Finite Group Databasebeta

← #13250698

Subgroups — #13250698

Order 1792.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2276C211yesmaximal, characteristic
2#11739893C211yesmaximal, characteristic
3#11729377C211yesmaximal, characteristic
4#11727247C211yesmaximal, characteristic
5#11727187C211yesmaximal, characteristic
6#11725427C211yesmaximal, characteristic
7#11725421C211yesmaximal, characteristic
8C2×C4×C7×Q8—21no—
9C2×C4×C7×Q8C2211yescharacteristic
10#90161C2211yescharacteristic
11C4×C7×C4⋊C4—21no—
12C4×C7×C4⋊C4—22no—
13C4×C7×C4⋊C4—21no—
14C4×C7×C4⋊C4C2211yescharacteristic
15C7×#376—21no—
16C7×#376C2211yescharacteristic
17#90000—21no—
18#89803C412yescomplemented
19#89802—22no—
20#89792C412yescomplemented
21#89743—21no—
22#89698C2211yescharacteristic
23#89698C2211yescharacteristic
24C4×#1136C2211yescharacteristic
25Q8⋊(C7⋊C8)—22no—
26#89318—22no—
27#89247—21no—
28#89247—21no—
29#9324—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C4×C7×Q8—28no—
32C4×C7×Q8—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8C2×C412yes—
35C4×C7×Q8D412yes—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C42×C7—22no—
41C2×C42×C7—21no—
42C2×C42×C7—21no—
43C2×C42×C7—22no—
44C2×C42×C7D412yes—
45C2×C42×C7C2311yescharacteristic
46C2×#1144—22no—
47#6489—21no—
48#6487—21no—
49#6487—21no—
50C2×#1136—21no—
51#6433—21no—
52#6432C2×C411yescharacteristic
53#6432C2×C411yescharacteristic
54C4×Dic14—21no—
55C4×Dic14—21no—
56C7×#95—22no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95D411yescharacteristic
60C7×#95D411yescharacteristic
61#6409—22no—
62C22⋊(C7⋊C8)—21no—
63C22⋊(C7⋊C8)—21no—
64#6384—22no—
65#6382—22no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)C2×C412yes—
68#6377—21no—
69#6377—21no—
70#6377—21no—
71C4×C7⋊C8—21no—
72#3113—71no—
73#2276D711yescharacteristic
74#1986—71no—
75#1506—71no—
76#1491—71no—
77#1300—71no—
78#1285—71no—
79C2×C7×Q8—24no—
80C2×C7×Q8—21no—
81C2×C7×Q8C22⋊C412yes—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7—21no—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C2×Dic14—21no—
91C2×Dic14—21no—
92C7×C4⋊C4—28no—
93C7×C4⋊C4—24no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4C22⋊C412yes—
98C42×C7—28no—
99C42×C7—24no—
100C42×C7—22no—
101C42×C7—22no—
102C42×C7—42no—
103C42×C7—82no—
104C42×C7C22×C411yescharacteristic
105C42×C7C2×D411yescharacteristic
106C42×C7#4111yescharacteristic
107C42×C7#4111yescharacteristic
108C22⋊Dic7—41no—
109#1144—44no—
110C4⋊Dic7—21no—
111C4⋊Dic7—21no—
112C4⋊Dic7—21no—
113C4⋊Dic7—21no—
114Dic7⋊C4—41no—
115Dic7⋊C4—41no—
116C4×Dic7—41no—
117#1136—22no—
118#1136—41no—
119C2×C7⋊C8—22no—
120C2×C7⋊C8—41no—
121C2×C7⋊C8—41no—
122#533—71no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2×D711yescharacteristic
125#483—141no—
126#475—72no—
127#462—72no—
128#451—141no—
129#441—142no—
130#432—71no—
131#432—71no—
132C4×M16—71no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2×D711yescharacteristic
137#376—21no—
138#376C2×D711yescharacteristic
139#339—142no—
140#328—141no—
141#328—141no—
142Q8⋊C8—142no—
143C23×C7#12011yescharacteristic
144C7×Q8—24no—
145C7×Q8—22no—
146C7×Q8#10414yes—
147C2×C4×C7—24no—
148C2×C4×C7—22no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—22no—
152C2×C4×C7—22no—
153C2×C4×C7—24no—
154C2×C4×C7—22no—
155C2×C4×C7—22no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7C2×C22⋊C411yescharacteristic
166C2×C4×C7C4×D411yescharacteristic
167C2×C4×C7C4×D411yescharacteristic
