Finite Group Databasebeta

← #13269408

Subgroups — #13269408

Order 1792.

670 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3509C211yesmaximal, characteristic
2#11729627C211yesmaximal, characteristic
3#11729625C211yesmaximal, characteristic
4#11729624C212yesmaximal
5#11729398C212yesmaximal
6C7×#557—22no—
7C7×#556—22no—
8C7×#552—24no—
9C7×#541—22no—
10C7×#536C2211yescharacteristic
11C7×#536C2211yescharacteristic
12C7×#533—22no—
13C7×#533C411yescharacteristic, complemented
14C2×C7×C4⋊Q8C411yescharacteristic, complemented
15C7×#519C411yescharacteristic, complemented
16C7×#519C411yescharacteristic, complemented
17C4×C7×#41C2211yescharacteristic
18#90012C2212yes—
19#90009C2212yes—
20#89343—22no—
21#89336—22no—
22#89334—24no—
23#89326—22no—
24#89318—22no—
25#12724—71nomaximal, Sylow
26C2×C7×#41C2311yescharacteristic
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×Q8C2×C411yescharacteristic
29C7×C4⋊Q8—22no—
30C7×C4⋊Q8—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8D412yes—
35C7×#126—24no—
36C7×#126—24no—
37C7×#125—24no—
38C7×#125—24no—
39C7×#125—24no—
40C7×#125—21no—
41C7×#124—21no—
42C7×#124D412yes—
43C7×#123—22no—
44C7×#123D414yes—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8—22no—
49C4×C7×Q8—24no—
50C4×C7×Q8—22no—
51C4×C7×Q8—21no—
52C4×C7×Q8—22no—
53C4×C7×Q8—22no—
54C4×C7×Q8—22no—
55C4×C7×Q8—22no—
56C4×C7×D4—22no—
57C4×C7×D4—21no—
58C7×#117—24no—
59C7×#117—22no—
60C7×#117—22no—
61C7×#117C2×C411yescharacteristic
62C7×#117C2×C411yescharacteristic
63C7×#117C2×C411yescharacteristic
64C7×#117D412yes—
65C2×C7×C4⋊C4—22no—
66C2×C42×C7C2×C411yescharacteristic
67C2×C42×C7D412yes—
68#6489—22no—
69C2×#1136—22no—
70#6410—22no—
71#6408—22no—
72#6408—22no—
73#6407—22no—
74#3509D711yescharacteristic
75#2135—71no—
76#2129—71no—
77#2123—72no—
78#1827—72no—
79C7×#41—42no—
80C2×C7×Q8—22no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×Q8—42no—
85C2×C7×Q8—42no—
86C2×C7×Q8C22⋊C412yes—
87C2×C7×Q8C2×D411yescharacteristic
88C2×C7×D4C2×D411yescharacteristic
89C2×C7×D4C2×D412yes—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7C22⋊C412yes—
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D412yes—
96C7×C4⋊C4—24no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—24no—
100C7×C4⋊C4—24no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—24no—
108C7×C4⋊C4—24no—
109C7×C22⋊C4—21no—
110C7×C22⋊C4—21no—
111C7×C22⋊C4—24no—
112C7×C22⋊C4—24no—
113C7×C22⋊C4—22no—
114C7×C22⋊C4C22⋊C412yes—
115C42×C7—24no—
116C42×C7—24no—
117C42×C7—21no—
118C42×C7—21no—
119C42×C7—22no—
120C42×C7C22⋊C411yescharacteristic
121C42×C7C22⋊C411yescharacteristic
122C42×C7C22⋊C412yes—
123C42×C7C22⋊C412yes—
124C22⋊Dic7—42no—
125C4⋊Dic7—42no—
126C4×Dic7—42no—
127#1136—42no—
128#1136—42no—
129C2×C7⋊C8—42no—
130#557—22no—
131#556—22no—
132#552—24no—
133#541—22no—
134#536C2×D711yescharacteristic
135#536C2×D711yescharacteristic
136#533—22no—
137#533Dic711yescharacteristic, complemented
138C2×C4⋊Q8Dic711yescharacteristic, complemented
139#519Dic711yescharacteristic, complemented
140#519Dic711yescharacteristic, complemented
141C4×#41C2×D711yescharacteristic
142#413—72no—
143#410—72no—
144#356—142no—
145#354—142no—
146#352—144no—
147#339—142no—
148H(3,ℤ/4)—142no—
149C23×C7C2×C22⋊C411yescharacteristic
150C23×C7#12012yes—
151C7×Q8—41no—
152C7×Q8—42no—
153C7×Q8—44no—
154C7×Q8—42no—
155C7×Q8—42no—
156C7×Q8—42no—
157C7×D4—42no—
158C7×D4—41no—
159C2×C4×C7—21no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—21no—
