Finite Group Databasebeta

← C7×#12233 #13189449

Subgroups — C7×#12233 #13189449

Order 1792.

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

normal subgroups only

This group has 370 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×#2835C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic
3C7×#1608C211yesmaximal, characteristic
4C7×#1602C211yesmaximal, characteristic
5C7×#1351C211yesmaximal, characteristic
6C7×#1344C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic
8C7×#543C2211yescharacteristic
9C7×#483—21no—
10C7×#451—21no—
11C7×#400—21no—
12C7×#400—21no—
13C7×#400—21no—
14C7×#397—21no—
15C7×#397C2211yescharacteristic
16C7×#395—21no—
17C7×#393C2211yescharacteristic
18C7×#385—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12233C711yesmaximal, 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—41no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4C2311yescharacteristic
47C7×#106—41no—
48C7×#103—41no—
49C7×#103—41no—
50C7×C22⋊C8—21no—
51C7×C22⋊C8—21no—
52C7×#95—21no—
53C7×#95—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#2835C2×C711yescharacteristic
64#2062C2×C711yescharacteristic
65#1608C2×C711yescharacteristic
66#1602C2×C711yescharacteristic
67#1351C2×C711yescharacteristic
68#1344C2×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—41no—
84C22×C4×C7—41no—
85C22×C4×C7—81no—
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—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—22no—
102C7×C22⋊C4—21no—
103C7×C22⋊C4—21no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—81no—
111#543C22×C711yescharacteristic
112#483—21no—
113#451—21no—
114#400—21no—
115#400—21no—
116#400—21no—
117#397—21no—
118#397C22×C711yescharacteristic
119#395—21no—
120#393C22×C711yescharacteristic
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×C7#13511yescharacteristic
136C23×C7#13711yescharacteristic
137C7×Q8—42no—
138C7×Q8—81no—
139C7×Q8—81no—
140C7×Q8—81no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
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—42no—
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—81no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C2×C4×C7—81no—
171C2×C4×C7—81no—
172C7×C8—81no—
173C7×C8—81no—
174C22×Q8C7×D411yescharacteristic
175C23×C4C7×D411yescharacteristic
176C2×Q16—41no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C7×D411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4C7×D411yescharacteristic
192C2×C22⋊C4C23×C711yescharacteristic
193#106—41no—
194#103—41no—
195#103—41no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—41no—
207#95C7×D411yescharacteristic
208#95C7×D411yescharacteristic
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—81no—
218C22×C7#45211yescharacteristic
219C22×C7#45611yescharacteristic
220C22×C7B(4,𝔽2)11yescharacteristic
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—81no—
224C4×C7—81no—
225C4×C7—81no—
226C4×C7—82no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7—81no—
231C4×C7—81no—
232C24C2×C7×D411yescharacteristic
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—81no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—81no—
248C22×C4C2×C7×D411yescharacteristic
249C22×C4C2×C7×D411yescharacteristic
250Q16—82no—
251C2×C8—41no—
252C2×C8—41no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—82no—
263C22⋊C4—22no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—81no—
273C2×C7—21no—
274C2×C7—21no—
275C2×C7—41no—
276C2×C7—41no—
277C2×C7#160311yescharacteristic
278C2×C7#219911yescharacteristic
279C2×C7#220111yescharacteristic
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23C7×#12011yescharacteristic
285C23C7×#13511yescharacteristic
286C23C7×#13711yescharacteristic
287Q8—42no—
288Q8—81no—
289Q8—81no—
290Q8—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
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—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C8—81no—
323C8—81no—
324C7#1223311yescharacteristic, direct product factor, Sylow
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22C7×#45211yescharacteristic
335C22C7×#45611yescharacteristic
336C22C7×B(4,𝔽2)11yescharacteristic
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C2—21no—
349C2—21no—
350C2—41no—
351C2—41no—
352C2C7×#160311yescharacteristic
353C2C7×#219911yescharacteristic
354C2C7×#220111yescharacteristic
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. #12233C7direct (N × H)
324. C7#12233direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28352
C7×#20622
C7×#16082
C7×#16022
C7×#13512
C7×#13442
C7×#13262
#122337