Finite Group Databasebeta

← #13509144

Subgroups — #13509144

Order 1792.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3164C211yesmaximal, characteristic, complemented
2#11736021C211yesmaximal, characteristic
3#11729137C212yesmaximal, complemented
4#11729136C212yesmaximal, complemented
5#11729107C212yesmaximal, complemented
6#11729105C212yesmaximal, complemented
7#11728408C211yesmaximal, characteristic, complemented
8#11728405C211yesmaximal, characteristic
9C2×#89287C211yesmaximal, characteristic, complemented
10#11728352C211yesmaximal, characteristic
11#11727635C211yesmaximal, characteristic, complemented
12C7×#544—21no—
13C2×#6472—21no—
14C22×Dic28C2211yescharacteristic
15C2×#6467C2211yescharacteristic
16#90217—21no—
17C7×#501—22no—
18C7×#500C2212yes—
19C7×#499C2212yes—
20C7×#428—21no—
21C7×#427C2211yescharacteristic
22C2×C7×#107C2211yescharacteristic
23C7×#405C2211yescharacteristic
24C2×#6386C2211yescharacteristic
25C22×C7⋊C16C2211yescharacteristic
26#89487—22no—
27#89471C2212yes—
28#89468C2212yes—
29#89287C2211yescharacteristic
30#89287C2211yescharacteristic
31#89287C2212yes—
32#89287C2212yes—
33#89287C2212yes—
34#89287C2212yes—
35#89287C2212yes—
36#89286C2211yescharacteristic
37#89286C2211yescharacteristic
38#89286C2212yes—
39#89256C2212yes—
40C4×C7⋊C16C2212yes—
41#33703—71nomaximal, Sylow
42C2×#1157—21no—
43C22×Dic14—21no—
44C2×C7×M16D411yescharacteristic
45C22×C7×C8C2311yescharacteristic
46C7×C4⋊Q8—22no—
47C7×C4⋊Q8—41no—
48C7×#125—22no—
49C7×#125—41no—
50C7×C22⋊Q8—42no—
51C7×#117—21no—
52C7×#117D411yescharacteristic
53C2×C7×C4⋊C4—21no—
54#6472—44no—
55C2×Dic28—22no—
56C2×Dic28—22no—
57C2×Dic28—21no—
58C2×Dic28—22no—
59C2×Dic28C2311yescharacteristic
60C2×Dic28C2312yes—
61#6467—22no—
62#6467—22no—
63C2×D56C2311yescharacteristic
64C2×#1132—21no—
65C2×#1132—22no—
66#6460—41no—
67#6460—41no—
68#6449—42no—
69#6438—22no—
70#6432—22no—
71C7×#107—22no—
72C7×#107—21no—
73C7×#107C2311yescharacteristic
74C7×#107C2312yes—
75C7×#107C2311yescharacteristic
76C7×#106—22no—
77C7×#106—21no—
78C7×#97D412yes—
79C4×C7×C8C2312yes—
80#6386—22no—
81C2×C7⋊C16—22no—
82C2×C7⋊C16C2311yescharacteristic
83C2×C7⋊C16C2311yescharacteristic
84C2×C7⋊C16C2312yes—
85#3164D711yescharacteristic, complemented
86#3156—71no—
87#2245—72no—
88#2243—72no—
89#2242—72no—
90#2241—72no—
91#2141—71no—
92#2139—71no—
93#2138—71no—
94C2×#358—71no—
95#2108—71no—
96C2×C7×Q8—42no—
97C22×C4×C7—41no—
98C22×C4×C7C2×D411yescharacteristic
99C2×C22⋊D7—41no—
100C22×Dic7—41no—
101#1157—42no—
102#1157—42no—
103C2×D28—21no—
104C2×C4×D7—41no—
105C2×Dic14—21no—
106C2×Dic14—22no—
107C2×Dic14—42no—
108C2×Dic14—42no—
109C2×Dic14—41no—
110C7×M16SD1614yes—
111C2×C7×C8C2×D411yescharacteristic
112C2×C7×C8C2×D412yes—
