Finite Group Databasebeta

← C7×#11900 #13189116

Subgroups — C7×#11900 #13189116

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×#2563C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1351C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#99—21no—
8C2×C7×C22⋊C8—21no—
9C7×#393C412yescomplemented
10C7×#393C2212yes—
11C7×#390—21no—
12C7×#382—21no—
13C7×#380—21no—
14C7×#380—21no—
15C7×#380—21no—
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11900C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C22×C7×C8—41no—
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4C2311yescharacteristic
40C2×C42×C7—21no—
41C2×C42×C7C2×C411yescharacteristic
42C7×#99—42no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—41no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—41no—
48C7×#95C2×C412yes—
49#2563C2×C711yescharacteristic
50#1799C2×C711yescharacteristic
51#1459C2×C711yescharacteristic
52#1351C2×C714yescomplemented
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4SD1612yes—
60C2×C7×D4Q1612yes—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C2×C7×C8—42no—
76C2×C7×C8—82no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—82no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—82no—
88C7×C22⋊C4—82no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—42no—
92C2×C4⋊Q8C4×C711yescharacteristic, complemented
93C2×C4×D4C22×C711yescharacteristic
94C2×#99—21no—
95C2×C22⋊C8—21no—
96#393C4×C712yescomplemented
97#393C22×C712yes—
98#390—21no—
99#382—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103C4×C4⋊C4C4×C711yescharacteristic, complemented
104#342C22×C712yes—
105#324—21no—
106#323C22×C712yes—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7#10312yes—
116C23×C7C2×C22⋊C411yescharacteristic
117C23×C7C2×SD1611yescharacteristic
118C23×C7C2×Q1611yescharacteristic
119C7×Q8—84no—
120C7×D4—42no—
121C7×D4—42no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7—82no—
142C2×C4×C7—82no—
143C2×C4×C7—82no—
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7#9911yescharacteristic
146C2×C4×C7#10312yes—
147C7×C8—82no—
148C22×Q8C2×C4×C712yes—
149C22×D4C7×D411yescharacteristic
150C23×C4C7×D411yescharacteristic
151C23×C4C7×D411yescharacteristic
152C22×C8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4×D4—41no—
156C4×D4—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C7×D411yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4C2×C4×C711yescharacteristic
166C2×C22⋊C4C23×C711yescharacteristic
167C2×C42—21no—
168C2×C42C2×C4×C711yescharacteristic
169#99—42no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172#95—22no—
173#95—22no—
174#95—41no—
175#95C2×C4×C712yes—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—22no—
180C22×C7—21no—
181C22×C7—42no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—81no—
189C22×C7—81no—
190C22×C7C2×#9911yescharacteristic
191C22×C7C2×#10311yescharacteristic
192C22×C7#42111yescharacteristic
193C4×C7—21no—
194C4×C7—22no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—82no—
199C4×C7—82no—
200C4×C7—162no—
201C4×C7#35412yes—
202C24C7×C22⋊C411yescharacteristic
203C24C2×C7×D411yescharacteristic
204C2×Q8—42no—
205C2×Q8—82no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C7×SD1612yes—
209C2×D4C7×Q1612yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C7×C22⋊C411yescharacteristic
220C22×C4C7×C22⋊C411yescharacteristic
221C22×C4C7×C22⋊C411yescharacteristic
222C22×C4C22×C4×C711yescharacteristic
223C22×C4C2×C7×D411yescharacteristic
224C2×C8—42no—
225C2×C8—82no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C42—21no—
239C42—21no—
240C42—42no—
241C2×C7—21no—
242C2×C7—21no—
243C2×C7—41no—
244C2×C7—42no—
245C2×C7—41no—
246C2×C7#152011yescharacteristic
247C2×C7#212211yescharacteristic
248C2×C7C2×#35411yescharacteristic
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C7×#10312yes—
258C23C2×C7×C22⋊C411yescharacteristic
259C23C2×C7×SD1611yescharacteristic
260C23C2×C7×Q1611yescharacteristic
261Q8—84no—
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4C7×#9911yescharacteristic
287C2×C4C7×#9911yescharacteristic
288C2×C4C7×#10312yes—
289C8—82no—
290C7#1190011yescharacteristic, direct product factor, Sylow
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×C7×#9911yescharacteristic
306C22C2×C7×#10311yescharacteristic
307C22C7×#42111yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—82no—
315C4—162no—
316C4C7×#35412yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2C7×#152011yescharacteristic
323C2C7×#212211yescharacteristic
324C2C2×C7×#35411yescharacteristic
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×#1351C2semidirect
5. C2×C7×C4⋊Q8C4semidirect
9. C7×#393C4semidirect
16. C4×C7×C4⋊C4C4semidirect
20. #11900C7direct (N × H)
52. #1351C2×C7semidirect
92. C2×C4⋊Q8C4×C7semidirect
96. #393C4×C7semidirect
103. C4×C4⋊C4C4×C7semidirect
290. C7#11900direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25632
C7×#17992
C7×#14592
C7×#13512
#119007