Finite Group Databasebeta

← C7×#11878 #13189094

Subgroups — C7×#11878 #13189094

Order 1792.

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

normal subgroups only

This group has 472 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×#2547C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1344C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×#124—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99—21no—
9C2×C7×C22⋊C8—21no—
10C7×#397C412yescomplemented
11C7×#397C2212yes—
12C7×#391—21no—
13C7×#380—21no—
14C7×#380—21no—
15C7×#380—21no—
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11878C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C22×C7×C8—41no—
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C7×#124—42no—
29C7×#124—42no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4C2311yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7C2×C411yescharacteristic
44C7×#99—42no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—41no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—41no—
50C7×#95C2×C412yes—
51#2547C2×C711yescharacteristic
52#1799C2×C711yescharacteristic
53#1459C2×C711yescharacteristic
54#1344C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—42no—
58C2×C7×Q8—42no—
59C2×C7×Q8—42no—
60C2×C7×D4—41no—
61C2×C7×D4—41no—
62C2×C7×D4D812yes—
63C2×C7×D4SD1612yes—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—22no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C2×C7×C8—42no—
79C2×C7×C8—82no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—82no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—44no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7—42no—
96C2×C4⋊Q8C4×C711yescharacteristic, complemented
97C2×#124—21no—
98C2×C4×D4C22×C711yescharacteristic
99C2×#99—21no—
100C2×C22⋊C8—21no—
101#397C4×C712yescomplemented
102#397C22×C712yes—
103#391—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4C4×C711yescharacteristic, complemented
108#342C22×C712yes—
109#324—21no—
110#323C22×C712yes—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7#10212yes—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7C2×D811yescharacteristic
122C23×C7C2×SD1611yescharacteristic
123C7×Q8—84no—
124C7×D4—42no—
125C7×D4—42no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7—82no—
146C2×C4×C7—82no—
147C2×C4×C7—82no—
148C2×C4×C7#9911yescharacteristic
149C2×C4×C7#9911yescharacteristic
150C2×C4×C7#10212yes—
151C7×C8—82no—
152C22×Q8—21no—
153C22×D4C7×D411yescharacteristic
154C23×C4C7×D411yescharacteristic
155C23×C4C7×D411yescharacteristic
156C22×C8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159#124—42no—
160#124—42no—
161C4×D4—41no—
162C4×D4—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C2×C4×C712yes—
167C2×C4⋊C4C7×D411yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4C2×C4×C711yescharacteristic
172C2×C22⋊C4C23×C711yescharacteristic
173C2×C42—21no—
174C2×C42C2×C4×C711yescharacteristic
175#99—42no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178#95—22no—
179#95—22no—
180#95—41no—
181#95C2×C4×C712yes—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—22no—
186C22×C7—21no—
187C22×C7—42no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—81no—
195C22×C7—81no—
196C22×C7C2×#9911yescharacteristic
197C22×C7C2×#10211yescharacteristic
198C22×C7#42111yescharacteristic
199C4×C7—21no—
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—82no—
205C4×C7—82no—
206C4×C7—162no—
207C4×C7#35412yes—
208C24C7×C22⋊C411yescharacteristic
209C24C2×C7×D411yescharacteristic
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C7×D812yes—
216C2×D4C7×SD1612yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C7×C22⋊C411yescharacteristic
227C22×C4C7×C22⋊C411yescharacteristic
228C22×C4C7×C22⋊C411yescharacteristic
229C22×C4C22×C4×C711yescharacteristic
230C22×C4C2×C7×D411yescharacteristic
231C2×C8—42no—
232C2×C8—82no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—82no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—44no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—42no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7—41no—
252C2×C7—42no—
253C2×C7—41no—
254C2×C7#151911yescharacteristic
255C2×C7#212311yescharacteristic
256C2×C7C2×#35411yescharacteristic
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23C7×#10212yes—
266C23C2×C7×C22⋊C411yescharacteristic
267C23C2×C7×D811yescharacteristic
268C23C2×C7×SD1611yescharacteristic
269Q8—84no—
270D4—42no—
271D4—42no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4C7×#9911yescharacteristic
295C2×C4C7×#9911yescharacteristic
296C2×C4C7×#10212yes—
297C8—82no—
298C7#1187811yescharacteristic, direct product factor, Sylow
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×C7×#9911yescharacteristic
314C22C2×C7×#10211yescharacteristic
315C22C7×#42111yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—162no—
324C4C7×#35412yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2C7×#151911yescharacteristic
331C2C7×#212311yescharacteristic
332C2C2×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×#1344C2semidirect
5. C2×C7×C4⋊Q8C4semidirect
10. C7×#397C4semidirect
16. C4×C7×C4⋊C4C4semidirect
20. #11878C7direct (N × H)
54. #1344C2×C7semidirect
96. C2×C4⋊Q8C4×C7semidirect
101. #397C4×C7semidirect
107. C4×C4⋊C4C4×C7semidirect
298. C7#11878direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25472
C7×#17992
C7×#14592
C7×#13442
#118787