Finite Group Databasebeta

← #13328549

Subgroups — #13328549

Order 1792.

978 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 207 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×#3501C211yesmaximal, characteristic
2#11740815C212yesmaximal, complemented
3#11739829C211yesmaximal, characteristic
4#11729647C218yesmaximal, complemented
5#11729589C212yesmaximal, complemented
6#11729355C211yesmaximal, characteristic
7C7×#554—22no—
8C7×#554—24no—
9C7×D4×Q8—22no—
10C7×#544—21no—
11C7×#543C2214yes—
12C7×#533C2212yes—
13C2×C7×C22⋊Q8—22no—
14C7×#518—21no—
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6432C2211yescharacteristic
17#89993—21no—
18#89984—22no—
19#89865—22no—
20#89864—22no—
21#89822C2218yes—
22#89803—24no—
23#89749—21no—
24#89719C2214yes—
25C4×C4⋊Dic7C2211yescharacteristic
26#89697C2212yes—
27#89333C2218yes—
28#89320C2214yes—
29#18390—71nomaximal, Sylow
30C22×C7×Q8—22no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7D412yes—
33C22×Dic14C2312yes—
34C7×C4⋊Q8—24no—
35C7×C4⋊Q8D412yes—
36C7×C4⋊Q8D412yes—
37C7×#125—24no—
38C7×#125—22no—
39C7×C22⋊Q8—24no—
40C7×C22⋊Q8—24no—
41C7×C22⋊Q8—28no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—28no—
45C7×C22⋊Q8C2314yes—
46C4×C7×Q8—22no—
47C4×C7×Q8D414yes—
48C4×C7×D4—24no—
49C4×C7×D4—24no—
50C4×C7×D4—22no—
51C4×C7×D4—22no—
52C7×#117—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4C2312yes—
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—24no—
60C2×#1144—28no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4×Dic7—21no—
65#6432—22no—
66#6432—22no—
67C7⋊#95—24no—
68C7⋊#95C2312yes—
69C22⋊(C7⋊C8)C2314yes—
70#6384—28no—
71C4⋊(C7⋊C8)—22no—
72C4×C7⋊C8—22no—
73#3501D711yescharacteristic
74#3209—72no—
75#2563—71no—
76#1602—78no—
77#1503—72no—
78#1460—71no—
79C24×C7C2×D412yes—
80C2×C7×Q8—24no—
81C2×C7×Q8—22no—
82C2×C7×Q8—84no—
83C2×C7×Q8C2×D414yes—
84C2×C7×D4—22no—
85C2×C7×D4—22no—
86C2×C7×D4Q1614yescomplemented
87C22×C4×C7—24no—
88C22×C4×C7—24no—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—44no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D412yes—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2411yescharacteristic
96C22×Dic7—22no—
97C22×Dic7—22no—
98C2×Dic14—24no—
99C2×Dic14—82no—
100C7×C4⋊C4—24no—
101C7×C4⋊C4—28no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—24no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—24no—
108C7×C4⋊C4C2×D414yes—
109C7×C22⋊C4—28no—
110C7×C22⋊C4—24no—
111C7×C22⋊C4—24no—
112C42×C7—24no—
113C42×C7C2×D412yes—
114C42×C7C2×D412yes—
115C22⋊Dic7—84no—
116#1144—88no—
117C4⋊Dic7—42no—
118C4⋊Dic7—41no—
119C4⋊Dic7—42no—
120C4⋊Dic7—41no—
121C4×Dic7—42no—
122C2×C7⋊C8—24no—
123#554—22no—
124#554—24no—
125D4×Q8—22no—
126#544—21no—
127#543C2×D714yes—
128#533C2×D712yes—
129C2×C4⋊Q8—71no—
130C2×C22⋊Q8—22no—
131#518—21no—
132C2×C4×D4C2×D711yescharacteristic
133#494—142no—
134#487—142no—
135#462—144no—
136#451—78no—
137#431—72no—
138#393—74no—
139#390—141no—
140#382—141no—
141#380—142no—
142C4×C4⋊C4—71no—
143#328—78no—
144#323—74no—
145C23×C7—22no—
146C23×C7—22no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7C2×Q1614yes—
150C23×C7C22×D411yescharacteristic
151C7×Q8—44no—
152C7×Q8—44no—
153C7×Q8—84no—
154C7×D4—28no—
155C2×C4×C7—22no—
156C2×C4×C7—24no—
157C2×C4×C7—24no—
158C2×C4×C7—22no—
159C2×C4×C7—41no—
160C2×C4×C7—44no—
161C2×C4×C7—42no—
162C2×C4×C7—82no—
163C2×C4×C7—84no—
164C2×C4×C7—84no—
165C2×C4×C7#12011yescharacteristic
166C2×C4×C7#12012yes—
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7C2×Q1612yes—
169C2×C4×C7#13712yes—
170C2×C4×C7C22×D412yes—