169C2×C4×C7—42no—
170C2×C4×C7—44no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7C2×C22⋊C411yescharacteristic
174C2×C4×C7C2×C22⋊C411yescharacteristic
175C2×C4×C7#12012yes—
176C2×Dic7—82no—
177C7⋊C8—82no—
178C2×#41C22×D711yescharacteristic
179C22×Q8C2×Dic711yescharacteristic
180C22×Q8C2×Dic711yescharacteristic
181C2×M16—142no—
182C4⋊Q8—22no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8C22⋊D712yes—
188#126—24no—
189#126—24no—
190#125—24no—
191#125—24no—
192#125—24no—
193#125—21no—
194#124—21no—
195#124C22⋊D712yes—
196#123—22no—
197#123C22⋊D714yes—
198C22⋊Q8—24no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C4×Q8—24no—
203C4×Q8—22no—
204C4×Q8—21no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×D4—22no—
210C4×D4—21no—
211#117—24no—
212#117—22no—
213#117—22no—
214#117—142no—
215#117C2×Dic711yescharacteristic
216#117C2×Dic711yescharacteristic
217#117C2×Dic711yescharacteristic
218#117C22⋊D712yes—
219C2×C4⋊C4—22no—
220C2×C42C2×Dic711yescharacteristic
221C2×C42C22⋊D712yes—
222#101—142no—
223#100—142no—
224#99—142no—
225#99—142no—
226C22×C7—22no—
227C22×C7—21no—
228C22×C7—41no—
229C22×C7—42no—
230C22×C7#37911yescharacteristic
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—21no—
234C4×C7—42no—
235C4×C7—41no—
236C4×C7—42no—
237C4×C7—42no—
238C4×C7—42no—
239C4×C7—42no—
240C4×C7#41012yes—
241Dic7—162no—
242#41—42no—
243C2×Q8—22no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8C22⋊Dic712yes—
250C2×Q8C2×C22⋊D711yescharacteristic
251C2×D4C2×C22⋊D711yescharacteristic
252C2×D4C2×C22⋊D712yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4C22⋊Dic712yes—
257C22×C4C22×Dic711yescharacteristic
258C22×C4C2×C22⋊D712yes—
259M16—282no—
260M16—282no—
261C2×C8—282no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—282no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—22no—
281C22⋊C4—282no—
282C22⋊C4C22⋊Dic712yes—
283C42—24no—
284C42—24no—
285C42—21no—
286C42—21no—
287C42—22no—
288C42—282no—
289C42C22⋊Dic712yes—
290C42C22⋊Dic711yescharacteristic
291C42C22⋊Dic711yescharacteristic
292C42C22⋊Dic712yes—
293C2×C7—21no—
294C2×C7—42no—
295C2×C7—41no—
296C2×C7#178211yescharacteristic
297C23C2×C22⋊Dic711yescharacteristic
298C23#651612yes—
299Q8—41no—
300Q8—42no—
301Q8—44no—
302Q8—42no—
303Q8—42no—
304Q8—42no—
305D4—42no—
306D4—41no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—562no—
322C2×C4C2×C22⋊Dic711yescharacteristic
323C2×C4C2×C22⋊Dic711yescharacteristic
324C2×C4#651612yes—
325C8—562no—
326C7#1272411yescharacteristic, complemented, Sylow
327C22—21no—
328C22—22no—
329C22—41no—
330C22—42no—
331C22#9001911yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—1122no—
342C4#9001212yes—
343C2—21no—
344C2—42no—
345C2—41no—
346C2#1172989311yescharacteristic
subgroupG/H
centerC2#11729893
commutatorC22×C4×C7C22×C4
FrattiniC2×#41C22×D7
FittingC7×#3509C2
socleC2×C7#1782
prime radical, cosocleC2×C7×#41C23
hypercenter#3509D7
hypocenterC7#12724
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ #3509

Complements

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

NH ≅ G/Ntype
13. C7×#533C4semidirect
14. C2×C7×C4⋊Q8C4semidirect
15. C7×#519C4semidirect
16. C7×#519C4semidirect
137. #533Dic7semidirect
138. C2×C4⋊Q8Dic7semidirect
139. #519Dic7semidirect
140. #519Dic7semidirect
326. C7#12724semidirect

Maximal subgroups

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

subgroupindex
C7×#35092
#117296272
#117296252
#117296242
#117293982
#127247