Finite Group Databasebeta

← C7×#12254 #13189470

Subgroups — C7×#12254 #13189470

Order 1792.

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

normal subgroups only

This group has 388 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×#2742C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic
3C7×#1614C211yesmaximal, characteristic
4C7×#1602C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic
6C7×#1351C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic
8C7×#543C2211yescharacteristic
9C7×#484—21no—
10C7×#451—21no—
11C7×#400—21no—
12C7×#399C2211yescharacteristic
13C7×#395—21no—
14C7×#393C2211yescharacteristic
15C7×#389—21no—
16C7×#388—21no—
17C7×#387—21no—
18C7×#385—21no—
19C7×#385C2211yescharacteristic
20C7×#380—21no—
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12254C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C2×C7×Q16—41no—
31C7×C22⋊Q8—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—41no—
34C7×C22⋊Q8—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—41no—
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×#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—41no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2742C2×C711yescharacteristic
64#2062C2×C711yescharacteristic
65#1614C2×C711yescharacteristic
66#1602C2×C711yescharacteristic
67#1352C2×C711yescharacteristic
68#1351C2×C711yescharacteristic
69#1326C2×C711yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×Q8—21no—
72C2×C7×Q8—21no—
73C2×C7×Q8—41no—
74C2×C7×Q8—81no—
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—21no—
82C22×C4×C7—21no—
83C22×C4×C7—42no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C7×Q16—82no—
89C2×C7×C8—41no—
90C2×C7×C8—41no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—81no—
99C7×C4⋊C4—81no—
100C7×C4⋊C4—81no—
101C7×C4⋊C4—82no—
102C7×C22⋊C4—21no—
103C7×C22⋊C4—22no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—81no—
109C7×C22⋊C4—81no—
110C7×C22⋊C4Q1612yescomplemented
111C42×C7—81no—
112#543C22×C711yescharacteristic
113#484—21no—
114#451—21no—
115#400—21no—
116#399C22×C711yescharacteristic
117#395—21no—
118#393C22×C711yescharacteristic
119#389—21no—
120#388—21no—
121#387—21no—
122#385—21no—
123#385C22×C711yescharacteristic
124#380—21no—
125#342—21no—
126#342C22×C711yescharacteristic
127#328—21no—
128#328—21no—
129#323C22×C711yescharacteristic
130#323C22×C711yescharacteristic
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7#12011yescharacteristic
136C23×C7C2×Q1611yescharacteristic
137C23×C7#13711yescharacteristic
138C7×Q8—42no—
139C7×Q8—81no—
140C7×Q8—81no—
141C7×Q8—81no—
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—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—42no—
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—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C2×C4×C7—81no—
171C2×C4×C7—81no—
172C2×C4×C7—81no—
173C7×C8—81no—
174C7×C8—81no—
175C22×Q8C7×D411yescharacteristic
176C23×C4C7×D411yescharacteristic
177C2×Q16—41no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C7×D411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C7×D411yescharacteristic
194C2×C22⋊C4C23×C711yescharacteristic
195C2×C42—21no—
196#107—41no—
197#103—41no—
198#103—41no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—41no—
208#95C7×D411yescharacteristic
209#95C7×D411yescharacteristic
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—81no—
219C22×C7#45111yescharacteristic
220C22×C7#45611yescharacteristic
221C22×C7B(4,𝔽2)11yescharacteristic
222C4×C7—42no—
223C4×C7—41no—
224C4×C7—42no—
225C4×C7—41no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7—82no—
229C4×C7—81no—
230C4×C7—81no—
231C4×C7—81no—
232C4×C7—81no—
233C24C2×C7×D411yescharacteristic
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—41no—
237C2×Q8—81no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C7×D411yescharacteristic
250C22×C4C2×C7×D411yescharacteristic
251Q16—82no—
252C2×C8—41no—
253C2×C8—41no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—82no—
265C22⋊C4—21no—
266C22⋊C4—22no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4C7×Q1612yescomplemented
274C42—81no—
275C2×C7—21no—
276C2×C7—21no—
277C2×C7—41no—
278C2×C7—41no—
279C2×C7#160511yescharacteristic
280C2×C7#219711yescharacteristic
281C2×C7#220011yescharacteristic
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23C7×#12011yescharacteristic
287C23C2×C7×Q1611yescharacteristic
288C23C7×#13711yescharacteristic
289Q8—42no—
290Q8—81no—
291Q8—81no—
292Q8—81no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
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—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C8—81no—
325C8—81no—
326C7#1225411yescharacteristic, direct product factor, Sylow
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22C7×#45111yescharacteristic
337C22C7×#45611yescharacteristic
338C22C7×B(4,𝔽2)11yescharacteristic
339C4—42no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—82no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2C7×#160511yescharacteristic
355C2C7×#219711yescharacteristic
356C2C7×#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. #12254C7direct (N × H)
110. C7×C22⋊C4Q16semidirect
273. C22⋊C4C7×Q16semidirect
326. C7#12254direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27422
C7×#20622
C7×#16142
C7×#16022
C7×#13522
C7×#13512
C7×#13262
#122547