Finite Group Databasebeta

← #13256198

Subgroups — #13256198

Order 1792.

554 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2568C211yesmaximal, characteristic, complemented
2#11740658C211yesmaximal, characteristic
3#11739886C211yesmaximal, characteristic
4#11729528C211yesmaximal, characteristic, complemented
5#11729375C211yesmaximal, characteristic, complemented
6#11727223C211yesmaximal, characteristic
7#11725426C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C2×#6505—21no—
11#90168C2211yescharacteristic
12C7×#400—22no—
13C7×#391—22no—
14C7×#391—21no—
15C7×#382—21no—
16C4×C7×C4⋊C4C2211yescharacteristic
17#89999—22no—
18#89857—21no—
19#89855—22no—
20#89853—21no—
21#89803—21no—
22#89800—22no—
23#89799—21no—
24#89738—21no—
25#89697C2211yescharacteristic
26#89697C2211yescharacteristic
27#89696C2211yescharacteristic
28#89341—22no—
29Q8⋊(C7⋊C8)—22no—
30#89246—22no—
31#10244—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D412yes—
34C2×#1157—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7—21no—
50C2×C42×C7C2311yescharacteristic
51C2×#1144—22no—
52#6505—42no—
53#6505—42no—
54C2×#1143—22no—
55C2×#1143—24no—
56#6493—21no—
57#6493—21no—
58C2×#1136—21no—
59#6438—21no—
60#6437D411yescharacteristic
61C4×D28—21no—
62C4×D28—21no—
63#6432D411yescharacteristic
64C7×#95—24no—
65C7×#95—24no—
66C7×#95D412yes—
67#6409—24no—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70#6384—22no—
71#6383—22no—
72C4⋊(C7⋊C8)—21no—
73C4⋊(C7⋊C8)—21no—
74C4⋊(C7⋊C8)D412yes—
75C4⋊(C7⋊C8)D412yes—
76C4×C7⋊C8—21no—
77C4×C7⋊C8—21no—
78#3206—71no—
79#3037—71no—
80#2568D711yescharacteristic, complemented
81#1548—71no—
82#1502—71no—
83#1479—71no—
84#1303—71no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—22no—
88C2×C7×Q8—21no—
89C2×C7×Q8—22no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—44no—
93C2×C7×Q8SD1612yescomplemented
94C22×C4×C7—21no—
95C22×C4×C7—22no—
96C22×C4×C7—21no—
97C22×C4×C7—22no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C2×C22⋊D7—41no—
102#1157—41no—
103#1157—41no—
104#1157—41no—
105#1157—41no—
106C2×D28—21no—
107C2×D28—21no—
108C2×D28—21no—
109C2×C4×D7—41no—
110C2×Dic14—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—42no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C42×C7—41no—
121C42×C7—41no—
122C42×C7—41no—
123C42×C7—41no—
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128#1144—44no—
129#1143—44no—
130#1143—84no—
131(C2×D7)⋊C4—41no—
132(C2×D7)⋊C4—41no—
133C4⋊Dic7—21no—
134C4⋊Dic7—21no—
135#1136—41no—
136#1136—41no—
137C2×C7⋊C8—22no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—41no—
140C2×#137—141no—
141#532—71no—
142C2×C4⋊Q8C2×D711yescharacteristic
143C2×C4×Q8C2×D711yescharacteristic
144#492—141no—
145#487—141no—
146#464—142no—
147#462—141no—
148#461—142no—
149#460—141no—
150#448—142no—
151#431—71no—
152#431—71no—
153#423—71no—
154#417—141no—
155#400—22no—
156#391—22no—
157#391—21no—
158#382—21no—
159C4×C4⋊C4C2×D711yescharacteristic
160#333—142no—
161#327—142no—
162Q8⋊C8—142no—
163C23×C7#12011yescharacteristic
164C22×D7—41no—
165C7×Q8—22no—
166C7×Q8—22no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—44no—
170C7×Q8—82no—
171C2×C4×C7—22no—
172C2×C4×C7—22no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—44no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7#12011yescharacteristic
184C2×C4×C7#12011yescharacteristic
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12711yescharacteristic
189C2×C4×C7C2×SD1611yescharacteristic
190C2×C4×C7#13511yescharacteristic
191C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C7#13711yescharacteristic
193C22⋊D7—81no—
194C22⋊D7—81no—
195C2×Dic7—41no—
196D28—41no—
197D28—41no—
198D28—42no—
199D28—81no—
200C4×D7—81no—
201C4×D7—81no—
202Dic14—41no—
203Dic14—41no—
204C7⋊C8—42no—
205C7⋊C8—42no—
206C7⋊C8—81no—
207C2×#41—141no—
208C22×Q8C22⋊D711yescharacteristic
209C22×Q8C22⋊D712yes—
210#137—282no—
