristic
146C23×C4C7×D411yescharacteristic
147C23×C4C23×C711yescharacteristic
148C2×Q16—22no—
149C22×C8—21no—
150C22×C8C7×D412yes—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C7×D412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165#107—22no—
166#103—24no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95C7×D414yes—
173C22×C7—22no—
174C22×C7—21no—
175C22×C7—22no—
176C22×C7—21no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—42no—
180C22×C7#39211yescharacteristic
181C22×C7#45112yes—
182C22×C7#45212yes—
183C22×C7#46711yescharacteristic
184C22×C7#47011yescharacteristic
185C4×C7—21no—
186C4×C7—22no—
187C4×C7—42no—
188C4×C7—42no—
189C4×C7—84no—
190C4×C7—82no—
191C4×C7—82no—
192C4×C7#45812yes—
193C24C2×C7×D411yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—44no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4C7×Q1612yescomplemented
209C22×C4C2×C7×D412yes—
210C22×C4C2×C7×D411yescharacteristic
211C22×C4C2×C7×D412yes—
212C22×C4C7×#4111yescharacteristic
213Q16—82no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—41no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—44no—
221C4⋊C4—48no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—41no—
229C42—42no—
230C42—41no—
231C2×C7—22no—
232C2×C7—22no—
233C2×C7—21no—
234C2×C7—21no—
235C2×C7#200111yescharacteristic
236C2×C7#201011yescharacteristic
237C2×C7#202211yescharacteristic
238C23—41no—
239C23—41no—
240C23C7×#12012yes—
241C23C7×#12111yescharacteristic
242C23C2×C7×Q1611yescharacteristic
243C23C7×#13511yescharacteristic
244C23C7×#13712yes—
245Q8—44no—
246Q8—82no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4C7×#12011yescharacteristic
269C2×C4C7×#12112yes—
270C2×C4C7×#12711yescharacteristic
271C8—42no—
272C8—81no—
273C7#1032511yescharacteristic, direct product factor, Sylow
274C22—22no—
275C22—21no—
276C22—22no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22C7×#39211yescharacteristic
282C22C7×#45112yes—
283C22C7×#45212yes—
284C22C7×#46711yescharacteristic
285C22C7×#47011yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—84no—
291C4—82no—
292C4—82no—
293C4C7×#45812yes—
294C2—22no—
295C2—21no—
296C2—21no—
297C2—22no—
298C2C7×#200111yescharacteristic
299C2C7×#201011yescharacteristic
300C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
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
21. #10325C7direct (N × H)
71. C22×C4×C7Q16semidirect
208. C22×C4C7×Q16semidirect
273. C7#10325direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31892
C7×#28552
C7×#18172
C7×#15202
C7×#12892
#103257