Finite Group Databasebeta

← C7×#12853 #13190069

Subgroups — C7×#12853 #13190069

Order 1792.

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

normal subgroups only

This group has 656 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×#2297C211yesmaximal, characteristic
2C7×#1785C211yesmaximal, characteristic
3C7×#1782C211yesmaximal, characteristic
4C7×#1390C214yesmaximal, complemented
5C22×C7×#41C2211yescharacteristic
6C2×C7×#117C411yescharacteristic, complemented
7C22×C42×C7C411yescharacteristic, 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—
18C4×C7×C22⋊C4—21no—
19C4×C7×C22⋊C4—21no—
20#12853C711yesmaximal, 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×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4C2×C412yes—
39C2×C42×C7—21no—
40C2×C42×C7—21no—
41C2×C42×C7—21no—
42C2×C42×C7—42no—
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#2297C2×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—44no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
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×C22⋊C4—44no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C42×C7—42no—
99C42×C7—42no—
100C42×C7—42no—
101C22×#41C22×C711yescharacteristic
102C2×#117C4×C711yescharacteristic, complemented
103C22×C42C4×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—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—42no—
126C23×C7C2×C22⋊C411yescharacteristic
127C23×C7#11712yes—
128C23×C7#12312yes—
129C7×Q8—44no—
130C7×D4—44no—
131C7×D4—44no—
132C7×D4—44no—
133C2×C4×C7—28no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—24no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—44no—
145C2×C4×C7—44no—
146C2×C4×C7—42no—
147C2×C4×C7—44no—
148C2×C4×C7—42no—
149C2×C4×C7—44no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—82no—
154C2×C4×C7—84no—
155C2×C4×C7#9912yes—
156C2×C4×C7#9912yes—
157C2×C4×C7#12312yes—
158C7×C8—82no—
159C2×#41—24no—
160C2×#41—42no—
161C2×#41—42no—
162C22×Q8C7×D411yescharacteristic
163C22×D4C2×C4×C711yescharacteristic
164C22×D4C7×D411yescharacteristic
165C22×D4C23×C711yescharacteristic
166C23×C4C2×C4×C711yescharacteristic
167C23×C4C2×C4×C712yes—
168C23×C4C7×D411yescharacteristic
169C23×C4C7×D411yescharacteristic
170C2×M16—22no—
171#117—42no—
172C2×C4⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C2×C4×C712yes—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—42no—
181#101—42no—
182#100—24no—
183#99—24no—
184#99—42no—
185C22⋊C8—42no—
186#95—22no—
187#95—22no—
188#95—42no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7—22no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—42no—
195C22×C7—42no—
196C22×C7—42no—
197C22×C7—42no—
198C22×C7—41no—
199C22×C7—42no—
200C22×C7—81no—
201C22×C7—81no—
202C22×C7#38111yescharacteristic
203C22×C7C2×#9911yescharacteristic
204C22×C7C2×#9911yescharacteristic
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—41no—
209C4×C7—44no—
210C4×C7—82no—
211C4×C7—162no—
212C24C7×C22⋊C411yescharacteristic
213C24C22×C4×C711yescharacteristic
214C24C2×C7×D411yescharacteristic
215#41—416no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4C7×#4114yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4C7×C22⋊C411yescharacteristic
243C22×C4C7×C22⋊C411yescharacteristic
244C22×C4C7×C22⋊C411yescharacteristic
245C22×C4C2×C7×D411yescharacteristic
246M16—42no—
247M16—42no—
248C2×C8—42no—
249C4⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C2×C7—21no—
259C2×C7—21no—
260C2×C7—42no—
261C2×C7—42no—
262C2×C7—81no—
263C2×C7#181911yescharacteristic
264C2×C7#212012yes—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23C2×C7×C22⋊C411yescharacteristic
276C23C7×#11712yes—
277C23C7×#12312yes—
278Q8—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4C7×#9912yes—
305C2×C4C7×#9912yes—
306C2×C4C7×#12312yes—
307C8—82no—
308C7#1285311yescharacteristic, direct product factor, Sylow
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22C7×#38111yescharacteristic
323C22C2×C7×#9911yescharacteristic
324C22C2×C7×#9911yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—44no—
330C4—82no—
331C4—162no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2—81no—
337C2C7×#181911yescharacteristic
338C2C7×#212012yes—
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×C42×C7C4semidirect
15. C7×#386C4semidirect
20. #12853C7direct (N × H)
54. #1390C2×C7semidirect
102. C2×#117C4×C7semidirect
103. C22×C42C4×C7semidirect
111. #386C4×C7semidirect
308. C7#12853direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#22972
C7×#17852
C7×#17822
C7×#13902
#128537