168C2×C4×C7#12111yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7#12311yescharacteristic
171C2×Dic7—41no—
172C2×Dic7—41no—
173Dic14—42no—
174Dic14—81no—
175C7⋊C8—42no—
176C7⋊C8—42no—
177C7⋊C8—81no—
178C22×Q8C22⋊D711yescharacteristic
179C2×Q16—142no—
180C2×M16—141no—
181C4⋊Q8—71no—
182C4⋊Q8—71no—
183#125—141no—
184C22⋊Q8—141no—
185C22⋊Q8—141no—
186C4×Q8—28no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—141no—
190C4×Q8—141no—
191C4×Q8C4×D712yes—
192C4×Q8D2812yes—
193#117—141no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊D711yescharacteristic
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42C22⋊D712yes—
203C2×C42C22×D711yescharacteristic
204#106—142no—
205C4⋊C8—72no—
206C4⋊C8—141no—
207#103—142no—
208#103—142no—
209C22⋊C8—141no—
210C22⋊C8—141no—
211#97—141no—
212#97—141no—
213#97—141no—
214C4×C8—141no—
215#95—22no—
216#95—21no—
217#95—21no—
218#95C22⋊D711yescharacteristic
219#95C22⋊D711yescharacteristic
220C22×C7—21no—
221C22×C7—21no—
222C22×C7#38611yescharacteristic
223C22×C7#45511yescharacteristic
224C22×C7#45611yescharacteristic
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—24no—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—44no—
232C4×C7—42no—
233C4×C7—42no—
234C4×C7—42no—
235C4×C7C2×#10412yes—
236Dic7—81no—
237Dic7—81no—
238C2×Q8—24no—
239C2×Q8—21no—
240C2×Q8—141no—
241C2×Q8—141no—
242C2×Q8(C2×D7)⋊C412yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4C2×C22⋊D711yescharacteristic
249C22×C4C2×C22⋊D711yescharacteristic
250C22×C4C2×C22⋊D711yescharacteristic
251Q16—284no—
252M16—142no—
253M16—281no—
254C2×C8—142no—
255C2×C8—281no—
256C2×C8—281no—
257C4⋊C4—28no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—281no—
267C4⋊C4—281no—
268C4⋊C4(C2×D7)⋊C412yes—
269C22⋊C4—281no—
270C42—28no—
271C42—24no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—82no—
276C42—281no—
277C42C2×C4×D711yescharacteristic
278C42C2×D2811yescharacteristic
279C42#115711yescharacteristic
280C42#115711yescharacteristic
281C2×C7—21no—
282C2×C7—21no—
283C2×C7#188211yescharacteristic
284C2×C7#189611yescharacteristic
285C2×C7#190511yescharacteristic
286C23#651611yescharacteristic
287Q8—24no—
288Q8—22no—
289Q8—282no—
290Q8—561no—
291Q8#637914yes—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—281no—
311C2×C4—281no—
312C2×C4C2×(C2×D7)⋊C411yescharacteristic
313C2×C4C4×C22⋊D711yescharacteristic
314C2×C4C4×C22⋊D711yescharacteristic
315C2×C4#649211yescharacteristic
316C2×C4#649311yescharacteristic
317C2×C4#649311yescharacteristic
318C8—282no—
319C8—282no—
320C8—561no—
321C7#932411yescharacteristic, complemented, Sylow
322C22—21no—
323C22—21no—
324C22#8998511yescharacteristic
325C22#9001411yescharacteristic
326C22#9001811yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—24no—
332C4—21no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—561no—
338C4—561no—
339C4C2×#637912yes—
340C2—21no—
341C2—21no—
342C2#1172928611yescharacteristic
343C2#1172941611yescharacteristic
344C2#1172942011yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2276C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#2276D7
hypocenterC7#9324
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2276

Complements

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

NH ≅ G/Ntype
18. #89803C4semidirect
20. #89792C4semidirect
321. C7#9324semidirect

Maximal subgroups

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

subgroupindex
C7×#22762
#117398932
#117293772
#117272472
#117271872
#117254272
#117254212
#93247