Finite Group Databasebeta

← #13257007

Subgroups — #13257007

Order 1792.

404 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11731403C211yesmaximal, characteristic
2C7×#1996C211yesmaximal, characteristic
3#11729443C211yesmaximal, characteristic
4#11729442C211yesmaximal, characteristic
5#11725432C211yesmaximal, characteristic
6#11725424C211yesmaximal, characteristic
7#11725422C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C7×#380C2211yescharacteristic
10C4×C7×C22⋊C4C2211yescharacteristic
11#89983C2211yescharacteristic
12C4×C22⋊Dic7C2211yescharacteristic
13#89979C2211yescharacteristic
14#89979C2211yescharacteristic
15#89758—21no—
16#89751—21no—
17#89720—21no—
18#89719—21no—
19#89427—21no—
20#89418—21no—
21C7×#342—21no—
22C7×#342—21no—
23#89333—21no—
24#89333—21no—
25#89333—21no—
26#89333—21no—
27#10383—71nomaximal, Sylow
28C22×C7×Q8Q811yescharacteristic, complemented
29C23×C4×C7C2311yescharacteristic
30C22×Dic14—21no—
31C7×C22⋊Q8—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4Q811yescharacteristic, complemented
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—
44C2×C22⋊Dic7—21no—
45C2×C4⋊Dic7—21no—
46C2×Dic7⋊C4—21no—
47C2×Dic7⋊C4—21no—
48C2×C4×Dic7—21no—
49C2×#1136—21no—
50C2×#1136—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95D411yescharacteristic
59C7⋊#95D411yescharacteristic
60C7⋊#95D411yescharacteristic
61C7⋊#95D411yescharacteristic
62C22⋊(C7⋊C8)—21no—
63C22⋊(C7⋊C8)—21no—
64C22⋊(C7⋊C8)—21no—
65C22⋊(C7⋊C8)—21no—
66#2665—71no—
67#1996D711yescharacteristic
68#1511—71no—
69#1508—71no—
70#1305—71no—
71#1297—71no—
72#1284—71no—
73C24×C7C2×D411yescharacteristic
74C2×C7×Q8—21no—
75C2×C7×Q8—21no—
76C2×C7×Q8—81no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—41no—
82C22×C4×C7—81no—
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×Q811yescharacteristic
86C22×C4×C7#4111yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C22×Dic7—21no—
90C22×Dic7—21no—
91C22×Dic7—21no—
92C22×Dic7—21no—
93C2×Dic14—42no—
94C2×Dic14—81no—
95C2×Dic14—81no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—81no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C42×C7—81no—
108C22⋊Dic7—42no—
109C22⋊Dic7—81no—
110C4⋊Dic7—81no—
111Dic7⋊C4—42no—
112Dic7⋊C4—81no—
113C4×Dic7—81no—
114#1136—81no—
115#1136—81no—
116C2×C7⋊C8—41no—
117C2×C7⋊C8—41no—
118C2×C22⋊Q8C2×D711yescharacteristic
119#407—71no—
120#407—71no—
121#400—141no—
122#396—141no—
123#395—141no—
124#393—141no—
125#391—141no—
126#387—141no—
127#381—71no—
128#380C2×D711yescharacteristic
129C4×C22⋊C4—71no—
130C4×C22⋊C4C2×D711yescharacteristic
131#342—21no—
132#342—21no—
133#328—141no—
134#328—141no—
135#328—141no—
136#328—141no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7#12011yescharacteristic
142C23×C7C22⋊Q811yescharacteristic
143C23×C7#12411yescharacteristic
144C7×Q8—81no—
145C7×Q8—81no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—81no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7—81no—
165C2×C4×C7C22⋊Q811yescharacteristic
166C2×C4×C7C22⋊Q811yescharacteristic
167C2×C4×C7#12411yescharacteristic
168C2×C4×C7#12411yescharacteristic
169C2×Dic7—22no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—41no—
181Dic14—84no—
182C7⋊C8—81no—
183C7⋊C8—81no—
184C22×Q8—141no—
185C22×Q8Dic1411yescharacteristic, complemented
186C23×C4C22×D711yescharacteristic
187C2×M16—141no—
