Finite Group Databasebeta

← #13251456

Subgroups — #13251456

Order 1792.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2544C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic, complemented
3#11729473C212yesmaximal, complemented
4#11729357C211yesmaximal, characteristic, complemented
5#11725427C212yesmaximal, complemented
6C2×C7×#127C411yescharacteristic
7C2×C7×#124C411yescharacteristic
8C2×C7×#124C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#394—21no—
11C7×#394—21no—
12C7×#391C411yescharacteristic
13C7×#390C411yescharacteristic
14C7×#386—21no—
15C7×#386—21no—
16C4×C7×C4⋊C4C2211yescharacteristic
17#89979—21no—
18C2×C22⋊(C7⋊C8)—21no—
19#89783—24no—
20#89698C2212yes—
21#89698C2212yes—
22#89335—24no—
23#89320—21no—
24#89320—21no—
25#9447—71nomaximal, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4—21no—
28C22×C7×D4—21no—
29C22×C7×D4C2×C411yescharacteristic
30C22×C7×D4C2×C411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C7×#127—42no—
34C7×#124—22no—
35C7×#124—22no—
36C7×#124—42no—
37C4×C7×D4—22no—
38C4×C7×D4—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C42×C7C2×C411yescharacteristic
50C2×C42×C7C2×C411yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C2×#1136—22no—
53C22×C7⋊C8—22no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C22⋊(C7⋊C8)—22no—
58C22⋊(C7⋊C8)—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C4⋊(C7⋊C8)—22no—
62C4⋊(C7⋊C8)—22no—
63#6377—22no—
64#6377D414yescomplemented
65#2973—71no—
66#2544D711yescharacteristic
67#1533—72no—
68#1466—71no—
69#1285—72no—
70C24×C7—21no—
71C24×C7C22⋊C411yescharacteristic
72C24×C7C22⋊C411yescharacteristic
73C2×C7×Q8—22no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×D4—22no—
77C2×C7×D4—22no—
78C2×C7×D4—22no—
79C2×C7×D4—41no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—42no—
85C2×C7×D4—42no—
86C2×C7×D4—81no—
87C2×C7×D4—81no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—81no—
96C22×C4×C7—81no—
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C7×C4⋊C4—22no—
103C7×C4⋊C4—42no—
104C7×C22⋊C4—22no—
105C7×C22⋊C4—22no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—81no—
112C7×C22⋊C4—81no—
113C42×C7—41no—
114C42×C7—41no—
115C42×C7C2×D412yes—
116C42×C7#4112yes—
117#1136—82no—
118C2×C7⋊C8—24no—
119C2×C7⋊C8—42no—
120C2×C7⋊C8—82no—
121C2×#127Dic711yescharacteristic
122C2×#124Dic711yescharacteristic
123C2×#124C2×D711yescharacteristic
124C2×C4×D4C2×D711yescharacteristic
125#435—144no—
126#432—72no—
127#432—72no—
128#407—141no—
129C2×C22⋊C8—141no—
130#394—21no—
131#394—21no—
132#391Dic711yescharacteristic
133#390Dic711yescharacteristic
134#386—21no—
135#386—21no—
136C4×C4⋊C4C2×D711yescharacteristic
137#329—144no—
138#323—141no—
139#323—141no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7#10012yes—
157C23×C7C2×C22⋊C411yescharacteristic
158C7×Q8—82no—
159C7×D4—44no—
160C7×D4—82no—
161C7×D4—84no—
162C7×D4—84no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—22no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7C4×D412yes—
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12112yes—
187C2×C4×C7#12311yescharacteristic
188C7⋊C8—44no—
189C7⋊C8—82no—
190C22×Q8C2×Dic711yescharacteristic
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2×Dic711yescharacteristic
194C22×D4C2×Dic711yescharacteristic
195C23×C4C22⋊D711yescharacteristic
196C23×C4C22⋊D711yescharacteristic
197C2×M16—142no—
198C22×C8—142no—
199#127—42no—
200#124—22no—
201#124—22no—
202#124—42no—
203C4×D4—22no—
204C4×D4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×Dic711yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4C22⋊D711yescharacteristic
214C2×C22⋊C4C22⋊D711yescharacteristic
215C2×C42C2×Dic711yescharacteristic
216C2×C42C2×Dic711yescharacteristic
217C2×C42C22×D711yescharacteristic
218C4⋊C8—142no—
219C4⋊C8—142no—
220C22⋊C8—142no—
221C22⋊C8—142no—
222C22⋊C8—142no—
223C22⋊C8—142no—
224#97—74no—
225#97—142no—
226#95—21no—
227#95—21no—
228#95—21no—
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—22no—
232C22×C7—42no—
233C22×C7—42no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7#38611yescharacteristic
249C22×C7C2×#10011yescharacteristic
250C22×C7#41311yescharacteristic
251C4×C7—22no—
252C4×C7—22no—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—42no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—81no—
259C4×C7—81no—
260C24—21no—
261C24C22⋊Dic711yescharacteristic
262C24C22⋊Dic711yescharacteristic
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—81no—
277C2×D4—81no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C22⋊Dic711yescharacteristic
288C22×C4C22⋊Dic711yescharacteristic
289C22×C4C22×Dic711yescharacteristic
290C22×C4C2×C22⋊D711yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292M16—562no—
293C2×C8—144no—
294C2×C8—282no—
295C2×C8—562no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C42—41no—
308C42—41no—
309C42D4×D712yes—
310C42#115912yes—
311C2×C7—21no—
312C2×C7—21no—
313C2×C7—42no—
314C2×C7—81no—
315C2×C7—161no—
316C2×C7#188411yescharacteristic
317C2×C7#190011yescharacteristic
318C2×C7#191111yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23#640712yes—
336C23C2×C22⋊Dic711yescharacteristic
337Q8—82no—
338D4—44no—
339D4—82no—
340D4—84no—
341D4—84no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4D4×Dic712yes—
364C2×C4#649811yescharacteristic
365C2×C4#650011yescharacteristic
366C2×C4#650212yes—
367C8—284no—
368C8—562no—
369C7#944711yescharacteristic, complemented, Sylow
370C22—21no—
371C22—21no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C2×#640711yescharacteristic
390C22#8999211yescharacteristic
391C22#9000911yescharacteristic
392C4—22no—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—161no—
406C2#1172943711yescharacteristic
407C2#1172957111yescharacteristic
408C2#1172960711yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2544C2
socleC22×C7#386
prime radicalC2×C7×#127C4
cosocleC2×C42×C7C23
hypercenter#2544D7
hypocenterC7#9447
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2544

Complements

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

NH ≅ G/Ntype
2. #11739747C2semidirect
3. #11729473C2semidirect
4. #11729357C2semidirect
5. #11725427C2semidirect
64. #6377D4semidirect
369. C7#9447semidirect

Maximal subgroups

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

subgroupindex
C7×#25442
#117397472
#117294732
#117293572
#117254272
#94477