Finite Group Databasebeta

← #13326861

Subgroups — #13326861

Order 1792.

974 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 203 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#11740813C212yesmaximal, complemented
3#11739833C211yesmaximal, characteristic
4#11729647C214yesmaximal, complemented
5#11729645C214yesmaximal, complemented
6#11729592C212yesmaximal, complemented
7#11729355C211yesmaximal, characteristic
8C7×#554—22no—
9C7×#554—24no—
10C7×D4×Q8—22no—
11C7×#544—21no—
12C7×#543C2214yes—
13C7×#533C2212yes—
14C2×C7×C22⋊Q8—22no—
15C7×#518—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×Dic14C2211yescharacteristic
18#89993—21no—
19#89982—22no—
20#89863—22no—
21#89862—22no—
22#89822C2214yes—
23#89816C2214yes—
24C4×#1144—24no—
25#89745—21no—
26#89721C2212yes—
27#89719C2212yes—
28#89708C2211yescharacteristic
29#89697C2212yes—
30#89333C2218yes—
31#89320C2214yes—
32#18213—71nomaximal, Sylow
33C22×C7×Q8—22no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7D412yes—
36C22×Dic14C2311yescharacteristic
37C7×C4⋊Q8—24no—
38C7×C4⋊Q8D412yes—
39C7×C4⋊Q8D412yes—
40C7×#125—24no—
41C7×#125—22no—
42C7×C22⋊Q8—24no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—28no—
45C7×C22⋊Q8—22no—
46C7×C22⋊Q8—24no—
47C7×C22⋊Q8—28no—
48C7×C22⋊Q8C2314yes—
49C4×C7×Q8—22no—
50C4×C7×Q8D414yes—
51C4×C7×D4—24no—
52C4×C7×D4—24no—
53C4×C7×D4—22no—
54C4×C7×D4—22no—
55C7×#117—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4C2312yes—
61C2×C42×C7C2311yescharacteristic
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64C2×#1144—24no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7C2311yescharacteristic
67C2×Dic7⋊C4—21no—
68C2×Dic7⋊C4—22no—
69C2×C4×Dic7—21no—
70C4×Dic14—22no—
71C4×Dic14—22no—
72#6409—24no—
73C7⋊#95—21no—
74C7⋊#95—22no—
75C7⋊#95C2312yes—
76C22⋊(C7⋊C8)C2314yes—
77#6384—24no—
78#6381—24no—
79C4⋊(C7⋊C8)—22no—
80C4×C7⋊C8—22no—
81#3501D711yescharacteristic
82#3104—72no—
83#2580—71no—
84#1614—74no—
85#1602—74no—
86#1518—72no—
87#1460—71no—
88C24×C7C2×D412yes—
89C2×C7×Q8—24no—
90C2×C7×Q8—22no—
91C2×C7×Q8—84no—
92C2×C7×Q8C2×D414yes—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4Q1614yescomplemented
96C22×C4×C7—24no—
97C22×C4×C7—24no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—82no—
101C22×C4×C7C2×D412yes—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2411yescharacteristic
105C22×Dic7—21no—
106C22×Dic7—22no—
107C22×Dic7—21no—
108C2×Dic14—22no—
109C2×Dic14—42no—
110C7×C4⋊C4—24no—
111C7×C4⋊C4—28no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—24no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—24no—
118C7×C4⋊C4C2×D414yes—
119C7×C22⋊C4—28no—
120C7×C22⋊C4—24no—
121C7×C22⋊C4—24no—
122C42×C7—24no—
123C42×C7C2×D412yes—
124C42×C7C2×D412yes—
125C22⋊Dic7—44no—
126C22⋊Dic7—82no—
127#1144—48no—
128C4⋊Dic7—22no—
129C4⋊Dic7—42no—
130Dic7⋊C4—41no—
131Dic7⋊C4—41no—
132Dic7⋊C4—82no—
133C4×Dic7—42no—
134C2×C7⋊C8—24no—
135#554—22no—
136#554—24no—
137D4×Q8—22no—
138#544—21no—
139#543C2×D714yes—
140#533C2×D712yes—
141C2×C22⋊Q8—22no—
142#518—21no—
143C2×C4×Q8—71no—
144C2×C4×D4C2×D711yescharacteristic
145#484—74no—
146#477—142no—
147#462—142no—
148#451—74no—
149C4×Q16—144no—
150#431—72no—
151#399—72no—
152#393—72no—
153#390—141no—
154#389—141no—
155#385—142no—
156#382—71no—
157#328—78no—
158#323—74no—
159C23×C7—22no—
160C23×C7—22no—
161C23×C7—42no—
162C23×C7—42no—
163C23×C7C2×Q1614yes—
164C23×C7C22×D411yescharacteristic
165C7×Q8—44no—
166C7×Q8—44no—
167C7×Q8—84no—
168C7×D4—28no—
169C2×C4×C7—24no—
170C2×C4×C7—24no—
171C2×C4×C7—22no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—44no—
175C2×C4×C7—42no—
176C2×C4×C7—82no—
177C2×C4×C7—84no—
178C2×C4×C7—84no—
179C2×C4×C7#12011yescharacteristic
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7#12012yes—
182C2×C4×C7C2×Q1612yes—
183C2×C4×C7#13712yes—
184C2×C4×C7C22×D412yes—
185C2×Dic7—22no—
186C2×Dic7—41no—
