Finite Group Databasebeta

← C7×#12273 #13189489

Subgroups — C7×#12273 #13189489

Order 1792.

364 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2586C211yesmaximal, characteristic
2C7×#2028C211yesmaximal, characteristic
3C7×#1611C211yesmaximal, characteristic
4C7×#1605C211yesmaximal, characteristic
5C7×#1354C211yesmaximal, characteristic
6C7×#1346C211yesmaximal, characteristic
7C7×#1319C211yesmaximal, characteristic
8C7×#544C2211yescharacteristic
9C7×#483—21no—
10C7×#451—21no—
11C7×#419—21no—
12C7×#398C2211yescharacteristic
13C7×#395—21no—
14C7×#395C2211yescharacteristic
15C7×#391—21no—
16C7×#391C2211yescharacteristic
17C7×#389—21no—
18C7×#387—21no—
19C7×#387—21no—
20C4×C7×C4⋊C4—21no—
21C7×#343C2211yescharacteristic
22C7×#339—21no—
23C7×#328—21no—
24C7×#328—21no—
25C7×#324C2211yescharacteristic
26C7×#324C2211yescharacteristic
27#12273C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×C7×Q8D411yescharacteristic
30C2×C7×Q16—41no—
31C2×C7×M16—21no—
32C7×C4⋊Q8—41no—
33C7×#125—41no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C7×#117—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C4⋊C4C2311yescharacteristic
44C2×C42×C7—21no—
45C2×C42×C7—21no—
46C2×C42×C7D411yescharacteristic
47C7×#106—41no—
48C7×#103—41no—
49C7×#103—41no—
50C7×#103—41no—
51C7×#103—41no—
52C7×C22⋊C8—21no—
53C7×C22⋊C8—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2586C2×C711yescharacteristic
64#2028C2×C711yescharacteristic
65#1611C2×C711yescharacteristic
66#1605C2×C711yescharacteristic
67#1354C2×C711yescharacteristic
68#1346C2×C711yescharacteristic
69#1319C2×C711yescharacteristic
70C2×C7×Q8—21no—
71C2×C7×Q8—21no—
72C2×C7×Q8—21no—
73C2×C7×Q8—21no—
74C2×C7×Q8—42no—
75C2×C7×Q8—41no—
76C2×C7×Q8—41no—
77C2×C7×Q8—41no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C7×Q16—82no—
88C7×M16—41no—
89C7×M16—41no—
90C2×C7×C8—41no—
91C2×C7×C8—41no—
92C2×C7×C8—41no—
93C7×C4⋊C4—24no—
94C7×C4⋊C4—21no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—42no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—81no—
106C7×C4⋊C4—81no—
107C7×C22⋊C4—41no—
108C42×C7—41no—
109C42×C7—42no—
110C42×C7—41no—
111C42×C7—81no—
112#544C22×C711yescharacteristic
113#483—21no—
114#451—21no—
115#419—21no—
116#398C22×C711yescharacteristic
117#395—21no—
118#395C22×C711yescharacteristic
119#391—21no—
120#391C22×C711yescharacteristic
121#389—21no—
122#387—21no—
123#387—21no—
124C4×C4⋊C4—21no—
125#343C22×C711yescharacteristic
126#339—21no—
127#328—21no—
128#328—21no—
129#324C22×C711yescharacteristic
130#324C22×C711yescharacteristic
131C23×C7#12011yescharacteristic
132C7×Q8—42no—
133C7×Q8—42no—
134C7×Q8—81no—
135C7×Q8—81no—
136C7×Q8—81no—
137C7×Q8—81no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—22no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7—22no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7#13511yescharacteristic
165C2×C4×C7#13711yescharacteristic
166C7×C8—81no—
167C7×C8—81no—
168C7×C8—81no—
169C22×Q8C7×D411yescharacteristic
170C22×Q8C7×D411yescharacteristic
171C2×Q16—41no—
172C2×M16—21no—
173C4⋊Q8—41no—
174#125—41no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177#117—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C7×D411yescharacteristic
184C2×C4⋊C4C23×C711yescharacteristic
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C7×D411yescharacteristic
188#106—41no—
189#103—41no—
190#103—41no—
191#103—41no—
192#103—41no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C7×D411yescharacteristic
203#95C7×D411yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7#45211yescharacteristic
207C22×C7#45511yescharacteristic
208C22×C7B(4,𝔽2)11yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—42no—
212C4×C7—42no—
213C4×C7—42no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—42no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7—81no—
220C4×C7—81no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C2×C7×D411yescharacteristic
236C22×C4C2×C7×D411yescharacteristic
237C22×C4C2×C7×D411yescharacteristic
238Q16—82no—
239M16—41no—
240M16—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C4⋊C4—24no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C22⋊C4—41no—
259C42—41no—
260C42—42no—
261C42—41no—
262C42—81no—
263C2×C7—21no—
264C2×C7—21no—
265C2×C7#160611yescharacteristic
266C2×C7#219711yescharacteristic
267C2×C7#220511yescharacteristic
268C23C7×#12011yescharacteristic
269Q8—42no—
270Q8—42no—
271Q8—81no—
272Q8—81no—
273Q8—81no—
274Q8—81no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4C7×#13511yescharacteristic
302C2×C4C7×#13711yescharacteristic
303C8—81no—
304C8—81no—
305C8—81no—
306C7#1227311yescharacteristic, direct product factor, Sylow
307C22—21no—
308C22—21no—
309C22C7×#45211yescharacteristic
310C22C7×#45511yescharacteristic
311C22C7×B(4,𝔽2)11yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C2—21no—
325C2—21no—
326C2C7×#160611yescharacteristic
327C2C7×#219711yescharacteristic
328C2C7×#220511yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
27. #12273C7direct (N × H)
306. C7#12273direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25862
C7×#20282
C7×#16112
C7×#16052
C7×#13542
C7×#13462
C7×#13192
#122737