Finite Group Databasebeta

← C7×#11899 #13189115

Subgroups — C7×#11899 #13189115

Order 1792.

452 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2487C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1460C211yesmaximal, characteristic
4C7×#1352C214yesmaximal, complemented
5C2×C4×C7×D4C2211yescharacteristic
6C22×C7×C4⋊C4C411yescharacteristic, complemented
7C7×#431—21no—
8C2×C7×#99—21no—
9C7×#400C412yescomplemented
10C7×#399C2212yes—
11C7×#389—21no—
12C7×#386—21no—
13C7×#385—21no—
14C7×#380—21no—
15C7×#380—21no—
16C7×#380C411yescharacteristic, complemented
17C7×#342C2212yes—
18C7×#323—21no—
19C7×#323C2212yes—
20#11899C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C4×C7×D4—41no—
26C4×C7×D4—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—41no—
31C2×C7×C4⋊C4—41no—
32C2×C7×C4⋊C4—41no—
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C7×C22⋊C4C2311yescharacteristic
40C2×C42×C7D411yescharacteristic
41C7×C4⋊C8—41no—
42C7×#99—42no—
43C7×C22⋊C8—22no—
44C4×C7×C8—41no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—41no—
48C7×#95C2×C412yes—
49#2487C2×C711yescharacteristic
50#1799C2×C711yescharacteristic
51#1460C2×C711yescharacteristic
52#1352C2×C714yescomplemented
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×D4—41no—
56C2×C7×D4—41no—
57C2×C7×D4SD1612yes—
58C2×C7×D4Q1612yes—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—82no—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C2×C7×C8—42no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—82no—
82C7×C4⋊C4—82no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—82no—
89C42×C7—21no—
90C42×C7—21no—
91C2×C4×D4C22×C711yescharacteristic
92C22×C4⋊C4C4×C711yescharacteristic, complemented
93#431—21no—
94C2×#99—21no—
95#400C4×C712yescomplemented
96#399C22×C712yes—
97#389—21no—
98#386—21no—
99#385—21no—
100#380—21no—
101#380—21no—
102#380C4×C711yescharacteristic, complemented
103#342C22×C712yes—
104#323—21no—
105#323C22×C712yes—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7#10312yes—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7C2×SD1611yescharacteristic
117C23×C7C2×Q1611yescharacteristic
118C7×D4—42no—
119C7×D4—42no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—22no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—81no—
139C2×C4×C7—84no—
140C2×C4×C7—82no—
141C2×C4×C7—82no—
142C2×C4×C7—82no—
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7#10312yes—
146C7×C8—82no—
147C22×D4C7×D411yescharacteristic
148C23×C4—21no—
149C23×C4C2×C4×C711yescharacteristic
150C23×C4C7×D411yescharacteristic
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4C2×C4×C712yes—
160C2×C4⋊C4C2×C4×C711yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4C7×D411yescharacteristic
165C2×C22⋊C4C23×C711yescharacteristic
166C2×C42C7×D411yescharacteristic
167C4⋊C8—41no—
168#99—42no—
169C22⋊C8—22no—
170C4×C8—41no—
171#95—22no—
172#95—22no—
173#95—41no—
174#95C2×C4×C712yes—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—22no—
179C22×C7—21no—
180C22×C7—42no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—42no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—81no—
189C22×C7C2×#9911yescharacteristic
190C22×C7C2×#10311yescharacteristic
191C22×C7#42111yescharacteristic
192C4×C7—21no—
193C4×C7—22no—
194C4×C7—41no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—82no—
198C4×C7—82no—
199C4×C7—162no—
200C4×C7#35212yes—
201C24C7×C22⋊C411yescharacteristic
202C24C2×C7×D411yescharacteristic
203C2×D4—41no—
204C2×D4—41no—
205C2×D4C7×SD1612yes—
206C2×D4C7×Q1612yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4C7×C22⋊C411yescharacteristic
219C22×C4C7×C22⋊C411yescharacteristic
220C22×C4C7×C22⋊C411yescharacteristic
221C22×C4C22×C4×C711yescharacteristic
222C22×C4C2×C7×D411yescharacteristic
223C2×C8—42no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C42—21no—
238C42—21no—
239C2×C7—21no—
240C2×C7—21no—
241C2×C7—41no—
242C2×C7—42no—
243C2×C7—41no—
244C2×C7#152011yescharacteristic
245C2×C7C2×#35211yescharacteristic
246C2×C7#212811yescharacteristic
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C7×#10312yes—
256C23C2×C7×C22⋊C411yescharacteristic
257C23C2×C7×SD1611yescharacteristic
258C23C2×C7×Q1611yescharacteristic
259D4—42no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C7×#9911yescharacteristic
285C2×C4C7×#9911yescharacteristic
286C2×C4C7×#10312yes—
287C8—82no—
288C7#1189911yescharacteristic, direct product factor, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×C7×#9911yescharacteristic
304C22C2×C7×#10311yescharacteristic
305C22C7×#42111yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—162no—
314C4C7×#35212yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2C7×#152011yescharacteristic
321C2C2×C7×#35211yescharacteristic
322C2C7×#212811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1352C2semidirect
6. C22×C7×C4⋊C4C4semidirect
9. C7×#400C4semidirect
16. C7×#380C4semidirect
20. #11899C7direct (N × H)
52. #1352C2×C7semidirect
92. C22×C4⋊C4C4×C7semidirect
95. #400C4×C7semidirect
102. #380C4×C7semidirect
288. C7#11899direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24872
C7×#17992
C7×#14602
C7×#13522
#118997