Finite Group Databasebeta

← C7×#12855 #13190071

Subgroups — C7×#12855 #13190071

Order 1792.

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

normal subgroups only

This group has 628 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×#2311C211yesmaximal, characteristic
2C7×#1785C211yesmaximal, characteristic
3C7×#1782C211yesmaximal, characteristic
4C7×#1390C214yesmaximal, complemented
5C22×C7×#41C2211yescharacteristic
6C2×C7×#117C411yescharacteristic, complemented
7C22×C7×C4⋊C4C411yescharacteristic, complemented
8C2×C7×#101—21no—
9C2×C7×#100C2212yes—
10C2×C7×#99—21no—
11C2×C7×#99C2212yes—
12C7×#407—21no—
13C7×#391—21no—
14C7×#390—21no—
15C7×#386C412yescomplemented
16C7×#386C2212yes—
17C7×#381—21no—
18C7×#380—21no—
19C7×#380—21no—
20#12855C711yesmaximal, characteristic, direct product factor, Sylow
21C2×C7×#41—24no—
22C2×C7×#41—42no—
23C2×C7×#41—42no—
24C22×C7×Q8D411yescharacteristic
25C22×C7×D4C2×C411yescharacteristic
26C22×C7×D4D411yescharacteristic
27C22×C7×D4C2311yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C412yes—
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C2×C7×M16—22no—
33C7×#117—42no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—42no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4C2×C412yes—
42C2×C42×C7—21no—
43C7×#101—42no—
44C7×#100—24no—
45C7×#99—24no—
46C7×#99—42no—
47C7×C22⋊C8—42no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—42no—
51#2311C2×C711yescharacteristic
52#1785C2×C711yescharacteristic
53#1782C2×C711yescharacteristic
54#1390C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C22×C411yescharacteristic
57C24×C7C2×D411yescharacteristic
58C7×#41—416no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8—41no—
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4#4114yes—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—22no—
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—82no—
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C7×M16—42no—
90C7×M16—42no—
91C2×C7×C8—42no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—42no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—82no—
99C7×C22⋊C4—82no—
100C42×C7—42no—
101C22×#41C22×C711yescharacteristic
102C2×#117C4×C711yescharacteristic, complemented
103C22×C4⋊C4C4×C711yescharacteristic, complemented
104C2×#101—21no—
105C2×#100C22×C712yes—
106C2×#99—21no—
107C2×#99C22×C712yes—
108#407—21no—
109#391—21no—
110#390—21no—
111#386C4×C712yescomplemented
112#386C22×C712yes—
113#381—21no—
114#380—21no—
115#380—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7C2×C22⋊C411yescharacteristic
130C23×C7#11712yes—
131C23×C7#12312yes—
132C7×Q8—44no—
133C7×D4—44no—
134C7×D4—44no—
135C7×D4—44no—
136C2×C4×C7—28no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—44no—
147C2×C4×C7—44no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—84no—
155C2×C4×C7—82no—
156C2×C4×C7—82no—
157C2×C4×C7—82no—
158C2×C4×C7#9912yes—
159C2×C4×C7#9912yes—
160C2×C4×C7#12312yes—
161C7×C8—82no—
162C2×#41—24no—
163C2×#41—42no—
164C2×#41—42no—
165C22×Q8C7×D411yescharacteristic
166C22×D4C2×C4×C711yescharacteristic
167C22×D4C7×D411yescharacteristic
168C22×D4C23×C711yescharacteristic
169C23×C4C2×C4×C711yescharacteristic
170C23×C4C2×C4×C712yes—
171C23×C4C7×D411yescharacteristic
172C23×C4C7×D411yescharacteristic
173C2×M16—22no—
174#117—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—42no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C2×C4×C712yes—
183C2×C42—21no—
184#101—42no—
185#100—24no—
186#99—24no—
187#99—42no—
188C22⋊C8—42no—
189#95—22no—
190#95—22no—
191#95—42no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—42no—
197C22×C7—42no—
198C22×C7—42no—
199C22×C7—42no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7#38111yescharacteristic
210C22×C7C2×#9911yescharacteristic
211C22×C7C2×#9911yescharacteristic
212C4×C7—22no—
213C4×C7—22no—
214C4×C7—22no—
215C4×C7—41no—
216C4×C7—82no—
217C4×C7—82no—
218C4×C7—162no—
219C24C7×C22⋊C411yescharacteristic
220C24C22×C4×C711yescharacteristic
221C24C2×C7×D411yescharacteristic
222#41—416no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4C7×#4114yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C7×C22⋊C411yescharacteristic
251C22×C4C7×C22⋊C411yescharacteristic
252C22×C4C2×C7×D411yescharacteristic
253M16—42no—
254M16—42no—
255C2×C8—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C42—42no—
265C2×C7—21no—
266C2×C7—21no—
267C2×C7—42no—
268C2×C7—42no—
269C2×C7—81no—
270C2×C7#181911yescharacteristic
271C2×C7#212211yescharacteristic
272C2×C7#212311yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C2×C7×C22⋊C411yescharacteristic
287C23C7×#11712yes—
288C23C7×#12312yes—
289Q8—44no—
290D4—44no—
291D4—44no—
292D4—44no—
293C2×C4—28no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—84no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4C7×#9912yes—
316C2×C4C7×#9912yes—
317C2×C4C7×#12312yes—
318C8—82no—
319C7#1285511yescharacteristic, direct product factor, Sylow
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22C7×#38111yescharacteristic
338C22C2×C7×#9911yescharacteristic
339C22C2×C7×#9911yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—82no—
345C4—82no—
346C4—162no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2C7×#181911yescharacteristic
353C2C7×#212211yescharacteristic
354C2C7×#212311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×#41C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1390C2semidirect
6. C2×C7×#117C4semidirect
7. C22×C7×C4⋊C4C4semidirect
15. C7×#386C4semidirect
20. #12855C7direct (N × H)
54. #1390C2×C7semidirect
102. C2×#117C4×C7semidirect
103. C22×C4⋊C4C4×C7semidirect
111. #386C4×C7semidirect
319. C7#12855direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23112
C7×#17852
C7×#17822
C7×#13902
#128557