188C2×M16—141no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—141no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4Dic1411yescharacteristic, complemented
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—141no—
202C2×C22⋊C4—141no—
203C2×C22⋊C4C22⋊D711yescharacteristic
204C2×C22⋊C4C22⋊D711yescharacteristic
205C2×C42—21no—
206C2×C42—141no—
207C22⋊C8—141no—
208C22⋊C8—141no—
209C22⋊C8—141no—
210C22⋊C8—141no—
211#95—21no—
212#95—21no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217#95—141no—
218#95—141no—
219#95—141no—
220#95—141no—
221#95—141no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—21no—
230C22×C7—81no—
231C22×C7—81no—
232C22×C7#39311yescharacteristic
233C22×C7#45511yescharacteristic
234C22×C7#45611yescharacteristic
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—81no—
240C4×C7—81no—
241C4×C7—82no—
242C4×C7—81no—
243C4×C7—81no—
244Dic7—42no—
245Dic7—81no—
246Dic7—81no—
247Dic7—81no—
248C24C2×C22⋊D711yescharacteristic
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—81no—
252C2×Q8—282no—
253C2×Q8—561no—
254C2×Q8—561no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—81no—
261C22×C4—141no—
262C22×C4—141no—
263C22×C4—141no—
264C22×C4—141no—
265C22×C4C2×Dic1411yescharacteristic
266C22×C4#115711yescharacteristic
267C22×C4D4×D711yescharacteristic
268C22×C4D4×D711yescharacteristic
269C22×C4#115911yescharacteristic
270C22×C4#115911yescharacteristic
271M16—561no—
272M16—561no—
273C2×C8—281no—
274C2×C8—281no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—282no—
281C4⋊C4—561no—
282C4⋊C4—561no—
283C22⋊C4—22no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—282no—
290C22⋊C4—561no—
291C42—81no—
292C42—561no—
293C2×C7—21no—
294C2×C7—21no—
295C2×C7—41no—
296C2×C7—41no—
297C2×C7#202811yescharacteristic
298C2×C7#203111yescharacteristic
299C2×C7#203211yescharacteristic
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23#648711yescharacteristic
305C23#649911yescharacteristic
306C23#650011yescharacteristic
307Q8—81no—
308Q8—81no—
309Q8—564no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—142no—
330C2×C4—281no—
331C2×C4—281no—
332C2×C4—281no—
333C2×C4—281no—
334C2×C4—281no—
335C2×C4—281no—
336C2×C4—281no—
337C2×C4—281no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4#644111yescharacteristic
342C2×C4#644111yescharacteristic
343C2×C4#644811yescharacteristic
344C2×C4#644811yescharacteristic
345C8—561no—
346C8—561no—
347C7#1038311yescharacteristic, complemented, Sylow
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#8971911yescharacteristic
359C22#8983311yescharacteristic
360C22#8986711yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—81no—
367C4—82no—
368C4—81no—
369C4—81no—
370C4—282no—
371C4—561no—
372C4—561no—
373C4—561no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#1172742711yescharacteristic
379C2#1172777911yescharacteristic
380C2#1172779211yescharacteristic
subgroupG/H
centerC22#89719
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#1996C2
socleC22×C7#393
prime radicalC24×C7C2×D4
hypercenter#1996D7
hypocenterC7#10383
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
28. C22×C7×Q8Q8semidirect
37. C2×C7×C4⋊C4Q8semidirect
185. C22×Q8Dic14semidirect
198. C2×C4⋊C4Dic14semidirect
347. C7#10383semidirect

Maximal subgroups

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

subgroupindex
#117314032
C7×#19962
#117294432
#117294422
#117254322
#117254242
#117254222
#103837