Finite Group Databasebeta

← C7×#10559 #13187775

Subgroups — C7×#10559 #13187775

Order 1792.

448 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 94 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×#1901C212yesmaximal
3C7×#1851C211yesmaximal, characteristic
4C7×#1508C212yesmaximal
5C7×#1293C211yesmaximal, characteristic
6C22×C7×M16C2211yescharacteristic
7C2×C7×C22⋊Q8C2212yes—
8C7×#484—22no—
9C7×#467—21no—
10C7×#467—21no—
11C7×#451—22no—
12C2×C7×C4⋊C8—21no—
13C2×C7×#101—22no—
14C7×#407C2212yes—
15C2×C7×C22⋊C8C2211yescharacteristic
16C4×C7×C22⋊C4C2211yescharacteristic
17C7×#343—22no—
18C7×#336—21no—
19C7×#328—22no—
20C7×#328—22no—
21#10559C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D412yes—
23C23×C4×C7C2311yescharacteristic
24C2×C7×Q16—21no—
25C2×C7×Q16—21no—
26C2×C7×M16—22no—
27C2×C7×M16—22no—
28C2×C7×M16—42no—
29C22×C7×C8—21no—
30C22×C7×C8D411yescharacteristic
31C22×C7×C8D411yescharacteristic
32C7×C22⋊Q8—22no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—22no—
35C7×C22⋊Q8—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4D412yes—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C42×C7—21no—
41C7×#107—21no—
42C7×#107—21no—
43C7×C4⋊C8—42no—
44C7×#103—22no—
45C7×#103—22no—
46C7×#101—44no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×#95D411yescharacteristic
51C7×#95D411yescharacteristic
52#3189C2×C711yescharacteristic
53#1901C2×C712yes—
54#1851C2×C711yescharacteristic
55#1508C2×C712yes—
56#1293C2×C711yescharacteristic
57C24×C7#4111yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—42no—
61C2×C7×Q8—42no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7C2×D412yes—
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7#4112yes—
72C7×Q16—81no—
73C7×Q16—81no—
74C7×M16—24no—
75C7×M16—42no—
76C7×M16—42no—
77C7×M16—42no—
78C2×C7×C8—22no—
79C2×C7×C8—21no—
80C2×C7×C8—21no—
81C2×C7×C8—41no—
82C2×C7×C8—41no—
83C2×C7×C8—41no—
84C2×C7×C8—41no—
85C2×C7×C8—42no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C42×C7—41no—
94C42×C7—41no—
95C22×M16C22×C711yescharacteristic
96C2×C22⋊Q8C22×C712yes—
97#484—22no—
98#467—21no—
99#467—21no—
100#451—22no—
101C2×C4⋊C8—21no—
102C2×#101—22no—
103#407C22×C712yes—
104C2×C22⋊C8C22×C711yescharacteristic
105C4×C22⋊C4C22×C711yescharacteristic
106#343—22no—
107#336—21no—
108#328—22no—
109#328—22no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7#12111yescharacteristic
115C23×C7#12312yes—
116C7×Q8—42no—
117C7×Q8—42no—
118C7×Q8—82no—
119C2×C4×C7—21no—
120C2×C4×C7—22no—
121C2×C4×C7—21no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—44no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—41no—
134C2×C4×C7—82no—
135C2×C4×C7#12011yescharacteristic
136C2×C4×C7#12112yes—
137C2×C4×C7#12411yescharacteristic
138C7×C8—42no—
139C7×C8—41no—
140C7×C8—41no—
141C7×C8—82no—
142C22×Q8C7×D412yes—
143C23×C4C23×C711yescharacteristic
144C2×Q16—21no—
145C2×Q16—21no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—42no—
149C22×C8—21no—
150C22×C8C7×D411yescharacteristic
151C22×C8C7×D411yescharacteristic
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C7×D412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C42—21no—
161#107—21no—
162#107—21no—
163C4⋊C8—42no—
164#103—22no—
165#103—22no—
166#101—44no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170#95C7×D411yescharacteristic
171#95C7×D411yescharacteristic
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7—81no—
178C22×C7—81no—
179C22×C7#39411yescharacteristic
180C22×C7#47311yescharacteristic
181C22×C7#47311yescharacteristic
182C4×C7—22no—
183C4×C7—21no—
184C4×C7—42no—
185C4×C7—81no—
186C4×C7—82no—
187C4×C7—82no—
188C4×C7B(4,𝔽2)11yescharacteristic
189C4×C7#45811yescharacteristic
190C24C7×#4111yescharacteristic
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—42no—
194C2×Q8—42no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4C2×C7×D412yes—
202C22×C4C2×C7×D411yescharacteristic
203C22×C4C2×C7×D411yescharacteristic
204C22×C4C7×#4112yes—
205Q16—81no—
206Q16—81no—
207M16—24no—
208M16—42no—
209M16—42no—
210M16—42no—
211C2×C8—22no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C42—41no—
227C42—41no—
228C2×C7—21no—
229C2×C7—21no—
230C2×C7—42no—
231C2×C7#204411yescharacteristic
232C2×C7#204711yescharacteristic
233C2×C7#205011yescharacteristic
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23C7×#12111yescharacteristic
239C23C7×#12312yes—
240Q8—42no—
241Q8—42no—
242Q8—82no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4C7×#12011yescharacteristic
260C2×C4C7×#12112yes—
261C2×C4C7×#12411yescharacteristic
262C8—42no—
263C8—41no—
264C8—41no—
265C8—82no—
266C7#1055911yescharacteristic, direct product factor, Sylow
267C22—21no—
268C22—21no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—81no—
273C22—81no—
274C22C7×#39411yescharacteristic
275C22C7×#47311yescharacteristic
276C22C7×#47311yescharacteristic
277C4—22no—
278C4—21no—
279C4—42no—
280C4—81no—
281C4—82no—
282C4—82no—
283C4C7×B(4,𝔽2)11yescharacteristic
284C4C7×#45811yescharacteristic
285C2—21no—
286C2—21no—
287C2—42no—
288C2C7×#204411yescharacteristic
289C2C7×#204711yescharacteristic
290C2C7×#205011yescharacteristic
subgroupG/H
center, socleC22×C7#394
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7#41
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. #10559C7direct (N × H)
266. C7#10559direct (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×#19012
C7×#18512
C7×#15082
C7×#12932
#105597