Finite Group Databasebeta

← #13257011

Subgroups — #13257011

Order 1792.

438 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 77 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×#2667C211yesmaximal, characteristic, complemented
2#11727285C211yesmaximal, characteristic
3#11727282C211yesmaximal, characteristic
4#11727198C211yesmaximal, characteristic
5#11725434C211yesmaximal, characteristic, complemented
6#11725432C211yesmaximal, characteristic, complemented
7#11725424C211yesmaximal, characteristic, complemented
8C2×#6493C2211yescharacteristic
9C7×#399—21no—
10C7×#398—21no—
11C7×#398—21no—
12C7×#396—21no—
13C7×#389—21no—
14C7×#387—21no—
15C7×#381C2211yescharacteristic
16C7×#380C2211yescharacteristic
17C4×C7×C22⋊C4C2211yescharacteristic
18C4×C22⋊Dic7C2211yescharacteristic
19#89979C2211yescharacteristic
20#89979C2211yescharacteristic
21C2×#6380—21no—
22C2×#6380—21no—
23#89246—21no—
24#89246—21no—
25#89246—21no—
26#89246—21no—
27#10384—71nomaximal, Sylow
28C23×C4×C7C2311yescharacteristic
29C22×C22⋊D7—21no—
30C22×D28Q811yescharacteristic
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C42×C7—21no—
43C2×C22⋊Dic7—21no—
44#6493—21no—
45#6493—21no—
46#6493—21no—
47#6493—21no—
48C2×(C2×D7)⋊C4—21no—
49C2×C4⋊Dic7Q811yescharacteristic
50C2×C4×Dic7—21no—
51C2×#1136—21no—
52C2×#1136—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95D411yescharacteristic
58C7×#95D411yescharacteristic
59C7×#95D411yescharacteristic
60C7×#95D411yescharacteristic
61C7⋊#95—21no—
62C22⋊(C7⋊C8)—21no—
63C22⋊(C7⋊C8)—21no—
64C22⋊(C7⋊C8)—21no—
65C22⋊(C7⋊C8)—21no—
66#6380—42no—
67#6380—42no—
68#2667D711yescharacteristic, complemented
69#1964—71no—
70#1510—71no—
71#1507—71no—
72#1305—71no—
73#1297—71no—
74#1284—71no—
75C24×C7C2×D411yescharacteristic
76C23×D7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—81no—
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×Q811yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C2×C22⋊D7—41no—
91C2×C22⋊D7—41no—
92C2×C22⋊D7—41no—
93C2×C22⋊D7—41no—
94C2×C22⋊D7—41no—
95C2×C22⋊D7—41no—
96C22×Dic7—21no—
97C22×Dic7—41no—
98C2×D28—21no—
99C2×D28—21no—
100C2×D28—81no—
101C7×C4⋊C4—42no—
102C7×C4⋊C4—42no—
103C7×C4⋊C4—81no—
104C7×C4⋊C4—81no—
105C7×C4⋊C4—81no—
106C7×C4⋊C4—81no—
107C7×C4⋊C4—81no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—81no—
112C42×C7—81no—
113C22⋊Dic7—22no—
114C22⋊Dic7—41no—
115(C2×D7)⋊C4—41no—
116(C2×D7)⋊C4—41no—
117C4⋊Dic7—21no—
118C4⋊Dic7—21no—
119C4×Dic7—81no—
120#1136—81no—
121#1136—81no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×#121—71no—
125C2×#99—141no—
126C2×#99—141no—
127#407—71no—
128#407—71no—
129#399—21no—
130#398—21no—
131#398—21no—
132#396—21no—
133#389—21no—
134#387—21no—
135#381C2×D711yescharacteristic
136#380C2×D711yescharacteristic
137C4×C22⋊C4—71no—
138C4×C22⋊C4C2×D711yescharacteristic
139#327—141no—
140#327—141no—
141#327—141no—
142#327—141no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7#12011yescharacteristic
148C23×C7C22⋊Q811yescharacteristic
149C23×C7#12411yescharacteristic
150C22×D7—41no—
151C22×D7—41no—
152C22×D7—41no—
153C22×D7—81no—
154C22×D7—81no—
155C2×C4×C7—22no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
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×C7C22⋊Q811yescharacteristic
180C2×C4×C7C22⋊Q811yescharacteristic
181C2×C4×C7#12411yescharacteristic
182C2×C4×C7#12411yescharacteristic
183C22⋊D7—81no—
184C22⋊D7—81no—
185C22⋊D7—81no—
186C22⋊D7—81no—
187C2×Dic7—42no—
188C2×Dic7—41no—
189C2×Dic7—41no—
190C2×Dic7—41no—
191C2×Dic7—81no—
192C2×Dic7—81no—
193D28—81no—
194D28—81no—
195C7⋊C8—81no—
196C7⋊C8—81no—
197C22×D4—71no—
198C22×D4—141no—
199C23×C4C22×D711yescharacteristic
200C2×M16—141no—
201C2×M16—141no—
202#121—141no—
203#121—141no—
204#121—141no—