171C2×Dic7—42no—
172C2×Dic7—42no—
173C2×Dic7—44no—
174C2×Dic7—42no—
175C2×Dic7—42no—
176Dic14—48no—
177C7⋊C8—44no—
178C22×Q8—22no—
179C22×Q8—72no—
180C22×D4C22⋊D711yescharacteristic
181C23×C4C22⋊D712yes—
182C2×Q16—148no—
183C4⋊Q8—24no—
184C4⋊Q8—142no—
185C4⋊Q8—142no—
186C4⋊Q8C22⋊D712yes—
187C4⋊Q8C22⋊D712yes—
188#125—24no—
189#125—22no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—28no—
193C22⋊Q8—22no—
194C22⋊Q8—24no—
195C22⋊Q8—28no—
196C22⋊Q8C22×D714yes—
197C4×Q8—22no—
198C4×Q8C22⋊D714yes—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203#117—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—141no—
207C2×C4⋊C4—141no—
208C2×C4⋊C4—141no—
209C2×C4⋊C4C22⋊D711yescharacteristic
210C2×C22⋊C4—24no—
211C2×C22⋊C4—144no—
212C2×C22⋊C4C22×D712yes—
213C2×C42—141no—
214C2×C42C22×D711yescharacteristic
215C4⋊C8—142no—
216#103—148no—
217C22⋊C8—74no—
218C4×C8—142no—
219#95—72no—
220#95—144no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—24no—
224C22×C7—44no—
225C22×C7—42no—
226C22×C7—42no—
227C22×C7—82no—
228C22×C7C2×#12011yescharacteristic
229C22×C7C22×Q1611yescharacteristic
230C22×C7C2×#13711yescharacteristic
231C4×C7—21no—
232C4×C7—22no—
233C4×C7—42no—
234C4×C7—44no—
235C4×C7—44no—
236C4×C7#45114yes—
237C4×C7#45512yes—
238Dic7—82no—
239Dic7—82no—
240C24C2×C22⋊D712yes—
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—84no—
244C2×Q8—144no—
245C2×Q8—562no—
246C2×Q8C2×C22⋊D714yes—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#114414yescomplemented
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—44no—
255C22×C4—142no—
256C22×C4—142no—
257C22×C4C2×C22⋊D711yescharacteristic
258C22×C4C2×C22⋊D712yes—
259C22×C4C2×C22⋊D711yescharacteristic
260C22×C4C23×D711yescharacteristic
261Q16—568no—
262C2×C8—144no—
263C4⋊C4—24no—
264C4⋊C4—28no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—282no—
272C4⋊C4—281no—
273C4⋊C4—282no—
274C4⋊C4—281no—
275C4⋊C4C2×C22⋊D714yes—
276C22⋊C4—28no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—564no—
280C42—24no—
281C42—282no—
282C42C2×C22⋊D712yes—
283C42C2×C22⋊D712yes—
284C2×C7—21no—
285C2×C7—21no—
286C2×C7—44no—
287C2×C7C2×#45111yescharacteristic
288C2×C7C2×#45511yescharacteristic
289C2×C7#302711yescharacteristic
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23C2×#114414yes—
295C23C22×C22⋊D711yescharacteristic
296Q8—44no—
297Q8—44no—
298Q8—84no—
299Q8—288no—
300D4—28no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—282no—
312C2×C4—284no—
313C2×C4—282no—
314C2×C4—282no—
315C2×C4—282no—
316C2×C4#650512yes—
317C2×C4C2×#114412yes—
318C2×C4#651611yescharacteristic
319C2×C4#651612yes—
320C2×C4#651611yescharacteristic
321C2×C4C22×C22⋊D712yes—
322C8—284no—
323C7#1839011yescharacteristic, complemented, Sylow
324C22—21no—
325C22—21no—
326C22—24no—
327C22—44no—
328C22—42no—
329C22—42no—
330C22—82no—
331C22C2×#650511yescharacteristic
332C22C22×#114411yescharacteristic
333C22C2×#651611yescharacteristic
334C4—21no—
335C4—22no—
336C4—42no—
337C4—44no—
338C4—44no—
339C4—562no—
340C4—562no—
341C4#9000014yes—
342C4#9001412yes—
343C2—21no—
344C2—21no—
345C2—44no—
346C2C2×#9000011yescharacteristic
347C2C2×#9001411yescharacteristic
348C2#1174157111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#3501D7
hypocenterC7#18390
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #11740815C2semidirect
4. #11729647C2semidirect
5. #11729589C2semidirect
86. C2×C7×D4Q16semidirect
249. C2×D4#1144semidirect
323. C7#18390semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117408152
#117398292
#117296472
#117295892
#117293552
#183907