113C2×C7×C8#4112yes—
114C2×C7×C8#4112yes—
115C2×C7×C8C2411yescharacteristic
116C7×C4⋊C4—21no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—42no—
121C7×C4⋊C4—42no—
122C7×C4⋊C4—42no—
123C7×C4⋊C4D811yescharacteristic
124C7×C4⋊C4D811yescharacteristic
125C7×C22⋊C4—41no—
126C7×C22⋊C4D812yes—
127C42×C7—41no—
128C42×C7C2×D412yes—
129(C2×D7)⋊C4—42no—
130(C2×D7)⋊C4—42no—
131C4⋊Dic7—42no—
132C4⋊Dic7—42no—
133Dic28—21no—
134Dic28—21no—
135Dic28—22no—
136Dic28—22no—
137Dic28—42no—
138Dic28—44no—
139D56—21no—
140D56—21no—
141#1132—42no—
142#1132—44no—
143C7⋊C16—21no—
144C7⋊C16—21no—
145C7⋊C16—22no—
146C2×#137—141no—
147C2×#135—71no—
148C22×Q16—71no—
149#544—21no—
150#536—141no—
151C2×M32—71no—
152C22×C16—71no—
153#501—22no—
154#500C2×D712yes—
155#499C2×D712yes—
156#497—142no—
157#495—72no—
158#494—72no—
159#428—21no—
160#427C2×D711yescharacteristic
161C2×#107C2×D711yescharacteristic
162#405C2×D711yescharacteristic
163#358—72no—
164#358—72no—
165#358—71no—
166#358—71no—
167#358—72no—
168#358—72no—
169#358—72no—
170#357—71no—
171#357—71no—
172#357—72no—
173#346—72no—
174C4×C16—72no—
175C23×C7C2×D811yescharacteristic
176C22×D7—41no—
177C7×Q8—162no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—161no—
182C2×C4×C7#12412yes—
183C2×C4×C7C2×D812yes—
184C2×C4×C7C2×D812yes—
185C2×C4×C7C2×D811yescharacteristic
186C2×C4×C7C2×SD1612yes—
187C2×C4×C7C22×D411yescharacteristic
188C22⋊D7—82no—
189C2×Dic7—41no—
190C2×Dic7—42no—
191C2×Dic7—82no—
192D28—41no—
193D28—41no—
194C4×D7—82no—
195Dic14—41no—
196Dic14—42no—
197Dic14—42no—
198Dic14—41no—
199Dic14—82no—
200C7×C8#12412yes—
201C7×C8C2×SD1614yes—
202C7×C8C2×#4111yescharacteristic
203C7×C8C2×#4111yescharacteristic
204C2×#41—141no—
205C22×Q8—141no—
206#137—284no—
207#135—142no—
208#135—142no—
209C2×Q16—72no—
210C2×Q16—71no—
211C2×Q16—142no—
212C2×Q16—142no—
213C2×Q16—141no—
214C2×Q16—142no—
215C2×SD16—141no—
216C2×SD16—142no—
217C2×D8—71no—
218C2×M16C22⋊D711yescharacteristic
219C22×C8C22×D711yescharacteristic
220C4⋊Q8—22no—
221C4⋊Q8—41no—
222C4⋊Q8—142no—
223#125—22no—
224#125—41no—
225#124—142no—
226#123—282no—
227C22⋊Q8—42no—
228C22⋊Q8—281no—
229C22⋊Q8—281no—
230#117—21no—
231#117C22⋊D711yescharacteristic
232C2×C4⋊C4—21no—
233M32—142no—
234C2×C16—71no—
235C2×C16—72no—
236C2×C16—71no—
237C2×C16—142no—
238#107—21no—
239#107—22no—
240#107C22×D712yes—
241#107C22×D711yescharacteristic
242#107C22×D711yescharacteristic
243#106—21no—
244#106—22no—
245#97C22⋊D712yes—
246C4×C8C22×D712yes—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7#48612yes—
250C22×C7C22×D811yescharacteristic
251C2×D7—81no—
252C2×D7—81no—
253C2×D7—81no—
254C4×C7—22no—
255C4×C7—22no—
256C4×C7—82no—
257C4×C7—81no—
258C4×C7—81no—
259C4×C7#48612yes—