205#121—141no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—71no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—141no—
217C2×C22⋊C4—141no—
218C2×C22⋊C4C22⋊D711yescharacteristic
219C2×C22⋊C4C22⋊D711yescharacteristic
220C2×C42—21no—
221C2×C42—141no—
222#99—282no—
223#99—282no—
224C22⋊C8—141no—
225C22⋊C8—141no—
226C22⋊C8—141no—
227C22⋊C8—141no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—141no—
233#95C22⋊D711yescharacteristic
234#95C22⋊D711yescharacteristic
235#95C22⋊D711yescharacteristic
236#95C22⋊D711yescharacteristic
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7#39311yescharacteristic
248C22×C7#45311yescharacteristic
249C22×C7#45411yescharacteristic
250C2×D7—81no—
251C2×D7—81no—
252C2×D7—81no—
253C2×D7—161no—
254C2×D7—161no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—42no—
260C4×C7—81no—
261C4×C7—81no—
262C4×C7—81no—
263C4×C7—81no—
264C4×C7—81no—
265Dic7—82no—
266Dic7—81no—
267C24—141no—
268C24C2×C22⋊D711yescharacteristic
269C2×D4—141no—
270C2×D4—141no—
271C2×D4—281no—
272C2×D4—281no—
273C2×D4—281no—
274C2×D4—281no—
275C2×D4—281no—
276C2×D4—281no—
277C2×D4—561no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—81no—
285C22×C4—141no—
286C22×C4—281no—
287C22×C4Q8×D711yescharacteristic
288C22×C4#116111yescharacteristic
289C22×C4#116111yescharacteristic
290C22×C4#116111yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292C22×C4C2×C22⋊D711yescharacteristic
293M16—561no—
294M16—561no—
295C2×C8—281no—
296C2×C8—281no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—141no—
305C4⋊C4—141no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C22⋊C4—142no—
311C22⋊C4—281no—
312C22⋊C4—281no—
313C22⋊C4—281no—
314C42—81no—
315C42—561no—
316C2×C7—21no—
317C2×C7—21no—
318C2×C7—41no—
319C2×C7—41no—
320C2×C7#202711yescharacteristic
321C2×C7#203111yescharacteristic
322C2×C7#203211yescharacteristic
323D7—161no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—41no—
328C23—281no—
329C23—281no—
330C23—281no—
331C23—561no—
332C23—561no—
333C23#650911yescharacteristic
334C23#651011yescharacteristic
335C23#651611yescharacteristic
336D4—561no—
337D4—561no—
338D4—561no—
339D4—561no—
340D4—561no—
341D4—561no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
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—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—282no—
367C2×C4—281no—
368C2×C4—281no—
369C2×C4—281no—
370C2×C4—561no—
371C2×C4—561no—
372C2×C4#650911yescharacteristic
373C2×C4#650911yescharacteristic
374C2×C4#651011yescharacteristic
375C2×C4#651011yescharacteristic
376C8—561no—
377C8—561no—
378C7#1038411yescharacteristic, complemented, Sylow
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—81no—
388C22—81no—
389C22—561no—
390C22—561no—
391C22—561no—
392C22—1121no—
393C22—1121no—
394C22#9000311yescharacteristic
395C22#9001311yescharacteristic
396C22#9001711yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—562no—
408C4—561no—
409C2—21no—
410C2—21no—
411C2—41no—
412C2—41no—
413C2—1121no—
414C2#1172959611yescharacteristic
415C2#1172965011yescharacteristic
416C2#1172965711yescharacteristic
subgroupG/H
centerC22#90003
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#2667C2
socleC22×C7#393
prime radicalC2×#6493C22
hypercenter#2667D7
hypocenterC7#10384
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2667

Complements

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

NH ≅ G/Ntype
1. C7×#2667C2semidirect
5. #11725434C2semidirect
6. #11725432C2semidirect
7. #11725424C2semidirect
68. #2667D7semidirect
378. C7#10384semidirect

Maximal subgroups

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

subgroupindex
C7×#26672
#117272852
#117272822
#117271982
#117254342
#117254322
#117254242
#103847