Finite Group Databasebeta

← C7×#12255 #13189471

Subgroups — C7×#12255 #13189471

Order 1792.

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

normal subgroups only

This group has 378 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×#2807C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic
3C7×#1614C211yesmaximal, characteristic
4C7×#1614C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic
6C7×#1352C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic
8C7×#543C2211yescharacteristic
9C7×#484—21no—
10C7×#484—21no—
11C7×#400—21no—
12C7×#399C2211yescharacteristic
13C7×#399C2211yescharacteristic
14C7×#398—21no—
15C7×#389—21no—
16C7×#389—21no—
17C7×#388—21no—
18C7×#385—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12255C711yesmaximal, characteristic, direct product factor, Sylow
28C23×C4×C7D411yescharacteristic
29C7×C22⋊Q8—21no—
30C7×C22⋊Q8—21no—
31C7×C22⋊Q8—41no—
32C7×C22⋊Q8—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C2×C42×C7—21no—
49C7×#107—41no—
50C7×#107—41no—
51C7×#103—41no—
52C7×#103—41no—
53C7×C22⋊C8—21no—
54C7×C22⋊C8—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—41no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2807C2×C711yescharacteristic
64#2062C2×C711yescharacteristic
65#1614C2×C711yescharacteristic
66#1614C2×C711yescharacteristic
67#1352C2×C711yescharacteristic
68#1352C2×C711yescharacteristic
69#1326C2×C711yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×Q8—21no—
72C2×C7×Q8—21no—
73C22×C4×C7—21no—
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×C7—21no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—81no—
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C2×C7×C8—41no—
87C2×C7×C8—41no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—81no—
97C7×C4⋊C4—81no—
98C7×C4⋊C4—81no—
99C7×C4⋊C4—81no—
100C7×C4⋊C4—82no—
101C7×C22⋊C4—21no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—81no—
109C7×C22⋊C4Q1612yes—
110C42×C7—81no—
111#543C22×C711yescharacteristic
112#484—21no—
113#484—21no—
114#400—21no—
115#399C22×C711yescharacteristic
116#399C22×C711yescharacteristic
117#398—21no—
118#389—21no—
119#389—21no—
120#388—21no—
121#385—21no—
122#385—21no—
123#385C22×C711yescharacteristic
124#342—21no—
125#342C22×C711yescharacteristic
126#328—21no—
127#328—21no—
128#323C22×C711yescharacteristic
129#323C22×C711yescharacteristic
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7#12011yescharacteristic
135C23×C7C2×Q1611yescharacteristic
136C23×C7#13711yescharacteristic
137C7×Q8—81no—
138C7×Q8—81no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
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—42no—
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—42no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—81no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C7×C8—81no—
171C7×C8—81no—
172C23×C4C7×D411yescharacteristic
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C7×D411yescharacteristic
185C2×C4⋊C4C7×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4C7×D411yescharacteristic
191C2×C22⋊C4C23×C711yescharacteristic
192C2×C42—21no—
193#107—41no—
194#107—41no—
195#103—41no—
196#103—41no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—41no—
205#95C7×D411yescharacteristic
206#95C7×D411yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—81no—
216C22×C7#45111yescharacteristic
217C22×C7#45611yescharacteristic
218C22×C7B(4,𝔽2)11yescharacteristic
219C4×C7—42no—
220C4×C7—42no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—81no—
224C4×C7—82no—
225C4×C7—81no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C24C2×C7×D411yescharacteristic
231C2×Q8—21no—
232C2×Q8—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—81no—
244C22×C4C2×C7×D411yescharacteristic
245C22×C4C2×C7×D411yescharacteristic
246C2×C8—41no—
247C2×C8—41no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—82no—
261C22⋊C4—21no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4C7×Q1612yes—
270C42—81no—
271C2×C7—21no—
272C2×C7—21no—
273C2×C7—41no—
274C2×C7—41no—
275C2×C7#160511yescharacteristic
276C2×C7#219811yescharacteristic
277C2×C7#220011yescharacteristic
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23C7×#12011yescharacteristic
283C23C2×C7×Q1611yescharacteristic
284C23C7×#13711yescharacteristic
285Q8—81no—
286Q8—81no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C8—81no—
319C8—81no—
320C7#1225511yescharacteristic, direct product factor, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22C7×#45111yescharacteristic
331C22C7×#45611yescharacteristic
332C22C7×B(4,𝔽2)11yescharacteristic
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—82no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C7×#160511yescharacteristic
349C2C7×#219811yescharacteristic
350C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
27. #12255C7direct (N × H)
320. C7#12255direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28072
C7×#20622
C7×#16142
C7×#13522
C7×#13262
#122557