187C2×Dic7—41no—
188C2×Dic7—41no—
189C2×Dic7—42no—
190C2×Dic7—42no—
191C2×Dic7—41no—
192C2×Dic7—41no—
193C2×Dic7—42no—
194Dic14—24no—
195Dic14—42no—
196C7⋊C8—44no—
197C22×Q8—22no—
198C22×Q8—71no—
199C22×D4C22⋊D711yescharacteristic
200C23×C4C22⋊D712yes—
201C2×Q16—144no—
202C4⋊Q8—24no—
203C4⋊Q8C22⋊D712yes—
204C4⋊Q8C22⋊D712yes—
205#125—24no—
206#125—22no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8—28no—
210C22⋊Q8—22no—
211C22⋊Q8—24no—
212C22⋊Q8—28no—
213C22⋊Q8C22×D714yes—
214C4×Q8—22no—
215C4×Q8—142no—
216C4×Q8—142no—
217C4×Q8C22⋊D714yes—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—22no—
221C4×D4—22no—
222#117—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—71no—
226C2×C4⋊C4—141no—
227C2×C4⋊C4—141no—
228C2×C4⋊C4—142no—
229C2×C4⋊C4C22⋊D711yescharacteristic
230C2×C22⋊C4—24no—
231C2×C22⋊C4—142no—
232C2×C22⋊C4—142no—
233C2×C22⋊C4C22×D712yes—
234C2×C42—141no—
235C2×C42C22×D711yescharacteristic
236#107—144no—
237C4⋊C8—142no—
238#103—144no—
239#103—144no—
240C22⋊C8—74no—
241C4×C8—142no—
242#95—72no—
243#95—141no—
244#95—142no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—24no—
248C22×C7—44no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7—82no—
252C22×C7C2×#12011yescharacteristic
253C22×C7C22×Q1611yescharacteristic
254C22×C7C2×#13711yescharacteristic
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—22no—
258C4×C7—42no—
259C4×C7—44no—
260C4×C7—44no—
261C4×C7#45114yes—
262Dic7—42no—
263Dic7—82no—
264Dic7—81no—
265C24C2×C22⋊D712yes—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—84no—
269C2×Q8—142no—
270C2×Q8—282no—
271C2×Q8C2×C22⋊D714yes—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4#114414yescomplemented
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—82no—
280C22×C4—141no—
281C22×C4—142no—
282C22×C4—141no—
283C22×C4C2×C22⋊D712yes—
284C22×C4C2×C22⋊D711yescharacteristic
285C22×C4C2×C22⋊D711yescharacteristic
286C22×C4C23×D711yescharacteristic
287Q16—288no—
288C2×C8—144no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—142no—
298C4⋊C4—282no—
299C4⋊C4—281no—
300C4⋊C4—281no—
301C4⋊C4—562no—
302C4⋊C4C2×C22⋊D714yes—
303C22⋊C4—28no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—284no—
307C22⋊C4—562no—
308C42—24no—
309C42—282no—
310C42C2×C22⋊D712yes—
311C42C2×C22⋊D712yes—
312C2×C7—21no—
313C2×C7—21no—
314C2×C7—44no—
315C2×C7C2×#45111yescharacteristic
316C2×C7#301911yescharacteristic
317C2×C7#302511yescharacteristic
318C23—22no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23C2×#114414yes—
323C23C22×C22⋊D711yescharacteristic
324Q8—44no—
325Q8—44no—
326Q8—84no—
327Q8—144no—
328Q8—282no—
329D4—28no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—142no—
341C2×C4—281no—
342C2×C4—281no—
343C2×C4—282no—
344C2×C4—281no—
345C2×C4—282no—
346C2×C4—281no—
347C2×C4—282no—
348C2×C4—281no—
349C2×C4#650512yes—
350C2×C4C2×#114412yes—
351C2×C4#651611yescharacteristic
352C2×C4#651611yescharacteristic
353C2×C4#651612yes—
354C2×C4C22×C22⋊D712yes—
355C8—284no—
356C7#1821311yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—24no—
360C22—44no—
361C22—42no—
362C22—42no—
363C22—82no—
364C22C2×#650511yescharacteristic
365C22C22×#114411yescharacteristic
366C22C2×#651611yescharacteristic
367C4—21no—
368C4—21no—
369C4—22no—
370C4—42no—
371C4—44no—
372C4—44no—
373C4—282no—
374C4—562no—
375C4—561no—
376C4#9000014yes—
377C2—21no—
378C2—21no—
379C2—44no—
380C2C2×#9000011yescharacteristic
381C2#1174154911yescharacteristic
382C2#1174157011yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#3501D7
hypocenterC7#18213
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. #11740813C2semidirect
4. #11729647C2semidirect
5. #11729645C2semidirect
6. #11729592C2semidirect
95. C2×C7×D4Q16semidirect
274. C2×D4#1144semidirect
356. C7#18213semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117408132
#117398332
#117296472
#117296452
#117295922
#117293552
#182137