211#137—282no—
212C2×Q16—142no—
213C2×SD16—142no—
214C2×SD16—144no—
215C2×M16—141no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—71no—
221#127—71no—
222#124—141no—
223#121—141no—
224#121—141no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×D4—141no—
230C4×D4—141no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22⋊D711yescharacteristic
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C22×D711yescharacteristic
239C4⋊C8—72no—
240C4⋊C8—72no—
241C4⋊C8—141no—
242C4⋊C8—141no—
243#103—142no—
244#103—144no—
245#102—142no—
246C22⋊C8—141no—
247C22⋊C8—141no—
248C4×C8—141no—
249C4×C8—141no—
250#95—24no—
251#95—24no—
252#95C22⋊D712yes—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7#39211yescharacteristic
256C22×C7#44811yescharacteristic
257C22×C7#44911yescharacteristic
258C2×D7—81no—
259C2×D7—81no—
260C2×D7—81no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—42no—
265C4×C7—82no—
266C4×C7—82no—
267C4×C7—81no—
268C4×C7#45011yescharacteristic
269C4×C7#45211yescharacteristic
270C4×C7#45512yes—
271C4×C7#46511yescharacteristic
272C4×C7#46811yescharacteristic
273Dic7—81no—
274#41—281no—
275#41—281no—
276#41—281no—
277#41—281no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—44no—
286C2×Q8—141no—
287C2×Q8#114312yescomplemented
288C2×D4—141no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—281no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—281no—
297C22×C4C2×C22⋊D711yescharacteristic
298C22×C4C2×C22⋊D711yescharacteristic
299C22×C4C2×C22⋊D711yescharacteristic
300Q16—284no—
301SD16—284no—
302SD16—564no—
303M16—281no—
304M16—281no—
305C2×C8—142no—
306C2×C8—142no—
307C2×C8—281no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—141no—
318C4⋊C4—141no—
319C22⋊C4—281no—
320C22⋊C4—281no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42D4×D711yescharacteristic
326C42D4×D711yescharacteristic
327C42D4×D711yescharacteristic
328C42#115911yescharacteristic
329C2×C7—21no—
330C2×C7—21no—
331C2×C7#200011yescharacteristic
332C2×C7#200411yescharacteristic
333C2×C7#200911yescharacteristic
334D7—161no—
335C23—281no—
336C23#651611yescharacteristic
337Q8—22no—
338Q8—22no—
339Q8—41no—
340Q8—41no—
341Q8—44no—
342Q8—82no—
343Q8—281no—
344Q8—281no—
345D4—281no—
346D4—281no—
347D4—282no—
348D4—561no—
349D4—561no—
350D4—561no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—281no—
364C2×C4—561no—
365C2×C4—561no—
366C2×C4#650011yescharacteristic
367C2×C4#650011yescharacteristic
368C2×C4#650111yescharacteristic
369C2×C4#650211yescharacteristic
370C2×C4#650211yescharacteristic
371C2×C4#650311yescharacteristic
372C2×C4C2×#114311yescharacteristic
373C2×C4#650511yescharacteristic
374C2×C4#651211yescharacteristic
375C2×C4#651311yescharacteristic
376C8—282no—
377C8—282no—
378C8—561no—
379C7#1024411yescharacteristic, complemented, Sylow
380C22—21no—
381C22—21no—
382C22—561no—
383C22—561no—
384C22—561no—
385C22#8999511yescharacteristic
386C22#8999911yescharacteristic
387C22#9001011yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—82no—
393C4—82no—
394C4—81no—
395C4—561no—
396C4#8981811yescharacteristic
397C4#8981911yescharacteristic
398C4#8982011yescharacteristic
399C4#8982111yescharacteristic
400C4#8983312yes—
401C2—21no—
402C2—21no—
403C2—1121no—
404C2#1172945311yescharacteristic
405C2#1172948511yescharacteristic
406C2#1172962311yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2568C2
socleC22×C7#392
prime radical#90168C22
hypercenter#2568D7
hypocenterC7#10244
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
1. C7×#2568C2semidirect
4. #11729528C2semidirect
5. #11729375C2semidirect
7. #11725426C2semidirect
80. #2568D7semidirect
93. C2×C7×Q8SD16semidirect
287. C2×Q8#1143semidirect
379. C7#10244semidirect

Maximal subgroups

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

subgroupindex
C7×#25682
#117406582
#117398862
#117295282
#117293752
#117272232
#117254262
#102447