Finite Group Databasebeta

← C7×#12258 #13189474

Subgroups — C7×#12258 #13189474

Order 1792.

352 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2612C211yesmaximal, characteristic
2C7×#2061C211yesmaximal, characteristic
3C7×#1617C211yesmaximal, characteristic
4C7×#1605C211yesmaximal, characteristic
5C7×#1354C211yesmaximal, characteristic
6C7×#1353C211yesmaximal, characteristic
7C7×#1324C211yesmaximal, characteristic
8C7×#544C2211yescharacteristic
9C7×#484—21no—
10C7×#451—21no—
11C7×#428—21no—
12C7×#400C2211yescharacteristic
13C7×#398—21no—
14C7×#395C2211yescharacteristic
15C7×#391—21no—
16C7×#389—21no—
17C7×#387—21no—
18C7×#387—21no—
19C7×#382C2211yescharacteristic
20C7×#376—21no—
21C7×#343C2211yescharacteristic
22C7×#339—21no—
23C7×#328—21no—
24C7×#328—21no—
25C7×#324C2211yescharacteristic
26C7×#324C2211yescharacteristic
27#12258C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C2×C7×Q16—41no—
30C2×C7×M16—21no—
31C7×C4⋊Q8—41no—
32C7×#125—41no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—21no—
35C7×#117—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4C2311yescharacteristic
43C2×C42×C7—21no—
44C2×C42×C7—21no—
45C2×C42×C7D411yescharacteristic
46C7×#107—41no—
47C7×#107—41no—
48C7×#106—41no—
49C7×#103—41no—
50C7×#103—41no—
51C7×C22⋊C8—21no—
52C7×C22⋊C8—21no—
53C7×#95—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#2612C2×C711yescharacteristic
64#2061C2×C711yescharacteristic
65#1617C2×C711yescharacteristic
66#1605C2×C711yescharacteristic
67#1354C2×C711yescharacteristic
68#1353C2×C711yescharacteristic
69#1324C2×C711yescharacteristic
70C2×C7×Q8—21no—
71C2×C7×Q8—21no—
72C2×C7×Q8—42no—
73C2×C7×Q8—41no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C7×Q16—82no—
84C7×M16—22no—
85C7×M16—41no—
86C2×C7×C8—41no—
87C2×C7×C8—41no—
88C2×C7×C8—41no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—42no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—81no—
102C7×C22⋊C4—41no—
103C42×C7—21no—
104C42×C7—42no—
105C42×C7—42no—
106C42×C7—41no—
107C42×C7Q1612yescomplemented
108#544C22×C711yescharacteristic
109#484—21no—
110#451—21no—
111#428—21no—
112#400C22×C711yescharacteristic
113#398—21no—
114#395C22×C711yescharacteristic
115#391—21no—
116#389—21no—
117#387—21no—
118#387—21no—
119#382C22×C711yescharacteristic
120#376—21no—
121#343C22×C711yescharacteristic
122#339—21no—
123#328—21no—
124#328—21no—
125#324C22×C711yescharacteristic
126#324C22×C711yescharacteristic
127C23×C7#12011yescharacteristic
128C7×Q8—42no—
129C7×Q8—81no—
130C7×Q8—81no—
131C7×Q8—81no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
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×C7C2×Q1611yescharacteristic
159C2×C4×C7#13711yescharacteristic
160C7×C8—42no—
161C7×C8—81no—
162C7×C8—81no—
163C22×Q8C7×D411yescharacteristic
164C2×Q16—41no—
165C2×M16—21no—
166C4⋊Q8—41no—
167#125—41no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170#117—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C7×D411yescharacteristic
176C2×C4⋊C4C7×D411yescharacteristic
177C2×C4⋊C4C23×C711yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42C7×D411yescharacteristic
181#107—41no—
182#107—41no—
183#106—41no—
184#103—41no—
185#103—41no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95C7×D411yescharacteristic
197#95C7×D411yescharacteristic
198C22×C7—21no—
199C22×C7—21no—
200C22×C7#45111yescharacteristic
201C22×C7#45611yescharacteristic
202C22×C7B(4,𝔽2)11yescharacteristic
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—42no—
206C4×C7—42no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—42no—
210C4×C7—81no—
211C4×C7—81no—
212C4×C7—81no—
213C4×C7—81no—
214C4×C7—81no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—42no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C2×C7×D411yescharacteristic
226C22×C4C2×C7×D411yescharacteristic
227C22×C4C2×C7×D411yescharacteristic
228Q16—82no—
229M16—22no—
230M16—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—81no—
247C22⋊C4—41no—
248C42—21no—
249C42—42no—
250C42—42no—
251C42—41no—
252C42C7×Q1612yescomplemented
253C2×C7—21no—
254C2×C7—21no—
255C2×C7#160511yescharacteristic
256C2×C7#219711yescharacteristic
257C2×C7#220411yescharacteristic
258C23C7×#12011yescharacteristic
259Q8—42no—
260Q8—81no—
261Q8—81no—
262Q8—81no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4C2×C7×Q1611yescharacteristic
290C2×C4C7×#13711yescharacteristic
291C8—42no—
292C8—81no—
293C8—81no—
294C7#1225811yescharacteristic, direct product factor, Sylow
295C22—21no—
296C22—21no—
297C22C7×#45111yescharacteristic
298C22C7×#45611yescharacteristic
299C22C7×B(4,𝔽2)11yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C2—21no—
313C2—21no—
314C2C7×#160511yescharacteristic
315C2C7×#219711yescharacteristic
316C2C7×#220411yescharacteristic
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. #12258C7direct (N × H)
107. C42×C7Q16semidirect
252. C42C7×Q16semidirect
294. C7#12258direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26122
C7×#20612
C7×#16172
C7×#16052
C7×#13542
C7×#13532
C7×#13242
#122587