260C4×C7C2×#12411yescharacteristic
261C4×C7C22×SD1611yescharacteristic
262Dic7—82no—
263Dic7—81no—
264#41—282no—
265#41—282no—
266C2×Q8—42no—
267C2×Q8—142no—
268C2×Q8—141no—
269C2×Q8—281no—
270C2×Q8—282no—
271C2×Q8—282no—
272C2×D4—141no—
273C2×D4—281no—
274C22×C4—41no—
275C22×C4—281no—
276C22×C4—281no—
277C22×C4C2×C22⋊D711yescharacteristic
278Q16—142no—
279Q16—142no—
280Q16—141no—
281Q16—141no—
282Q16—284no—
283Q16—282no—
284SD16—284no—
285SD16—282no—
286D8—141no—
287D8—141no—
288M16#114314yes—
289C2×C8#116112yes—
290C2×C8#116112yes—
291C2×C8C2×C22⋊D712yes—
292C2×C8C2×C22⋊D711yescharacteristic
293C2×C8C23×D711yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—282no—
302C4⋊C4—282no—
303C4⋊C4#114111yescharacteristic
304C4⋊C4#114111yescharacteristic
305C22⋊C4—41no—
306C22⋊C4—282no—
307C22⋊C4—282no—
308C22⋊C4#114112yes—
309C42—41no—
310C42C2×C22⋊D712yes—
311C16—141no—
312C16—141no—
313C16—142no—
314C2×C7—22no—
315C2×C7C2×#48611yescharacteristic
316C2×C7#341611yescharacteristic
317C2×C7#341711yescharacteristic
318D7—161no—
319C23—281no—
320C23C2×#114111yescharacteristic
321Q8—162no—
322Q8—282no—
323Q8—281no—
324Q8—281no—
325Q8—282no—
326Q8—562no—
327D4—281no—
328D4—281no—
329D4—562no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—161no—
334C2×C4—282no—
335C2×C4—281no—
336C2×C4—562no—
337C2×C4—562no—
338C2×C4C2×#114112yes—
339C2×C4C2×#114112yes—
340C2×C4C2×#114111yescharacteristic
341C2×C4C2×#114312yes—
342C2×C4#651012yes—
343C2×C4C22×C22⋊D711yescharacteristic
344C8C2×#114314yes—
345C8#651012yes—
346C8C2×#116111yescharacteristic
347C8C2×#116111yescharacteristic
348C7#3370311yescharacteristic, complemented, Sylow
349C22—21no—
350C22—21no—
351C22—561no—
352C22—561no—
353C22—561no—
354C22#8986012yes—
355C22C22×#114111yescharacteristic
356C4—22no—
357C4—22no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—561no—
362C4—562no—
363C4#8986012yes—
364C4C22×#114311yescharacteristic
365C4C2×#651011yescharacteristic
366C2—22no—
367C2—1121no—
368C2C2×#8986011yescharacteristic
369C2#1174145311yescharacteristic
370C2#1174146211yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#3164C2
socleC22×C7C22×D8
prime radicalC2×#6467C22
hypercenter#3164D7
hypocenterC7#33703
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
1. C7×#3164C2semidirect
3. #11729137C2semidirect
4. #11729136C2semidirect
5. #11729107C2semidirect
6. #11729105C2semidirect
7. #11728408C2semidirect
9. C2×#89287C2semidirect
11. #11727635C2semidirect
85. #3164D7semidirect
348. C7#33703semidirect

Maximal subgroups

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

subgroupindex
C7×#31642
#117360212
#117291372
#117291362
#117291072
#117291052
#117284082
#117284052
C2×#892872
#117283522
#117276352
#337037