Finite Group Databasebeta

← #13256976

Subgroups — #13256976

Order 1792.

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

normal subgroups only

This group has 360 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×#2605C211yesmaximal, characteristic
2#11727320C211yesmaximal, characteristic
3#11727247C211yesmaximal, characteristic
4#11727187C211yesmaximal, characteristic
5#11725439C211yesmaximal, characteristic
6#11725427C211yesmaximal, characteristic
7#11725421C211yesmaximal, characteristic
8#90161C2211yescharacteristic
9C7×#400—21no—
10C7×#395—21no—
11C7×#391—21no—
12C7×#391—21no—
13C7×#387—21no—
14C7×#387—21no—
15C7×#383C2211yescharacteristic
16C4×C7×C4⋊C4C2211yescharacteristic
17C7×#376C2211yescharacteristic
18#89698C2211yescharacteristic
19#89698C2211yescharacteristic
20C4×#1136C2211yescharacteristic
21#89318—21no—
22#89318—21no—
23#89247—21no—
24#89247—21no—
25#89247—21no—
26#89247—21no—
27#10379—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C42×C7—21no—
34C2×C42×C7D411yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C42×C7C2311yescharacteristic
37#6489—21no—
38#6487—21no—
39#6487—21no—
40C2×#1136—21no—
41#6433—21no—
42#6432Q811yescharacteristic, complemented
43#6432Q811yescharacteristic, complemented
44C4×Dic14—21no—
45C4×Dic14—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95D411yescharacteristic
55C7×#95D411yescharacteristic
56C7×#95D411yescharacteristic
57C7×#95D411yescharacteristic
58C22⋊(C7⋊C8)—21no—
59C22⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)—21no—
62#6377—21no—
63#6377—21no—
64#6377—21no—
65C4×C7⋊C8—21no—
66#2605D711yescharacteristic
67#1986—71no—
68#1515—71no—
69#1506—71no—
70#1300—71no—
71#1288—71no—
72#1285—71no—
73C2×C7×Q8—42no—
74C2×C7×Q8—42no—
75C2×C7×Q8—81no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C2×Dic14—21no—
86C2×Dic14—21no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—81no—
91C42×C7—41no—
92C42×C7—41no—
93C42×C7—42no—
94C42×C7C2×Q811yescharacteristic
95C42×C7#4111yescharacteristic
96C42×C7#4111yescharacteristic
97C42×C7#4111yescharacteristic
98C22⋊Dic7—41no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103Dic7⋊C4—41no—
104Dic7⋊C4—41no—
105C4×Dic7—41no—
106#1136—22no—
107#1136—41no—
108C2×C7⋊C8—41no—
109C2×C7⋊C8—41no—
110C2×C7⋊C8—41no—
111#533—71no—
112#432—71no—
113#432—71no—
114C4×M16—71no—
115#400—21no—
116#395—21no—
117#391—21no—
118#391—21no—
119#387—21no—
120#387—21no—
121#383C2×D711yescharacteristic
122C4×C4⋊C4C2×D711yescharacteristic
123#376C2×D711yescharacteristic
124#339—141no—
125#339—141no—
126#328—141no—
127#328—141no—
128#328—141no—
129#328—141no—
130C23×C7#12011yescharacteristic
131C7×Q8—84no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
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×C7C22⋊Q811yescharacteristic
155C2×C4×C7C22⋊Q811yescharacteristic
156C2×C4×C7C22⋊Q811yescharacteristic
157C2×C4×C7#12411yescharacteristic
158C2×C4×C7#12411yescharacteristic
159C2×C4×C7#12411yescharacteristic
160C2×Dic7—41no—
161C2×Dic7—41no—
162Dic14—81no—
163Dic14—81no—
164C7⋊C8—42no—
165C7⋊C8—81no—
166C7⋊C8—81no—
167C22×Q8—21no—
168C2×M16—141no—
169C4⋊Q8—71no—
170C4⋊Q8—71no—
171#125—141no—
172C22⋊Q8—141no—
173C22⋊Q8—141no—
174C4×Q8—141no—
175C4×Q8—141no—
176#117—141no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C42—21no—
182C2×C42C22⋊D711yescharacteristic
183C2×C42C22⋊D711yescharacteristic
184C2×C42C22×D711yescharacteristic
185C4⋊C8—141no—
186C4⋊C8—141no—
187C22⋊C8—141no—
188C22⋊C8—141no—
189#97—141no—
190#97—141no—
191#97—141no—
192C4×C8—141no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95C22⋊D711yescharacteristic
202#95C22⋊D711yescharacteristic
203#95C22⋊D711yescharacteristic
204#95C22⋊D711yescharacteristic
205C22×C7—21no—
206C22×C7—21no—
207C22×C7#39311yescharacteristic
208C22×C7#45511yescharacteristic
209C22×C7#45611yescharacteristic
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—42no—
217C4×C7—42no—
218C4×C7—81no—
219C4×C7—81no—
220C4×C7—81no—
221C4×C7—81no—
222Dic7—81no—
223Dic7—81no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—81no—
227C2×Q8—141no—
228C2×Q8—141no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C2×C22⋊D711yescharacteristic
236C22×C4C2×C22⋊D711yescharacteristic
237C22×C4C2×C22⋊D711yescharacteristic
238M16—142no—
239M16—281no—
240C2×C8—281no—
241C2×C8—281no—
242C2×C8—281no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—281no—
252C4⋊C4—281no—
253C22⋊C4—281no—
254C42—41no—
255C42—41no—
256C42—42no—
257C42—281no—
258C42Q8×D711yescharacteristic
259C42#116111yescharacteristic
260C42#116111yescharacteristic
261C42#116111yescharacteristic
262C2×C7—21no—
263C2×C7—21no—
264C2×C7#202811yescharacteristic
265C2×C7#203111yescharacteristic
266C2×C7#203111yescharacteristic
267C23#651611yescharacteristic
268Q8—84no—
269Q8—561no—
270Q8—561no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4#650911yescharacteristic
296C2×C4#650911yescharacteristic
297C2×C4#650911yescharacteristic
298C2×C4#651011yescharacteristic
299C2×C4#651011yescharacteristic
300C2×C4#651011yescharacteristic
301C8—282no—
302C8—561no—
303C8—561no—
304C7#1037911yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22#9000311yescharacteristic
308C22#9001411yescharacteristic
309C22#9001811yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—561no—
323C4—561no—
324C2—21no—
325C2—21no—
326C2#1172959611yescharacteristic
327C2#1172959611yescharacteristic
328C2#1172965911yescharacteristic
subgroupG/H
centerC22#90003
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2605C2
socleC22×C7#393
prime radicalC23×C7#120
hypercenter#2605D7
hypocenterC7#10379
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2605

Complements

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

NH ≅ G/Ntype
42. #6432Q8semidirect
43. #6432Q8semidirect
304. C7#10379semidirect

Maximal subgroups

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

subgroupindex
C7×#26052
#117273202
#117272472
#117271872
#117254392
#117254272
#117254212
#103797