Finite Group Databasebeta

← #13272463

Subgroups — #13272463

Order 1792.

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

normal subgroups only

This group has 488 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×#2579C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11725443C212yesmaximal
5#11725443C212yesmaximal
6C2×C4×C7×D4C2211yescharacteristic
7C22×C7×C4⋊C4C411yescharacteristic, complemented
8C7×#398—21no—
9C7×#396—21no—
10C7×#387—21no—
11C7×#386C2211yescharacteristic
12C7×#386C2211yescharacteristic
13C7×#385—21no—
14C7×#385C411yescharacteristic, complemented
15C7×#385C411yescharacteristic, complemented
16C7×#380C411yescharacteristic, complemented
17C2×#6408C2212yes—
18C2×#6407C2212yes—
19C4×C22⋊Dic7—21no—
20#89696—21no—
21#89325—21no—
22#89321—21no—
23#13297—71nomaximal, Sylow
24C22×C7×D4C2311yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C4×C7×D4—41no—
30C4×C7×D4—41no—
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—41no—
37C2×C7×C4⋊C4—41no—
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C22⋊Dic7—22no—
47C2×C4×Dic7—41no—
48C2×#1136—22no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54#6408—22no—
55#6408Q814yes—
56#6407—24no—
57C7⋊#95—41no—
58C4⋊(C7⋊C8)—41no—
59C4⋊(C7⋊C8)—41no—
60#2579D711yescharacteristic
61#1800—71no—
62C4×#99—71no—
63#1390—72no—
64#1390—72no—
65C24×C7C22×C411yescharacteristic
66C24×C7C2×D411yescharacteristic
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4C2×Q812yes—
70C2×C7×D4#4112yes—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—42no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7—81no—
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×Dic7—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—81no—
97C7×C4⋊C4—81no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—81no—
104C42×C7—21no—
105C42×C7—21no—
106C22⋊Dic7—24no—
107C22⋊Dic7—42no—
108C4×Dic7—82no—
109#1136—42no—
110#1136—42no—
111C2×C7⋊C8—42no—
112C2×C4×D4C2×D711yescharacteristic
113C22×C4⋊C4Dic711yescharacteristic, complemented
114#423—141no—
115C2×#100—72no—
116C2×#99—72no—
117#398—21no—
118#396—21no—
119#387—21no—
120#386C2×D711yescharacteristic
121#386C2×D711yescharacteristic
122#385—21no—
123#385Dic711yescharacteristic, complemented
124#385Dic711yescharacteristic, complemented
125#380Dic711yescharacteristic, complemented
126C4×C22⋊C4—141no—
127#343—141no—
128#342—141no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7C2×C22⋊C411yescharacteristic
138C23×C7C4×Q812yes—
139C23×C7C22⋊Q812yes—
140C7×D4—42no—
141C7×D4—42no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—22no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
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—81no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7—82no—
170C2×C4×C7#9911yescharacteristic
171C2×C4×C7#9911yescharacteristic
172C2×C4×C7#12411yescharacteristic
173C2×C4×C7C4⋊Q811yescharacteristic
174C2×Dic7—44no—
175C2×Dic7—82no—
176C2×Dic7—82no—
177C7⋊C8—82no—
178C22×D4C22×D711yescharacteristic
179C23×C4C2×Dic711yescharacteristic
180C23×C4C2×Dic711yescharacteristic
181C23×C4C2×Dic711yescharacteristic
182C23×C4C22⋊D711yescharacteristic
183C2×M16—142no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4C22⋊D711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—142no—
197C2×C22⋊C4C2×Dic711yescharacteristic
198C2×C22⋊C4C2×Dic711yescharacteristic
199C2×C22⋊C4C2×Dic711yescharacteristic
200C2×C22⋊C4C22⋊D711yescharacteristic
201C2×C42—281no—
202C2×C42C22⋊D711yescharacteristic
203C4⋊C8—281no—
204C4⋊C8—281no—
205#100—144no—
206#99—74no—
207#99—142no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—281no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—42no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—81no—
226C22×C7#39111yescharacteristic
227C22×C7C2×#9911yescharacteristic
228C22×C7#41011yescharacteristic
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—41no—
234C4×C7—42no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—81no—
238C4×C7—81no—
239C4×C7—81no—
240Dic7—84no—
241C24C22×Dic711yescharacteristic
242C24C2×C22⋊D711yescharacteristic
243C2×D4—41no—
244C2×D4—41no—
245C2×D4Q8×D712yes—
246C2×D4#116112yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—282no—
262C22×C4C22⋊Dic711yescharacteristic
263C22×C4C22⋊Dic711yescharacteristic
264C22×C4C22⋊Dic711yescharacteristic
265C22×C4C22⋊Dic711yescharacteristic
266C22×C4C2×C22⋊D711yescharacteristic
267M16—282no—
268M16—282no—
269C2×C8—282no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—81no—
283C22⋊C4—144no—
284C22⋊C4—282no—
285C42—21no—
286C42—21no—
287C42—562no—
288C2×C7—21no—
289C2×C7—21no—
290C2×C7—42no—
291C2×C7—42no—
292C2×C7#199611yescharacteristic
293C2×C7#212011yescharacteristic
294C2×C7#212111yescharacteristic
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23Q8×Dic712yes—
304C23#650912yes—
305C23C2×C22⋊Dic711yescharacteristic
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—284no—
337C2×C4—562no—
338C2×C4—562no—
339C2×C4#640811yescharacteristic
340C2×C4#640811yescharacteristic
341C2×C4#650711yescharacteristic
342C2×C4#651011yescharacteristic
343C8—562no—
344C7#1329711yescharacteristic, complemented, Sylow
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22C2×#640811yescharacteristic
358C22#9000211yescharacteristic
359C22#9001211yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—564no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#1172945511yescharacteristic
377C2#1172947011yescharacteristic
378C2#1172960111yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2579C2
socleC22×C7C2×#99
prime radical, cosocleC22×C7×D4C23
hypercenter#2579D7
hypocenterC7#13297
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
7. C22×C7×C4⋊C4C4semidirect
14. C7×#385C4semidirect
15. C7×#385C4semidirect
16. C7×#380C4semidirect
113. C22×C4⋊C4Dic7semidirect
123. #385Dic7semidirect
124. #385Dic7semidirect
125. #380Dic7semidirect
344. C7#13297semidirect

Maximal subgroups

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

subgroupindex
C7×#25792
C4×#64082
#117293712
#117254432
#132977